Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

The canonical commutation relation, [Q, P ] = i , stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (a.k.a. contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations, but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following: The simultaneous P &Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a 7-dimensional manifold, a universal covering group we call the Instrumental Weyl-Heisenberg Group, IWH. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of IWH, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument. After World War II, theoretical quantum physics became dominated by the design of quantum field theory. There were three branches of physics that stemmed from this: High-Energy, Condensed-Matter, and Atomic-Molecular-Optical (AMO.) Although incredibly developed as predictive methods, quantum field theory in all three of these branches has left some very basic ideas of quantum observation underdeveloped. That this is indeed still the case is evident in how the coherent-state resolutions of the identity or "overcomplete bases" [1][2][3][4] are usually finessed: for bosonic amplitudes (where Z is an arbitrary complex scalar) [1,[5][6][7][8], and 1 (spin) j = (2j + 1) |j,n j,n| (1.2) for fermionic spins (where 2j is an integer) [2][3][4]9]. These overcomplete bases are both key to establishing functional integration [1,[10][11][12][13][14][15][16][17][18][19] and to understanding the observation of energy quanta [20]. Yet these overcomplete bases do not fit the most basic idea of a Hermitian eigenbasis and as such are often not considered a serious form of observation. By itself, this inattention to how one could observe in these overcomplete bases leaves a missing piece at the foundation of quantum mechanics and quantum field theory.
Meanwhile, an understanding of the overcomplete bases as a bona fide means of observation has been slowly coming to light. First, the idea of observables matured into the general mathematical theories of instruments and operations [21][22][23][24][25][26][27][28][29][30][31][32]. In this more contemporary language, the overcomplete bases of equations 1.1 and 1.2 are called positive-operator-valued measures or POVMs and are understood to be informationally complete, meaning the distribution observed by any one of these POVMs is enough to reconstruct the quantum state. With this theoretical technology, it has further been discovered that the overcomplete bases correspond to various forms of continual (or continuous) simultaneous observation. The standard coherent-state POVM (equation 1.1) was discovered to be the effect of simultaneously observing both quadratures of a leaky cavity, a form of observation we will call Goetsch-Graham-Wiseman (GGW) heterodyne detection [33][34][35][36][37]. Then the spin-coherent POVM of equation 1.2 was discovered to be the effect of simultaneously observing the three orthogonal spin components, a form of observation we call the spin-isotropic measurement [38][39][40].
Before proceeding, we caution that this paper uses a mathematical apparatus not familiar to most physicists and quantum scientists. This apparatus is introduced here naturally, as it becomes both desirable and necessary. Readers who are made uncomfortable by this apparatus are urged to consult the companion paper [40], which attempts to persuade the reader that the unfamiliar mathematical concepts and techniques are essential tools-a new way of thinking and doing-and then introduces these tools as gently as possible.
GGW heterodyne detection and the spin-isotropic measurement work in a very similar way, but they are different in one very important respect. While GGW heterodyne detection assumes energy-conserving system-meter interactions, the spin-isotropic measurement assumes Hermitian meter displacements, −iH (iso) dt/ = − √ κ dt J k ⊗ 2σ∂ q , where J k is an orthogonal spin component of the system, q is the meter register, σ is the width of the meter pointer, and κ is the measurement rate. In both cases, the measuring instrument consists of Kraus operators defined by a time-ordered exponential over the duration T of the measurement. For GGW heterodyne detection, the Kraus operators are [37] where a = (Q + iP )/ √ 2 is the usual complex-amplitude operator and dw t = (dW q t + idW p t )/ √ 2 is the registered complex Wiener path. For spin-isotropic measurement, the Kraus operators are [39,40] where J = (J x , J y , J z ) is the triple of orthogonal spin-component observables and d W t = (dW x t , dW y t , dW z t ) is the registered 3-vector of Wiener paths. The most striking feature about the instruments defined by equations 1.3 and 1.4 is that they can be integrated universally-that is, independent of matrix representation. The difference between the two cases can now be summarized as the following: integrating equation 1.4 defines a 7-dimensional manifold that requires the theory of symmetric spaces, whereas integrating equation 1.3 defines a 3-dimensional manifold that is much more straightforward.
This paper is an analysis of the quadrature analog of the spin-isotropic measurement, a form of observation we call the Simultaneous P &Q Measurement or SPQM. The name SPQM is our homage to Alberto Barchielli [41,42], who appears to be the first to have considered and analyzed this problem [43]. SPQM generates a measuring instrument with Kraus operators [40] (1.5) where 2H o ≡ P 2 +Q 2 and P and Q are (dimensionless) canonical momentum and position (or the conserved quadrature components of a harmonic oscillator). This time-ordered exponential defines another fundamental 7-dimensional manifold, the universal covering group, which for SPQM we call the Instrumental Weyl-Heisenberg Group, G = IWH. The universal covering group is defined by a map γ with the universal property that for any Hilbert space H carrying the paths of Kraus operators L (SPQM) : C T /dt −→ GL(H), there exists a unique representation R such that This universal way of considering G = IWH essentially amounts to suspending the choice of H and therefore , but it is very important to appreciate that the measuring instrument is, in fact, fundamentally independent of the Hilbert space and therefore . The same is true for the spin-isotropic measurement, except that different irreducible representations don't amount to choices of , but rather to choices of the total angular momentum number j.
The universal covering group IWH can be navigated in much the same way as can be done for semisimple Lie groups, with the use of right-invariant vector fields and decompositions similar to those of Cartan and Harish-Chandra. In particular, the sample-paths defined by x(t) = γ[dw [0,t) ] diffuse according to a Fokker-Planck-Kolmogorov equation, where D T (x) ≡ Dµ dw [0,T ) δ x, γ dw [0,T ) (1.8) is the Kraus-operator distribution function of the SPQM instrument with respect to the Haar measure [18,[44][45][46] of IWH and H o ← − , Q ← − , and P ← − are right-invariant derivatives tangent to IWH. We regard "Kraus-operator distribution function," "Kraus-operator distribution," and "Kraus-operator density" as interchangeable, despite subtle differences some might attribute to these usages. We abbreviate Kraus-operator distribution function as KOD to invite the reader to use any of these terms. The KOD can be considered the universal unraveling of the total (or unconditional) operation (a completely positive, trace-preserving superoperator), where Dµ dw [0,T ) is the Wiener path measure, d 7 µ(x) is the Haar measure, and O · (L) ≡ L ⊙ L † . The technology of right-invariant differentiation [47][48][49] will not be familiar to most quantum physicists and information scientists. Introducing SPQM, IWH, the concept of right-invariant motion, and the KOD is the subject of section II. Section III translates the coördinate-independent formulation of section II to forms that physicists are more likely to recognize. Indeed, what the aforementioned decompositions do is to coördinate the points of IWH [2,3,[50][51][52][53]. The decomposition of IWH similar to the Harish-Chandra [53][54][55] (a.k.a. "Gauss" [2,52,56]) decomposition is with purity coördinate r ∈ R, which we will call the ruler, central coördinate z = −s+iψ ∈ C, phase-space coördinates ν, µ ∈ C, and Ω = 1 H . The decomposition of IWH similar to the Cartan decomposition [51][52][53]55] is with different central coördinates φ, ℓ ∈ R, the same ruler r, and phase-space coördinates β, α ∈ C appearing in the conventional displacement operator D α ≡ e a † α−aα * . Introducing these decompositions and using them to transform equation 1.5 into standard Itô-form stochastic differential equations (SDEs) [57][58][59][60][61] and to transform equation 1.7 into a coördinate Fokker-Planck-Kolmogorov (FPK) diffusion equation [60,61] is the subject of section III, which also solves those SDEs and (mostly) solves the FPK diffusion equation. As a function of the registers of SPQM, we find that the ruler satisfies r T = 2κT . (1.12) For the remaining Harish-Chandra coördinates, we find the phase points to follow Ornstein-Uhlenbeck [60][61][62] and GGW processes, where T − ≡ T − dt, and the center to follow a quadratic functional process, where used is the sign function, sgn(u) = u/|u| for u = 0 and sgn(0) = 0. The Cartan phase-space coördinates follow linear functionals, The Cartan central coördinates ℓ and φ follow from the coördinate transformation between Harish-Chandra and Cartan coördinates (equations B13, B14, B17, and B18), but we do not write those solutions explicitly here. As for the KOD D t (x), we will not be able to solve analytically for the distribution over all seven dimensions. Summing over the center, however, Z ≡ e 1z : z ∈ C ⊳ IWH , (1.16) gives a reduced SPQM unraveling of the total operation, where the integral is over the quotient group IWH/Z, which we call the Reduced Instrumental Weyl-Heisenberg Group (RIWH). The reduced Kraus-operator distribution (RKOD), is a marginal over the center that includes the Cartan center factor e −2ℓ . We call C T (Zx) the Cartan-section reduced distribution function, and we are able to solve for it from its FPK diffusion equation. The solution is a Gaussian with ill-defined normalization, where the mean-square distance between the two phase points is given by The normalization factor 2 sinh 2κT is particularly interesting, as the POVM completeness relation for the SPQM instrument boils down to (assuming = 1) 21) and this can be recognized as equivalent to the result of energy quantization, . (1.22) This demonstrates that the KOD can be considered an alternative to energy quantization. Indeed, the operator H o does not appear in SPQM as an energy observable, but rather as a term required by the positivity of sampling measurement records. The late-time limit of the completeness relation is of particular interest: when T ≫ 1/κ, showing that the SPQM POVM elements approach Glauber coherent states; the completeness relation shows that it does so uniformly, thus giving the coherent-state resolution of the identity. Everything in our analysis, the path integrals, the KODs, the FPK diffusion equations, the SDEs-everything!follows from the path integral 1.9 for the overall quantum operation Z (SPQM) T , which integrates over the sample-paths as they appear in the time-ordered Kraus operator L (SPQM) dw [0,T ) . Yet as the analysis develops in sections II and III, there emerges a disconnect between the reduced distribution C T (Zx), which expresses the completeness relation, and the SDEs and stochastic integrals for the phase-point coördinates: C t (Zx) has ill-defined normalization, and it is not the weight function whose moments are those of the Cartan phase-point variables, β and α, as they are expressed in the stochastic-integral solutions 1.15. The disconnect is all about the real center term, e −Ωℓ , which scales the Kraus operators when they are written in Cartan coördinates.
The three faces of the KOD stochastic trinity-path integrals, diffusion equations, and SDEs-having been sundered in section III, we re-unite them in section IV. The vehicle for the reunion is the Harish-Chandra-section reduced distribution function, where −s is the real part of the Harish-Chandra center coördinate z. The distribution B T (Zx) can be determined in two equivalent ways: first, from the FPK diffusion equation for B T (Zx) and, second, from applying the stochastic integrals for the Harish-Chandra phase-plane variables, ν and µ, to the above path-integral expression for B T (Zx). The stochastic trinity thus restored, one returns to completeness through the relation where f (Zx) is a quadratic function of the phase-plane variables that comes directly out of the coördinate transformation between Cartan and Harish-Chandra coördinates (equations B9 and B17); e 2f (Zx) can be regarded as a positive gauge transformation [19]. Section V concludes with musings on the stochastic trinity and the Lie-group manifolds that house instrument evolution.

