Global and local gauge symmetries beyond Lagrangian formulations

What is the structure of general quantum processes on composite systems that respect a global or local symmetry principle? How does the irreversible use of quantum resources behave under such symmetry principles? Here we employ an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has a highly rigid decomposition in terms of the flow of symmetry-breaking degrees of freedom between each subsystem and its environment. The decomposition has a natural causal structure that can be represented diagrammatically and makes explicit gauge degrees of freedom between subsystems. The framework also provides a novel quantum information perspective on lattice gauge theories and a method to gauge general quantum processes beyond Lagrangian formulations. This procedure admits a simple resource-theoretic interpretation, and thus offers a natural context in which features such as information flow and entanglement in gauge theories and quantum thermodynamics could be studied. The framework also provides a flexible toolkit with which to analyse the structure of general quantum processes. As an application, we make use of a `polar decomposition' for quantum processes to discuss the repeatable use of quantum resources and to provide a novel perspective in terms of the coordinates induced on the orbit of a local process under a symmetry action.

What is the structure of general quantum processes on composite systems that respect a global or local symmetry principle? How does the irreversible use of quantum resources behave under such symmetry principles? Here we develop an information-theoretic framework to address these questions and show that every symmetric quantum process on a system has an essentially unique decomposition in terms of the flow of symmetry-breaking degrees of freedom between each subsystem and its environment. The decomposition has a natural causal structure that can be represented diagrammatically and makes explicit gauge degrees of freedom between subsystems. Our framework also provides a novel quantum information perspective on lattice gauge theories and a method to gauge general quantum processes beyond Lagrangian formulations. This procedure admits a simple resource-theoretic interpretation, and thus offers a natural context in which features such as information flow and entanglement in gauge theories could be studied. The framework also provides a flexible toolkit with which to analyse the structure of general quantum processes. As an application, we make use of a 'polar decomposition' for quantum processes to discuss incompatibility in the use of quantum resources and to provide a novel perspective in terms of the geometry induced on the orbit of a local process under a symmetry action.

I. INTRODUCTION
Symmetry plays a central role in establishing the fundamental structure of physical theories and provides an invaluable tool to analyse physics in general terms. In quantum physics it finds numerous applications in high energy physics [1], condensed matter [2], quantum optics [3], open system dynamics [4,5], metrology [6], and in more recent times in the expanding field of quantum information science [7][8][9][10][11][12][13][14][15]. Typically symmetry principles are associated with reversible dynamics, where they are fundamentally linked with conservation laws. However they also arise in situations in which there is some form of irreversibility present [5,[16][17][18][19][20]. In such regimes it has been shown that there is a break-down between symmetry principles and conservation laws [5], and novel information-theoretic measures come into play [21][22][23][24][25][26].
Traditionally, such principles in physics are most easily described for reversible dynamics, where they have been vastly successful -they have provided invaluable results in crystallography, deep insights into fundamental particle physics, fundamental spacetime structure and the importance of gauge symmetries, to name a few. However there are increasing motivations to extend these concepts beyond Lagrangian and state formulations into a more general setting [27][28][29]. This is not only for the sake of greater abstraction and unity, but also to connect with the large array of results that have been developed recently in quantum information theory, which are framed in the more general terms of completely-positive tracepreserving (CPTP) operations [30][31][32]. The present work seeks to contribute to this aim.
The central questions we address here are: 1. What is the structure of general quantum processes on a composite quantum system that respect a global or local gauge symmetry?
2. Under such symmetry principles, how does irreversible quantum information processing occur in the use of symmetry-breaking degrees of freedom?
We tackle these questions within the context of quantum information theory, and develop a novel "process mode" formalism for general quantum processes 1 . In particular in Section IV we analyse how the dynamics of a quantum system with global symmetry constraints arises from local exchange of symmetry-breaking resources across any bipartite split. Previous work [36,37] focused on resource states that break a symmetry, and the resulting framework has provided a number of significant applications [16,25,38,39]. Here we also deal with localized symmetry-breaking degrees of freedom, but instead focus on the structure of the quantum processes that act on them. We find that bipartite quantum processes that are globally symmetric must satisfy a rigid structure in order to respect underlying symmetries. An advantage of our analysis is that it describes the causal structures at play, without looking at state properties. We also show that this decomposition of symmetric quantum processes has a natural gauge degree of freedom. In Section II we show that this freedom has a simple interpretation in terms of a local 'process orbit', while in Section III we use this process orbit setting to consider potential incompatibility in the use of symmetry-breaking quantum resources for local information-theoretic tasks. This provides a clear physical explanation of recent results on quantum coherence [17,[40][41][42][43] framed in simple geometric terms.
Finally in Section V we apply our process mode formalism to the problem of gauging a global symmetry principle for a general quantum process to a local one. We provide an information-theoretic perspective on the gauging procedure in terms of concepts from the field of quantum reference frames, and so enables the application of ideas from one area into the other. Our gauging procedure for quantum processes neither assumes a Lagrangian formulation, nor places restrictions on the existence of 'classical regimes' in the form of macroscopic reference frames. To demonstrate consistency with traditional gauge theories we describe how our procedure coincides with the gauging of unitary dynamics on a lattice model. We also describe how this approach provides a simple interpretation of Gauss' law and gauge dynamics from a resource-theoretic perspective, and discuss future directions to be explored.

A. Notation and conventions
We fix a symmetry group G, and assume it to be a discrete group or a compact Lie group. For any quantum system A, its Hilbert space will be denoted H A and is assumed to carry a representation of the group g → U (g) which associates a unitary U (g) to every group element g ∈ G. Operators on H A in turn transform under the adjoint action. Specifically, given a linear operator X ∈ B(H A ) we have X → U (g)XU (g) † for all g ∈ G. In what follows we will use the abbreviation U g (X) := U (g)XU (g) † to simplify expressions. We also note that U † g = U g −1 . A quantum process E is a linear map E : B(H A ) → B(H A ) that is both completely positive and tracepreserving, and takes states of a system A into states of a system A . The set of all superoperators from B(H A ) into B(H A ) will be denoted T (A, A ). We assume that H A carries a representation denoted by g → U (g) with adjoint action U of B(H A ). Any superoperator Φ ∈ T (A, A ) has a natural group action on it given by Φ → U g • Φ • U † g . We will use the terms "symmetric" (also called "covariant" in the literature) when referring to operators or superoperators which are invariant under group actions. States which are not invariant under the group action are called "asymmetric" in that they possess symmetry-breaking degrees of freedom.
FIG. 1: Structure of the paper. Our results follow naturally from the key concept of process modes {Φ λ k }, which are a collection of superoperators on the system that transform irreducibly under the symmetry action. On an individual system, these lead (in Section II) to the notion of a process orbit M(G, E) (lower figure), while for multipartite systems (Section IV) they lead to exchange diagrams (top figure) for quantum systems. Finally we describe how general multipartite processes can be gauged (Section V), and also provide geometric perspective on irreversibility in the use of symmetry-breaking quantum resources (Section III).

II. CORE SYMMETRY-BREAKING STRUCTURE FOR QUANTUM PROCESSES
What does it mean for a quantum system described by Hilbert space H to carry a particular type of symmetry principle? Mathematically it is often the case that we ascribe a priori to H a particular type of symmetry given by a unitary representation U of some group G. In other words for any element of the group g ∈ G there are unitary transformations U (g) acting on the Hilbert space H that have the same multiplicative properties as the group. In reality the symmetry constraints may arise from either a particular type of dynamics that is invariant under the symmetries of the group, from a lack of a shared coordinates (a reference frame) [7] or imposed by the geometry of the physical system (such as the case in crystallography or quantum chemistry [44]). Other times the origin of the symmetries involved is deeper and points to fundamental laws of physics [1].
A. Relation to quantum resources, quantum coherence and quantum thermodynamics With the development of quantum information theory came the simple yet powerful notion of a quantum resource, which is a physical property not described by a hermitian observable, but whose presence is required in order for some kind of task (physical transformation or information-processing task) to be performed. Quantum entanglement is the canonical example of a quantum resource [45]. It does not correspond to any hermitian observable, however it is a property that is needed for quantum teleportation and quantum computing to be possible. The resource theory approach provides a cohesive general framework [46,47] to analyse a variety of physical phenomena such as athermality [48][49][50], coherence [51][52][53], non-gaussianity in quantum optics [54,55], magic states in quantum computation [56,57]. Moreover, seemingly disconnected resource theories are found to share many common structural features [46,58,59].
Of direct relevance to our work is the theory of quantum asymmetry [5,21,23,36,37] which was developed through the study of quantum reference frames [7,22]. In this theory the quantum resource is the symmetrybreaking degrees of freedom (or "asymmetry") of a state. This resource has been referred to as "unspeakable" quantum information [60,61], and the virtue of the framework is that it allows one to make clear statements about how useful the system is to perform tasks forbidden by a symmetry principle.
A key formulation of this theory was provided in the work of Marvian and Spekkens, whose analysis is based on modes of asymmetry [36,37], in which a quantum state ρ is analysed in terms of irreducible components. Specifically, one can write ρ = λ,k ρ λ k , where the operators ρ λ k form irreducible tensor operators under the group action, namely U g (ρ λ k ) = j v kj (g)ρ λ k with v λ (g) kj being the matrix components of λ-irrep of the group G.
A simple example is provided by quantum coherence in a quantum system S between eigenstates of the number operator N = n≥0 n|n n|. The system S has a U (1) group action U (θ) := e iθN corresponding to phase shifts. An operator X is a mode of coherence if it transforms irreducibly under the group. For U (1) this simply means that U (θ)XU (θ) † = e ikθ X for some integer k ∈ Z. A coherent state such as ρ = |ψ ψ|with |ψ = 1 √ 2 (|a + |b ) has two non-zero off-diagonal terms |a b| and |b a|. Under the group action |a b| → e iθN |a b|e −iθN = e i(a−b)θ |a b| (1) and therefore is a mode of coherence with k = (a − b), while |b a| is a k = (b−a) mode. In contrast the diagonal terms |a a| and |b b| are each k = 0 modes. By adopting a resource-theoretic perspective with these asymmetry modes, it is possible to encode symmetry principles in general processes, and to quantify their effects in a rigorous manner. This has provided a range of novel insights; in particular the structure has recently provided a natural, explanatory framework for quantum thermodynamics [41,49,[62][63][64][65][66][67][68][69][70][71][72][73][74][75]. It has been used to prove that no formalism based solely on free energy functions can fully describe coherence in the thermodynamics of extreme quantum regimes [67]. The modes perspective also makes explicit general upper and lower bounds for quantum coherence [16,76], however at present a complete specification of fully coherent thermodynamics is lacking.

B. Mode decompositions for quantum processes
The theory in [36,37] provides an elegant way in which to describe how general quantum states behave under a symmetry principle. The starting point of our work is to generalize this approach to the level of superoperators, and to explore the resulting structures. We will find that instead of asymmetry modes that occur in a quantum state, the generalization for quantum processes has a natural causal structure to it that describes the flow of symmetry-breaking resources. Specifically, the formalism has, as basic building-blocks, process terms that can be represented as three-legged objects (see Figure 2) labelled with an "in-going" mode a that evolves into an "out-going" modeã by way of an interaction with an external degree of freedom λ.

FIG. 2:
The structure of process modes. A diagrammatic representation of a process mode {Φ λ k } transforming between state mode a in the input system A and state modeã in the output system A . Time runs up the page. The horizontal leg is labelled by the irrep label λ for the process mode and corresponds to symmetry-breaking degrees of freedom required for the process. For the particular case of the process being state preparation, we have a = 0 andã = λ only. This recovers the modes of asymmetry decomposition.
More explicitly, these three-legged objects are obtained from the following notion of a process mode.
Definition II.1. Given an input quantum system A, and output quantum system A , a set {Φ λ k } of superoperators Φ λ k : B(H A ) → B(H A ) is called a process mode if we have that U g • Φ λ k • U † g = j v λ kj (g)Φ λ j for all g ∈ G, where (v λ kj ) are the matrix components of the λ-irrep of G.
The intuition for this is simply that every quantum process is composed of "pure note" components, that respond to the symmetry in a simple way. Specifically, the space of superoperators T (A, A ) splits up into subspaces spanned by sets of process modes and as such, the consider two state transformations. The first involves the ground state of a quantum system |E 0 E 0 | thermalising to a Gibbs state at temperature T = 1/β under a quantum process E 1 . The inverse transformation under any map E 2 is only possible thermodynamically if a non-zero amount of ordered energy (in the form of mechanical work) is utilized. (Bottom figure -symmetry principles) Process modes have a built-in directionality: the process E 1 on the left takes a spin polarized along some axisn in the state |n n| into a completely rotationally symmetric state 1 d 1, and it can be composed of λ = 0 modes. In contrast any quantum process E 2 taking 1 d 1 into the state |n n| must possess non-trivial λ = 0 process modes, corresponding to the avoidable need for symmetry-breaking resources. Note that in both cases the processes may generically transform between systems of different dimension.
process modes can be viewed as the elementary units of any quantum process with respect to a symmetry group G. More technically, a process mode can be described as an irreducible tensor superoperator (ITS) for the system under the group action. This generalises the more familiar notion of irreducible tensor operators (ITO) [77] (e.g. for rotations we have the angular momentum operators J x , J y , J z on a spin system). Given this definition, a canonical set of process modes can be built out of coupling incoming and outgoing state-modes (described by ITOs) according to Clebsch-Gordan rules [77,78]. Moreover, their construction is independent of any choice of basis for the input and output quantum systems. One of the main advantages is that they provide a natural basis for T (A, A ) that admits a simple diagrammatic representation (given in Figure 2) for the underlying symmetry properties of the process. A straightforward but tedious algorithm to construct these process modes for a general system is provided in sections B 3 and B 4 of the Supplementary Material.
Simple examples of process modes are easily constructed in the case of the rotational group on a single spin-1/2 system. For this the irrep label λ is an angular momentum label, and the set of quantum processes involve only spin-0, spin-1 and spin-2 contributions. The triple of spin-1 terms {Φ 1 k } arising from incoming and outgoing a =ã = 1 modes are given by where (σ x , σ y , σ z ) are the Pauli spin observables and σ ± = 1 √ 2 (σ x ± iσ y ) are the raising and lowering operators between spin-up and spin-down states. A derivation of this example is provided in the Supplemental Material, and that it obeys the required conditions is seen by inspection.
Since {Φ λ k } λ,k form a complete basis for T (A, A ) one can decompose any quantum process E : B(H A ) −→ B(H A ) according to the symmetry imposed on the input and output spaces. We have that for some coefficients α λ,m λ ,k ∈ C, and the label m λ denotes the multiplicity of the λ-irrep. Once we have fixed a basis for H A , H A the process is then fully specified by the coefficients α λ,m λ ,k .
While irreducible tensor operators (ITOs) are familiar from quantum physics, it is far from obvious what these process modes physically correspond to for a quantum system. In the next section we provide an answer to this through the introduction of a process orbit for a quantum process E, and we show that the data {α λ,m,k } have a natural physical interpretation in terms of the specific reference frame data required to perform E subject to an underlying symmetry constraint.
C. Local coordinates from quantum and classical reference frames Given a symmetry principle, a core question is how quantum processes local to some region A can arise dy-namically through interactions with an ambient environment B. If these interactions are constrained by underlying symmetry principles then the ambient environment must function so as to generate a set of local "coordinates" {x A i } with respect to which a quantum process E at A is induced. For example, a time coordinate {x i } = {t} for the case of using B as a quantum clock with which to performed timed operations on A, or angular data {(θ, φ)} when we want to use a quantum system to break rotational symmetry on A. The coordinates {x A i } required depend on both G and the quantum process E, and are described by a process orbit M(G, E). More precisely is the orbit of E within the space of superoperators, under the symmetry action. Given this, we discuss in the following subsections the use of this process orbit.

