Currencies in Resource Theories

Abstract

How may we quantify the value of physical resources, such as entangled quantum states, heat baths or lasers? Existing resource theories give us partial answers; however, these rely on idealizations, like perfectly independent copies of states or exact knowledge of a quantum state. Here we introduce the general tool of “currencies” to quantify realistic descriptions of resources, applicable in experimental settings when we do not have perfect control over a physical system, when only the neighbourhood of a state or some of its properties are known, or when slight correlations cannot be ruled out. Currencies are a subset of resources chosen to quantify all the other resources—like Bell pairs in LOCC or a lifted weight in thermodynamics. We show that from very weak assumptions in the theory we can already find useful currencies that give us necessary and sufficient conditions for resource conversion, and we build up more results as we impose further structure. This work generalizes axiomatic approaches to thermodynamic entropy, work and currencies made of local copies. In particular, by applying our approach to the resource theory of unital maps, we derive operational single-shot entropies for arbitrary, non-probabilistic descriptions of resources.

1. Introduction

Resource theories are frameworks that allow agents to model the world where they move in a subjective, operational way: they describe the systems available to the agents, and the constraints limiting their actions. Using only these simple ingredients, resource theories enable us to derive rules for resource transformations—what agents can do, and how they may act efficiently. Resource theories can be used to model all kinds of constraints on the actions of an agent, ranging from physical laws to technical limitations or even rules of a game. Because of this, they have proven successful models in many different contexts, from physics to computer science and cryptography. For example, in entanglement theory, the resource theory of local operations and classical communications (LOCC) [10,11,12,13] has provided powerful tools to quantify entanglement. In quantum thermodynamics, the resource theories of noisy and thermal operations [14,15,16,17,18,19] have shed light on irreversibility and the second law, while resource theories of asymmetry and reference frames [20,21,22,23,24,25,26], coherence [27,28,29,30,31] and quantum control [32] explore the role of other resources in quantum information theory.

Here, we extend the range of applicability of resource theories, by showing that we do not need most of the assumptions of usual approaches in order to derive useful tools to quantify resources.

1.1. Quantifying Resources

Resource theories start from a set of allowed transformations that an agent is able to implement, and derive a structure that encodes which resources can be converted into which other resources under such transformations. Formally, a resource theory gives rise to a pre-order → on the set of resources, i.e., a reflexive ($A\to A$, “we can always build a resource A starting from itself, e.g., by doing nothing”) and transitive ($A\to B\wedge B\to C\Rightarrow A\to C$) relation. Note that the set of resources need not be totally ordered (that is, given two resources, it is not necessary that either $A\to B$ or $B\to A$), and that unlike a partial order, there can be several equivalent resources, satisfying $A\to B\wedge B\to A$ (for example, in LOCC, different Bell pairs are equivalent resources). While this pre-order structure carries all the information needed to characterize the resource theory, it is often difficult to compute relevant properties. For example, it can be hard to decide for any two given resources A and B whether or not $A\to B$. Similarly, if it is not the case that $A\to B$, which additional resources would the agent have to supply in order to generate the resource B? Or if it is the case that $A\to B$, what kind of additional resources could be extracted along the way?

This is where the concept of assigning value to resources comes in. The idea is to quantify in a simple way how much can be done with a particular resource, and how difficult it is to transform one resource into another. This concept is hence central to resource theories: if we can determine a good characterization of value, then we can answer important questions in the theory. In equilibrium thermodynamics, for example, the free energy of a state quantifies the work that can be extracted from it, and is also a monotone under isothermal transformations. More generally, we need more than one monotone to characterize the structure of a resource theory.

1.2. Currencies

Looking at particular resource theories, we can find a common and powerful tool that is used to characterize the value of resources in an operational way. Namely, there is usually a special class of standard or reference resources that one can measure or quantify more easily, and that can be used to determine the value of other resources. What is more, these special resources are usually universal in the sense that they allow an agent to generate any other resource from them. Such resources are then particularly useful to measure the value of other resources, since we can always convert back and forth between the resources of interest and the universal reference resources. We call such reference resources a currency—they provide a natural operational interpretation of value.