II. IWH AND COÖRDINATE-FREE RIGHT-INVARIANT MOTION
This section defines the Simultaneous P&Q Measurement (SPQM) process and presents the Instrumental Weyl-Heisenberg Group, G = IWH, as the universal covering group of SPQM.
Section II A introduces Kraus operators and the concept of observables generating positive transformations instead of unitary transformations. Section II B introduces the SPQM process and the group IWH. Section II C explains how the SPQM instrument is universal and defines a Kraus-operator distribution (or density) (KOD) over IWH. Section II D explains how the KOD diffuses over time with the introduction of right-invariant derivatives, a differentialgeometric technology that will be unfamiliar to most physicists and quantum scientists. Section II E explains how the sample-paths of the Kraus-operator diffusion are described with the introduction of the right-invariant one-forms, which are dual to right-invariant derivatives.
The diffusion equation and stochastic differential equations in sections II D and II E will be solved in section III.

A. Observables and Infinitesimal Positive Transformation
While observables are often considered to be infinitesimal generators of unitary transformations, they can also generate positive transformations. Let X be a Hermitian observable, κ be a real number with units of inverse time, and dW be a standard Wiener increment [10-12, 15, 16, 19, 60, 61], which has measure Unitary transformations can be infinitesimally generated either deterministically or stochastically, such as Positive transformations on the other hand are fundamentally stochastic. In addition, the infinitesimal generators of positive transformations are not of the canonical form, with a single parameter conjugate to the infinitesimal generator. The positive transformations we will be interested in do not involve jump operators, as in photodetection [37,63], but rather are differential, with infinitesimal generators of the form [40,[64][65][66] As a set, these positive transformations define a measuring instrument, complete over the Hilbert space, H, according to the relation, the operators L X (dW ) are known as Kraus operators [29,31], and the set of elements, {dµ(dW )L X (dW ) † L X (dW )}, is known as a positive-operator-valued measure or POVM. We will call the positive transformations of equation 2.3 differential Kraus operators and their set a differential instrument, both in anticipation of the (multi-dimensional) differential geometry coming up and to contrast these Kraus operators with jump operators. The form of these infinitesimally generated positive transformations comes from the requirement that the total operation be completely positive and trace preserving. In particular, define the superoperator A ⊙ B by Then the aforementioned Kraus operators define a total operation, where defined is the adjoint superoperator In this context, the infinitesimal generator or Lindbladian, defines X as a Lindblad operator [28].
Kraus operators can be interpreted as an indirect measurement [29,31,40,67,68], where in the differential case a meter with initial meter wavefunction, is displaced by the system according to the interaction which in turn registers some "position" q, fixing (for further details, see [40]). An irresistible sidenote, developed more generally and in more detail in [40], is that the stochastic unitary transformations of equation 2.2 follow from the same meter interaction 2.11, but with registration of the meter momentum p, instead of its position q. As such, the stochastic unitary transformations have a total operation identical to equation 2.7, This alternative unraveling of the total operation corresponds to a symmetry of the general Lindbladian, so that −iX is an alternative Lindblad operator of the total operation.

B. SPQM and the Group IWH
The subject of this paper is the continual (or continuous) simultaneous observation of the canonical observables P and Q, defined by the canonical commutation relations, As a Lie algebra of infinitesimal generators, these observables are usually considered to generate unitary displacement operators, which together define the 3-dimensional Unitary Weyl-Heisenberg Group, If the generators operate irreducibly on the Hilbert space H, then by Shur's lemma Ω = 1 H for some ∈ R (because gΩg -1 = Ω for all g ∈ UWH.) Assuming that is finite, that is = 0, all such representations are essentially equivalent because the observables can always be rescaled so as to make the choice = 1. Therefore, it is usually assumed that We shall now assume = 1, but we will continue nonetheless to use Ω for the infinitesimal generator to emphasize its existence is not defined by the Hilbert space, but instead by the canonical commutation relations. In particular, what is defined by the Hilbert space is the relation Ω 2 = Ω, which is associative algebraic and not Lie algebraic. For quantization, the Unitary Weyl-Heisenberg Group is supplemented with the unitary group generated by defining the 4-dimensional Dynamical Weyl-Heisenberg Group, Here, the use of "dynamical" refers to the analogies between H o and the classical Hamiltonian of a simple harmonic oscillator, which quantum mechanics was originally founded upon.
The observables P and Q can be measured simultaneously in the sense that the positive transformations they generate commute infinitesimally (to order dt), (2.27) so long as the Wiener outcome increments, dW q and dW p , are independent, their joint Wiener measure being Here we switch to using a complex Wiener increment Continually repeating this simultaneous measurement for a finite amount of time, T , defines the overall Kraus operators, where T denotes a time-ordered product. It is important to appreciate that while the Kraus operators commute infinitesimally, they do not commute over finite amounts of time. Finally, these Kraus operators are accompanied by the Wiener path measure, which is written here in terms of the complex Wiener increments. In summary, we have defined a time-dependent instrument with Kraus operators where T exp is the time-ordered exponential. This is the Simultaneous P&Q Measurement (SPQM.) It is worth repeating that H o appears here due to the form of the differential positive transformations of equation 2.3 and in this context is not a Hamiltonian because it is not generating unitary transformations. While SPQM has been considered as far back as [43], it has not been fully solved before. There are two other ways of measuring P and Q simultaneously that are important to distinguish from SPQM: the Arthurs-Kelly measurement [68,69] and the Goetsch-Graham-Wiseman (GGW) model of heterodyne detection [33][34][35][36][37]. The Arthurs-Kelly measurement has the same system-meter interaction as the SPQM process, but is different in that Arthurs and Kelly imagine continually interacting the same two meters with the system until the measurement is terminated, whereas in the SPQM process the system interacts with many pairs of meters successively, registering the complex Wiener path dw [0,T ) . The GGW model of heterodyne detection has the same many-meter model as the SPQM process, but each system-meter interaction is energy conserving (the so-called leaky cavity), whereas the system-meter interaction of the SPQM process is the meter displacement of equation 2.11.
The total operation of the SPQM process is a Wiener-like path integral, which is absolutely trivial to solve, The interest of this article, however is entirely in the manifold diffusion process defined by SPQM, where equation 2.32 is understood to define sample-paths in a finite-dimensional manifold. The infinitesimal generators of these samplepaths are Q, P , and H o . By simply considering their Lie brackets to first order, There is a fourth and final Weyl-Heisenberg group worth defining, the 6-dimensional Complex Weyl-Heisenberg Group, CWH ≡ e Qq e P p e −Ωℓ D α e iΩφ : q, p ∈ R, α ∈ C, φ, ℓ ∈ R , (2.43) which is a maximal normal subgroup of IWH, called the derived subgroup. In Lie-group terminology, IWH is said to be solvable while CWH is said to be nilpotent or unipotent [52,53]. In particular, the derived series of G is We have Note also that the displacement operator, has the ordered forms, 2 Ω e −aα * e a † α , (2.51) which will prove useful in relating coördinate systems and in evaluating right-invariant derivatives. The first form in equation 2.51 is usually called normal ordering, and the second form is called antinormal ordering.