D. Local simulation via symmetry-breaking resources
We first show that the relative alignment between A and its environment B corresponds to choosing an "origin" on the process orbit, and any changes in relative alignment (due to passive or active transformations) of the systems correspond to translations of this point. If the underlying dynamics of A and B is symmetric, then any non-symmetric process E on A can only arise from there being a symmetry-breaking state σ B on B, together with globally symmetric dynamics V : . Assuming no correlations between A and B, we therefore have that where [U g ⊗ U g , V] = 0 for all g ∈ G. However, the symmetry representation U (g) ⊗ U (g) depends on the synchronisation of the group action on A and B. One is free to shift this convention via a constant group transformation on B and still have a global symmetry constraint. For fixed h ∈ G we replace σ → U h (σ) and have that Therefore a shift in the relative alignment of A and B corresponds to the group action on the process E → U h • E • U † h , and so amounts to a translation on the space M(G, E).
In particular whenever the induced map E is symmetric the process orbit M(G, E) reduces to a single point and an extension of Stinespring dilation theorem [79,80] ensures that E can be achieved by a global unitary symmetric dynamics on system A and an environment initialised in a symmetric state. More generally the mode properties of the initial state σ B of the environment B are transferred into the symmetry-breaking simulated map E on system A. The connection between modes of asymmetry [36,37] for resource states (which we shall call state-modes to contrast with process modes) and process modes can be made more explicit in this context. We can decompose the state σ B as σ B = λ σ λ k , where σ λ k is a λ-asymmetry mode in the state. Then it is easy to verify that the map is a particular process mode at A with irrep label λ. However, the point of defining process modes purely local to the system A is that it avoids the need to specify the form of external states, or the interacting dynamics that accesses this asymmetry. In particular it should be emphasized that there is no direct link between the process data {α λ,m,k } and asymmetry measures of σ, since the particular dynamics in V must also be taken into account.
We shall adopt a notational convention in the coming sections. We shall absorb the multiplicity label m into λ to simplify our expressions, and simply write α λ,k instead of α λ,m,k . This is because the multiplicity label plays no role in the analysis. We now state the link between M(G, E) and process modes. For clarity we first present a result for the specific case in which E is an axial process. The more general statement is provided in Theorem (II.4) is a simple generalization of the axial case.
Definition II.2. Given a global symmetry action for G = SU (2), a quantum process E : B(H A ) → B(H A ) is said to be an axial process if it is symmetric under the action of a subgroup H ∼ = U (1) of SU (2). Namely, we . Simply put, the set of axial quantum processes comprises of CPTP maps that break the full rotational symmetry group, but still have a residual U (1)-symmetry in some direction. Such maps are abundant throughout quantum physics -for example: dephasing a qubit about an axis, preparation of a pure, polarized spin state, measurements along a particular axis, unitary rotations that leave a fixed axis invariant -and therefore form a convenient set of quantum processes to illustrate structures. We can establish the following basic result, which is easily generalised.
Theorem II.3. Let E : B(H A ) → B(H A ) be an axial process. The process orbit M(G, E) for E under the group action is a sphere S 2 and with E corresponding to the pointn on S 2 . Then there exists a basis of pro- : 'Polar-decomposition' of a general quantum process. Given a symmetry G the decomposition of quantum processes gives rise to process modes {Φ λ k }. The local simulation of a symmetry-breaking process E on a quantum system A requires specific resources in the environment B. These are encoded by data {α λ,k (x)}, which are un-normalised wavefunctions on a process orbit M(G, E). The local process is given by For the case of E being a symmetric process the process orbit M(G, E) collapses to being a single point and so has no structure.
that under the group action the components α jk are unnormalised wavefunctions on the sphere given by where Y jk (θ, φ) are spherical harmonics and (θ, φ) are the angular coordinates of the pointn on the sphere. The complex coefficients a j are independent of k and constant for all processes in the orbit of E.
The exact form that the spherical harmonics take is expanded upon in Section B 6 b of the Supplementary Material. The core point of this result is that it separates the process resource requirements {α j,k } into local demands, given by a set of invariant resource demands (a j1 , a j2 , . . . ), from the purely relational aspects of how B is aligned relative to A. More explicitly, any axial process E is fully specified by the numbers {α j,k }. These can be further decomposed into quantities (a j ) that are independent of the relative alignment of A and its environment, together with a choice of coordinates (θ, φ) on M(G, E) that specify the relative alignment of A and B. The proof of the theorem relies on the basic notion of a homogeneous space 2 and is provided in section B 5 of the Supplementary Material. 2 A manifold M is said to be a homogeneous space if there is a While axial processes are natural and intuitive, the above construction can be extended easily to a general statement for any quantum process E that has a particular symmetry sub-group H ⊂ G. This can be viewed as a form of polar-decomposition for the process E into parts independent of lab alignments and those parts that specify these alignments.
Theorem II.4. Let G be a symmetry group that acts on quantum systems A and A . Let E : B(H A ) → B(H A ) be a quantum process that is H-symmetric under some subgroup H ⊂ G. The orbit of E under the symmetry action is the homogeneous space M(G, E) ∼ = G/H and denote the point corresponding to process E by x ∈ M(G, E). Then there is a basis of process modes {Φ λ k } for which where Y λ,k are harmonic wavefunctions 3 on the homogeneous space M(G, E) and the complex coefficients a λ are constant for all processes in the orbit of E.
Technical constraints on the residual symmetry subgroup H and a proof of the result can be found in section B 6 c of the Supplementary Material To illustrate the core ingredients of this process description, we describe the set of single-qubit quantum processes under the symmetry group G = SU (2) in more detail. For the case of qubit-to-qubit processes the construction of process modes is straightforward, and involves computing Clebsch-Gordan couplings for H ⊗4 , where H is a qubit Hilbert space with the usual SU (2) group action U (g) = e i(aσx+bσy+cσz) , where (σ x , σ y , σ z ) are the Pauli operators and a, b, c ∈ R specify the group element g.
An integer spin value j labels each process mode, and it is found that for a qubit it only takes on values j = 0, 1 and 2. The j = 0 process mode components by definition transform trivially under the SU (2) action and it is easy to see that the most general form is group G that acts transitively on it via a mapping (x, g) → xg ∈ M, for all x ∈ M and g ∈ G, which respects the group structure. Transitivity here simply means that given x, y ∈ M there is a g ∈ G such that xg = y, and thus all points in M can be reached from x via the action of the group. 3 In the case that H is a massive subgroup these are "associated spherical harmonics". We expand on this in section B 5 b of the Supplementary Material, and an in-depth analysis of harmonic functions on homogeneous spaces can be found in [78,81] Quantum process (a 0 , a 1 , a 1 , a 2 ) Dephasing process: Rotation about an axis: Depolarising process: FIG. 5: Axial maps and resource demands. Any process E admits a natural decomposition under a symmetry group G, however for axial maps this decomposition takes on a simple and intuitive form as in Theorem II.3. The table shows the decomposition of axial maps on a single qubit into invariant resource demands. Here Π 0 = |n n| is the projector onto then direction, while Π 1 = 1 − Π 0 is the projection onto the −n direction. The data for each process are invariants for the group orbit of that process, and together with the choice of relative alignment to external references (in terms of the location on M(G, E)) fully specify the particular process. The coefficients correspond to monopole (a 0 ), dipole (a 1 and a 1 ) and quadrupole (a 2 ) contributions -no higher orders are needed for qubit to qubit processes.
which in the case of quantum processes gives a directionless depolarization process on the qubit. These are the only SU (2)-symmetric processes on a qubit. The j = 1 process mode terms transform under the symmetry as a vector (Φ 1 1 , Φ 1 0 , Φ 1 −1 ) of process components. There are precisely two such process modes that arise and are given by and (σ x ±iσ y ) are raising/lowering operators. Finally, for the j = 2 case, there is a single process mode term, which transforms as a 5-dimensional of process components. These are given by (σ x tr(σ x ρ) + σ y tr(σ y ρ) − 2σ z tr(σ z ρ)) .
In terms of a general qubit axial process E that fixes the axisn with spherical polar coordinates (θ, φ), each of the above process modes are coupled to the dual spherical harmonics Y 00 (θ, φ), Y 1m (θ, φ) and Y 2m (θ, φ) with prefactors given by (a 0 , a 1 , a 1 , a 2 ), to yield where we denote Φ id (ρ) := 1 2 for one qubit system. As we range over all g ∈ SU (2) the processes thus obtained will correspond to displacing the directionn = (θ, φ) along all points on the sphere (which acts as the process orbit). Most importantly we emphasise again that any process in the orbit of E which is generically given by U g • E • U † g for some g ∈ SU (2) will necessarily have the same invariant coefficients (a 0 , a 1 , a 1 , a 2 ). In Figure 5 we give the values for these invariant resource demands for various examples of one-qubit processes that leave invariant an unspecified, but fixed directionn on the Bloch sphere. A few points warrant mention regarding this perspective. Firstly, note that extreme cases are easy to state. For the case that E is symmetric the set M(G, E) is a single point and so has no structure to it. In the case that the quantum process E lacks any residual symmetry, the space M(G, E) is the full group manifold G (quotiented by any phase symmetries e iθ(g) in the representation), and the realisation of E under a symmetry constraint can be interpreted in terms of encoding a group element g in some token quantum system as g → σ g ∈ B(H B ) followed by a symmetric state discrimination to access this group element. However the group element encoding perspective ignores the fact that not all group elements are equal in the realisation of E in A. In contrast, the homogeneous space M(G, E) boils things down to the essentials and provides uniformity in the data required.
Secondly, transformations of B relative to A correspond to displacements on M(G, E), and therefore the choice of origin on M(G, E) is an arbitrary one that simply encodes how the physics in A is related to the physics in B. It is therefore a gauge choice in the sense that the physics local to A does not care about how the symmetry action on B is defined relative to A. A redundancy exists in our description of the composite system that leaves observed physics invariant. The decomposition of a quantum process under the asymmetry analysis therefore can be phrased schematically as We justify this use of this terminology more in Section V.
Finally, in Section III we will be interested in saying when fundamental irreversibility occurs in the use of a symmetry-breaking resource to induce a quantum process E on a subsystem A. This statement must explicitly depend on both the symmetry group and the particular quantum process involved. We shall argue that the geometry induced on M(G, E) provides a natural perspective on whether a resource state can be used in a repeatable manner without degrading it. From the perspective of asymmetry the realisation of any quantum process on a quantum system A, subject to a symmetry constraint, in the presence of a perfect "classical reference" can be viewed abstractly as a mapping M(G, E) → T (A, A ) with x → E. The role of M(G, E) is to provide an asymptotic, classical set of coordinates demanded under the symmetry constraint. However if one attempts to realise E using a bounded non-classical system, then we still have a mapping from M(G, E) into T (A, A ) however now the resolving power on M(G, E) is "blurred" due to the non-classical resources used.

III. IRREVERSIBILITY IN SYMMETRY-BREAKING DEGREES OF FREEDOM.
A symmetry principle in a quantum system need not correspond to a conservation law [5]. In the case of symmetric unitary dynamics we do have that conservation of charges (corresponding to hermitian observables) hold, and that any symmetry-breaking degrees of freedom in quantum states (which may include a property not described by a hermitian observable) is also conserved.
However, more generally there is a disconnect between symmetry principles and conservation laws [5]. For general symmetric quantum processes the expectation values of the generators of the symmetry can both increase and decrease, and a proper account must be supplemented with information-theoretic measures. In such cases quantum incompatibility [82] is expected to give rise to irreversibility in the symmetry-breaking degrees of freedom of a quantum system. For example, a quantum system that acts as a clock functions to break time-translation symmetry. However its use in say quantum thermodynamics may result in a back-action that distorts its subsequent ability to function as a clock.
One might generally expect globally symmetric quan- , such that the state σ B breaks the symmetry in a much weaker form than the original state σ B and is therefore less useful as a result. This constitutes an irreversibility under the symmetry constraint, however it could arise due to the particular interactions used -might it be possible to use the state more wisely and not suffer such irreversibility?
There is a range of notions related to reversibility and irreversibility. In the simplest case an isolated symmetric, unitary evolution preserves all symmetry-breaking properties and conserves charges. There is also the notion of a catalytic use of a symmetry-breaking resource σ B in which a quantum process is performed In general theories of quantum resources (for example entanglement theory) such use of catalysts can have non-trivial effects and enlarge the set of accessible transformations on A.
However the above notions do not exhaust the possibilities. In [40] a phenomenon called catalytic coherence was discovered by Johan Åberg in which quantum coherence resources can be re-used in such a way that the state of the resource constantly changes, however its ability as a resource for inducing processes on multiple independent systems remains unchanged. The core setting involves a U (1) symmetry constraint associated to a 'number' operator N , with integer eigenvalues n ∈ Z, on quantum systems [16]. A coherent state σ B on a 'ladder' system, with Hilbert space H ladder spanned by the eigenstates {|n } n∈Z of N , is present and used to induce some target map E on a system A through the interaction HereẼ is an approximation of some target map E on the primary system. V is a bipartite unitary that respects the global U (1) symmetry constraint and takes the general form where forms an orthonormal basis for system A such that it transforms under the U(1) action as U (θ) |φ m = e imφ |φ m , the operators ∆ n−m are displacement operators on the ladder system ∆ n = j∈Z |j + n j| and U mn denotes the matrix entries of some arbitrary unitary with dimension dim(A). More generally, we will refer to any protocol that implements E of the formẼ (ρ) = n,i tr(∆ n σ)K n,i ρK † n,i (16) for operators {K n,i } on A, as simply a catalytic coherence protocol, without any further qualifications.
The state on B evolves non-trivially under the above V as σ B → σ B , however if one reuses B on another quantum system under precisely the same protocol V it was found that its ability to lift the symmetry constraint is undiminished (see [40] for more details). In what follows we shall use the term 'repeatability' to cover the above three distinct concepts, and which will be defined in the next subsection.
With these subtleties in mind we now study, in general terms, the use of symmetry-breaking resources and when they may be repeatedly used without degrading. In doing so we make use of the process mode framework, and demonstrate its utility for the analysis of such questions.
A. The use of symmetry-breaking resources and local simulation of quantum processes We assume the simulation of a quantum process E : B(H A ) → B(H A ) locally on A using a state σ B on the environment B, takes the form where V : is symmetric under the group action of G, and which for simplicity can be assumed to be an isometry. We are interested in the use of some resource σ B to induce E, independent of the state ρ.
More generally we might wish to simulate a set of local processes {E k } on the system A, which for simplicity we assume is a discrete set labelled by k. The general task is to devise a protocol that tries to achieve any one of these target maps when presented with an arbitrary quantum system B that is prepared in an unknown state σ B . Abstractly, given (E k , B), a protocol must specify a symmetric process V k : such that the approximate process induced using σ B , de-notedẼ k = P(k, B, σ B ), is given bỹ Generally, the performance of the protocol P is then determined by how closeẼ k is to the target process E k (using e.g. the diamond norm ||Ẽ k − E k || ) for given k and σ B . It is also natural to assume a general protocol has a perfect classical limit, in the sense that as the environment B becomes sufficiently large, and we are provided a state σ B that encodes group elements (asymptotically) perfectly, then the protocol P provides P(k, B, σ B ) = E k exactly for all k.
Since we are interested in studying irreversibility in the use of B, we define the back-action on the environment, FIG. 6: Repeatable use of symmetry-breaking system B. A system B is used sequentially to induce otherwise an inaccessible process on systems A 1 , A 2 , A 3 . . . A n . If the induced processes are identical for all n ∈ N then the system B is used with arbitrary repeatability, even if the state of B changes in time.
. Given these details we state a precise a notion of repeatability as follows.
Definition III.1. Let E : B(H A1 ) → B(H A 1 ) be a quantum process on A 1 , and let B be any other quantum system. We say that a protocol P for E is 2-repeatable if given any system A 2 isomorphic to A 1 it specifies symmetric processes tr A2,B (V(ρ 1 ⊗ ρ 2 ⊗ σ B )) =Ẽ(ρ 1 ).
, andẼ being the approximation to E using σ B .
An elementary aspect of the repeatable use of some symmetry-breaking resource σ B to induce a map E on a system A is that a subsequent use will also result in exactly the same quantum process. One can easily extend the above definition to n-repeatability where the same process E is induced on n identical systems using the same initial system B and we expand upon this in section E 2 of the Supplementary Material. This scenario is also depicted in Figure 6 in which a sequence or "circuit" of symmetric interactions V 1 , V 2 , . . . V n are performed on B so as to induce local processes on subsystems A 1 , A 2 , . . . A n .
We wish to study when the state σ B can be used in an arbitrarily repeatable way, namely n-repeatable for any n ∈ N. To determine this we can consider for any fixed total state on the systems A 1 , . . . A n , the induced channel F k under the protocol map V = V n • · · · • V 1 from the system B into any system A k , which describes the transfer of reference frame data needed to induce the local process on A k . The induced process for fixed input state ρ 1...n on A 1 , . . . , A n is given by where tr k denotes discarding all systems except A k . However for the particular case of ρ 1...n = ρ ⊗n , the nrepeatability implies that F k = F for all k = 1, . . . n, and so in this case the protocol results in the same channel from the reference frame B into each of the subsystems.
In entanglement theory one has the notion of an nextendible state, which gives a simple measure of the entanglement in the state. However this notion can be generalised to quantum processes, and relates directly to our present discussion.
Therefore we see that n-repeatability of a protocol involving B to simulate E implies the induced channel F from B to the output system A must be n-extendible. Given this, we can establish the following general constraint on any protocol that admits arbitrarily repeatable use of a resource σ B .
The proof of this is provided in section E 2 of the Supplementary Material. This places a strong constraint on the repeatable use of an environment B to induce maps on other systems. The content is easy to understand -if the environment acts as a reference system for an arbitrary number of systems then the only information that can be used must be classical information [83,84].
It is important to emphasize that this result applies to a circuit that induces a single quantum process E on the system A -simply interrogate the system B once, copy the measurement information and propagate it to an unbounded number of systems to induce E. This says nothing about whether the system B (which might be finite dimensional) suffers irreversibility in the process. In order to determine this we must consider protocols that use B for a second, independent quantum process E .
In the next subsection we use our earlier results on the decomposition of quantum processes to analyse coherence protocols and provide an account of how in the case of a non-commutative symmetry G that fundamental incompatibility in the use of symmetry-breaking resources is expected to arise.
B. Process mode picture: The repeatable use of quantum coherence.
We can now apply the process mode formalism to the question of the repeatable use of symmetry-breaking resources. We first look at a quantum subsystem A, and the task of inducing a target quantum process E on that system under a U (1) quantum coherence constraint. We shall determine precisely when we can induce E in an arbitrarily repeatable protocol using a coherent environment B. As discussed, the orbit M(G, E) of E under the group action encodes the reference frame data required from the rest of the global system. For the case of coherence G = U (1) it is clear that M(G, E) is either a point, for E being a symmetric process on A, or is a circle, when U g •E •U † g = E. The set of processes decompose under U (1) into a basis of process modes {Φ λ }, which are one-dimensional since the group is abelian. Again we absorb any multiplicities into the λ label for clarity of the exposition.
We are free to pick any point on M(G, E) which corresponds to a reference quantum process E 0 , and decompose this map in terms of E 0 = λ α λ (E 0 )Φ λ , with data {α λ (E 0 ) ∈ C} for the reference operation E 0 . With respect to E 0 the target map E is obtained from E 0 via some group transformation θ ∈ U (1), as E = U θ • E 0 • U † θ , and so To implement any such target map one needs to specify the group element θ and the constant invariant data α λ (E 0 ). Note that a perfect classical reference frame for the group U(1) is provided by the infinite dimensional space of wavefunctions on a circle L 2 (S 1 ). Using such a system for the environment B means that an arbitrarily repeatable protocol can perfectly achieve the target E by performing a measurement that estimates the group element and transmitting the classical result to arbitrary many systems. The non-trivial issue is to determine what form such protocols can take under a U(1) global symmetry constraint. This is our aim for the current section.
In particular we restrict to the case when B is an infinite-dimensional ladder system H ladder for two reasons. First there is a fundamental connection between L 2 (S 1 ) and H ladder such that the latter can also perfectly encode group elements θ ∈ U (1) into quantum states |θ ∈ H ladder that can be perfectly discriminated. This has technical subtleties for the continuous group (since we work in a separable Hilbert space), but in terms of the eigenstates {|n } n∈Z we can consider the following set of orthonormal 'states' that encode any θ ∈ U (1) which should be understood as being meaningful in a distributional sense. In terms of L 2 (S 1 ) this amounts to viewing the Dirac delta distribution δ(x − θ) to be a normalized 'wavefunction' on S 1 . We will refer to the set of all {|θ } θ∈U (1) as asymptotic reference frames. For an in depth analysis of this connection and further useful properties of these states we refer the reader to the excellent book [85]. The second reason why we restrict B to be H ladder is that we want to have an asymptotic classical limit in the environment such that an initial state σ B = |θ θ| induces the target operation E = U θ • E 0 • U † θ that is associated with the point θ on the process orbit M(E 0 , G).
We can now state the main result of this section.
Theorem III.4. If a protocol P to induce a local process E on A using a ladder system B satisfies: (a) Global U(1) symmetry.
(c) Asymptotic reference frames on B are not disturbed.
(d) Asymptotic reference frames on B yield perfect simulations. then P is a catalytic coherence protocol.
This provides a clear physical interpretation of the repeatable use of quantum coherence in simple physical terms. Note it does not imply that the system B is in some perfectly coherent state, or that the state of B stays the same -the repeatability holds irrespective of the state on B. The proof of this result is straightforward using process modes, and is given as follows.
Proof. From (b) we have that if the protocol P for target map E ∈ T (A, A ) as in Equation (22) is arbitrarily repeatable on system B then this implies from theorem III.3 that the induced maps take the form of a measure and prepare channel on B such thatẼ σ (ρ) = a tr(M a σ)Φ a (ρ). However we can decompose each Φ a into the complete process modes basis as Φ a = λ c λ,a Φ λ for constants c λ,a resulting in the induced target mapsẼ Since there is a global U (1) constraint, the action of the symmetry group on σ will generate the orbit of E σ . More concretely for any σ ∈ B(H B ): Now we substitute equation (24) into (25) to get that The process modes form a complete orthonormal set and transform as U θ • Φ λ • U † θ = e iλθ Φ λ . Therefore the coefficients associated to each Φ λ in the above must be equal and we have that for all λ-irreps and all θ ∈ U (1) Using cyclicity of the trace in the left-hand side of the above we can move the group action U θ on to the POVM element instead to obtain In particular because we are ranging over the whole Hilbert space σ ∈ B(H B ) the above implies that the following must hold for all θ ∈ U (1) We simplify the above equation using the notation X λ := a c λ,a M a and then the constraint in Equation (29) reduces to U θ (X λ ) = e −iλθ X λ for all λ and θ ∈ U (1). Using the above simplified notation the induced maps are given by:Ẽ Then assumption (d) implies that if σ = |0 0| then the reference map E 0 is obtained perfectlyẼ |0 0| = E 0 . Observe that it follows from Equation (23) that the asymptotic reference frames satisfy |θ = U (θ) † |0 . In particular orthonormality of the process modes implies that for all θ ∈ U (1) we must have that Assumption (c) implies that the POVMs must commute with the self adjoint operatorΦ associated with the asymptotic reference frames {|θ } θ∈U (1) , given bŷ Φ := 2π 0 θ |θ θ| d θ. This means that [M a ,Φ] = 0, and therefore M a (and each X λ ) will be diagonal in the asymptotic reference frame basis. Finally, we can write this as X λ = α λ (E 0 )e −iλθ |θ θ| d θ. However (see [85]) the displacement operators can be written as ∆ λ = e iλΦ . This implies that X λ = ∆ −λ and the maps induced by the protocol must take the form of : This implies the protocol must be a catalytic coherence protocol as defined above, which completes the proof.
For the case where B is a finite-dimensional system, a similar result can be established, but with additional qualifications. The back-action R on B must (in general) map into a slightly larger system C in order to maintain the repeatability condition (or otherwise it can only be n-repeatable for some finite n). This is discussed in section E 3 of the Supplementary Material. For finite ddimensional subsystems A, we also see that the POVM required only involves modes no larger than d. For example for the case of d = 2 we have that where x a , y a , z a ∈ R and we have ∆ ±1 = cosφ ± i sinφ being the only interaction term required between A and the environment.
More generally, from the perspective of M(G, E) this corresponds to the fact that the target map E only requires a resolution of the target point on M(G, E) to an angular scale at worst δθ ∼ 2π d , and so is as efficient as possible in the use of the reference. We can therefore view the protocol as involving a coarse-grained measurement ofφ, which can be made without disturbing a subsequent measurement.
In the case of coherence there is a single coordinateφ that must be extracted from the environment up to some resolution, and therefore the simulations of different coherent maps {E 1 , E 2 , . . . } on A are essentially equivalent. This is no longer true for more general symmetry groups, as we discuss shortly.