In real life, for instance, money constitutes a currency, since we can use it to trade against essentially any resource. For a more physical example, note that we can also identify a currency in two-party LOCC: maximally entangled pairs of qubits (Bell pairs) can be used to teleport half of any quantum state [10], and so the value of resources like bipartite states can be defined in terms of how many Bell pairs are needed to create, or how many can be extracted from, a resource. These considerations yield the well-known concepts of entanglement of formation and distillable entanglement. Similarly, work in traditional thermodynamics can be considered a currency that allows to implement essentially any transformation if given enough of it. Finally, the concept of currencies is also underlying the ideas in Lieb and Yngvason’s approach to defining entropy in thermodynamics [2,3,4,5], following earlier axiomatic work on thermodynamics [33,34]; there, the set of equilibrium states forms a currency.

Now, if a resource theory has a currency, we can use it to find necessary and sufficient conditions for resource transformations. For instance, if the yield of “selling” the initial resource is larger than the cost of creating the final one, then the transformation is possible: we simply sell the initial resource and use some of the coins to buy the final one. In addition, the cost and yield of resources are also monotones on the pre-order → of the theory. As they have to be decreasing along the pre-order, this yields additional necessary conditions for resource transformations. For a further discussion on monotones and their connection to currencies, see Appendix D.

1.3. Previous Approaches

While currencies serve the same purpose in many resource theories, they are built on a number of different assumptions. For example, Bell pairs in LOCC rely on the concept of copies of many resources, while work in (quantum) thermodynamics can be modelled as a pure state on an energy ladder, often satisfying a requirement of translational invariance [35,36,37,38]. In contrast, the currency or “reference states” in the axiomatic approach to thermodynamics of [2,3,4,5] satisfy special comparability and scalability properties.

But which assumptions are essential to guaranteeing necessary or sufficient conditions for resource transformations? Which assumptions do we need in order to characterize the cost of transformations? In this work, we will take a more abstract view on currencies and discuss more precisely which properties allow us to make which particular statements. Preceding this article, there has been some work towards abstract characterizations of resource theories and common tools like monotones and conversion rates therein [7,8,9]. However, these works and the very concepts they seek to describe rely on idealizations such as exact knowledge of the state on the relevant systems and complete absence of correlations between subsystems, which cannot always be guaranteed in practice. In quantum theory, exact and perfectly uncorrelated states are the equivalent of spherical cows in a vacuum: idealizations that ease the maths but cannot be implemented in the lab (for example, it is impossible to control the conditions of an experiment so well as to claim that a quantum system is in a given pure state with probability one). Indeed, such abstractions may be of little use to experimentalists wanting to apply a physical theory to the resources at hand. To be relevant, resource theories must realistically model the limitations of real agents, both in their descriptions of resources and in the constraints in action [39].

In [1], we have developed a general framework for resource theories that allows us to overcome these issues, but we left the question open as to whether it is possible to recover the useful tools of traditional resource theories without the idealizations of having access to pure states, i.i.d. copies of a resource, or being able to model an agent’s knowledge as a precise probability distribution.

1.4. Contribution of This Work

In Section 2, we provide an abstract characterization of currencies and identify minimal key assumptions needed to derive useful notions of value of resources and transformations. We show that we can make relevant statements without many of the properties that are taken for granted in other works.

In Section 3, we use the framework of [1] to apply these tools to general settings where we may not have a known subsystem structure, a composition operation or a precise description of resources. This paves the way to studying currencies that are themselves made up of more general resources, such as approximate Bell pairs or systems where slight correlations cannot be ruled out, and that lack properties like scalability as assumed in [2,3,4,5]. Then we present a concrete application of our tools to the resource theory of unital maps, where we quantify the cost, yield and balance for general resources and transformations. These quantities correspond to notions of entropy for specifications, and yield direct results for the questions of work extraction and transformation balance for general resources.

2. Types of Currencies

We split this discussion into three stages, starting from a basic definition of currencies and adding in stronger conditions as we progress. We outline the properties and results that follow from the assumptions made up to each stage. Technical definitions, statements and proofs can be found in Appendix A.

2.1. Stage I: A Universal Standard

At a basic level, currencies have two defining properties: order and universality for a given target. We show that such a basic notion of currencies already allows for a definition of cost and yield of all resources, from which we may derive necessary and sufficient conditions for general resource transformations.