C. Haar Measure, Dirac Delta, and Kraus-Operator Distribution Function
Many readers will interpret the groups defined in the previous section as matrix groups, where the observables are quantized in the usual way. The exponentials can be understood more abstractly, however, as generating pathconnections, and this way of thinking gives rise to what is called the universal covering group [40,53,[70][71][72][73]. We will now start to consider more seriously the points of IWH in this universal fashion and think of the instrument of SPQM as a representation of G = IWH. The map L : C T /dt −→ GL(H) from the set of paths, C T /dt , to the operator space, GL(H), can be factored into two maps, where γ : C T /dt −→ G maps Wiener paths to the universal covering group and R : G −→ GL(H) is the representation, mapping the universal cover to the space of linear operators on H [53]. We have denoted a sample-path by dw [0,T ) , and we now start denoting elements of the instrumental group by x. To drive the notation home, we note that the instrumental group element and Kraus operator associated with a sample path are denoted by The distinction between L and γ emphasizes that the time-ordered exponential of equation 2.32 actually defines a diffusion problem on the instrumental group that is independent of the spectral information inherent in the definition of a linear operator. In particular, this means the entire analysis of this article is independent of whether or how H o is quantized. (Remember, H o is quantized in the usual way the moment H is assumed to be irreducible and to have a ground state.) As does every finite-dimensional Lie group, G = IWH has a right-invariant Haar measure [18,[44][45][46], As is not always the case for (solvable) Lie groups [52], it turns out that this right-invariant measure is also equal to the left-invariant measure, and this left invariance will turn out to be important for the interpretation of the SPQM process as a diffusion, as will be seen in the very next subsection. Comments about the existence and uniqueness of the Haar measure will be given in appendices D and E, but first it is worth taking it for granted and appreciating what can be done with it.
With the Haar measure, we can introduce the accompanying singular distributions or "Dirac deltas," defined by the property (sometimes called reproduction [4,74]) that for any function f of G = IWH, (2.56) From the invariance properties of the Haar distribution, the Dirac delta distributions inherit the corresponding covariance properties, With the Haar measure and the Dirac delta, we can define a universal instrument by adding up all of the Wiener paths that end at the same Kraus operator, starting from the origin. This becomes visible by considering the total operation, In summary, the SPQM process defines a universal instrument, which unravels the total operation, according to a Haar-based Kraus-operator distribution function (KOD), which is defined by a Wiener path integral [10-12, 15, 16, 19]. The total operation is a completely positive, trace-preserving superoperator. The trace-preserving property is equivalent to saying that the POVM elements, , satisfy a completeness relation, The term "universal" refers to the fact that this description of the instrument is common to every representation and comes from the concepts of universal covering group and universal enveloping algebra [40,72,75]. It is important to understand that the universal covering group, IWH, is defined purely by the local structure (that is, the Lie algebra) of the observables and the quadratic term, here H o , which accompanies the observables due to the nature of differential positive transformations. In particular, this means G = IWH is not defined by the Hilbert space of states. This ability to describe the measuring instrument without reference to a Hilbert space is so striking that we give it a name: universal instrument autonomy [37,40]. SPQM is in a very special class of universal instruments for which the universal covering group is finite dimensional; we dub such instruments principal instruments [40].

D. Diffusion Equation in Terms of Right-Invariant Derivatives
The definition of the Kraus-operator distribution function 2.63 can be thought of as a Feynman-Kac formula [15,16,19] for the solution of a Fokker-Planck-Kolmogorov (FPK) diffusion equation [60,61]. This FPK diffusion equation can be obtained easily with the help of the so-called right-invariant derivatives [39,40,[47][48][49], which can be seen to have commutators [40] X This negative sign is why left-invariant derivatives are usually considered instead of right-invariant ones. Nevertheless, because the convention is to consider operators as acting to the right we are more-or-less stuck with having to consider a right-invariant basis of local transformations. With the right-invariant derivatives, D t+dt can then be expanded about t in the standard way. We start with is the purely group-theoretic version of L(dw t ) of equation 2.23, that is, Here we also define the forward generator δ t : γ(dw t ) = e δ t is the fundamental differential positive operator for SPQM, and the forward generator δ t is thus the core mathematical object for the theory of the SPQM instrument. Continuing with equation 2.68, we have is the FPK forward generator. Equation 2.70 is where the left invariance of the Haar measure is used. Equation 2.71 is the analog of a Chapman-Kolmogorov equation for the distribution function [60,61]. Equation 2.73 moves e δ t outside the argument of the KOD to become an exponential of the right-invariant derivative which we call the vector-valued SPQM increment. Equation 2.74 Taylor expands the distribution function to second order, that is, to order dt, as required by the Itô rule for the Wiener outcome increments. The remaining step to the FPK forward generator ∆ involves averaging over the Wiener distribution dµ(dw t ); in this averaging the deterministic contributes a first-derivative term to the FPK forward generator, whereas the Wiener outcome increment terms in δ t ← − contribute second-derivative diffusion terms.
In summary, the KOD of SPQM evolves according to the FPK diffusion equation, where defined is the group identity which is the origin of IWH. Equation 2.78 will be (mostly) solved in section III E after having established two coördinate systems. The subtlest thing about equation 2.78 is remembering that the Lie algebra of the three directions apparent in the equation means that the motion beyond first order is actually 7-dimensional. Before proceeding to SDEs, we draw attention to an important property of right-invariant derivatives. The reader might already be thinking about this property by wondering why we didn't write δ t ← − and ∆ in terms of the rightinvariant derivatives associated with a and a † . The reason is that the map from operators to right-invariant derivatives, X −→ X ← − , is only R-linear and not C-linear [40]. This means that Q ← − , P ← − , iQ ← − , and −iP ← − are R-linearly independent, with Q ← − and P ← − displacing Kraus operators in positive directions on IWH and iQ ← − and −iP ← − displacing in unitary directions. The right-invariant derivatives a ← − , a † ← − , −ia ← − , and ia † ← − each represent a different way of combining equally displacements in the positive and unitary directions. In view of this, it is instructive to note that the vector-valued SPQM increment (2.77) has the form, This means, in particular, that

E. Sample-Path SDEs in Terms of Right-Invariant One-Forms
As has been mentioned, the time-ordered exponential of the SPQM process (equation 2.32) can be interpreted as defining sample-paths in the 7-dimensional manifold G = IWH. Sample-paths are usually described by stochasticdifferential equations (SDEs). We finish this section by explaining how such SDEs can be expressed in terms of the right-invariant structure.
The basis of right-invariant derivatives, defines a dual basis of right-invariant one-forms, In terms of the right-invariant one-forms, the Haar measure has a simple expression in terms of wedge products, Also in terms of the right-invariant one-forms, the SDEs equivalent to equation 2.32 are given by reading off the coefficient conjugate to the corresponding generator in the vector-valued SPQM increment 2.77, These SDEs can be broken into two types: the first-order SDEs, and the Pfaffians, These equations will be solved in section III D after having established a coördinate system. The stochastic equations 2.93 and 2.94 are almost obvious by definition, but there is a subtlety that requires attention. The right-invariant derivatives and one-forms live in the tangent and cotangent spaces to the group manifold G and thus are based on linear transformations that use the chain rule of ordinary calculus. Hence the stochastic equations that come from the right-invariant one-forms are Stratonovich-form SDEs [60,61]-this means mid-point evaluation of coefficients of stochastic increments-and should be converted to the Itô-form SDEs in which coefficients are evaluated at the beginning of the increment. In the context of IWH, the only place this subtlety makes a difference is in the SDEs that come from θ iΩ and θ −Ω . Jackson and Caves [39,40] introduced the modified Maurer-Cartan stochastic differential (MMCSD) as a way to get to the Itô-form equations directly. The MMCSD is an example of the Itô correction in SDEs [60,61], specifically the Itô correction that arises when one transforms between a stochastic variable and the exponential function of that variable, as occurs in the forward generator 2.69. In this paper we get directly to Itô-form SDEs in a different way, when we consider the Harish-Chandra decomposition in section III B.
A further subtlety about the right-invariant one-forms is that they have "curl" in the same sense as Gibbs would have defined. In the standard language of forms, this is because the exterior algebra of forms is equivalent to the Lie algebra of derivatives [47], This equivalence is standard in modern differential topology, but an introduction is included in appendix A; though no use will be made of the "curls" dθ λ in this article, they are given for completeness in equations A15-A21.