C. Interpretation of coherent protocol as broadcasting of reference frame data
One can understand this result from another informal perspective, which perhaps helps clear up some confusion that might exist on catalytic coherence. The system B can be continually reused, and its state will change continually under the protocol. Despite this, its ability to function as a coherence reference remains the same. One might feel that this clashes with cloning intuitionnamely quantum resources cannot be copied in general. However the coherence protocol should not be viewed as a cloning of reference frame data, but as the broadcasting of reference frame data to multiple systems. Broadcasting is a mixed state version of cloning in which one wishes to copy unknown quantum states {ρ 1 , . . . , ρ n } to multiple other parties. In the single copy case a state ρ k is transformed to a bipartite σ AB , such that the marginals are σ A = ρ k and σ B = ρ k . It is known [86] that a set of quantum states {ρ k } may be broadcast perfectly if and only if [ρ i , ρ j ] = 0 for all i, j.
The relevance for us here is that the coherent properties of the environment B are fully described by the expectation values ∆ k := tr[∆ k σ B ], and so we need only consider these degrees of freedom. However [∆ k , ∆ j ] = 0 for all j, k and so a state of the form σ B = 1 d (1 + k c k ∆ k + other terms) can have the ∆ k components of the state broadcast in the sense described. What is non-trivial to establish, is that this can be done under a global symmetry constraint. The classicality of the underlying data is the key point, and explains how the protocol works from a different perspective. This is also consistent with our analysis in the next section concerning the structure of symmetric bipartite processes.
D. Irreversibility under general symmetry constraints and a geometric perspective.
Given the analysis we have provided for quantum coherence, we might wonder if a similar construction applies for more general groups. For such cases, there is one simple way in which an environment B can be used in a repeatable way -namely we can embed the system's Hilbert space into the space of wavefunctions on G and perform the measurement that estimates groups elements {|g g|} on this infinite dimensional space. Since this extracts all the reference data from B into a classical form it can be copied and repeatedly used. However this assumes a very particular interaction, and that B can physically be embedded in the required infinite-dimensional system (which is a non-trivial assumption).
We can therefore ask if repeatability can occur for a general group G and a finite dimensional system B? For simplicity we can restrict to G = SU (2) and consider just the set of all axial processes as our target quantum processes. As already described, the process or-bit M(G, E) for these quantum processes is the 2-sphere S 2 , with coordinates (θ, φ). Now if arbitrary repeatability is present then we have by the same analysis that E = a tr(M a σ)E a where the POVM elements on B must supply the coordinates on M via the condition where {a j (E)} are the invariant data for the process orbit, and c j,a are the coefficients of E a in the SU (2) process mode basis. However, now an important distinction is made with the U (1) coherence case. The POVM that extracts the reference data from B must estimate a point on a sphere. In the classical limit one can have perfect resolution of any point (θ, φ) ∈ S 2 , however for finite dimensional B it is impossible to provide a perfect encoding on the point. Moreover, we know that quantum mechanics on S 2 is a phase space and so in the case that B is finite dimensional there will be a non-trivial uncertainty relation present. If, for example, B is a d-dimensional spin, then one has oper-atorsX i := r d 2 −1 J i for B that constitute non-commuting coordinates such thatX 2 1 +X 2 2 +X 2 3 = r 2 . This defines the so-called "fuzzy sphere" [87] in non-commutative geometry where one has a discrete representation of spherical geometry. In the d → ∞ limit this coincides with classical geometry, however for finite d has a fundamental lower bound on resolution and complementarity in measurements.
Therefore if one is using the system B within some globally symmetric process to represent M(G, E) ∼ = S 2 , then the complementary in measurements on this phase space will imply incompatibility in the use of symmetrybreaking resources. This incompatibility is not present for coherence, since essentially onlyφ is needed to supply the reference data.
More generally, the process orbit perspective suggests a form of quantum-mechanical irreversibility in the use of symmetry breaking resources that depends on whether the geometry that can be induced on M(G, E) by B is non-commuting or not. This is consistent with the asymptotic limit of classical reference frames in quantum theory for an arbitrary group G, and also with the case of quantum coherence, however we must leave any further analysis to later work.

IV. GLOBALLY SYMMETRIC QUANTUM PROCESSES
The analysis of II and III explains the physical significance of the process mode decomposition, and provides a compact perspective on the role of quantum reference systems for the implementation of a quantum process on a system. However it does not tell us how these resources and global processes are constrained under a global symmetry. So far we have only described how local quantum processes on a subsystem A decompose in the demands they place on B, which serves to encode reference data M(G, E). As mentioned, the choice of origin on M(G, E) is a gauge freedom corresponding to how A and B are jointly described. We now build on this and specify the structure of global quantum processes that respect the symmetry principle. This will lead in turn to the issue of gauging a multipartite quantum process from a global to a local symmetry, which includes traditional lattice gauge theories as a special case.
To begin with, we consider a bipartite split of the full quantum system into A and B. Some more notation is needed. For an quantum system with input Hilbert system H A and output Hilbert space H A we define Irrep(A, A ) to be the set of irreps of G in the product Moreover, given an irrep λ for a group G, we denote the dual irrep as λ * , where the dual Given this, we have the following basic result for how symmetric quantum processes for a group G decompose.
Theorem IV.1. Let G be a finite or compact Lie group and consider an input quantum system with Hilbert space H in = H A ⊗ H B and an output quantum system with Hilbert space H out = H A ⊗ H B , which carry unitary representations of G given by U A ⊗ U B and U A ⊗ U B respectively. Then every symmetric process E can be decomposed into symmetric operations The proof of this is provided in section C of the Supplementary Material. The result highlights the rigid structure of symmetric quantum processes, and physically states that the bipartite process is composed of invariant process modes, which involve internal exchange of asymmetry between A and B in a balanced way.
It also allows a diagrammatic representation of the components of such a quantum process E AB in terms of Φ (λ,m,m ) . In particular we can construct a natural basis of process modes from coupling input state modes with output state modes for each subsystem. To make this clearer we package this mode data in terms of a diagram label θ = [(a,ã) λ −→ (b,b)], and simply write Φ θ to denote where a,ã, b,b label in-coming and out-going states modes. To every such symmetric superoperator Φ θ there is the associated θ-diagram that has the following representation.
FIG. 8: A generic diagram for symmetric bipartite processes. Basis of superoperators for the space of symmetric, bipartite quantum processes T (H in , H out ). The λ-irrep arrow is associated to a (directed) flow of quantum information. For an abelian group G, this is a one-dimensional degree of freedom and so corresponds to classical data (e.g. can be cloned, as in the case of coherence). The diagram is represented mathematically in terms of incoming and outgoing asymmetry modes .
We adopt a convention with the direction of arrows in a diagram such that the dual diagram θ * has the reverse arrows to those in diagram θ.
Using this notation, we then have the following basic constraints, valid for any symmetric quantum process on a bipartite system.
non-trivial subsystems A, B, A , B , (b) globally symmetry, and (c) complete-positivity, then E must take the form where x θ ∈ C and where θ * is the dual diagram to θ. Moreover trace-preservation implies The proof is straightforward in the Choi-Jamiolkowski picture, and is provided in section C of the Supplementary Material .
Remark: It is interesting to note that the diagram θ = [(λ, 0) λ −→ (λ, 0)] is always forbidden within quantum processes. However it can be shown that such diagrams do contribute in process matrices that do not possess a definite global causal order [88].
A. Classes of diagrams -local, injection and relational.
We need to distinguish several types of irreps for what follows. For every quantum system the identity operator 1 ∈ B(H) is always a trivial state-mode (U g (1) = 1 for all g) that is required for a state to be normalised. We denote with state mode via a = 0. Any other trivial state mode is then a ∼ = 0. For the process mode label we need only specify whether it is a trivial irrep or not, and therefore write λ ∼ = 0 and λ ∼ = 0 respectively. For general bipartite scenarios we may decompose the set of all diagrams into three natural types: 1. The first class of diagrams are local diagrams for which λ ∼ = 0 and so involve no symmetry-breaking interactions. These are therefore symmetric under the local symmetry action.
2. The second class of diagrams we distinguish are injection diagrams for which λ ∼ = 0, and eitherã = 0 orb = 0. These describe the transference of asymmetry degrees of freedom from one side to the other. For example, these diagrams are required to take a symmetric state at A to a non-symmetric state via the injection of asymmetry from B.
3. The final class of diagrams we distinguish are relational diagrams for which λ ∼ = 0, andã,b = 0. These evolve the relational asymmetry degrees of freedom between A and B. As such, these diagrams only contribute when both the inputs at A and B carry symmetry-breaking degrees of freedom. In particular, they have no local affects at either A or B.
The first class is easy to justify, while the other two classes are best seen by example. In the next section we illustrate these features in the elementary case of processes on 2 qubits.

B. Illustrative examples -The set of 2-qubit symmetric quantum processes
To illustrate this we can consider the group G = SU (2) and analyse the set of 2-qubit symmetric quantum processes (we assume the input and output spaces coincide). Consider a bipartite system AB consisting of two qubits each of which carry the spin-1/2 irrep of SU (2). It is easy to see that Irrep(A, A ) = Irrep(B, B ) = {0, 1, 2} with multiplicities two for λ = 0, three for λ = 1 and one for λ = 2.
The set of all diagrams from which all 2-qubit symmetric processes are built, are shown in Figure (IV B).

FIG. 9:
The set of 2-qubit SU (2)-diagrams. All possible diagrammatic terms allowed for a 2-qubit symmetric quantum process. The space of valid processes is 13-dimensional.