2.1.1. Ordered Set and Value

We have argued above that currencies are a collection of ‘standard’ or ‘reference’ resources relative to which we can quantify other resources. Formally, we can define a currency as a subset $\mathcal{C}$ of resources equipped with a well-defined value function $Val:\mathcal{C}\to {\mathbb{R}}_{+}$ that satisfies

$$\begin{array}{cc}\hfill C\to C^{\prime }\iff & Val\left(C\right)\ge Val\left(C^{\prime }\right),\hfill \end{array}$$

for all currencies $C,C^{\prime }\in \mathcal{C}$. (Note: we consider functions to the positive real numbers; we assume the currency to have a finite least valuable resource, a bottom (largely motivated by the framework of [1], where it corresponds to ‘knowing nothing’). Negative values can then be eliminated without loss of generality through an additive constant.) In particular, the set $\mathcal{C}$ of currency resources must be totally ordered, up to equivalences (e.g., two different Bell states have the same value under LOCC, and in general $C\rightleftharpoons C^{\prime }\iff Val\left(C\right)=Val\left(C^{\prime }\right)$). This condition guarantees that a currency of high value is really strictly more powerful than one with a lower value (Proposition A1).

While this may seem like a strong requirement at first, we observe that essentially all of the examples of currencies we find in resource theories satisfy this order requirement. Our order condition is also not unusual: it is in fact reminiscent of the comparability requirement in axiomatic approaches to thermodynamics [2,3,4,5,34], where it is demanded for the set of reference states (‘equilibrium states’ or ‘entropy meter’). Finally, note that we can always start from the whole pre-order structure of the resource theory and build a currency by selecting a maximal subset of resources that is totally ordered (plus equivalences); any monotone function to ${\mathbb{R}}_{+}$ on this set is a valid value function.

2.1.2. Universality

We have mentioned that a currency is typically universal in the sense that any resource can be generated from the currency if given enough of it, and any resource can be “sold” for some currency resource (which could, however, be of zero value). While we usually find that this holds for all resources, we relax this condition slightly to hold for a target set $\mathcal{S}$ of resources. Formally, we can then write the property of universality as follows: for any resource A in a target set $\mathcal{S}$, there is a currency resource $C\in \mathcal{C}$ such that $C\to A$ and a currency resource $C^{\prime }\in \mathcal{C}$ such that $A\to C^{\prime }$.

In the axiomatic approach to thermodynamics of [4], this property corresponds to an extended comparability axiom for states out of thermodynamic equilibrium with equilibrium states.

2.1.3. Cost and Yield of Resources

Order and universality of a currency allow us to define the cost and yield of any target resource $A\in \mathcal{S}$:

$$\begin{array}{cc}\hfill Cost{\left(A\right)}_{\mathcal{C}}& =\underset{C\in \mathcal{C}}{inf}(ValC:C\to A)\hfill \\ \hfill Yield{\left(A\right)}_{\mathcal{C}}& =\underset{C^{\prime }\in \mathcal{C}}{sup}(Val{C}^{\prime }:A\to C^{\prime }).\hfill \end{array}$$

The cost and yield of resources are monotones—and so we immediately obtain the following necessary conditions for resource transformations in the target $\mathcal{S}$ (Theorem A1):

$$\begin{array}{cc}\hfill A\to B& \Rightarrow Cost{\left(A\right)}_{\mathcal{C}}\ge Cost{\left(B\right)}_{\mathcal{C}},\hfill \\ \hfill A\to B& \Rightarrow Yield{\left(A\right)}_{\mathcal{C}}\ge Yield{\left(B\right)}_{\mathcal{C}}.\hfill \end{array}$$

We also get a sufficient condition based on the cost and yield of resources:

$$Yield{\left(A\right)}_{\mathcal{C}}>Cost{\left(B\right)}_{\mathcal{C}}\Rightarrow A\to B.$$

These conditions correspond to the necessary and sufficient conditions obtained in [4] for thermodynamics.

2.1.4. Tight Currencies

For any resource A in the target, we can show (Proposition A2) that

$$Cost{\left(A\right)}_{\mathcal{C}}\ge Yield{\left(A\right)}_{\mathcal{C}}.$$

We can then look at the special case when the cost and the yield of a resource A are equal, $Cost{\left(A\right)}_{\mathcal{C}}=Yield{\left(A\right)}_{\mathcal{C}}$. When they are furthermore achievable by a currency resource C, that is, when both $C\to A$ and $A\to C$ for some $C\in \mathcal{C}$, we call the currency tight for the resource A. We will then also denote the set of such resources for which the currency is tight by ${\mathcal{S}}_{=}$.