III. IWH AND TWO COÖRDINATE SYSTEMS
Having introduced the Instrumental Weyl-Heisenberg Group, G = IWH, a coördinate system needs to be established so that we can locate the sample-paths of the SPQM process and follow their propagation. If the concept of a universal covering group introduced in the previous section is unclear, seeing how equation 2.32 is equivalent to a set of coördinate SDEs should help to appreciate that G = IWH and the SPQM process are independent of matrix representation.
We will use two coördinate systems, analogous to what are called Harish-Chandra [53-55] (a.k.a. "Gauss" [2,52,56]) decompositions and Cartan decompositions [51][52][53]55]. These decompositions were originally designed in the context of semisimple Lie groups [50,52,53,76,77], of which IWH is quite the opposite (in the sense of the Levi-Malcev decomposition). In spite of this distinction, it is very useful to think of IWH in many ways as if it were semisimple. The Harish-Chandra decomposition is easier to prove first and will allow us to make connections between the SPQM process and two processes more familiar to physicists, the Ornstein-Uhlenbeck process and the Goetsch-Graham-Wiseman (GGW) heterodyne measuring process. The Cartan decomposition is better suited for considering the POVM.
Section III A identifies the analogs of the various elements of semisimple group theory. Section III B proves the Harish-Chandra decomposition of IWH in a way that also produces the corresponding Itô-form coördinate SDEs, which are immediately recognized and solved. Section III C introduces the Cartan decomposition and the transformations to Harish-Chandra coördinates and presents the right-invariant derivatives and one-forms in both coördinate systems. Section III D solves the SDEs of the SPQM process in both Cartan and Harish-Chandra coördinates. Section III E solves most of the FPK diffusion equation of the SPQM process in Cartan coördinates, by which we mean introducing the Cartan-section reduced distribution function and solving for it. Section III F explains how the solution of the FPK diffusion equation means that the POVM of the SPQM process offers an alternative perspective on the meaning of energy quantization.

A. Usual Elements of Semisimple Lie Group Theory
As introduced in the previous section, the Lie group of interest is the so-called Instrumental Weyl-Heisenberg Group Although G is literally solvable, with derived series, and center, it can be navigated in much the same way as a semisimple group. While a bit of the terminology [53] will be used here, the theory of semisimple groups will be more-or-less glossed over. The purpose of this subsection is basically to label the various subgroups that will prove to be both meaningful and useful for navigating IWH and therefore understanding SPQM. The significance of these subgroups should become apparent as they are applied. The map from the SPQM instrument to the SPQM POVM, defines the SPQM POVM as similar to a symmetric space, albeit a nonRiemannian one, with Cartan group involution The subgroup of transformations that are even under the Cartan involution is the usual Unitary Weyl-Heisenberg Group of equation 2.19: On the other hand, the remainder of G displaces from the origin of the symmetric space, Considering the conjugation action of K on E, almost all of the K-conjugacy classes can be parameterized by the Cartan subgroup, and regular POVM elements are invariant under the commutant, Thus the K-conjugacy classes have the topology of the familiar phase space, Indeed, it is the "almost all" feature where G and E depart from semisimple groups and Riemannian symmetric spaces, since positive transformations of the form e Qq+P p are characteristically not in the conjugacy classes of A (see appendix B for additional perspective). Finally, important also are the maximal nilpotent (or unipotent) subgroup, and perhaps the most important, the Borel subgroup, This group-theoretic context now in hand, we stress that the most important groups for what follows are G = IWH itself, the center Z of G, and the quotient group G/Z = IWH/Z. The center Z contains a phase, which is of no importance, and a normalization, which is the main source of difficulty in analyzing SPQM. The cosets Zx ∈ G/Z are parametrized by what we call the ruler, r, and by two complex phase-space parameters. One of these complex phase-space parameters is associated with the POVM, and the other parametrizes a post-measurement displacement operator. We call G/Z the Reduced Instrumental Weyl-Heisenberg Group (RIWH). It is isomorphic to the adjoint group of G, but given the way we will use multipliers on the cosets Zx, we prefer to think of G as a central extension of G/Z.

B. Harish-Chandra Decomposition and SDEs as a Proof by Transfinite Induction
The 7-dimensional Instrumental Weyl-Heisenberg Group G affords a Harish-Chandra decomposition, G = N † M AN , where every element can be decomposed into the form defining seven Harish-Chandra coördinates (ν, r, z, µ) ∈ C×R×C×C. We often break the complex coördinates into real and imaginary parts, 14) The coördinate r we call the ruler, ν and µ are the postmeasurement and POVM phase-plane coördinates, and z is the IWH center coördinate. A proof that this decomposition exists for every element of the SPQM process (equation 2.32) is not difficult if we allow ourselves to apply transfinite induction: At time t = dt (the first infinitesimal increment), it is easy to see that the decomposition exists because (see equation 2.69) Trivially, this also means that the decomposition exists for any finite integer n and infinitesimal time t = ndt simply because the one-parameter subgroups commute to infinitesimal order, so long as the Wiener increments are independent. Now for the transfinite step. If we assume that the decomposition holds for a finite time t, then an increment later in the SPQM process we have This concludes the proof of the Harish-Chandra decomposition for the SPQM process and G = IWH.
A consequence of the proof is that the SPQM process in Harish-Chandra coördinates is equivalent to the Itô-form SDEs [57][58][59][60][61], Although these are Itô-form SDEs, notice that we did not use the Itô rule in deriving them; in particular, we do not set |dw t | 2 = dt in the SDE for the center coördinate z. We solve these equations for the initial condition r 0 = ν 0 = µ 0 = z 0 = 0. These initial values are chosen so that x 0 = 1, in agreement with the δ-function initial condition 2.79 for the KOD. It is straightforward to see that the first three SDEs have as solution, where T − ≡ T − dt. The fourth equation is solved by plugging the solution for ν t into the equation for z t and integrating, with the result that where is the sign function. There being subtleties in deriving and interpreting this solution, it is worked out carefully in appendix C. The solutions for the real and imaginary parts of z are The complementarity in time of the OU and GGW processes (equations 3.28 and 3.29) is also interesting: whereas the post-measurement variable ν T of equation 3.28 depends only on the end of the outcome register, the POVM variable ν T of equation 3.29 depends only on the beginning of the register. It is thus reasonably clear that the POVM of the SPQM process culminates in the usual "measurement in the coherent-state basis," where d 2 µ = 1 2 dµ 1 dµ 2 . This is just like GGW heterodyne detection, except that the post-measurement state is not vacuum but instead scrambled over phase space. That the POVM variable µ T ceases to evolve after a few e-foldings seems to be an important feature in the interpretation of SPQM as a measuring process [67]. All this said, there remains the elephant in the room that prompts us to label these conclusions as "reasonably clear": the elephant is the factor that scales the long-time Kraus