Local processes on two qubits
The simplest bipartite symmetric processes are those for which λ ∼ = 0 in all process diagrams. In other words, no asymmetry flows between A and B. For these processes E is symmetric on both A and B separately, and form a two-parameter family of processes given by the product of partial depolarising processes These processes involve only two process modes locally, id(ρ) = ρ and Φ 0 (ρ) = σ x ρσ x + σ y ρσ y + σ z ρσ z .
2. Asymmetry injection processes on AB and simulation of processes at A.
The next simplest bipartite processes to consider are those globally symmetric quantum processes built from diagrams in the second class mentioned above. We can characterise the general set of processes that made use of asymmetry in the input state at system B and transfer it to the subsystem A. These are significant in being the only terms of relevance in any protocol where we wish to simulate or induce a target quantum process at A via asymmetry resources at B. Specifically, we would consider using a state σ B at B under a bipartite symmetric process V AB on AB as It can be proven (see Supplemental Material) that only diagrams in class-1 and class-2 contribute to E A , and since class-1 diagrams are purely local, the class-2 are precisely the diagrams that contribute to the injection of asymmetry from B into A or vice versa. The general structure of these processes are where E 0 is built solely from class-1 diagrams, and for any ρ we have tr B [E in,A (ρ)] = 0, and tr A [E in,B (ρ)] = 0 and so these components describe the injection into A and B respectively. In this we also include diagrams that maintain the local Bloch vectors (which are class-1 diagrams) since we are interested in inducing local processes. Each of these two terms form a 3-parameter family of maps given by for coefficients x, y, z, x , y , z chosen such that E is a valid quantum process. The diagrams involved are given by for simulation at A, and by for simulation at B. We can also describe the action of these processes via their action on a general 2-qubit state. We use the canonical form: for such a state, where local Bloch vectors a and b together with the correlation matrix T ij are chosen such that ρ AB is a positive matrix. Given the above parameterization of the general process E, we have that where the local Bloch vectors of the output states are given byã Here T the vector with components T k := i,j kij T ij . The geometric significance of this can be seen for the case of initial product states ρ AB = ρ A ⊗ ρ B with local Bloch vectors a and b then the correlation matrix takes the form of T ij = a i b j and therefore since kij a i b j = (a × b) k and so the vector T = a × b is cross-product between the input Bloch vectors at each site. More generally T is a vector component that describes the joint asymmetry of A and B, in contrast to a and b, which are purely local terms.
We may now set x = y = z = 0, and study the set of all processes involved in simulation at A. This is a 3-parameter family in (x, y, z) and so we can plot the allowed region in 3-D. The set of all such quantum processes is given by the convex set bounded by the paraboloid and the plane √ 3(1 + y) + x = 0. We can make a change of coordinates for which the quartic boundary (paraboloid) reduces to one of the 17 standard forms. Let Then the region of parameters (X, Y, Z) is given by the three dimensional convex set bounded by an elliptic paraboloid described by the equation, and the plane 2 + X − Y = 0. In Figure 10 we show this parameter region while highlighting the points corresponding to distinguished extremal processes. In particular, we find that the vertex of the paraboloid at (X, Y, Z) = (0, 0, 0) corresponds to the quantum process which is the result of discarding the input at A, performing an approximate Universal NOT gate on system B, and then injecting this into the output system at A. This approximate U-NOT turns out to actually be the optimal "spin inversion" that is allowed by quantum mechanics [89]. The intersection of the two boundary regions is an ellipse that we parametrise by a single coordinate φ. In terms of the parameters we have And gives rise to the one-parameter family of processes with The set of induced channels on A via a globally symmetric process on AB. Shown is the allowed parameter region for the rotated coefficients corresponding to each class-2 diagram appearing in a general quantum process that injects asymmetry. The boundary is described by the intersection of a plane and an elliptic paraboloid.
For the case of product input states with local Bloch vectors a and b at A and B respectively, we see that the set of accessible points form an ellipse displaced from the origin by 1 2 (a + b) with orientation defined by 1 2 (b − a) and 1 2 a × b. Also note that for φ = π 2 we have that the output on A is the input state on B. Therefore the line joining E UNOT and E φ= π 2 is the set of general depolarization processes on B with output sent to A.
We therefore find that the set of all processes using a qubit B to induce a non-symmetric process on A is to a good approximation given by the convex hull of the optimal U-NOT gate and the set of processes E φ with 0 ≤ φ ≤ 2π.

Purely relational processes
There are 5 diagrams in total in the class-3 resulting in the most general quantum processes that involve these type of diagrams taking the form of: Any such quantum process which contains only class-3 diagrams has the property that the output state E(ρ) always has maximally mixed marginals for all initial states. More precisely, for any ρ ∈ B(H A ⊗ H B ) we have that tr A (E(ρ)) = tr B (E(ρ)) = 1 2 1. This implies that we must for some correlation matrix R ij that depends on both ρ AB and the particular relational process. We can make more precise the contribution of each diagram in class-3 to the tensor R ij .
where we have denoted by R θm to be the correlation matrix of the output under applying the superoperator Φ θm to ρ AB . In other words, Φ θm (ρ AB ) = i,j R θm ij σ i ⊗ σ j . To explore such processes, we restrict to those that are invariant under swapping A and B. For this, the most general form is given by for real parameters x, y, z. Imposing that E is a valid quantum process implies the 3-d convex region of parameters with boundary surfaces given by the following quartics: In other words the boundary is the intersection of an elliptic cone, a parabolic cylinder, two intersecting planes and a hyperbolic cylinder respectively.
There are distinguished simple processes that correspond to points on the surface boundary. For instance the point (1, 0, 0) is the unique intersection of the two intersecting planes, the parabolic cylinder, the elliptic cone. It corresponds to the singlet preparation process E singlet (ρ) = |ψ − ψ − | for any 2-qubit state ρ. In addition there are two points that lie at the intersection of the elliptic cone the parabolic cylinder and the hyperbolic cylinder and each of them are in one of the two planes and correspond to imposing that x = 0. The characterising feature of the processes for which x = 0 is that they do not displace the maximally mixed state and are therefore unital processes. Corresponding to the two distinguished points we get the following quantum processes where E 1 corresponds to the point lying in the plane y = 1 2 (x − 1) and all the other three surfaces and E 2 corresponds to the point lying in the plane y = 1 2 (x + 1) and the non-degenerate three surfaces.
Both are only sensitive to the T ij components of the input state ρ. Without loss of generality, we look at how they act on states of the form: Moreover, up to local unitaries any such state can be brought to a canonical form . The range of these parameters lie in a tetrahedron whose vertices correspond to the Bell states.
The action of these processes on the Bell states φ ± , ψ ± is given by: and Since the Bell states are extremal the convex hull of these images give the action in the more general case. The image of the tetrahedron of state is graphically displayed in Figure 12. The preceding analysis can be used Under the extremal processes E 1 and E 2 the set of T-states is mapped into the (blue) triangle and inner (brown) tetrahedron respectively. on more general bipartite quantum systems, where it allows a compact book-keeping for the analysis of quantum processes and simplifies the analysis.

Symmetric unitary processes on two qubits
For the same of completeness, we briefly describe one more kind of symmetric process -the SU (2)-symmetric unitaries on two qubits. Since V = exp[itH] for some Hamiltonian H, the problem reduces to computing the allowed Hamiltonians. The symmetry of V implies that U g (H) = H for all g ∈ G and so H is an invariant operator under the group action. The space of invariant hermitian observables is spanned by 1 and σ x ⊗ σ x + σ y ⊗ σ y + σ z ⊗ σ z , and therefore the symmetric unitaries on the system is a two-parameter family given by The first term is a phase term and so V (t) = e it(σx⊗σx+σy⊗σy+σz⊗σz) is the only non-trivial unitary interaction present. The quantum process E(ρ) = V (t)ρV (t) † has a mode decomposition as shown in Figure 13. Note that because V is symmetric under swapping A and B, we have this additional symmetry reflected in the diagram contributions. The expansion in terms of process modes shows that the global unitary has a non-trivial structure under the symmetry action, which is perhaps not surprising since the process is perfectly reversible and keeps all symmetry properties in the state constant.

V. FROM GLOBAL TO LOCAL GAUGE SYMMETRIES FOR QUANTUM PROCESSES.
Gauge symmetries have played a deep and important role in modern quantum physics. However, traditional undergraduate accounts often phrase this gauging process in slightly opaque terms: it is first observed that the wavefunction is invariant under changes in global phases ψ(x) → e iα ψ(x). This invariance is then promoted somehow to a "local" invariance under changes of phase, so that ψ(x) → e iα(x) ψ(x). It is then observed that terms in the Lagrangian involving derivatives do not respect this local symmetry. A minimal coupling prescription is is a gauge potential coupling to a charge q, which transforms together with ψ(x) as This potential A(x) turns out to not be an observable, however one may form gauge invariant quantities, which are observables (for example the magnetic field in electromagnetism).
However, quantum states are not wavefunctions ψ(x), but (convex mixtures of) projectors ρ = |ψ ψ| on the system's Hilbert space H, and it is therefore automatically true that ρ is invariant under changes of phase |ψ → e iα |ψ , or any more general local changes of phase |ψ → k e iα(x k ) |ψ . The invariance of a quantum state ρ under changes in the particular ray representative |ψ is not gauge symmetry. Instead gauge symmetry, in the traditional sense, is a statement about a redundancy in the system's dynamics. In what follows we shall make use of the process mode formalism to provide an informationtheoretic account of gauge symmetries. In particular this account makes no requirement of a Lagrangian description, or that the dynamics is reversible.
A. An information-theoretic perspective In the previous section we analysed the structure of bipartite processes that are symmetric under the action of a group, via an action U g ⊗ U g . As mentioned, implicit in this symmetry action is a relative alignment of the systems, which is encoded in the choice of tensor product ⊗ for states on AB. We have also shown that this gauge freedom corresponds to an arbitrary choice of origin for the process orbit M(G, E). We also showed that the structure of bipartite processes is naturally analysed in terms of diagrams θ = [(a,ã) λ −→ (b,b)], which have a similar group-theoretic structure to Feynman diagrams for particles interacting via gauge bosons (e.g. electrons scattering via photons) [90]. Given these aspects it is therefore natural to ask if the freedom in choice of origin in M(G, E) coincides in a way with the more traditional notion that arises in gauge theories. To analyse this we describe how one gauges a general quantum process on a multipartite system from a global symmetry to a local symmetry.
Of relevance is recent work on gauging quantum states within the context of Tensor Networks [91][92][93][94]. Since unitary dynamics and state preparation are particular instances of quantum processes, our results can be viewed as generalizations that include both cases within a single setting -here our goal is to derive the structure of gauge theories using only primitive information-theoretic concepts such as quantum reference frames, and quantum processes on multipartite systems, and with as few assumptions as possible.
B. Gauging global symmetries for quantum processes -The core recipe.
We shall set the scene with a high-level account of gauging quantum processes on a multipartite system. We do not consider continuous quantum systems here, however it is natural to expect that these can be approximated through a lattice formulation. The basic setting is of a system comprised of subsystems A 1 , A 2 , . . . A n . A global symmetry action is present and takes the form U (g) ⊗ · · · ⊗ U (g) on the subsystems. In contrast, a fully local symmetry action would take the form U (g 1 ) ⊗ U (g 2 )⊗· · ·⊗U (g n ), with distinct group elements for each subsystem. We are given a process E on these subsystems that is symmetric under the global, but not the local action, and we wish to extend it to a processẼ on a larger system that is symmetric under the fully local symmetry. An informal algorithm that describes our gauging of a globally symmetric quantum process to a local one is as follows: 1. (Background systems) We define an array of quantum reference frames that function to encode relational data.
2. (Background dynamics) We define a quantum process for the collection of reference frames that is symmetric under the global symmetry.
3. (Gauging of symmetry) We discard our access to the relational data between subsystems, via a uniform average over the local symmetry group.
The convenience of this protocol is that it spells out the information-theoretic components involved in the gauging of general dynamics, and allows for generalizations. It shows that the gauge systems encode quantum information about the relative alignment of subsystems, and that the gauge interactions generated under this prescription depend on the information-theoretic properties of the quantum reference frame states, as we shall describe below. This however raises the question of how to interpret such quantum reference frames. One can view the relational data as merely a book-keeping device without physical content, however "information is physical" and so ultimately this data must be encoded in some physical degrees of freedom, which may or may not be background classical coordinates. However this general approach raises the possibility of analysing gauge-invariant scenarios that describe relational quantum physics, with no need for such a background classical regime.

C. Gauging 2-symmetric quantum processes
We have a quantum process E on a multipartite system A 1 , . . . A n , which is symmetric under a global symmetry U (g) ⊗ · · · U (g), and wish to gauge this to a process that is symmetric under local symmetry action U (g 1 ) ⊗ · · · ⊗ U (g n ). To make notation simple, we shall use g := (g 1 , . . . , g n ) to specify a group element in G ×n , the local symmetry group for the composite system. We write U (g) for the action on H A1...An and U g to denote the corresponding action on operators in B(H A1...An ). Then we have U g •E •U † g = E for all g = (g, g, . . . , g) with g ∈ G, and seek to extend this to a processẼ on A 1 . . . A n and some additional system R, such that U g •Ẽ • U † g =Ẽ for all g ∈ G ×n . For compactness we shall also use the notation U g [E] := U g • E • U † g . For simplicity we now assume that the multipartite process has a particular structure. Let Γ = (V, E) be the graph obtained by associating each subsystem to a vertex x ∈ V , and E denotes the set of all edges linking each subsystem. To each link l = [xy] joining x and y we pick an arbitrary but fixed choice of orientation.
Definition V.1. A quantum process on quantum systems A 1 , . . . , A n , with E : with c {(l k ,θ k )} ∈ C, and where we range over all ordered links l = [xy] ∈ E, between A x and A x , and where Φ (l,θ) is a θ-diagram term on A x and A y .
This definition has a simple physical interpretation. For example, a special case is if we have a spin lattice model with a Hamiltonian H involving pairwise Heisenberg interactions, and expand the unitary exp[itH] in powers of H, then we will have non-trivial terms acting on multiple systems, but they will take the form of being pairwise symmetric. This clearly generalises in an obvious way to 3-symmetric (and beyond), where we would consider not just directed links but also oriented triangles (or simplices) on the total graph. Including this would obscure the core ideas and also require a generalisation of Theorem IV.1, so instead we focus on the case of 2symmetric quantum processes for simplicity.

The inclusion of background reference frame systems
We first introduce an array of quantum reference frames that behave trivially under the global group action. Specifically to every link l ∈ E we place a quantum system, with Hilbert space H l whose principal role is to encode the relative alignment of the end-points of the link. This relative alignment is fully determined by a single group element in h ∈ G. More explicitly, since for h = g y g −1 x ∈ G, the fully local action on A x ⊗ A y differs from a global one by a single group element degree of freedom (on either subsystem).
The reference frame on l functions so as to record this relative alignment through an encoding h → σ h ∈ B(H l ). In order to be consistent with Equation (69) the action of the local symmetry group on a state σ h of l is given by This defines the symmetry action on the reference system l. For an initialisation of l in the state σ e we have that FIG. 14: Gauging a quantum process. Given a number of subsystems A 1 , . . . A n , we associate to each directed link l k a quantum reference frame (green ellipses) that encodes the relative orientation of the subsystems at its end-points. For the class of 2-symmetric processes the array of systems {l k } suffices to gauge the dynamics. For 3-symmetric processes, one must consider plaquette terms (yellow curved region), or equivalently the relative alignments of triples such as (l 3 , l 4 , l 5 ). The properties of the quantum reference frames determine the interactions between subsystems.
U g (σ e ) = σ e for a global action g = (g, g, . . . , g), while for a more general action the reference l encodes the relative alignment via e → g x g −1 y , as required. We do not need to make any assumption as to how well such an encoding can be done, however, modulo technical aspects, there always exists a classical encoding in which one has a set of perfectly distinguishable of states {|h } for a reference frame system, which carries a well-defined group action given on the basis via U (g)|h := |g x hg −1 y .

Specifying dynamics for the reference frame systems
Now crucially the quantum reference frames on the links become dynamical objects, and themselves must be subject to a quantum process. However we require the the total quantum process, on subsystems and reference frames, to be invariant under the full local group action. Moreover, we wish that any changes in the relative alignments of systems be encoded in the reference frames. Therefore we must define interactions between subsystems and reference frames that act non-trivially so as to accomplish this.
One could introduce arbitrary couplings between subsystems and reference frames and deduce how well they perform, however the simplest construction is to define couplings that naturally mirror with the process modes that we have introduced. These objects are defined as follows.
Definition V.2. A process gauge coupling, {A jk (l,λ) } for a quantum reference frame on a link l is a set of super- under the local symmetry action, and where x and y are the endpoints of the directed link l.
These process gauge couplings are essential for gauging the global symmetry to a local one, and if one views process modes Φ λ k as comprising a vector Φ λ = (Φ λ 1 , . . . , Φ λ d ) T of terms that transform irreducibly, then a process gauge coupling {A ij } can be viewed as comprising a matrix of process terms for which a local symmetry transformation on subsystem A x corresponds to left multiplication by the d × d matrix v(g −1 x ) of irrep components (for the irrep λ with dim(λ) = d), a symmetry transformation on subsystem A y corresponds to right multiplication by v(g y ), while the diagonal components of A are each invariant under global actions.
The definition ensures that any interactions with the subsystems has the correct form so as to encode relative alignments. However, as Proposition V.1 below shows, this makes no assumption about how well the gauge couplings actually encode the relative alignments.
Since we have restricted to processes that are 2symmetric, it suffices to describe the construction for a general bipartite superoperator term Φ (l,θ) = j Φ λ x,j ⊗ Φ λ * y,j with the link l joining A x and A y . The promotion of the globally symmetric process E to a locally symmetric oneẼ is implemented by first making explicit the background process. Since U g [A j,j (l,λ) ] = A j,j (l,λ) for g being a global symmetry action it is a "background scalar" under the global action and so can be included into Φ (l,θ) without affecting any symmetry properties.
The superoperator Φ (l,θ) acts on the subsystems in exactly the same way as Φ (l,θ) under the global group action -we have simply made explicit the background degrees of freedom. While the above describes the process couplings to the system, we must also ensure that the full process is completely positive and trace-preserving. In particular it is insufficient to include only interaction terms on the reference frames -there must be purely local terms on the reference frames so as to ensure trace-preservation. The details of this are not needed for our present analysis.