We show in Theorem A2 that for the set ${\mathcal{S}}_{=}$, the currency in fact gives a simple necessary and sufficient condition for resource transformations,

$$A\to B\iff Cost{\left(A\right)}_{\mathcal{C}}\ge Cost{\left(B\right)}_{\mathcal{C}}$$

when $A,B\in {\mathcal{S}}_{=}$ (and of course similarly for the yield). This then also means that on the set ${\mathcal{S}}_{=}$, the resource theory provides a full order (up to equivalences). In particular, currency resources are also in ${\mathcal{S}}_{=}$ if they are in the target, and in turn, resources in ${\mathcal{S}}_{=}$ could always be added to the currency without changing the theory.

Note that most quantum resource theories do not have a known tight currency for the whole target (that would have made things too easy). However, an example for tightness is given by the resource theory of noisy operations (or unital maps) when restricting to uniform states on some support (that is, states for which all non-zero eigenvalues are equal) [40]. Similarly, assuming that any state transformation in macroscopic thermodynamics can be implemented reversibly, work in equilibrium thermodynamics can be considered a tight currency [2,3,5].

2.2. Stage II: Independent Currency

In addition to defining the cost and yield monotones on resources, it would be useful to use the currency to determine the balance of general resource transformations: if resource A is more valuable than B, how much currency can be extracted in the process $A\to B$? Or how much currency needs to be supplied to transform B into A?

2.2.1. Independence between Currency and Target

In order to answer these questions and formulate concepts such as “adding a currency C to a resource A”, we need to introduce a notion of composing resources. For now we represent an arbitrary notion of composition simply as $(A,C)$; in Section 3 we show that there is always a natural way to formalize it. It is also essential to be able to address the currency and the target resources independently. For this, it might help to think that we keep the currency in an independent wallet, but we will see that this notion is more general in Section 3.

In quantum theory, for example, this is guaranteed if the currency and the target resources live in different subsystems which are composed via the tensor product, and the allowed transformations in the theory let us address their respective degrees of freedom individually. In real life, this independence is often an approximation: for example, in an optics experiment we might not be able to individually address an atom or transition without slightly disturbing the others.

We can formulate the independence condition as

$$C\to C^{\prime }\Rightarrow (A,C)\to (A,C^{\prime })$$

for all target resources A and currencies C. This condition guarantees that a higher valued currency resource is really more powerful than a lower valued one in facilitating state transformations (Proposition A3). In Section 3 we show how to apply this conditions to realistic settings, where for example slight correlations between target and currency cannot be ruled out.

2.2.2. Balance of Resource Conversions

Composition of resources allows us to define the balance of a transformation from resource A to resource B in the target, conditioned on available currency C as

$$\begin{array}{cc}\hfill & Balance{(A\to B|C)}_{\mathcal{C}}\hfill \\ \hfill & =\underset{C^{\prime }\in \mathcal{C}}{sup}(Val\left(C^{\prime }\right)-Val\left(C\right):(A,C)\to (B,C^{\prime })).\hfill \end{array}$$

In particular, if the balance is negative, it corresponds to minimal cost of transforming A into B given access to a currency C. Of course, this quantity is only defined if the value of C is actually large enough to afford the transformation. The independence condition ensures that the balance is meaningful, because it guarantees that the value of currencies is connected to how helpful they are at facilitating transitions in the target (Proposition A3). Note: The symmetric independence condition $A\to B\Rightarrow (A,C)\to (B,C)$ would guarantee that $A\to B\Rightarrow Balance{(A\to B|C)}_{\mathcal{C}}\ge 0$, but does not seem to have a large impact otherwise.

Note that in general, this definition will depend on the available currency resource C at the start of the transaction. This dependency could in principle be arbitrary: on the one hand, having access to additional currency in the wallet might facilitate the transformation and act partially like a catalyst, such that the balance of a transformation becomes larger the more currency is used to implement the transformation. In real life, for example, a client with more money may receive a special discount for a transaction (e.g., offered in the hope of acquiring a returning, rich loyal customer). On the other hand, having additional money might occasionally make the transaction more expensive: if instead of paying the exact amount of coins, one tries to pay with a larger note, one might end up paying more for an object or service (e.g., if the selling party cannot give change).