C. Cartan Decomposition and Various Transformations
As far as what the SPQM process is ultimately doing in time, section III B in many ways says it all-except for that elephant in the room. We now turn our attention to a detailed understanding of the measuring process at finite times, which requires addressing the elephant. Key to this is the realization that the Harish-Chandra decomposition is not well suited to telling us the behavior of the POVM. Rather, a Cartan decomposition of the Instrumental Weyl-Heisenberg Group, G = KAK, works best for this purpose. A straightforward calculation (left to appendix B) shows that almost every group element-and every Kraus operator-with a Harish-Chandra decomposition also affords a Cartan decomposition, and therefore the POVM elements decompose as where the Cartan coördinates, (β, φ, r, ℓ, α) ∈ C×R×R×R×C, can be obtained by the coördinate transformations (the ruler r is shared by the two coördinate systems), (3.41) Here f and ξ are functions of the ruler and the phase-plane coördinates, that is, functions on the RIWH G/Z, (3.43) Notice the singularity in Cartan coördinates at r = 0, about which there is further discussion throughout the remainder of the paper and, in particular, in appendix B.
As we wish to solve equation 2.78 and equations 2.86-2.92, more to our purpose are the transformations from the right-invariant moving frame to the Cartan coördinate frame. A calculation of these frame transformations can be found in appendix D, but we include them here for continuity; we also include the transformations to the Harish-Chandra coördinate frame, which are worked out in appendix E. For the FPK diffusion equation 2.78, we require the transformations of the derivatives, where The two coördinate systems share the coördinate r, but the partial derivative with respect to r is, of course, different in the two systems. In the above equation we distinguish ∂ r in the two systems, but we do this nowhere else because it is always clear in which coördinate system we are operating. It is worth recording for working in terms of the complex phase-space variables that For the SDEs 2.86-2.92, we require the transformations of the one-forms, With the one-form transformations in hand, the Haar measure 2.85 in the two coördinate systems is Here and elsewhere complex phase-plane measures are denoted by d 2 β = 1 2 dβ 1 dβ 2 . The factors of 1/π in the Cartan measure are conventional in quantum optics and ultimately come from the coherent-state completeness relation 3.35. The factors of 1/2π in Harish-Chandra coördinates then follow from the transformation from Cartan to Harish-Chandra coördinates. The left invariance of these measures is discussed in appendices D and E. While we are considering measures, we should record the reduced measure on the 5-dimensional group RIWH = IWH/Z: Here dµ(Z) = dφ dℓ = dψ ds is the measure on the center Z.
Accompanying the coördinate Haar measure d 7 µ(x) are the coördinate forms of the conjugate δ-function, There are obvious coördinate forms for the δ-function δ(Zx, Zx ′ ) that is conjugate to d 5 µ(Zx): We are especially interested in δ(x, 1). Because the identity 1 has Harish-Chandra coördinates φ = ℓ = r = 0 and ν = µ = 0, we have δ(x, 1) = δ(φ) δ(s) δ(r) 2πδ 2 (ν) 2πδ 2 (µ) . (3.55) The Cartan form of δ(x, 1) requires more attention because of the coördinate singularity at r = 0 (see appendix B for discussion). The singularity is about more than just the 1/ sinh 2 r in the Cartan form of the δ-function, although that singularity is the root of the difficulties that require attention. We provide the necessary attention in appendix F.
dν + e r dµ + ν dr , (3.57) and the four Pfaffians 2.94 give where the evaluation of the coördinate one-forms on the vector-valued SPQM increment, δ t ← − , is no longer denoted. As was discussed in section II E, these are Stratonovich-form SDEs, which means that the coefficients are evaluated at the midpoint, t + 1 2 dt, of the increment-technically the midpoint has no status in the stochastic calculus, so one can regard midpoint evaluation as a t+dt/2 = 1 2 (a t + a t+dt ) = a t + 1 2 da t -but for these SDEs, that evaluation does not produce an Itô correction (because dr t = 2κdt has no stochastic term), and the equations can be read as Itô-form SDEs, with the coefficients evaluated at the beginning of the increment. When r and dr are substituted into these SDEs, the equations for the Harish-Chandra complex phase-point coördinates, ν and µ, become those of equations 3.24 and 3.25, with solutions 3.28 and 3.29 for initial conditions ν 0 = 0 and µ 0 = 0.
The SDEs for Cartan phase-space points are For integrating, these SDEs are more profitably written as and the reason is that these are the same as integrating the SDEs for the Harish-Chandra phase-plane coördinates. The solutions, satisfying initial conditions ν 0 = µ 0 = 0, are those obtained in equations 3.28 and 3.29 for the Harish-Chandra phase-space points. Summarizing, we have that the solutions for the ruler and the Cartan phase-space coördinates are The Stratonovich-form SDE for the Harish-Chandra center coördinate z follows from the SDEs 3.59 and 3.60, Converted to Itô form, this equation is where all the coefficients are evaluated at the midpoint t+ 1 2 dt. Converted to equivalent Itô-form SDEs, these equations become Though more complicated, these equations have a similar character to the SDEs for the Harish-Chandra center coördinate z = −s + iψ. The Itô correction for dℓ is the last term in equation 3.74, and it becomes the coth term at the beginning of equation 3.75, whereas the Itô correction for dφ at the end of equation 3.76 vanishes. We could solve these equations directly, but it is both easier and more productive to combine the solution for z with the coördinate transformation to Cartan coördinates, thus finding where s T and ψ T are the Harish-Chandra solutions 3.32 and 3.33 and f T and ξ T are the functions 3.42 and 3.43 with all the coördinates evaluated at time T .

E. Solving Most of the FPK Diffusion Equation
Section II D left off showing that the sample-paths of SPQM diffuse according to the KOD D t (x), which satisfies the FPK equation 2.78 with initial condition D 0 (x) = δ(x, 1). The crucial mathematical object in the diffusion equation is the FPK forward generator ∆, which is written in terms of right-invariant derivatives in equation 2.76.
With the frame transformations of equation 3.44 at hand, it is easy to express the three pieces of the forward generator in Cartan coördinates, where we introduce the quantities which are independent of φ and commute with ∂ φ . The φ-derivative terms quickly disappear from the analysis, ultimately because φ is irrelevant to the instrument elements as a consequence of the symmetry O · (e iΩφ L) = O · (L). Putting these expressions together, the FPK forward generator 2.76 in Cartan coördinates is where ∇ and ∇ * , defined in equations 3.46 and 3.47, are derivatives with respect to complex phase-space coördinates. Because of the cubic and quartic nature of the last few terms in equation 3.91, we do not hope to find a complete analytic solution to equation 2.78. However, "5/7-ths" of the distribution can be analyzed quite easily. Remember that we are interested in the instrument, and observe that the instrument elements can be partitioned by reconsidering the total operation 2.62 as where we use the coset measure d 5 µ(Zx) of equation 3.50 and define the Cartan-section reduced distribution function, Readers uncomfortable with the coset notation can think that in this equation, x = e iΩφ e −Ωℓ D β e −H o r D † α and Zx = D β e −H o r D † α . Even more prosaically, one can regard D T as being a function of all seven Cartan coördinates and C T as being a function of five of them, the ruler r and the complex phase-space coördinates β and α. Our excuse-quite a good excuse, really-for using the coördinate-independent coset notation is that we will elaborate on this distribution function and another one in section IV, but working there mainly in Harish-Chandra coördinates. Equation 3.95 is a new unraveling of the SPQM instrument, which we call the reduced SPQM instrument, with instrument elements in which the Cartan reduced distribution C T (Zx) is conjugate to the operation O · (D β e −H o r D † α ). If H o is quantized in the standard way, we have at late times, This makes clear that C T (Zx) is the elephant in the room hinted at in section III B: it is the phase-space weighting function that is crucial for the POVM completeness relation, which takes the form As an elephant, however, C t (Zx) is not normalized to unity-indeed, its normalization is ill-defined. Moreover, C t (Zx) is not the weight function whose moments are those of the Cartan phase-point variables β and α according to the SDE solutions 3.69. The distribution function that does give these moments is the straight marginal of D T (x) over the center Z, This distribution is normalized and has finite moments, those coming from the SDE solutions 3.69. For those very reasons, however, D T (Zx) cannot possibly give rise to a POVM completeness relation; it will not be seen again in this paper.
To derive an evolution equation for C t (Zx) of equation 3.96, one takes its time derivative, substitutes the FPK equation 2.78 into the integral, and pushes the FPK forward generator ∆ of equation 3.91 through the center integrals by integrating by parts. Integration by parts on φ gets rid of the derivatives with respect to φ, and integration by parts on ℓ translates to substituting ∂ ℓ → 2, resulting in the partial differential equation (PDE), where recall that ∇ * ∇ = 1 2 (∇ 2 1 + ∇ 2 2 ). This PDE is ballistic in the ruler r-solution proportional to δ(r − 2κt)-and Gaussian preserving in the phase-space variables, but as a consequence of the −2 coth r term, the PDE does not preserve normalization [15,16,19].
To solve for C t (Zx) requires knowing D dt (x), which is, when dt → 0, the δ-function δ(x, 1). This can be done fairly easily by evaluating the Cartan-coördinate solutions 3.69 and 3.78 at T = dt. We perform that task in appendix F, where we also identify all the δ-function forms for initial conditions. The result for D dt (x) is The distinctive feature of this distribution is the wide, normalized Gaussian in β + α, which limits to a uniform distribution in β + α as dt → 0. What the wide Gaussian is about is that the identity is represented by Cartan coördinates φ = ℓ = r = 0 and β = α, with β + α free to take on any complex value. With this expression, it is easy to see that The integral of this distribution has a zero from the 1/r behavior multiplying the sinh 2 r in the measure d 5 µ(Zx) and an infinity from the uniformity in β + α; therefore, the normalization is ill defined. This ill-defined normalization is, however, exactly what is needed to give a well-defined POVM. For these reasons, C dt (Zx) does not limit to δ(Zx, Z1) as dt → 0, for which see appendix F. The initial condition C dt (Zx) is independent of β + α, and the distribution C t (Zx) remains so under the PDE 3.101. To see the consequences most clearly, it is useful to transform to sum and difference variables, (3.104) in which the covariant derivative ∇ of equation 3.46 becomes Therefore the weight function evolves according to the PDE, The width of the difference in phase points, Σ T , satisfies the differential equation, dΣ t /dt = κ tanh 2 κt, with solution, given initial condition Σ 0 = 0, In summary, the SPQM instrument can be considered as the reduced-SPQM-instrument unraveling, with instrument elements (3.110) where the width Σ T of the difference in phase points is given by equation 3.108. There are four notable features in the temporal behavior of the reduced SPQM instrument: 1. The ruler r (or purity parameter) is ballistic, which means that e −H o r collapses exponentially to e −κT |0 0| in the standard quantization. More generally, D β e −rH o D † α collapses exponentially at late times to an outer product of coherent states, e −κT |β α|.
2. The dependence on the future and past phase-space parameters, β and α, is only in their difference.
3. The distribution of the difference spreads out very slowly for small times, as Σ T ∝ T 3 , and then for long times becomes normal diffusion, with Σ T ∝ T .
4. There is a center normalization, 2 sinh 2κT , that increases over time.
This center normalization is the focus of the next section, which finds that the elephant provides an alternative perspective on the quantum.