Gauging the process to a local symmetry
Having made explicit the background reference frame and process gauge couplings we promote the global symmetry to a local one by discarding relative alignments. This is done by averaging over all independent local group actions. The gauge-invariant process components are now obtained via G-twirling the superoperator Φ (l,θ) over the full local group G ×n , and are given by (74) By construction, this is invariant under the local symmetry group. Abstractly in the vector-matrix form described in the previous section, we can denote this as with the dot denoting summation over the adjacent indices. This makes the construction more transparent and allows us to write it in a compact form. It also highlights that the gauging procedure can be viewed as transforming a bilinear form (δ ij ) → A = (A ij ) defined independently on each link. The fully local process is theñ and the invariance of each term implies we have gauged the globally symmetric dynamics to a process with local gauge symmetry. We also note that in our construction there was no need for E to necessarily be a quantum process -the argument works equally well if E is any superoperator on A 1 , . . . A n that is 2-symmetric under the global symmetry. For example it might be more convenient to consider a Lindbladian L on the system that describes the dynamics throughρ = L(ρ), and perform the gauging on L instead of the resulting dynamics. We shall see an example of this in the next section.

D. Illustrative example: lattice gauge theory
We can now illustrate the previous procedure within the traditional context of unitary dynamics for a lattice gauge theory. Nothing we say in what follows is new in content -the purpose is to highlight an alternative information-theoretic perspective and to show consistency.
We follow the Kogut-Susskind Hamiltonian approach [95], where one retains a continuous time but discretizes space as a lattice, and consider a non-relativistic setting based on a two dimensional square lattice Γ = (V, E), which is a graph that consists of a collection of vertices/sites V connected by edges E, we assume the lattice spacing is of scale . We assume that every site x in the lattice Γ has associated to it a d-dimensional quantum system, with Hilbert space H On this Hilbert space we assume an irreducible unitary group action U of G, with matrix components (u kj (g)). The full Hilbert space H is the Fock space obtained by acting on a fiducial (vacuum) state |Ω with the set {a † k (x)} k,x∈V . Then N (x) := k a † k (x)a k (x) is the local particle density observable for the site x. In contrast, a 'kinetic' term is simply one that acts to decrease the particle content at one site, while simultaneously increasing it at another. We can consider a nearest neighbour hopping from a site x to a neighbouring one x+ , described by the hermitian operator The G = SU (d) group action on H x induces an action on the operators a k (x) and a † k (x), and sends a † k (x) → j u kj (g x )a † j (x). Therefore the operators {a k (x)} k form an ITO at every vertex x on the lattice. As before, we denote by U g the adjoint action under local symmetry transformations, where g := (g x1 , g x2 , . . . ) is an enumeration of the group elements at all sites. We then have Now if we have a local group action U (g x ) acting at x then N (x) is invariant and does not require any relational data to describe how the system at x is related to the system at other sites x . The hopping term however is not invariant under a purely local group action -although it is invariant under the global action, corresponding to the global symmetry. The total Hamiltonian for the lattice model is H = where ∼ x denotes that points to a nearest neighbour to x, and the resulting unitary evolution is symmetric under the global group action. Given any state ρ ∈ B(H) is given by E(ρ) = e −itH ρe itH and we have that U g [E] = E for all global actions g = (g, g, . . . , g). (79) We now wish to make purely local dynamical statements by gauging this unitary process. As discussed we introduce a quantum reference frame to every link l ∈ E in the lattice. Moreover we assume that each quantum system l carries a representation of G, which can perfectly encode the group element for the relative alignment of adjacent sites on the lattice. Specifically we have σ h = |h h| with {|h h|} being perfectly distinguishable set of pure states, and We must now construct process gauge couplings A (l,λ) = {A j,j (l,λ) } that encode the relative alignment of subsystems into the reference frames. This is done at the level of the Hamiltonian.
To derive these operators we consider the interaction term k a † k (x + ) ⊗ a k (x) and write it as (81) This is invariant under the global action. Finally, we discard the relative orientations of the two sites via averaging over the full local group.This correlates the subsystems to the reference frame and results (since G-twirling is a projection) in a locally invariant interaction. We have where we have generated a link operator L mn (l), that has the following properties L mn (l) = dhu mn (h)|h l h| (83) The second equation describes the gauge transformation of this degree of freedom. For simplicity we gauge the generator L(ρ) = −i[H, ρ] of the unitary dynamics. We have thaṫ The first term is invariant under the gauging, while the second is gauged as K →K = x, ∼x K(x, ), with the inclusion of the link operator given bỹ Therefore the gauged dynamics is given by the generator This describes the dynamics of the systems at the vertices -on top of this one must include kinetic terms for the links. A full treatment of this would be beyond the aims of the present work, and so we refer the reader to [96][97][98][99][100].
E. Fixing a gauge -from local to global symmetry.
In this context, we can also consider the opposite direction, namely how to go from a local gauge symmetry to a global one. We restrict our discussion to the case in which the reference frame can perfectly encode group elements in a basis {|g }.
The way in which this gauge fixing can be done is simply by pre-and post-selecting the reference frames onto particular group elements. This breaks the the local symmetry G ×n down to a particular global G symmetry.
Again, it suffices to consider gauging the two site case. The local symmetry is U (h,g) , which we wish to fix to a global action U (h(g),g) where we assume h(g) = wgw −1 , for some w ∈ G, and which defines the way in which the action at A 2 is related to that at A 1 .
The gauge-fixing is achieved as a pre-and postselecting of the form where Π h (σ) = |h h|σ|h h|, is the projection onto the pure state |g . The projection id ⊗ Π h breaks the G ×2 symmetry action U (h,g) to the global symmetry action U g := U (hgh −1 ,g) , for any g ∈ G. Note that and so the passage between global and local symmetry coincides with the degree of freedom discussed in section II, for the relative alignment of two subsystems. More details on this gauge-fixing can be found in section D 2 of the Supplementary Material 4 .

F. A resource theory perspective on gauge dynamics and Gauss' Law
Having described how the gauging procedure coincides with the traditional unitary dynamics on a lattice approach, we can briefly discuss how the gauging procedure looks from the perspective of quantum resource theories.
We have presented a general construction in terms of a quantum process and quantum reference frames in which a symmetry principle is promoted from a global action to a local one. In the resource theory of asymmetry, and quantum reference frames, this symmetry defines the freely preparable states (or 'free states') of the theory. In particular, under the full local symmetry constraint we have the elementary information-theoretic result that any composite state ρ cannot be distinguished from This fact can be used within the resource-theoretic approach to determine the observables that can be measured under the symmetry constraint [101][102][103][104]. In the language of gauge theories however these observables are called "physical observables" and the states for which G[ρ] = ρ are called the "physical states" of the theory. SinceẼ is symmetric under the local symmetry group we have thatẼ(G(ρ)) = G(Ẽ(ρ)) = G(Ẽ(G(ρ)). Therefore the dynamics preserve the set of all symmetric states, which is a minimal requirement for consistency. In the language of asymmetry resource theory, these gaugeinvariant processes are precisely the free operations of the theory. Now, the states for which ρ = G[ρ] are convex mixtures of pure states with support in the eigenspaces of commuting observables {O k } built from the generators {J c } of the group action U (g) = e i c θcJc , and where θ c are local group parameters. However it turns out this condition is a generalized form of Gauss' Law.
We can outline why this is true for the lattice gauge system. The local group representation is g → U (g) and has independent group parameters g x defined at each site x on the lattice. Therefore the representation can be written as where, local to each site x, we have θ c (x) ∈ R and with the operators {J c (x)} being the local generators of the group action. However it is important to note the the operators J c (x) act non-trivially only on the vertex x quantum system and also on the quantum systems residing on the four adjacent links around x. The "physical Hilbert space of states" is defined as the span of the gauge invariant vectors |ψ that obey O k (x)|Ψ = s k (x)|Ψ all x and for a maximal commuting subset of observables {O k } obtained from the generators [96][97][98][99]. The eigenvalues {s k (x)} are called "static charges", since they are constants of any gauge-invariant evolution. Typically it is assumed that there are no static charges and so the physical space of states is the null space of the above observables.
For the case of G = U (1) we simply have a scalar number degree of freedom at each site, and a single generator J(x) at each site. Denoting the lattice vectors as in the horizontal direction, and in the vertical direction for a 2-d square lattice. It turns out (see [96][97][98][99] or the recent review [100]) that this decomposes into a term q(x) that is purely local to x, and link operators E(x + y) acting on the directed link joining x to y. More explicitly, it takes the form (94) Thus in the limit | |, | | → 0 the set of physical states are required to obey which is Gauss' Law for the electric field E(x) at the point x in terms of the local charge density q(x), as claimed. Therefore from the resource-theoretic perspective, the Gauss law coincides with the condition that we can only freely prepare states for which G[ρ] = ρ, and refer the reader to the literature for more lattice details [96][97][98][99]. For convenience we summarize this resourcetheoretic perspective.
Result V.1. In the resource theory of asymmetry for a local gauge group G, the free states of the theory coincide with the set of all convex mixtures of pure quantum states |Ψ Ψ| that obey a generalized Gauss' Law. The set of free operations within the resource theory coincide with the set of all locally gauge-invariant processes.
Note that one can define observables on the reference frames via Wilson loop operators, which are also fully invariant under the local group action. The basic loops are around individual plaquettes of the lattice and give rise to terms where l 1 , l 2 , l 3 , l 4 form the boundary of the plaquette p, and there is an implicit summation and trace over the m, n indices of L mn (l) viewed as a matrix of operators. It is readily seen that U g [W p ] = W p for all g ∈ G ×n in the local symmetry group. We leave a more detailed analysis to later work, but highlight that viewing the gauge symmetry from a resource-theoretic perspective is convenient if one is interested in studying entanglement in gauge theories.
To finish off this section, we can illustrate the role of the reference frames via the following simple result that describes the extremal situation in which the reference frame dynamics completely fails to encode any relative alignment. The statement is as follows.
Proposition V.1. Given any 2-symmetric multipartite process E on subsystems A 1 , . . . , A n and an associated array of reference frames as described above, with process gauge couplings {A (l,λ) }. If it is the case that U g [A m,n (l,λ) ] is independent of either g x or g y for all g ∈ G ×n , and for all l, m, n, then the corresponding gauged processẼ takes the form E = k1,...,kn+1 c k1,...,kn+1ẼA1,k1 ⊗Ẽ A2,k2 ⊗. . .Ẽ An,kn ⊗Ẽ R,kn+1 , where c k1,...,kn+1 ∈ C,Ẽ Ai,j is a local map symmetric under the group action on A i for all i = 1, . . . n, and E R,j is symmetric on the link reference frames.
Proof. The gauging of the global process E involves Gtwirling the subsystem process mode terms and the process gauge couplings. If U g [A m,n (l,λ) ] is independent of g x , then we havẽ where the last line follows from the Schur orthogonality theorem applied to the integration over g x . Therefore all non-symmetric interaction terms inẼ between the subsystems must vanish, and the only terms that can remain form locally symmetric terms {Ẽ k } as claimed.

VI. DISCUSSION
We started with two core questions, concerning symmetry principles in general quantum processes. We introduced the basic notion of process modes {Φ λ k }, and demonstrated how they allow an intuitive way of decomposing general processes and to describe the flow of symmetry-breaking degrees of freedom.
The process mode formalism leads naturally to several useful features, such as the process orbit, which provided a means to analyse the use of quantum resources to induce otherwise forbidden processes on a subsystem. The formalism also provided a means by which one can extend techniques used in Lagrangian formulations to more general quantum processes. In particular, we described the gauging of a globally symmetric quantum process to one with purely local symmetries in terms of process modes and process couplings. This gauging reproduces traditional results in lattice gauge theories, however it allows one to go beyond such regimes.
Although not discussed, the formalism also reproduces recent constructions in Tensor Networks that involve gauging quantum states [105]. This can be seen by noticing that a quantum state can itself be viewed as a completely positive trace preserving map from C into B(H), with 1 → ρ. The gauging of this under the procedure described will reproduce the gauged quantum state result in [94].
As mentioned, a future topic of interest within this formalism is entanglement in gauge theories [106][107][108]. However entanglement theory is best described in terms of the resource theory of Local Operations and Classical Communications (LOCC) [45]. This setting does not readily admit a Lagrangian description and so one might expect that the formalism that we have presented would be ideally suited for tackling such features in systems with gauge symmetry.

VII. ACKNOWLEDGEMENTS
We would like to thank Iman Marvian, Matteo Lostaglio and Kamil Korzekwa for useful discussions on these topics. CC is supported by EPSRC through the Quantum Controlled Dynamics Centre for Doctoral Training. DJ is supported by the Royal Society.
We use H A to denote the Hilbert space associated to a quantum system A, and B(H A ) to denote the set of (bounded) linear operators on H A . A quantum process E : B(H A ) → B(H A ) is a completely-positive tracepreserving superoperator taking states ρ A ∈ B(H A ) into states E(ρ A ) ∈ B(H A ) for an output system A . We denote the space of superoperators Φ : B(H A ) → B(H A ) by T (A, A ).
By Wigner's theorem, a symmetry on a system A is represented by either a unitary or anti-unitary action on H A . In this work we consider only unitary actions. Associated to a symmetry group G we have a unitary representation U : G → B(H A ), with U (g) being unitary on H A for all g ∈ G that respects the usual group composition rules.
Since we work at the level of density operators and processes, it is convenient to use additional notation. For any X ∈ B(H A ) we denote the adjoint action as U g (X) := U (g)XU (g) † . In a similar way we can define a group action on superoperators We also use the short-hand U g [Φ] := U g • Φ • U † g . We make use of vectorization of linear operators extensively, and use a modified version of the notation in [109]. Given a linear map L : H A → H B we can define its vectorization, denoted |vec(L) which is a vector in H B ⊗ H A , by the following method. For L = |a b|, with {|a } and {|b } being computational bases for the two spaces, we define |vec(L) := |a ⊗ |b . (A1) The vectorization of a more general linear map L = a,b L ab |a b| with L a,b ∈ C is then fully specified by demanding linearity hold: |vec(L 1 + L 2 ) = |vec(L 1 ) + |vec(L 2 ) for all linear maps L 1 , It is then easy to verify the following two central properties of vectorization: for all linear maps between the appropriate spaces. The first relation is powerful in the context of entangled bipartite quantum systems, while the second simply says that the mapping vec is an isometry between the Hilbert space H B ⊗ H A and the space of linear maps from H A to H B with the Hilbert Schmidt inner product L, M := tr(L † M ). The application of these relations make the following easy to establish Lemma A.1. Given two quantum systems A and B that are isomorphic we have that for all L, M ∈ B(H A ) and for all unitaries U ∈ B(H A ), and where F := |vec(1) vec(1)| T B = |ab ba| is the swap operator on AB.
These relations generalise to the case where A and B are not isomorphic, and where we allow M to map into a different space, by observing that the smaller system, B say, has H B isomorphic to a strict subspace of H A .

Representations of superoperators
Given a superoperator Φ ∈ T (A, A ) we can represent it in a number of different ways. The Choi representation with inverse relation given by for any X ∈ B(H A ). The Kraus decomposition of Φ is given by are the set of Kraus operators. This automatically implies that the corresponding Choi operator is given by The vectorization map gives another representation K(Φ) ∈ B(H A ⊗ H A ) via the expression K(Φ) : |vec(X) → |vec(Φ(X)) for all X. It is easy to verify that We also have that Φ is a quantum process if and only if A k = B k for all k and k A † The Steinspring dilation (V, H B , σ B ) provides a final representation for a quantum process Φ ∈ T (A, A ) given by where V : H A ⊗ H B → H A ⊗ H C is an isometry (V † V = 1), and σ B is a fixed quantum state on an auxiliary system B, which can be taken to be pure.
Appendix B: Decomposition of quantum processes

Representations and tensor product representations
Given a fixed group G one can usually classify and construct every irreducible representation for that particular group. These are exactly those representations which do not have a proper subrepresentation and therefore they contain no subspace invariant under the action of all group elements. We will be dealing with compact Lie groups G and for these types of groups all their irreducible representations are finite dimensional. We denote byĜ the set of all irreducible representations of G. Each irreducible representation is uniquely determined in a canonical way by a distinguished vector which we generically denote by λ ∈Ĝ and is called the heighest weight vector. A λ-irrep acts on an dim(λ) vector space V λ with an irreducible representation v λ : G −→ GL(V λ ) that has matrix coefficients v (λ) mm (g) determined by some fixed basis choice for V λ . In particular they satisfy Schur's orthogonality relations (which are valid for any compact group) for any λ, µ ∈Ĝ: For any unitary representations U A : G −→ B(H A ) the Hilbert space H A has a canonical decomposition into subspaces on which the group acts irreducibly. Formally we can write where α is a multiplicity label counting the number of times an irreducible representation appears in the decomposition of H A . The symmetry of the system A which manifests itself through the unitary representation U is the only property that dictates which irreps and corresponding multiplicities appear in the decomposition. Since we will be interested in bipartite systems H A ⊗ H B we want to know how one can decompose this space into irreducible components. Suppose that U B : G −→ B(H B ) is a unitary representation of H B then there is a tensor product representation acting on the compos- For example in the case of SU(2) the irreps are labelled by positive half-integers j ∈ {0, 1/2, 1, 3/2, ...} and have dimension 2j + 1. The tensor product representation of two irrep j 1 ⊗ j 2 decomposes into irreducible components according to the Clebsch-Gordan series j 1 ⊗ j 2 = |j 1 − j 2 | ⊕ ... ⊕ j 1 + j 2 . These correspond physically to the possible total angular momentum values that arise when coupling a particle with spin j 1 with another with spin j 2 . Notice how there is only one configuration for each value of the total angular momentum meaning that each irrep in the decomposition appears with multiplicity one. While this is not necessarily the case for general compact groups G similar techniques can be applied there to obtain the canonical decomposition of tensor product representations. We summarise below how these apply generally and refer to [78] for a detailed analysis.

a. Detour into generalised Clebsch-Gordan coefficients
Let U µ and U ν be two irreducible representations of G and assume these are realised on the vector spaces V µ and V ν respectively where µ, ν ∈Ĝ. Under the tensor product representation the space V µ ⊗V ν decomposes into irreducible components: where m λ is the multiplicity of the λ-irrep. This implies that the product of representations U µ ⊗ U ν is unitarily equivalent to a block decomposition where each block is an irreducible representation of the group. One can write that for all g ∈ G for some unitary matrix C which represents nothing more than a change of basis in V µ ⊗V ν from the tensor product basis to a basis that achieves the decomposition. The entries of this matrix are what we call the Clebsch Gordan coefficients (CGC) and provide a generalisation to arbitrary compact groups G of the coefficients that appear when coupling angular momentum states. where the coefficients µ, m; ν, n|λ, α, k represent entries for the unitary matrix C. The CGCs depend on the choice of orthonormal basis in the spaces V µ , V ν and V λ,α . Beyond orthonormality relations inherited from the unitarity of C, the generalised Clebsch-Gordan coefficients posses many different types of permutation symmetries and they are non-zero when particular types of relations hold. Within quantum mechanics these relations are exactly the ones that give the selection rules.
The problem of determining the multiplicity m λ of each irrep in B3 for the general linear group of fixed dimension n is sharpP -complete and it can be approximated with a randomized polynomial time algorithm [110]. The problem of determining CGCs is in the NP-complete class [111].