As a last remark, note that we could have defined an analogous notion of balance,

$$\begin{array}{cc}\hfill & {Balance}^{*}(A\to B|C)\hfill \\ \hfill & =\underset{C^{\prime }\in \mathcal{C}}{sup}(Val\left(C\right)-Val\left(C^{\prime }\right):(A,C^{\prime })\to (B,C)),\hfill \end{array}$$

where instead of conditioning on a starting resource C in the currency, we demand that a final resource C is reached at the end of the transformation. While we work with the first definition of balance in this paper, all the results could also be formulated with respect to this adapted notion.

2.3. Stage III: Fair Currency

In Stage II, the balance of a resource transformation can in principle depend on the available currency C in the wallet. If that is not the case, we say that the currency is fair, a property composed of two aspects. On the one hand, it implies that there are no discounts for the poor, colloquially speaking: a transformation does not become cheaper to implement just because one has less available currency. On the other hand, having access to more currency does not make a transformation cheaper either—in other words, the currency does not act as a catalyst. We analyse the two conditions and their implications separately. In both cases the starting point is the same: suppose that we have resources A and B in the target, and we can perform the transformation

$$(A,C_1)\to (B,C_2)$$

for some currency $C_1,C_2\in \mathcal{C}$, with

$$\Delta :=Val\left(C_2\right)-Val\left(C_1\right).$$

We will now see what may happen if we start from a different currency $C_3$.

2.3.1. Having Less Does Not Help

Fairness in this direction means that if $C_3$ is more valuable than $C_1$, we can always find a final $C_4\in \mathcal{C}$ with the same difference $Val\left(C_4\right)-Val\left(C_3\right)=\Delta$ that achieves the transformation $(A,C_3)\to (B,C_4)$ (and vice versa: we can first fix $C_4$ with $Val{C}_4\ge Val{C}_2$ and look for an appropriate $C_3$). Note: This condition is required to hold within suitable bounds, so that we do not hit the top boundary of the currency (if it exists), that is $Val\left(C_3\right)<{sup}_{C\in \mathcal{C}}ValC-\Delta$.

Naturally, this condition implies that the balance of a transformation cannot decrease given access to more currency, that is (Proposition A4),

$$\begin{array}{cc}\hfill & ValC\ge Val{C}^{\prime }\hfill \\ \hfill & \Rightarrow Balance{(A\to B|C)}_{\mathcal{C}}\ge Balance{(A\to B|C^{\prime })}_{\mathcal{C}},\hfill \end{array}$$

In many familiar resource theories, currency resources can be composed to yield resources of higher value, such as an increasing number of Bell pairs in LOCC or a higher number of coins in real life. When these individual resources can be acted upon without disturbing the others, we can always ignore some of the currency and bring it back after the transaction. In these cases, more currency can never make a transaction more expensive; if anything, it can be cheaper when having access to more. Formally, what guarantees fairness in this direction in such cases is a condition of independence, like the one we introduced between currency and target, but extended to individual subsystems within the currency. For example, this is implied when currency resources are seen as a collection of objects in the paradigm of symmetric monoidal categories (one can understand copies of Bell pairs in LOCC as an example, see e.g., [8]). There, transformations are defined on individual objects and when two objects $A,B$ are composed, local transformations $f,g$ can also be combined and applied to the composed object such that $\left(f\right(A),g(B\left)\right)=(f,g)(A,B)$ (see e.g., [7]). Choosing one of the functions as the identity, this ensures that we can put extra currency on the side for the purpose of a transformation and re-introduce it again later.

In resource-theoretical approaches to macro and microscopic thermodynamics, we also find this condition: it is expressed through the composition axiom in [4], and is assumed in some models for quantum thermodynamics where work is stored in pure states on a number of individual qubits [17]. In fact, because this assumption is so common in resource theories, it is rarely questioned or exposed in this way. However, we would like to work with more general currencies than tensor product copies of individual states, and so we make this aspect of fairness explicit.