F. POVM as an Alternative Perspective on the Quantum
The center normalization just mentioned is remarkable in that it is equivalent to traditional energy quantization. Specifically, the completeness relation 3.99 for the SPQM process is It is important to appreciate that for late times T ≫ 1/κ, when e −H o 4κT collapses to e −2κT |0 0| in the standard quantization, this completeness relation becomes the coherent-state resolution of the identity of equation 3.35: The completeness relation says much more, however, when considered for arbitrary times T . If H is an irreducible representation, Schur's lemma says that The trace, which one recognizes as a partition function, is defined within the representation and is evaluated using traditional energy quantization as The center normalization in the POVM completeness relation thus evaluates the partition function of e −H o 4κT without using traditional energy quantization. This is to be expected in view of the Stone-von-Neumann theorem, but expected though it is, please appreciate how different the setting of this paper is from the original ideas of energy quantization and thermal equilibrium. Remember that here the operator H o comes from the trace-preserving character of the instrument. It is not the energy of the system; indeed we have explicitly eschewed any notion of system energy or Hamiltonian. The operator H o plays the role of a "dissipator," specifically a dissipator that damps the POVM to the coherent states, but there is no notion of energy associated with this dissipation. The coherent states and H o arise within a group structure constructed solely from the measured observables, Q and P . Moreover, the parameter conjugate to H o , the ruler r, is quite literally time, rather than an inverse temperature. It seems remarkable that this result holds, from the completeness of the SPQM POVM, without any assumption of a Hamiltonian, a ground state, or thermal equilibrium.

IV. REDUCED DISTRIBUTION FUNCTIONS AND FEYNMAN-KAC PATH INTEGRALS
This section further considers reduced distribution functions, their path-integral expressions and diffusion equations, and their relation to SDEs. The path-integral expression, is generally considered to be a Feynman-Kac formula [15,16,19] for the associated diffusion equation. Our analysis is rooted in the path integral 2.33 for the overall SPQM quantum operation, Section IV A reviews the Cartan-section reduced distribution C T (Zx), introduces the Harish-Chandra-section reduced distribution B T (Zx) of Eq. (1.24), and shows that these two are related by a positive gauge transformation. Section IV B defines a normalized version of the Harish-Chandra reduced distribution, denoted byB T (Zx), and finds its path-integral expression in terms of a modified path-integration measure in which the outcome increments are correlated. Section IV C formulates the diffusion equations for B T (Zx) andB T (Zx), and section IV D solves the path integral forB T (Zx) using the stochastic integrals for the Harish-Chandra phase-space coördinates.

A. Feynman-Kac Formulas
The ur KOD of equation 4.1, unravels Z T over the universal domain of G = IWH, Reduced distributions are defined on G/Z = RIWH. The first of these reduced distributions, introduced in equation 3.96 as the Cartan-section reduced distribution, (4.5) can also be defined by the Feynman-Kac path integral, where ℓ[dw [0,T ) ] = ℓ T is the solution of the SDE 3.75 for the Cartan center coördinate ℓ, with the notation here emphasizing that this solution is a functional of the sample-path of Wiener outcome increments. As was noted in equation 3.78, one can use the transformation from Harish-Chandra coördinates to write where f T is the function of G/Z = RIWH given in equation 3.42, 2f (Zx) = e −r/2 sinh(r/2) |β + α| 2 + e −r/2 cosh(r/2) |β − α| 2 , (4.9) with the ruler and the phase-plane coördinates evaluated at time T , and is the stochastic integral 3.32 for the Harish-Chandra center coördinate s (derived in appendix C). The function C T (Zx) unravels Z T onto RIWH = G/Z as in equation 3.95, (4.11) and C T (Zx) satisfies the FPK equation 3.101, There is another natural reduced distribution, conjugate to the Harish-Chandra section, This Harish-Chandra reduced distribution function, B T (Zx), unravels Z T as The two reduced distribution functions are equivalent to one another by a positive gauge transformation [19], In the lingo of Feynman-Kac formulas [19], 2f (Zx) is the "convective pressure." Section III E introduced the Cartan reduced distribution C T (Zx) and showed that it is the distribution that expresses POVM completeness. The price for relevance to POVM completeness is that C T (Zx) has ill-defined normalization and is disconnected from the stochastic-integral solutions for the phase-space variables. The next three sections investigate the Harish-Chandra reduced distribution B T (Zx). It is clear that B T (Zx) is not the right distribution for addressing POVM completeness, because of the Gaussian gauge function e −2f (Zx) , but this Gaussian gauge transformation is just what is needed to get a Gaussian distribution function that can be normalized and whose normalized version can be evaluated from the moments of the phase-space variables, albeit as we shall see, moments defined relative to a modified path-integration measure.  The quadratic functional in the exponential can be written as When converting between stochastic integrals and sums, we use t k = k dt (t N = T = N dt) and dw k = dw kdt = dw t k . Now define the real, symmetric, and positive N × N matrix M T , whose matrix elements are It is elegant for various formal expressions to introduce a continuous version of M T , but we do not bother with that here, since we work with the sums that the stochastic integrals represent. Notice that M T is a Toeplitz matrix, that is, M k+j,l+j = M kl . Putting this together, we have which integrates over the Wiener outcome paths to This prompts us to define a new (normalized, zero-mean) Gaussian measure on the Wiener outcome paths, Relative to this measure, the outcome increments are correlated, Notice that the increment correlation dw * k dw l M , with k and l held fixed, changes as the total time T changes. The inverse matrix M -1 T matrix is real, symmetric, and positive, all properties inherited from M T . The inverse does not inherit the Toeplitz property of M T . That M T is Toeplitz implies that it is persymmetric, that is, symmetric across the anti-diagonal; M -1 T does inherit the persymmetry, which turns out to have an important consequence in section IV D.
Returning to the Harish-Chandra reduced distribution 4.13, we see that its normalization can be written in several ways: The normalized version of the Harish-Chandra reduced distribution is Section IV C finds the diffusion equations satisfied by the reduced Harish-Chandra distributions, and section IV D uses the path integral for the normalized distribution, (4.33) to evaluateB T (Zx) from the SDE solutions for the Harish-Chandra coördinates.