Irreducible tensor operators
The structure of H A provided by the symmetry carries over to higher-level Hilbert spaces such as B(H A ) and T (H A , H A ) in such a way that it respects their algebraic structure. The mathematical construction that will allow us to upgrade the decomposition of the Hilbert space H A into irreducible components to the decomposition of B(H A ) are called irreducible tensor operators.
Definition B.1. Let G be a compact group and U a unitary representation of G on the Hilbert space H A . Then for every irreducible representation λ ∈Ĝ define the irreducible tensor operators (ITO) to be the set of operators {T in B(H A ) such that for all g ∈ G: kj are matrix coefficients of the λ-irrep and ranges over all irreps in the decomposition of the representation U ⊗ U * .
The action of the group G on the space of operators B(H A ) is given by the adjoint action U. Therefore there is a canonical decomposition for B(H A ) into irreducible components such that U acts like an irrep when restricted to each subspace. There is a natural isomorphism between B(H A ) and H A ⊗ H * A but since we can identify any Hilbert space with its dual we can identify the space of operators with two copies of H A carrying the representation given by U ⊗ U * . This means that all irreps that appear when decomposing B(H A ) into irreducible subspaces under U are exactly those that appear when decomposing H A ⊗ H A into irreducible subspaces under U ⊗ U * . The following lemma makes this point precise and shows that the set of all ITOs forms an orthonormal basis for B(H A ). for λ ranging over all irreps (including multiplicities) that appear in the representation U ⊗U * . For every λ the set of ITOs under the adjoint action transform irreducibly. Particularly U g acts on the O λ,α := span{T λ,α k : 1 ≤ k ≤ dim(λ)} in the same way as does the irreducible representation of highest weight λ. This corresponds to the λ-irreducible component of multiplicity α in the decomposition of B(H A ). Then the space of operators splits into: Since there is clearly an underlying choice of basis for the irreducible tensor operators there is a sense in which the above decomposition is not entirely unique. However at the high level of the structure of the decomposition there is no freedom to mix operators belonging to different irreducible components. Denote the λ-mode by A λ = α O λ,α the full λ-irreducible component where we have summed over all copies of the λ irrep that appear in the decomposition of B(H A ). Therefore the space decomposes in a unique way into subspaces: Then given any density matrix ρ ∈ B(H A ) we can effectively decompose it into modes of asymmetry according to: where each of the ρ λ ∈ B(H A ) represents the orthogonal projection of ρ onto the λ-mode, that is onto the subspace A λ of B(H A ). While indeed some of the projectors above can be zero, the decomposition of ρ into asymmetry modes will be unique because the coarse grained structure of B(H A ) given by the unitary representation U g is rigid and always fixed by the symmetries of the underlying Hilbert space.

a. Uniqueness of the ITOs and asymmetry modes
In order to construct a fixed set of ITOs for the space of operators B(H A ) there are two underlying choice of basis: i) the basis for the Hilbert space H A and ii) a basis for each irreducible representation resulting in a fixed set of matrix coefficients v λ kj (g). More specifically the mode decomposition (the coarse grained structure) B(H A ) ∼ = λ A λ is always unique and depends upon the symmetry of the Hilbert space so only on the unitary representation U . Once we have fixed a basis for the underlying Hilbert space then the finer-grained decomposition B(H A ) ∼ = λ,α O λ,α becomes unique. Finally whenever we have fixed a basis for the irrep v λ kj this implies that we fix the ITOs all together and particularly we fix the vector-component label associated to that particular irrep.

Irreducible tensor superoperators
A similar type of structure we find when dealing with the space of superoperators T (A, A ) and we can further upgrade the irrep decomposition at the level of superoperators by defining the analogue of ITOs: in T (A, A ) that transforms under the group action as: where v λ kj are matrix coefficients of the λ-irrep and ranges over all irreps in the decomposition of U ⊗ U * ⊗ U ⊗ U * .
In the main text we use the term process modes for these ITS.
The space of superoperators T (A, A ) has a similar structure in the sense that can be decomposed into irreducible components according to the underlying symmetries. In particular the set of all ITS forms a basis that achieves this decomposition. Therefore when we take into account the possibility of multiplicities to write: where we sum over all λ-irreps and corresponding multiplicities α. The following lemma gives a rigorous proof of this: forms an ITS for T (A, A ) then for each λ-irrep in the decomposition of The above lemma allows us to construct ITSs form ITOs which in turn can be obtained by vectorising basis vectors for irreducible components appearing under the decomposition of the required representations. In light of Lemma B.4 we can define an inner product on T (A, A ) as ) . This ensures that the complete set of ITS for T (A, A ) forms an orthonormal basis.
Similarly to our previous discussion for ITOs the ITS also rely on particular basis choices: i) for the input and output Hilbert spaces H A and H A ii) for the matrix coefficients/ irreducible representations and therefore are not uniquely determined by how they transform under the group action. However the coarse-grained decomposition of T (A, A ) into irreducible components is unique in the sense that for any E ∈ T (A, A ) the orthogonal projection onto the λ-irrep isotypical component (i.e including multiplicities) given by α span{Φ λ,α k : k = 1, ...dim(λ)} does not depend on the underlying choice of basis: where tr(v λ (g)) is just the character of the λ-irrep and it is independent on the choice of basis that give the matrix coefficients. Furthermore fixing a basis only for irreps and consequently their matrix coefficients then we obtain the asymmetry modes E λ k of the superoperator E and these are the projection onto the subspaces α span{Φ λ,α k } for any fixed k and λ: Therefore any E ∈ T (A, A ) can be written as: where the projection on any λ-isotypical component is independent on the choice of basis that fixes the ITS and E λ k = α Φ λ,α k E, Φ λ,α k depends only on the choice of basis for the v (λ) irrep.

Constructing a natural basis of ITS for T (A, A )
By building upon notions of ITOs we provide a way to construct an orthogonal basis of ITSs for the space of superoperators. While the construction does not a priori assume any choice of basis for the underlying Hilbert spaces or irreps -in practice for computational purposes these will be hidden behind the ITOs and determining these requires basis choices.
The irreducible representations that appear in the decomposition of T (A, A ) are exactly those found in the product representation U ⊗ U * ⊗ U ⊗ (U ) * and we have denoted the set of all such irreps by Irrep(A, A ). Each λ ∈ Irrep(A, A ) can be thought of as arising in the tensor product of a a-irrep in the decomposition of U ⊗ U * with aã-irrep in the decomposition of U ⊗ (U ) * . This represents a useful way to keep track of multiplicities of each irrep in T (A, A ) which takes into account the natural way in which superoperators act on the input and output systems. This is becauseã and a also labels irreps in the decomposition of the output space B(H A ) respectively the input space B(H A ). We then say that the λ-irrep has an associated multiplicity label that we denote by m λ = (ã, a). While in practiceã and a-irrep themselves can carry multiplicities as well as giving rise to more than one λ-irrep in the tensor productã ⊗ a we will often suppress this for simplicity of notation. We only make it explicit when it becomes a relevant issue. For instance one should keep in mind that in the case of SU(2) the tensor product of two irreps does not contain irreps with multiplicity greater than one in their decomposition. Proof. We just need to check that under the group action where the matrix coefficients for the λ-irrep need to be consistent with the choice of basis assumed by the CGCs. Since {Tã m } in B(H A ) and {T a n } in B(H A ) are ITOs then they will transform as: and therefore the set of superoperators defined in the Equation B15 will transform as: nn Tã m tr(T a n ρ) (B17) Taking into account the definition of CGC and the fact that they represent a unitary change of basis then they must satisfy: nn . For more details on the relations between CGCs and matrix coefficients we refer the reader to [78]. Finally by substituting this into Equation B17 we obtain kk (g)Tã m tr(T a n ρ) In light of the above theorem we can identify a special type of ITSs that have support only on a single irreducible component in the input space and map it to a single irreducible component in the output space.
Definition B.6. We define a complete set of primitive ITS in T (A, A ) to be the ITS set {Φ λ,m λ k } k,λ such that for any λ ∈ Irrep(A, A ) and any corresponding multiplicity m λ there exists irreps in the input and output spaces labelled by a andã respectively such that for all k the ITS Φ λ,m λ k : V a −→ V ã is supported only on the a-irrep subspace of B(H A ) and with range included only in thẽ a-irrep subspace of B(H A ) While every primitive ITS will take the form given by Theorem B.5 for some choice of ITOs in the input and output spaces it is generally not true that any ITS  FIG. 16: The edges label asymmetric resource species while the vertex represents the interaction of asymmetric resources. Mathematically it is a diagramatic representation of an intertwiner between couplings of irreducible representations. Physicallydue to the directionality provided by the quantum operations -it has the interpretation that the input asymmetric mode interacts with some external asymmetry to produce the output asymmetric mode. We can associate one such diagram to every irreducible tensor superoperator that decomposes the space can take this form. However the primitives are building blocks that allow us to construct any general ITS and therefore any basis that achieves the decomposition of T (A, A ) into irreducible components.
Corollary B.7. Any λ-irrep ITS in T (A, A ) can be written as a linear combination of λ-irrep primitive ITSs.
Moreover we draw attention on the fact that the construction of primitive ITS does not rely on a particular choice of basis for the input or output states since the only defining property is that it acts non-trivially only on a particular irreducible subspace and transforms it into another irreducible subspace.

Homogeneous spaces
Given a topological space M a group action of G on M is defined formally as · : G × M −→ M such that it maps for every g ∈ G a point x ∈ M into g · x ∈ M and satisfies i) g 1 g 2 · x = g 1 · (g 2 · x) and ii) e · x = x where e is the identity element in G. A transitive group action is one that allows to obtain every point on the manifold from an arbitrary initial point by acting with some group element: ∀x, y ∃ g ∈ G such that g·x = y. A homogeneous space is a space that has a transitive group action. Moreover all of these can be realised as quotient spaces G/H = {gH : g ∈ G} for some closed subspace H of G carrying the subspace topology.

a. Spherical Harmonics
The spherical harmonics form a basis for the space L 2 (S 2 , C) of complex wavefunctions on the sphere. These are picked out through the decomposition of the space intro irreps. For spaces of functions this is conveniently expressed in terms of Lie derivatives. The Lie derivatives corresponding to the Lie algebra generators are given by L z and L ± = L x ± iL y , and are obtained from the angular momentum operators (L x , L y , L z ) defined in spatial coordinates as L i = jk ijk x j ∂ j . Going from cartesian to spherical coordinates (θ, φ) it is easy to check that • In terms of these differential operators we require that It can be shown that imposing orthonormality implies that where P jm (x) are Legendre functions, with

. Harmonic analysis on homogeneous spaces
We will be concerned with functions on homogeneous spaces and generalisations of spherical harmonics. We denote the space of square integrable functions on the compact 5 group G by by L 2 (G) = {f : G −→ C : It carries a natural group action given by the left regular representation for all g, h ∈ G. From the perspective of group theory function spaces are useful objects because they encompass the representation theory structure of the respective group. In particular every λ-irrep of G is realised in L 2 (G) under the above group action with multiplicity equal to dim(λ). This is the content of Peter-Weyl theorem which importantly leads to the Fourier series decomposition of functions when the group G is abelian. One of the consequences of this result is that any function f ∈ L 2 (G) can be written as linear combinations of matrix coefficients ranging over all irreps for some complex coefficientsf (λ, i, j) ∈ C that can be recovered via the integral formula:f (λ, i, j) = G f (g)(v λ i,j (g)) * d g which follows directly from orthonormality of matrix coefficients. For abelian groups G = U (1) the irreps are one-dimensional and the matrix coefficients are given by the usual exponentials (which also correspond to the characters of the irreps) e iλg for g ∈ U (1). Then Equation B25 becomes the Fourier series decomposition of f ∈ L 2 (U (1)).
Spherical harmonics arise naturally when we consider functions on homogeneous spaces G/H and at the heart of it lies a generalisation of Peter-Weyl's theorem to such spaces. A function on a homogeneous space G/H is an L 2 (G) function that is constant on the left cosets. This means that for all g ∈ G and h ∈ H f (gh) = f (g). For these types of function the decomposition into irreducible matrix coefficients as given in Equation B25 simplifies. In particular only those matrix coefficients that are constant on the left cosets appear in the expansion. In defining spherical harmonics we require the subgroup H to be a massive subgroup. A subgroup H is called massive whenever for any λ-irrep given by v λ if there exists a non-zero H-invariant vector |n in the λ-irrep carrier space such that v λ (h) |n = |n for all h ∈ H then it is unique (up to scalar multiples). This ensures that each λ-irrep in the decomposition of L 2 (G/H) into irreducible components has multiplicity at most one. 6 In particular each such unique irreducible subspace is spanned by an orthonormal set of functions on G/H which are exactly the spherical harmonics. The associated spherical harmonics give an explicit such basis with respect to some fixed orthonormal basis {|e k } for the λ-irrep carrier space where we identify |e 1 := |n with the unique H-invariant vector for the λ-irrep and are given by: for k = 1, ..., dim(λ) and gH a coset representing an element in G/H. It is straightforward to check that v λ k1 (g) are invariant on left cosets and therefore the above are well defined (orthonormal) functions on L 2 (G/H) that span the λ-irrep subspace.
Spherical harmonics have many useful properties and there are different equivalent viewpoints to the above: through the Lie derivatives, as eigenfunctions of invariant differential operators. However we will be mostly concerned with how they transform under the action of the left regular representation. From the above definition it is straightforward to establish that they transform similarly to irreducible tensor operators. Therefore for any g ∈ G, λ-irrep appearing in L 2 (G/H) and k = 1, ..., dim(λ) A rigorous account of spherical harmonics on general homogeneous spaces can be found in Chapter 2 of [81].