2.3.2. Having More Does Not Help

Fairness in the other direction means that if the initial currency $C_3$ is less valuable than $C_1$, we can again always find a $C_4\in \mathcal{C}$ with the same difference $Val\left(C_4\right)-Val\left(C_3\right)=Val\left(C_2\right)-Val\left(C_1\right)$ that achieves the transformation $(A,C_3)\to (B,C_4)$ (and also vice versa: we can first fix $C_4$ with $Val{C}_4\le Val{C}_2$ and look for an appropriate $C_3$). Note: we assume that$Val\left(C_3\right)\ge Val\left(C_1\right)-Val\left(C_2\right)$ so that we can actually afford the transformation with $C_3$.

Naturally, this condition implies that the balance of a transformation cannot increase given access to more currency, that is (Proposition A5),

$$\begin{array}{cc}\hfill & ValC\le Val{C}^{\prime }\hfill \\ \hfill & \Rightarrow Balance{(A\to B|C)}_{\mathcal{C}}\ge Balance{(A\to B|C^{\prime })}_{\mathcal{C}}.\hfill \end{array}$$

Operationally, this direction of fairness makes sure that transformations are not easier to implement just because one has access to more currency—that is, the currency does not act like a catalyst. This would be guaranteed for example in a theory in which one is always able to ‘borrow’ extra currency for free. Since this is in general not the case (for example due to finite-size effects), fairness in this direction is more common to fail than in the other.

2.3.3. Both Directions

In case fairness holds in both directions, it follows that the balance of a transition does not depend on the initial currency C,

$$\begin{array}{cc}\hfill Balance{(A\to B|C)}_{\mathcal{C}}& =Balance{(A\to B|C^{\prime })}_{\mathcal{C}}\hfill \\ \hfill & =:Balance{(A\to B)}_{\mathcal{C}}\hfill \end{array}$$

for any $C,C^{\prime }$ within suitable boundaries (Proposition A6).

For this new single notion of balance, we can also show that (Theorem A4)

$$Balance{(A\to B)}_{\mathcal{C}}\gtrsim Yield{\left(A\right)}_{\mathcal{C}}-Cost{\left(B\right)}_{\mathcal{C}}.$$

For resources $A,B$ for which $\mathcal{C}$ is tight, furthermore

$$Balance{(A\to B)}_{\mathcal{C}}=Yield{\left(A\right)}_{\mathcal{C}}-Cost{\left(B\right)}_{\mathcal{C}}.$$

In many familiar resource theories in physics, fairness of the currency is either taken for granted or imposed as a fundamental restriction on good currencies. In thermodynamics, for example, it appears under the name of translational invariance: transformations are not allowed to depend on the initial state of the energy storage system [5,35,36,37,38]. These storage systems are modeled as harmonic oscillators that mimic a classical weight; work, or the balance of a transformation, is counted as the difference between the energy of initial and final states. The underlying assumption required is that the weight system has many evenly spread energy levels and is far from the ground state.

2.4. Pathological Cases

For Stage I currencies, we have seen that all resources A in the target satisfy $Cost{\left(A\right)}_{\mathcal{C}}\ge Yield{\left(A\right)}_{\mathcal{C}}$. We can read this as an impossibility statement that says that we cannot increase the amount of currency for free in the process of buying and re-selling a resource A—this would collapse the order in the currency and render the theory trivial. In a Stage II currency, there might, however, exist resources for which $Balance{(A\to A|C)}_{\mathcal{C}}>0$ for some $C\in \mathcal{C}$, that is, resources A which allow us to generate a little bit of currency for free if we have access to A and some particular value C in the currency. Then, A would essentially act like a catalyst in a process on the currency that is otherwise forbidden.

The existence of such a resource A does not make the theory trivial because the condition $Balance{(A\to A|C)}_{\mathcal{C}}>0$ depends on the exact currency C we start with. For example, it could require a valuable currency resource C, or could hold only for a very expensive resource A that is hard to buy in the first place. In real life, for example, a house that is rented to a reliable tenant generates rent money every month, but requires a large investment to start with. However, if the currency is fair (in the direction that more does not help), we can show that such a pathological resource would again collapse the order in the currency. Pathological cases are discussed in detail in Appendix C.