C. Diffusion Equation for Harish-Chandra Reduced Distribution Function
The easiest way to get to the diffusion equation for B t (Zx) is to return to the FPK equation 2.78 for D t (x), to write the FPK forward generator ∆ in Harish-Chandra coördinates, and then to marginalize over the center to get a PDE for B t (Zx).
By using the frame transformations 3.44, it is easy to express the terms of the forward generator in Harish-Chandra coördinates, where the terms

44)
are independent of ψ and commute with ∂ ψ . Putting these expressions together, the FPK forward generator 2.76 in Harish-Chandra coördinates is . To derive an evolution equation for B t (Zx) of equation 4.13, one follows the procedure outlined for C t (Zx) at equation 3.101, using here the rules that integration by parts on s and ψ make the substitutions ∂ s → 2 and ∂ ψ → 0. The resulting PDE for B t (Zx) is where for brevity, we define a reduced generator for the phase-space-variable derivatives, Converting fully to complex phase-space coördinates puts the PDE in the form This PDE is ballistic in the ruler r-solution proportional to δ(r − 2κt)-and Gaussian preserving in the phase-space variables, but as a consequence of the term −3 + |ν| 2 , it does not preserve normalization [15,16,19]. The effect of the positive gauge transformation from C t (Zx) to B t (Zx) is two-fold: (i) the norm-nonconserving "potential" term changes character, from a ruler-dependent −2 coth r in the PDE 4.12 for C t (Zx) to a term −3 + |ν| 2 in the PDE 4.52 for B t (Zx), which depends on the posterior phase-space variable ν; (ii) there are first-derivative, "vector-potential" terms in the PDE for B t (Zx), corresponding to the Ornstein-Uhlenbeck behavior 3.24 of ν, whereas there are no such terms in the PDE for C T (Zx).
We turn now to the task of converting the PDE 4.52 to the normalized distributionB t (Zx). The initial condition for the PDE 4.52 comes from inserting the Harish-Chandra D dt (x) of equation F20 into B dt (Zx) as it is expressed in the integral 4.13 specialized to T = dt. Taking the limit dt → 0, one finds the expected result that B 0 (Zx) is the δ-function on G/Z of equation F23: This is expected because the identity is represented uniquely in Harish-Chandra coördinates by ψ = s = r = 0, ν = µ = 0. The initial condition means that B t (Zx) is initially normalized to unity; thus the normalization factor N t has initial value N 0 = 1, and the normalized distribution has the same initial condition, The normalization factor, satisfies the differential equation, where the reader should notice that integration by parts on the ruler becomes the rule ∂ r → −2. This differential equation assumes the form where is the second moment of ν relative to the normalized distributionB t (Zx). We place a subscript M on this moment because we can use the path-integral expression 4.33 forB T (Zx) to reëxpress it as where ν[dw [0,T ) ] = ν T is the stochastic-integral solution 3.28 for ν. Once the stochastic integral is plugged into this equation, the correlations of the Wiener increments are evaluated according to the modified measure Dµ M dw [0,T ) , that is, as in equation 4.27.
The PDE for the normalized reduced distributionB t (Zx) of equation 4.32 now follows as Inserting equation 4.58 gives It is easy to see how this equation preserves normalization. The presence of the moment n t , essential for normalization, makes the equation nonlinear, but it can still be solved easily.
To solve forB t (Zx), one notes that the PDE is ballistic in the ruler r and Gaussian preserving in the phasespace variables. Thus the solution has the formB t (Zx) = e −2r δ(r − 2κt)Φ t (Zx), where Φ t (Zx) is a normalized, zero-mean Gaussian in the phase-space variables ν and µ. The derivatives in the PDE 4.62 are invariant under complex conjugation and under simultaneous rephasing of the phase-space variables, that is, ν → νe iχ and µ → µ iχ ; it is productive to think of the invariance under complex conjugation as invariance under the change of phase-space coördinates ν ↔ ν * and µ ↔ µ * . It is useful to appreciate that all the diffusion equations in this paper share these invariance properties. The invariance under simultaneous rephasing implies that Φ t (Zx) is a zero-mean Gaussian, since it starts from a zero-mean δ-function initial condition. It further implies that Φ t (Zx) is determined by the three nonzero second moments of the phase-space variables: n t of equation 4.59 and with the invariance under complex conjugation implying that q t is real. This form of the solution forB t (Zx) in hand, one derives from the PDE 4.62 first-order temporal differential equations for the second moments n t , m t , and q t . Just as the ordinary differential equation (ODE) for the normalization factor involves a second moment, so the equations for the second moments involve fourth moments. The Gaussian form of the solution relates the fourth moments to second moments, thus closing the system of differential equations. The resulting three ODEs, for the derivatives of n t , m t , and q t , have terms that are constant, linear, and quadratic in the moments; the presence of the quadratic terms makes these ODEs (coupled) Riccati equations. The last step is to solve the three Riccati equations, with initial conditions n 0 = m 0 = q 0 = 0, which are implied by the δ-function initial condition forB 0 (Zx). This gives the solution forB t (Zx). By integrating to find N T , using the solution for n t , one can backtrack to find B T (Zx) and C T (Zx) from equation 4.32.
We have carried out this procedure of deriving the Riccati equations from the PDE 4.62, but we do not present that derivation in this paper, preferring instead to use a different method, which derives the Riccati equations from the path integral forB T (Zx). Implementing that method is the final task of this paper, carried out in the next section.

D. Normalized Harish-Chandra Reduced Distribution Function from its Path Integral
The path integral 4.33 forB T (Zx) can be written explicitly in terms of the δ-function in Harish-Chandra coördinates, where r T = 2κT , ν T = ν[dw [0,T ) ], and µ T = µ[dw [0,T ) ] are the solutions 3.27-3.29 to the SDEs for the ruler and the Harish-Chandra phase-space coördinates. The measure for the path integral is a (normalized, zero-mean) Gaussian measure in the outcome increments dw t ; thus the path integral gives a normalized, zero-mean Gaussian in ν and µ, which is determined by the three (real) moments introduced in the previous section: In this context, that the first moments and all the other second moments of the phase-space variables are zero follows from the fact that the measure is invariant under simultaneous rephasing of all the outcome increments; the reality of q T follows from the fact that the measure is unchanged under the transformation dw [0,T ) → dw * [0,T ) . These properties come from the fact that M T is real and symmetric.
Plugging in the stochastic-integral solutions for ν T and µ T puts these moments into the following form: In the final form of q T , it is evident that q T is real. Notice that these expressions satisfy the zero initial conditions. That M T is a Toeplitz matrix introduces an additional, quite important symmetry. The inverse matrix does not inherit the Toeplitz property of M , but it does inherit a less restrictive property. That M T is Toeplitz implies that it is symmetric about the anti-diagonal, that is, M kl = M N −1−l,N −1−k . A matrix that is symmetric about the antidiagonal is called persymmetric. It is easy to show that the inverse of a persymmetric matrix is persymmetric, so M -1 Persymmetry has a major consequence for the three moments, which comes from manipulating m T : We can set e 4κ dt = 1 and thus conclude that The complementarity in time of ν T and µ T was discussed in section III B: the post-measurement variable ν T of equation 3.28 depends exponentially on the end of the outcome register, and the POVM variable µ T of equation 3.29 depends exponentially on the beginning of the register. The persymmetry of M T and M -1 T expresses that the beginning and end of the record look the same statistically, so it is not surprising that the persymmetry implies that m T = n T . The next step, deriving Ricatti equations for the three moments, involves incrementing the moments from T to T + dT . The tedious part of this task is determining how M -1 T increments, that is, finding M -1 T +dT , and that can be done using the Schur complement. We relegate this entire task to appendix G and here skip directly to the coupled Ricatti ODEs, taken from equations G31-G33: With the zero initial conditions, these have the solutions, It is quite instructive to notice that the equality n T = m T and the reality of q T together imply that the sum and difference Harish-Chandra phase-space variables are uncorrelated, with second moments, where n T + q T = 2e −κT sinh κT , (4.86) The width Σ T = κT − tanh κT was introduced in equation 3.108. It is worth noting the early-and late-time behavior of the various moments: At early times, ν and µ are tightly correlated, with their sum undergoing standard diffusion; at late times, they become uncorrelated, and their second moments saturate at 1. The sum and difference variables being uncorrelated, the Gaussian path integral 4.65 has the normalized solution, . The unnormalized distribution is therefore The second line transforms to Cartan phase-space coördinates. Unnormalizing changes the Gaussian's prefactor; transforming to Cartan coördinates changes the Gaussian. The final step is to undo the gauge transformation to get back to the Cartan reduced distribution, which matches the solution 3.107 obtained from the PDE 3.101. It is fair to ask whether the point of this section is just to provide a different, more complicated route to the solution for C T (Zx)? We think it is more than that, and here's why. It all comes back to the elephant in the room, that is, how one handles the normalization or scaling of the Kraus operators that comes from the center Z. The Cartan reduced distribution, defined in equation 3.96 and determined from the diffusion equation 3.101, succeeds in representing POVM completeness by marginalizing the ur -distribution D T (x) over the Cartan-center normalization, e −2l = e 2f (Zx) e −2s ; this gives a distribution uniform in the Cartan sum variable β +α and thus spread over all of phase space in a way that gives POVM completeness. As a consequence, however, the moments of C T (Zx) are not those of the stochastic integrals for the phase-plane variables. The Harish-Chandra reduced distribution B T (Zx), defined in equation 4.13, marginalizes D T (x) over the Harish-Chandra-center normalization, e −2s . After normalization to unity by the factor N T , the normalized distributionB T (Zx) is determined from the diffusion equation 4.62 or by applying the path-integral expression 4.33, with its modified path measure, to the stochastic integrals for the Harish-Chandra phase-space variables. The route fromB T (Zx) to POVM completeness runs backwards through the normalization factor N T and the anti-Gaussian gauge transformation e 2f (Zx) and arrives at C T (Zx). The point of this section, one might say, is to find and reveal these connections among path integrals, diffusion equations, and SDEs; discovering these connections, driven in this paper by a combination of necessity and opportunity, allows us to re-unite the three faces of the stochastic trinity. Accessing the entire stochastic trinity by using a positive gauge transformation that is grounded in a problem's Lie group-this, we hope, might be generally applicable to Feynman-Kac formulas for non-normalization-preserving diffusion equations.