Axial quantum operations
Before we prove the main theorem regarding axial operations we will take a closer look at the technical aspects involved in defining the process orbit. Given a fixed operation E ∈ T (A, A ) with input and output spaces carrying representations of the general compact group G one can construct the stabiliser (or isotropy) group consisting of all those elements h ∈ G that leave the operation unchanged: Stab(E) = {h ∈ G| U h [E] = E}. This also defines an equivalence relation between the group elements by identifying any two that belong to Stab(E) and therefore we can construct the quotient space which for us corresponds to the process orbit M(G, E) := G/Stab(E). By the orbit-stabiliser theorem the process orbit is homeomorphic (for compact groups) to the orbit of E under the group action and therefore M(G, E) is a homogeneous space. The orbit of E under the group action is given by: The space of all superoperators T (A, A ) splits up into disjoint orbits as ∪ E Orb(E) = T (A, A ). One can uniquely specify any operation by identifying a choice of origin in a particular orbit together with a set of coordinates on the process orbit. In general not all orbits will be isomorphic. This means that we might need two different types of process orbits to describe two quantum processes acting on the same spaces. In particular every fully symmetric operation lies in a single point orbit and the process orbit is simply a point. The invariant data of quantum processes can be directly associated with the choice of origin which gives a distinguished operation for each orbit. a. Paradigmatic example: One qubit unitary axial operation under SU (2) Before giving the general statement we look at an illustrative example of the decomposition of unitary axial operation on a qubit into irreducible tensors. This will help to clarify all the core ingredients. Let V = e in·σ with corresponding one qubit process V that takes the form V = j,k α jk Φ j k where without loss of generality assume the ITSs are orthonormal. This is an axial map that leaves invariant any qubit with Bloch vector aligned along the n axis.
Denote by J j k := J[Φ j k ] the Choi representation of the Φ j k superoperator. The Choi operator of the unitary process V is given by J[V] = |vec(V ) vec(V )|. Therefore we can write the alpha-coefficients in the form: There is a natural group action on the space of functions on a sphere α ∈ L 2 (S 2 ) which for any g ∈ SU (2) is given by: where the action of the group element g ∈ SU (2) on the vector in S 2 is defined by (g · n) · σ := g(n · σ)g † . By a simple direct calculation we can check that the set of alpha-coefficients α jk for fixed j transform under this group action exactly like the spherical harmonics. We get that: Since the group action of SU(2) on the sphere is transitive this means that by fixing a representative process in the Orb(V) with some initial fixed Bloch vector then from this and the above relation we obtain the alphacoefficients for any process in Orb(V). Moreover we argue that they will be proportional to the dual spherical harmonics on the sphere. The spherical harmonics Y jk ∈ L 2 (S 2 ) form a complete orthonormal basis for square integrable functions on a sphere. They achieve the decomposition of L 2 (S 2 ) into irreducible components and are simultaneous eigenfunctions for all rotationally invariant differential operators. Under the group of rotations for any R ∈ SO(3) the standard spherical harmonics transform according to: where D j are the Wigner matrix for the j-irrep of SO(3). As a homogeneous space the sphere is diffeomorphic to the quotient spaces S 2 ∼ = SU (2)/U (1) ∼ = SO(3)/SO (2). This leads to a left regular representation of SU(2) and SO(3) on the space L 2 (S 2 ) that decomposes the space into multiplicity free irreducible components. While for SO(3) the decomposition will be achieved by the spherical harmonics and every possible j-irrep appears, for SU (2) only the odd-dimensional irreps appear and these are exactly the ones isomorphic under the double cover to those of SO (3). Suppose that the double cover map is given by H : SU (2) −→ SO(3) and associates an element H(g) = R ∈ SO(3) for some g ∈ SU (2). We choose the matrix coefficients v j kk for the j-irrep of SU (2) to be the Wigner matrix i.e v j kk (g) = D j kk (R) (this in turn means that we construct the ITSs with respect to this choice of basis as well). Then the alpha coefficients α jk transform like the dual spherical harmonics Y j * k = (−1) k Y j,−k in their standard form using the Condon-Shortly phase. Therefore they live in the irreducible subspace spanned by these and consequently can be written as linear combinations of Y j * k for k = −j, ..., j. Furthermore transitivity of the group action and some algebraic manipulation detailed in the main proof will result in the following form for the coefficients: for some complex number a j independent on the vector component label k and which depends on the choice of origin for the process orbit or equivalently a representative axial map in Orb(V).
In conclusion any one-qubit unitary axial map with fixed direction n can be decomposed as: which only assumes that the choice of basis for the ITS is aligned with the choice of basis for the spherical harmonics. However it remains a basis-independent statement since only the relative alignment is fixed and not the particular basis choice. The following sections give the general results.

b. SU-2 symmetry
Axial operations E ∈ T (A, A ) under SU(2) are those that are not invariant under the full group but have some residual symmetry given by a subgroup H such that U h [E] = E for all elements of h ∈ H. By the orbitstabilizer there is a natural homeomorphism between the homogeneous space SU (2)/H and Orb(E) which we denote by T : SU (2)/H −→ Orb(E) and explicitly given by for all g ∈ G where gH denotes a coset in SU (2)/H. 7 Whenever H is isomorphic to a U(1) subgroup of SU(2) then the process orbit M(E, G) will be diffeomorphic to a 2-sphere S 2 . This means we can associate to any axial quantum process some directionn.
Theorem B.8. Let E : B(H A ) → B(H A ) be an axial process associated to some directionn, that takes states of a quantum system A into states of a system A . Then the orbit of E under the symmetry action is a 2-sphere S 2 and there exists a basis of irreducible tensor superoperators {Φ j k } with E(ρ) = j,k α j,k Φ j k (ρ) such that the components α jk are un-normalised wavefunctions on the sphere given by wheren = (θ, φ), a j is the norm of α jk that is independent of k, and Y jk (θ, φ) are spherical harmonics.
Proof. Because SU (2)/U (1) ∼ = S 2 means that the process orbit is a sphere so we the orbit will be homeomorphic to S 2 . We denote this by f : Any operation in the Orb(E) is an axial operation since it will be also fixed by H. Suppose we denote by x ∈ S 2 such that f (x) = E. The coefficients when decomposing E into ITSs will depend on E and therefore on x and take the form E = j,k α j,k (x)Φ j k . This means that any axial operation in Orb(E) will decompose in terms of coefficients that we can view as complex-valued functions on S 2 such that α j,k : where x depends on the fixed g. The transitive group action on the orbit induces a group action on S 2 given by x : In turn since the coefficients α j,k ∈ L 2 (SU (2)/U (1)) we have an induced group action on the function space given by g · α j,k (x) = α j,k (g −1 · x). Moreover this group action will be equivalent to the left regular representation since we can identify the orbit with the coset space.
As E transforms under the group action it generates only operations in Orb(E) so we can explicitly show how the coefficients transform under the induced group action by expressing α j,k (x ) = g · α j,k (x) in terms of only the set of coefficients evaluated at x i.e α j ,k (x) for all j and k . We show the following: Claim 1 The alpha-coefficients will transform under the induced group action as: for any g ∈ SU (2) and any irrep j ∈ Irrep(A, A ) with v j * kk matrix coefficients of the dual irrep j * . Proof of claim: We have that: action of H we must have equivalently that g −1 g ∈ H which means that the group elements generate the same coset gH = g H. Finally SU(2) acts transitively on Orb(E) so any operation in the orbit will be reached by some g ∈ G We want to express α j,k (x ) in terms of α j,k (x) to explicitly show how the alpha coefficients transform under the group action. Since Φ j k transform as irreducible tensor superoperators we have that: The ITS sets Φ j k form a complete basis for the space of superoperators and in particular they are linearly independent. Then equating the two different expressions for U g [E] and using orthogonality leads to Since x = g·x we can re-write the above as g −1 ·α j,k (x) = j,k v j k k (g)α j,k (x). However there is nothing special in our choice of the particular element g and corresponding point x so it turns out that this occurs for all elements g ∈ SU (2). Because we can use (v j kk (g)) * = v j k k (g −1 ) and (v j kk (g)) * = v j * kk (g) where j * is the dual irrep this leads to the desired transformation This ends our proof of Claim 1.
It is immediate to check that α j,k ∈ L 2 (S 2 , C). However square integrable functions on the sphere decompose into a complete set of spherical harmonics. It is important to mention at this point that the irrepdecomposition of L 2 (S 2 , C) is multiplicity-free and each j-irrep component is spanned by a complete basis of orthonormal functions Y jk for k = −j, ...j which correspond to the spherical harmonics. The fact that the isotypical decomposition is multiplicity free is exactly what allows us to define spherical harmonics for this homogeneous space in the first place. Since α jk are functions on the manifold Orb(E) and transform according to B38 then they lie in the j * -irrep component of L 2 (S 2 , C) that is spanned by spherical harmonics Y j * k . Therefore we can write them in terms of a spherical harmonics basis such that: for some complex coefficients a k jk depending on some fixed j and k. We show that a k jk = δ kk a j for some complex number a j that is independent on the vector component k. To do so note that the above should hold for all x ∈ S 2 . So we have: and similarly we have that the alpha coefficients transform as: (B41) Using orthonormality of spherical harmonics we can equate the two different forms for the transformed alphacoefficients to obtain that for all m the following holds: Now we can use orthonormality of matrix coefficients for the j * irrep to multiply both sides by (v j * ls ) * and integrate over all group elements to get k δ kl δ k s a jl which means that the coefficients are all independent on the vector components and we have that α jk = a j Y j * k where a j = a (k) jk for any k. For the case of SU(2) the irreps sand their duals are isomorphic. Moreover under the choice of Condon-Shortly phase the spherical harmonics for the dual representation are related to the usual spherical harmonics by Y j * k = (−1) k Y j,−k . Therefore under some basis choice for the ITS/spherical harmonics the alpha-cofficients will take the form: for any x ∈ S 2 and some a j independent on the vector component labelled by k.

c. Axial operations under general symmetry
The results in the previous subsection can be generalised to arbitrary compact groups G.
Theorem B.9. Let G be a compact group with unitary representations U A and U A on the Hilbert spaces H A and H A . Consider operations E in T (A, A ) that have an isotropy group H subgroup of G. Then any such operation can be decomposed into irreducible tensor superoperators with corresponding coefficients proportional to harmonics functions on the homogeneous space G/H. That is: where x ∈ G/H correspond to the point in the process orbit associated with E and the complex coefficients a λ ∈ C correspond to invariant data that are fixed for any operation in Orb(E).
Proof. Any E can be decomposed in terms of irreducible superoperators as E = λ,k α λ,k Φ λ k . However since E is also an axial operation with stabilizer (or isotropy) group H = {g ∈ G : U g [E] = E}. Moreover any operation in Orb(E) will also be an axial operation with the same stabilizer group. By the orbit-stabilizer theorem there is an homeomorphism Orb(E) ∼ = G/H. Clearly G/H is a homogeneous space because G acts transitively on any single orbit. Therefore we can associate to any point x ∈ G/H an axial operation in Orb(E), which also was previously referred to as the process orbit M(G, E). This in turn implies that we can view the corresponding alphacoefficients for each such axial operation as functions on the process orbit. In particular we can show that α λ,k ∈ L 2 (G/H). From orthonormality of ITS we have that for any λ, k α λ,k (x) = Φ λ k , E with x being the associated coordinate point of E on the process orbit. Because the group acts transitively for any y ∈ G/H there exists g ∈ G such that g · x = y. Therefore: , E and hence we obtain that (B47) which by substituting in the above integral we get straight away from orthonormality of matrix coefficients that G/H |α λ,k (y)| 2 d y < ∞ and therefore α λ,k ∈ L 2 (G/H). By a repeat of the argument for the normalization in for axial maps, we find that the norm of α λ,k is independent of k and so we can write α λ,k (x) = a λ Y λ * ,k (x) as claimed.
In the case that H is a massive subgroup, the space L 2 (G/H) decomposes under the left regular representation into irreducible subspaces all appearing with multiplicity one. Unlike the decomposition of L 2 (G) not all irreps of G appear in the decomposition of L 2 (G/H). However each such λ irreducible subspace has a basis of spherical harmonic functions Y λ,k with the vector label component ranging from k = 1, ...dim(λ). However the alpha-coefficients transform like the λ * -irrep under the left regular representation and hence they can be written in terms of the corresponding spherical harmonics Y λ * k .

One qubit operations under SU(2) symmetry
Consider a one qubit system H ∼ = C 2 which carries the 1/2-irreducible representation of SU (2). This is the fundamental representation of SU(2) whose action is given by matrix multiplication. The space of operators B(H) decomposes according to this symmetry into two orthogonal subspaces carrying the trivial 0-irrep respectively the 1-irrep. An analogous way of expressing this statement is through the Bloch-vector representation of a qubit state. Any valid density matrix ρ ∈ B(H) can be written as ρ = 1 2 (1 + r · σ) where σ = (σ x , σ y , σ z ) is the vector of Pauli matrices and r represents the Bloch vector associated with ρ. The Pauli matrices transform non-trivially under the adjoint action of any SU(2) matrix and they span the 1-irrep subspace whereas the identity is invariant under the group action and corresponds to the trivial 0-irrep subspace.
According to this symmetry any one-qubit quantum process can be decomposed into ITSs -each of which will act non-trivially only on one of the two modes a ∈ {0, 1} and transform them into a different modeã ∈ {0, 1} according to the λ-irrep label that captures how the ITS behaves under the group action. To each ITS we associate a particular diagram label θ = (a,ã) λ −→ where λ ∈ {0, 1, 2} with multiplicity two, three and one respectively. The diagram (1, 0) 1 −→ is unphysical and therefore does not appear in the decomposition of any valid quantum process. Out of the total five remaining possible ITSs, two of them (0, 0) 0 −→ and (0, 1) 1 −→ give an output independent of the Bloch vector of the initial state ρ ∈ B(H). The first corresponds to the superoperator Φ 0 (ρ) = 1 2 and the second span a 3-dimensional superoperator subspace that displace the maximally mixed state and therefore appear only in non-unital quantum processes. The latter we denote by Φ 1,m1 k (ρ) = σ k where the multiplicity labelm 1 specifies that this particular 1-irrep ITS arrises from coupling (0, 1) 1 −→ and k corresponds to the vector component. The space of invariant onequbit superoperators is two dimensional and the most general such quantum process is a depolarised process. It is spanned by Φ 0 and the ITS given by diagram (1, 1) 0 −→ which we explicitly denote by Φ 0,m0 (ρ) = r·σ √ 3 = ρ − 1 2 . This leaves only two non-trivial diagrams corresponding to both input and output mode carrying asymmetry. The remaining 1-irrep ITS corresponds to coupling (1, 1) 1 −→ which we label with multiplicity m 1 will be given by: Similarly for the 5-dimensional subspace spanned by superoperators corresponding to the diagram (1, 1) 2 −→ will be described by the following (where we suppress the multiplicity label since there is a single component for The most general axial quantum process on one qubit with fixed axis n = (x, y, z) takes the diagramatic form: (B50)

Appendix C: Biparitite symmetric
In this section we will be concerned with fully characterising bipartite symmetric operations under general symmetry constraints.
where in the last equation we only used the fact that {A λ k } and {B λ * k } form complete sets of ITSs. Observe that there is an underlying assumption that both sets of ITS transform relative to the matrix coefficients computed with respect to the same basis. Moreover the standard λ-irrep matrix is unitary. Therefore where the last equality comes from unitarity and have used the dual representation is given by v λ * (g) = v λ (g −1 ) T . This implies that for any λ and multiplicities m and m we have that 8 : This implies that each of the superoperators Φ λ,m,m are invariant under the group action. Finally all we have left to prove is they span the whole invariant subspace of superoperators on the bipartite system considered. To do so we need to characterise the trivial subspaces and their multiplicity in the decomposition of T (A ⊗ B, A ⊗ B ) into irreducible components. All these irreducible components are exactly the ones that appear in the decomposition of the tensor product of representa- The irreducible components remain the same under swaping the tensors in the previous representation. Therefore any irrep in T (A ⊗ B, A ⊗ B ) arises in decomposing the tensor product coupling of an irrep λ A ∈ Irrep(A, A ) and λ B ∈ Irrep(B, B ). This accounts for all irreps in the space of superoperators on the bipartite system. In particular a classical result in representation theory says that there is at most one trivial 0-irrep subspace in the decomposition of λ A ⊗ λ B and appears if and only if λ B ∼ = λ * A .
Therefore as we range over all λ ∈ Irrep(A, A ) with multiplicity m and λ * ∈ Irrep(B, B ) (if it exists) with multiplicity m we generate all possible trivial components. Each of them will be distinct because they arise in orthogonal subspaces. Most easily this can be seen by the orthogonality relations: where in the last equation we used the fact that ITS at T (A, A ) and T (B, B ) are orthogonal (and we also assume without loss of generality that they are normalised).
While the above theorem holds for general local ITS at system A and B we will be interested in decomposing general symmetric operations in terms of primitive local ITS. This means that the complete basis of symmetric superoperators can be labelled by diagrams ] which reflect what are the local ITS couplings that make up each basis element. In particular in terms of the allowed symmetry-breaking properties of the input and output states it has the following mathematical form: By ranging over all possible diagrams Equation C3 gives an orthogonal complete basis of symmetric bipartite superoperators. We make several remarks on the generality of such a construction: i) There is no a priori choice of basis for the underlying Hilbert spaces and there is freedom in choosing a basis for the local input and output states. ii) There is however an assumption in lifting the choice of basis from the local Hilbert space to basis of local 8 Remark: Dealing with matrix coefficients gives a false sense of simplicity. Orthonormality of matrix coefficients is actually a powerful result that can be linked to Peter-Weyl's theorem and Schur's lemma. These are technical statements which are central to many important results in group theory. The core underlying result of the main theorem we prove is a consequence of these: The tensor product of two irreducible representations λ ⊗ µ contains at most one trivial irrep and this occurs if and only if µ ∼ = λ * (and global) superoperators. In particular the choice of basis for superoperators is such that it maps input basis states to output basis states, but generally there are valid orthonormal superoperator basis that do not act in this way. The reason for this construction is purely operational: it allows us to analyse both local symmetry-breaking properties of the bipartite states and of the global symmetric processes as well as how these two concepts interact with one another.
iii) The notation for each diagram θ and underlying irrep couplings does not explicitly include multiplicities. However it should be understood that when we identify a particular diagram it arises from particular couplings of input and output irreps having specific multiplicities.
Lemma C.2. Given E ∈ T (AB, A B ), if E has (a) non-trivial subsystems A, B, A , B , (b) globally symmetry, and (c) complete-positivity, then E must take the form where x θ ∈ C and where θ * is the dual diagram to θ.
Moreover trace-preservation implies where A λ k and B λ k are ITOs for the input system at A and B respectively. However due to orthonormality of ITOs in any linear combination of such diagrams for which θ x θ tr A B (J[Φ θ ]) = 1 AB only the coefficient associated to θ = [(0, 0) 0 −→ (0, 0)] is nonzero. Since the ITOs are also normalised this results in a fixed value for the associated non-zero coefficient given by Appendix D: Gauging processes