V. CONCLUDING REMARKS. THE STOCHASTIC TRINITY
We set out on the project of analyzing simultaneous measurements of noncommuting observables [39,40] with the goal of showing that such measurements end up with a POVM in the overcomplete coherent-state basis. We now think that we have uncovered something more ambitious, a distinctive new window into the space of quantum dynamics. The formulation of the problem of continual, differential measurements invites one-compels one, really-to think in terms of the paths of Wiener outcome increments. These outcome paths, as sample-paths drawn from the Wiener measure, know nothing about a space in which they are wandering. When they are instantiated in the exponents of Kraus operators, however, the time-ordered products of the differential Kraus operators generate a (complex) Lie-group manifold, the instrumental Lie group, in which the Kraus operators are, in the way of groups, both the transformations and the moving points. The motion in the instrumental Lie-group manifold is described by Krausoperator SDEs or by the diffusion of the KOD, as embodied in a FPK diffusion equation. The continuous, but not differentiable paths are handled effortlessly by the Itô calculus of the outcome increments, with its terms of order √ dt and dt. This requires getting just beyond the linear structure of vector fields (right-invariant derivatives) and one-forms (right-invariant one-forms) on the Lie-group manifold, as is evident from our discussion of Stratonovich vs. Itô SDEs and in the derivation of the FPK diffusion equation, where right-invariant derivatives end up as diffusive second derivatives. We end up in a very comfortable place, working in all three corners of the stochastic trinity: path integrals, FPK diffusion equations, and SDEs, all three describing motion, equivalently, on the instrumental Lie-group manifold.
Why don't others find the same comfort in all three faces of the trinity? Field theorists, interested in the propagator of closed-system dynamics, have a different way of handling the continuous, but not differentiable paths, coming from a Stratonovich-calculus way of dealing with the temporal derivatives in the kinetic terms in a Lagrangian. While they usually have an equivalent Schrödinger-like equation for the propagator, they do not have the analog of SDEs, even though the problem often undergoes a Wick rotation into "Euclidean spacetime." Open-systems theorists, both in condensed-matter physics and in quantum optics, generally work with master equations or stochastic master equations for the evolution of quantum states and sometimes with diffusion equations for a probability distribution associated with the states. Those who start with diffusion equations can avail themselves of a Feynman-Kac formula for a path integral, but the connection to SDEs has generally not been made. The reason for not using all three faces of the stochastic trinity is, we think, a failure to identify the appropriate Lie-group manifold; this failure is connected to the emphasis on the evolution of quantum states, which obscures nearly entirely the Lie-group manifold on which the open-system dynamics is occurring.
Suffice it to say that we think we have found something: the home of quantum dynamics, the Lie-group manifold that supports all three faces of the trinity. The exhausted reader who has survived to read this concluding sentence of a very long paper might be pleased-or so we hope-to learn that our ambition is larger than was evident at the beginning.

ACKNOWLEDGMENTS
CSJ thanks Mohan Sarovar for all of the helpful discussions and financial support through Sandia National Laboratories and thanks CMC for the incredibly fruitful collaboration. CMC supported himself and is eternally grateful for the opportunity to work with CSJ.
This work was supported in part by the Center for Quantum Information and Control at the University of New Mexico and in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under the Quantum Computing Application Teams (QCAT) program. Sandia National Laboratories is a multimission laboratory managed and operated by NTESS, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE's NNSA under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. The Lie algebra of right-invariant derivatives is equivalent to the exterior algebra of right-invariant one-forms. This can be seen by considering the second exterior differential of a scalar function, which by linear independence means that Specifically, if we write out all of the different Lie brackets, we can read off all of the one-form differentials or "curls", Identifying the parameters of the last line gives the coördinate transformation, are functions of the RWIH G/Z. The inverse coördinate transformation is It is quite useful to notice that the sum and difference phase-space coördinates simply rescale under this transformation, We also have It is crucial to appreciate that the transformation from Harish-Chandra to Cartan coördinates is singular at r = 0, with the consequence that positive transformations of the form e Qq+P p are not represented in Cartan coördinates. At r = 0 the Harish-Chandra decomposition becomes x r=0 = e Ωz e a † ν e aµ * = e Ω(z− 1 2 νµ * ) e a † ν+aµ * = e Ω(z− 1 2 νµ * ) D (ν−µ)/2 e a † 1 2 (ν+µ)+a 1 2 (ν+µ) * .
Each displacement operator and each positive transformation of the form e Qq+P p is represented uniquely in the 4plane of ν and µ coördinates; in particular, the identity operator has coördinates r = 0 and z = ν = µ = 0. At r = 0 the Cartan decomposition reduces to x r=0 = e iΩφ e −Ωℓ D β D −α = e iΩ[φ+(β * α−βα * )/2i] e −Ωℓ D β−α . (B20) The Cartan coördinate singularity at r = 0 is that x r=0 does not depend on β+α. A displacement operator D β−α = D τ is represented by a plane of β + α values, specifically, by all the coördinates satisfying β − α = τ , r = 0, ℓ = 0, and φ = β + α) * τ − (β + α)τ * /4i. Most importantly, the identity 1 is represented by the plane of Cartan coördinate values satisfying β = α and r = φ = ℓ = 0. Positive transformations of the form e Qq+P p are not represented at all in Cartan coördinates; this is not a problem because these positive operators lie on a boundary that is not accessible to the Kraus operators of SPQM. To solve the SDE 3.26 for the Harish-Chandra center coördinate z, it is best to work with the sums over Wiener increments that underlie the Itô stochastic integrals. Thus we begin by writing the solution 3.28 for the postmeasurement Harish-Chandra coördinate ν as where dw k = dw kdt and t k = k dt (t N = T = N dt). Notice that the initial condition ν 0 = 0 is enforced by having no terms in the sum for N = 0. The solution for z is where we omit the k = 0 term on the second line since ν 0 = 0 and where we can insert the absolute value because k > l. Notice that, as for ν, the way the initial condition z 0 = 0 is enforced is that there are no terms in the sums when N = 0. Now define as the Heaviside step function with the choice H(0) = 1 2 . This appendix is really an exercise in relating the real and imaginary parts of z to its symmetric and antisymmetric parts and, in the process, getting the weighting of the diagonal (k = l) term right-equivalently, making the right choice for the u = 0 value of the sign and Heaviside functions. The incremental Itô calculus, with its quite explicit diagonal terms 1 2 κ|dw k | 2 , leaves no doubt about the right choice.
It is easy to show that the left-invariant derivatives and one-forms can be obtained from the right-invariant quantities by changing the sign of the quantities associated with anti-Hermitian operators and transforming the coördinates according to φ ↔ −φ and β ↔ α. The sign changes don't change the Haar measure, and the measure is unchanged by the coördinate transformation, which shows that the Haar measure is both right-and left-invariant.
coördinate partial derivatives and one-forms in the two coördinate systems. The relations between coördinate partial derivatives and one-forms can, of course, be derived straighforwardly from the global coördinate transformations in appendix B. We do not record these relations because the expressions in terms of the right-invariant quantities are more useful for our purposes. We remind the reader that even though the ruler coördinate r is shared between Harish-Chandra and Cartan coördinates, the partial derivative ∂ r is different between the two systems because partial derivatives are defined by holding the other coördinates constant.
The Haar measure 2.85 in Harish-Chandra coördinates is d 7 µ(x) = dψ ds d 2 ν 2π dr e 2r d 2 µ 2π . (E33) The factors of 1/2π in the Harish-Chandra phase-plane measures follow from transforming the Cartan Haar measure to Harish-Chandra coördinates. Just as for Cartan coördinates, it is easy to show that the left-invariant one-forms yield the same measure.