From global to local symmetries
In the following we illustrate the gauging procedure for 2-symmetric quantum processes by looking at the symmetric bipartite superoperator associated with diagram θ and link l connecting local system A x with A y . Our main Theorem C.1 together with linearity of the gauging map ensures that the same procedure works for any 2-symmetric quantum process. More precisely we denote the gauging map for superoperators by: where as in the main text H l is the quantum reference frame corresponding to link l and encodes the group elements. The action of the gauging map is such that it promotes a global symmetry to a local symmetry. Specifically any Φ θ ∈ T (A x A y , A x A y ) which is symmetric under the global representation (i.e U g [Φ θ ] = Φ θ for all g ∈ G) will be mapped to Gauge(Φ θ ) which is symmetric under the local representation. This means that We provide an explicit gauging map that satisfies these requirements. First we explicitly add in the background degrees of freedom that are encoded in the quantum reference frame initialised with a gauge coupling that is symmetric under the global symmetry. The resulting process on the local systems and the link reference frame will be invariant under the global representation. Then we promote the global symmetry to a local symmetry by averaging over the local independent degrees of freedom and therefore removing all the relative alignments.
Recall that the globally symmetric superoperator given by diagram θ decomposes into local ITS as: are process gauge couplings such that U (g,g) [A j,j (l,λ) ] = A j,j (l,λ) for all g ∈ G. Clearly Φ (l,θ) is invariant under the global representation: We have that: Because the Haar measure for a compact group is both left and right invariant it is straightforward to check that the operation constructed above is symmetric under the local de-synchronised group action. Therefore we have that U gx ⊗ U (gx,gy) ⊗ U gy [Gauge(Φ (l,θ) )] = Gauge(Φ (l,θ) ) for all g x , g y ∈ G. However we know how the local ITS and the process gauge couplings transform under group actions on their respective systems and therefore can obtain a compact form for the gauging map. In particular we have that: We can combine all of these to obtain the form of the gauging map. The matrix coefficients satisfy v λ jm (g x )v λ m j (g −1 x ) = δ mm and also from unitarity we get that v λ jn (g y )(v λ jn (g y )) * = v λ jn (g y )(v λ (g y ) † ) nj = v λ jn (g y )(v λ (g −1 y )) nj = δ nn . Therefore the action of the gauging map is given by: a. Example: U(1) symmetry We consider the situation when the symmetry group is U (1) and suppose that the local ITS on system A x and A y are given by Φ λ x and Φ λ y . Under the group action for each element φ ∈ U (1) they transform according For abelian groups all irreducible representations are one-dimensional and therefore each θ-diagram corresponding to λ-irrep symmetry breaking carrier takes the form of: Φ (l,θ) = Φ λ x ⊗ Φ λ * y . This is invariant under the global group action. We add in the quantum reference frame that encodes group elementsin this case angles. Consider H l to be the Hilbert space of an infinite ladder system with equally spaced energy eigenstates labelled by {|m } m∈Z . The coherent states on the circle are defined in H l for each element φ ∈ U (1) as: They form an orthonormal set of eigenvectors for the selfadjoint operatorφ = φ |φ φ| d φ the canonical conjugate of angular momentum in the z-direction. Therefore these states perfectly encode all group elements of U (1). The group action on these states will be given by: for all g x , g y ∈ U (1). In particular |0 0| is invariant under the global representation i.e whenever g x = g y . Therefore adding in the globally symmetric background degrees of freedom: Φ (l,θ) −→ Φ λ x ⊗ Φ λ * y ⊗ |0 0| and therefore the gauging procedure maps: ) However invariance of Haar measure means that: Using the defining decomposition of |φ in terms of the energy eigenstates of the ladder then we obtain the gauged operation that has a local symmetry: (D6) Remark: In the above we assume that the gauging processes are given by states on the reference frame system H l . This is a particular simplified scenario to illustrate the gauging procedure.

Fixing a gauge: connections with pre and post selection with a group element
In here we demonstrate a particular way in which the gauge fixing for general quantum processes is achieved via pre and post selection with group elements. In doing so we also underline the physical interpretation of the polar decomposition and the role of the process orbit in providing the relative alignment between subsystem and environment.
Suppose that the quantum processẼ ∈ T (H Ax ⊗H Ay ⊗ H l ) that acts on systems A x , A y and the reference frame H l situated on the link between x and y is invariant under the local group action. This means that U (gx,gy) [Ẽ] =Ẽ for all group elements g x , g y ∈ G. Suppose that we post select with group element h 2 ∈ G and pre select with h 1 ∈ G then the resulting operation will be given by: where Π h (σ) = |h h|σ|h h|, is the projection onto the pure state |h . In general, the processẼ h1,h2 will not remain locally invariant since the measurements with group elements will break that symmetry. One can check that E h1,h2 now transforms under the local group action according to This means that the process resulting after pre and postselection with a group elements h 1 and h 2 is transformed under the de-synchronised local group action with elements (g x , g y ) into the process corresponding to pre and post selection with group elements g x h 1 g −1 y and g x h 2 g −1 y respectively. We show this result by checking directly: In the previous calculation we have only used thatẼ is invariant under the local group action U (gx,gy) .
To establish how much the local symmetry has been broken by the pre and post selection we look at the set of all group elements (g x , g y ) under whichẼ h1,h2 remains an invariant process. First note that the the reference frame perfectly encodes group elements such that the pure states |g are orthonormal (and hence perfectly distinguishable). Therefore using the above result we have that U (gx,gy) [Ẽ h1,h2 ] =Ẽ h1,h2 holds if and only if g x h 1 g −1 y = h 1 and g x h 2 g −1 In particular this is clearly satisfied whenever h 1 = h 2 = h for some h ∈ G . In this case the pre and post selection with group element h breaks the local de-synchronised invariance resulting in a process that is invariant under the global action U g := U (hgh −1 ,g) = U (h,e) • U (g,g) • U (h −1 ,e) . Therefore we can view U (h −1 ,e) as a local change of basis that aligns system A x and A y . Perfect alignment means that the process on the bipartite system is globally symmetric i.e invariant under the global representation. In other words the reference frame on the link encodes the group element h required to align the two systems. Proof. A bipartite state is n-extendible for all finite n it must be a separable state. Whenever F is n-extendible for all n then its corresponding Choi operator J[F] is also n-extendible for all finite n and therefore is separable. A Choi operator is separable if and only if the the corresponding process is entanglement breaking. Moreover an entanglement-breaking process has the form of a measure and prepare and therefore F takes the form stated above.

Repeatability and arbitrary repeatability
Definition E.3. Let A 1 , ...A n be n isomorphic systems A i ∼ = A and E : B(H A ) −→ B(H A ) a target channel. We say that the protocol P for E using system B is nrepeatable if it specifies a circuit of symmetric operations for all i initially acting on B 0 = B such that for any σ ∈ H B and any k: tr k,Bn (W(ρ 1 ⊗ ρ 2 ⊗ ... ⊗ ρ n ⊗ σ)) =Ẽ(ρ k ) (E3) the induced processẼ is the same on all subsystems A k and is an approximation of E using σ. In particular E will depend on σ but not on k.
The above definition for n-repeatability along with the definition of an n-extendible quantum channel will directly result in the following lemma: Lemma E.4. Suppose that system B admits an nrepeatable use on systems A 1 , ...A n then for any fixed ρ the resulting process on B given by where E is the induced process on each system A as defined in Equation E3 Proof. The previous Lemma E. 4 implies that the effective process on the environment F ρ : B(H B ) −→ B(H A ) given for any fixed ρ by F ρ (σ B ) = tr k,Bn (W(ρ ⊗ σ B ) is nextendible for all finite n. This statement is independent on the initial state on system A (although the process F ρ generally may not be so). Moreover F ρ is a valid CPTP map as it arises as a composition of CPTP maps. By lemma E.2 every n-extendible quantum operation for all finite n must take the form of a measure and prepare process. Therefore for each fixed ρ ∈ B(H A ) there exists a POVM {N i } on H B and quantum states ρ i ∈ B(H A ) such that: Note that there could be dependence on ρ in either ρ i or N i , however this can be simplified by noting that one can decompose any POVM into a convex combination of extremal POVMs. We can write where M k = (M k,i ) is an extremal POVM for each k, and p k is a probability distribution. This implies that tr k,Bn (W(ρ ⊗n ⊗ σ)) = i,k where Φ k,i (ρ) := p k ρ i is a completely-positive linear map on ρ (since it always returns up to normalization a valid quantum state), and which implies that the POVM acting on B is independent of the input state ρ on A.
Introducing the single index a = (k, i) completes the proof.

Recovering catalytic coherence protocol-finite dimensional setting
We first observe that Theorem E.5 does not imply arbitrary repeatability of the reservoir whenever the induced process on the reservoir is measure and prepare. In particular for the catalytic coherence protocol every preparation of the induced process on a new qubit system has the effect of spreading the coherence levels in the state of the reservoir. Therefore in this particular protocol an infinite dimensional reservoir is required for arbitrary repeatabiliy. Generally we conjecture that non-trivial arbitrary repeatability cannot occur for a finite dimensional reservoir.
In the following we analyse a finite ladder system in terms of a single repeatable use of states under global symmetric dynamics. For simplicity we assume that system A on which we perform the target operations consists of a single qubit with basis states {|↑ , |↓ } and U(1) group action U A (θ) |↓ = |↓ and U A (θ) |↑ = e iθ |↑ . The finite d-dimensional ladder has basis states {|1 , |2 , ... |d } and the U(1) group action U (θ) |m = e imθ |m for all m. 9 A generic symmetric operator V therefore takes the form V AB = |↑ ↓| ⊗ ∆ + |↓ ↑| ⊗ ∆ (−1) + + |↑ ↑| ⊗ ∆ (0) + |↓ ↓| ⊗∆ (0) (E10) where ∆ (λ) is an arbitrary linear combination of terms |e i e j | for which (e i − e j ) = λ. Therefore we can write: for arbitrary coefficients c j , d j , f k andf k . The aim of this section is to deduce the form of the symmetric operation V AB by specifying the constraints on the coefficients that are imposed when we impose repeatability. For finite-sized environment we do not expect arbitrary repeatability and even n-repeatability for finite n to occur in general for all states in the environment σ ∈ B(H E ). Therefore we only require repeatability on a subset of all states on B. We denote by T the isometry that embeds this subset into the ladder system H B .
To summarise, we investigate the form of the symmetric operator V under the assumptions: i) The environment system B consists of a finite ddimensional ladder system with the specified U(1)-group action. ii) For any state σ within some fixed support set T (B(H B )) ⊂ B(H B ) the environment B admits a 2repeatable use with a circuit consisting of (unitary) symmetric operator V applied to system A 1 ∼ = A and environment B in one step and to system A 2 ∼ = A and environment B in the second step. iii) We also assume a slightly stronger version of 2repeatability in the form of : However we will fully motivate how this constraint arises from our definition of 2-repeatability in the following subsection.
One should emphasise that the set of assumptions above is much weaker than imposing arbitrary repeatability. For arbitrary repeatability to be possible with a finite ladder system even on some restricted support of environment states this would necessarily imply that the global symmetric operation V acts trivially i.e it is a product of locally symmetric operations. In other words arbitrary repeatability for finite-sized environment system is too demanding to investigate reversible non-trivial use of the symmetry breaking properties. However the catalytic coherence protocol can be restricted to finite ladder system case in which all of the requirements i) to iii) are met providing a non-trivial example of how the asymmetry of the state in a finite-sized environment is repeatably (or reversibly) used.
The following analysis shows how the catalytic coherence protocol for finite ladder arises form assumptions i)-iii) and moreover show it is one of only two possible forms that V can take under these circumstances.
For our scenario since V is a symmetric operation on the one qubit and ladder system then the general form of such operator and demanding repeatability in the form of equation E15 we obtain the equivalent set of conditions on the environment B: If we were to look only at the commutation relations above (and ignore for a moment the projection onto some support) then we would in general get terms which for example may be found in ∆ a ∆ −a but not in ∆ −a ∆ a and since these terms would be orthogonal with any other term in the expansion into modes we get that the only way to satisfy the above conditions is if those "boundary" terms will be mapped into 0 by the isometry T . In other words we will require that the boundary will not lie in the support. Therefore define the boundary states to be the union B = B up ∪ B down of following: B up = {|e j : k such that e k = e j + a as weights} B down = {|e j : k such that e k = e j − a as weights}.
In particular whenever the system B consists of a finite-dimensional ladder then the boundary corresponds to the lowest and highest energy levels. We have also seen that repeatability is not compatible with a support that contains the boundary states.That means the top state |e n and the bottom state |e 1 will form the boundary set. The maximal support on which we assume repeatability holds (one step repeatability in the sense of the commutativity condition) will therefore be given by the set of states {|e 2 , ..., |e n−1 }.
We want to investigate what conditions does repeatability impose on the coefficients for each ITO that makes up the quality parameters defined above. We have:  There are two different cases we can distinguish for equation E20: Case 1 All c j and d j are non-zero. This implies that for all j we have f j = f j−1 . In other words this means that the zero mode ∆ 0 behaves like a multiple of the identity on the support. Similarly for the zero mode∆ 0 . Immediately this recovers the catalytic coherence protocol for finite dimensions. Case 2 There is a j 0 such that c j0 = 0 and for which f j0 − f j0−1 = 0. First this also implies that d j0 = 0 from the other relation of equation E20. Addtionally this also implies that c j d j = 0 for all j which in turn gives either c j = 0 or d j = 0 for all j. This means also that ∆ a ∆ −a = ∆ −a ∆ a = 0 so the repeatability conditions are trivially satisfied. Both must hold whenever we have that f j −f j−1 = 0 orf j −f j−1 = 0. Intuitively this means that the zero mode ITO given by ∆ 0 behaves as a multiple of the identity on all those states in the support which are activated within the non-trivial ITO modes ∆ a and ∆ −a . All of this means the case then reduces to showing that the set of states(labels) for which we have the coefficients f j − f j=1 = 0 must form a non-trivial subspace and therefore we can always restrict our support to this particular scenario.
In a very strict sense this does not result in the catalytic coherence protocol but it is the case when if S = {j : c j = d j = 0} and S C = {j : c j = 0 and d j = 0} and S D = {j : d j = 0 and c j = 0} where S C ∩ S D = 0 and S and S C ∪ S D are disjoint sets. Then this case results in the following forms: ∆ 0 = j∈S f j |e j e j | + f j∈S C ∪S D |e j e j |, ∆ a = j∈S c c j |e j+1 e j |, ∆ −a = j∈S D d j |e j e j+1 |. Moreover imposing unitarity for U AB implies in particular that |c j | = 1 and |d j | = 1 for all j and f =f = 0 and f j =f j = 1 for all j. This means that each mode on the environment always acts on disjoint energy levels so there is no interaction between different symmetry breaking terms.
a. Re-casting 2-repeatability into a commutativity statement as in Equation E15 Repeatability as we have defined it in terms of circuits of symmetric operations can be equivalently re-written in the following way. For any ρ A1 ∈ B(H A1 ), ρ A2 ∈ B(H A2 ) and σ ∈ T (B(H B )) in the support of the isometry T we have that: where V A1B and V A2B are the adjoint actions of the unitaries V A1B and V A2B which act trivially on A 2 and A 1 respectively while applying B to the other two systems. There is however nothing special about system A 1 or system A 2 . In fact they must both be isomorphic and behave exactly the same way in this process. Indeed swapping the order of the two system A 1 ↔ A 2 implies that the relation tr A1B ([V A1B , V BA2 ](ρ A1 ⊗ σ ⊗ ρ A2 )) = 0 (E23) also holds for all ρ A1 ∈ B(H A1 ), ρ B ∈ B(H B ) and σ in the support of T . There is a sense in which the repeatability process is not affected when we swap systems A 1 and A 2 . This means that there is a bit more structure hiding behind the definition of repeatability. The particular way that composite states evolve before tracing out system the rest of the systems represents a key feature of repeatability. Correlations between system A 1 and system A 2 appear solely as a result of their interaction through the common reservoir B. Therefore tracing out over the environment B and the auxiliary system A 2 also destroys the relational properties between A 1 and A 2 . In light of this we can define a slightly stronger notion where instead of discarding both system A 2 and B we only trace out the reservoir B which we refer to as strong repeatability. The more general notion also encapsulates the ability to perform particular types of joint operation on system A 1 and A 2 given a fixed set of resources and free unitary operation. Clearly strong repeatability implies 2-repeatability but they are not equivalent notions. However note that the scenario described by Aberg is also strongly repeatable so it corresponds to a particular example when the two definitions overlap.
Definition E.6. Strong repeatability We say that unitary V is strongly repeatable if when we apply it to an environment B and system A through U AE and then to the same environment E and an isomorphic system B the following holds for all ρ A ∈ B(H A ), ρ B ∈ B(H B ) and σ in the support of Π which is included in B(H E ). In the above U AE and U BE correspond to the adjoint actions of the unitary operators U AE and U BE respectively.
Finally it turns out that we are able to use an equivalent definition for strong repeatability instead of the above. Indeed the freedom in the Stinespring dilations of symmetric operations will imply that the above formulation is equivalent: U is strongly repeatable if there is a symmetric isometry W E on E such that We argue however that any symmetric operation on the environment does not bring in any additional asymmetry resources so we should be able to assume without loss of generality that W E = 1 E hence leading to the previously given definition.