3. Application to Realistic Resources

Now we show how to apply those ideas to explicit descriptions of arbitrary physical resources. The following approach allows us to model approximate transformations and generalize the idea of composition to a natural concept that applies for example when correlations between subsystems cannot be excluded, or when the subsystem structure is not clear. For details on this framework, we refer to [1]; for the purposes of this work, the following summary will suffice.

3.1. Setup for Resource Theories

Before we start, note that the formalism of currencies developed in this paper can also be applied to “traditional” resource theories, for example where resources are represented by density operators (in the quantum case), as opposed to the specifications over sets of states that we are about to introduce. We choose to present the more general version here.

3.1.1. Realistic Descriptions of Resources

We may see the state space Ω of a theory as the language that an agent uses to describe resources: for example, in quantum theory Ω could be the set of all density operators over a global Hilbert space; in traditional thermodynamics the set of all distinguishable macrostates. Realistically, agents may not know the exact state of a system, and may instead describe resources through more coarse descriptions, like the $\epsilon$-neighbourhood of a state (obtained after tomography) or the specification of a few measurement outcomes. Such specifications are simply subsets of the state space, like ${\mathcal{B}}^{\epsilon }\left(\rho \right)$, composed of all states that are compatible with the agent’s knowledge.

Specifications are useful also when talking about target states: often we do not need to end up precisely in state $\rho$, but it suffices to get sufficiently close to it—this applies to cryptographic tasks, when in order to ensure $ϵ$-security of a key distribution protocol, for example, two agents need only share a state close enough to a Bell pair, and make measurements sufficiently close to the ideal projective measurements. These specifications help us characterize the robustness of protocols; realistic scenarios where we are happy to reach the neighbourhood ${\mathcal{B}}^{\epsilon }\left(\rho \right)$ of a state give rise for example to single-shot smooth entropy measures (see [41] for a review). Indeed smooth entropies can be seen as monotones—derived from currencies, for the appropriate resource theories and target resources; more on this in the following sections.

Another example of a specification is a local state like ${\rho }_A$ within a global state space ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$. This reduced description can be seen in the global view as the specification “I know that the joint state can be any state whose marginal in ${\mathcal{H}}_A$ is ${\rho }_A$”, or in other words, the set of states $\{{\sigma }_{AB}:{Tr}_B\left({\sigma }_{AB}\right)={\rho }_A\}$. We will develop this idea further ahead.

Specifications have a natural partial order: if a description V is more specific than another, W, we simply have $V\subseteq W$, for example ${\mathcal{B}}^{\epsilon }\left(\rho \right)\subseteq {\mathcal{B}}^{\epsilon +\delta }\left(\rho \right)$. Together, the subsets of $\mathsf{\Omega }$ form the specification space $S^{\mathsf{\Omega }}$—the space of realistic resources.

3.1.2. Transformations

Physical actions implemented by an agent transform resources into resources ($f:S^{\mathsf{\Omega }}\to S^{\mathsf{\Omega }}$), in such a way that if an agent is unsure about the underlying state, this uncertainty carries through the transformation. For example, if the agent knows that the state of a system is either $\rho$ or $\sigma$, then after applying a transformation f, her knowledge is updated as $f\left(\right\{\rho \left\}\right)\cup f\left(\right\{\sigma \left\}\right)$. In general, transformations $f\in \mathcal{T}$ need to act element-wise, that is for any specification $V\subseteq \mathsf{\Omega }$,

$$f\left(V\right)=\bigcup _{\nu \in V}f\left(\left\{\nu \right\}\right).$$

This formalism also allows us to model cases where there is uncertainty about the exact effect of a transformation: for example, after performing process tomography of an experimental procedure f, we may learn only that f is in a neighbourhood of a trace preserving completely positive (TPCP) map g. We could model this procedure as $f:\left\{\rho \right\}\mapsto {\mathcal{B}}^{\epsilon }\left(g\left(\rho \right)\right)$.

3.1.3. Resource Theories

As discussed in the introduction, a resource theory is defined by a set $\mathcal{T}$ of allowed transformations available to the agent, which act on the space of resources—the specification space $S^{\mathsf{\Omega }}$ that describes the resources from the point of view of an agent. The set of allowed transformations induces a pre-order → of accessibility on $S^{\mathsf{\Omega }}$, which encodes whether or not a resource $V\in S^{\mathsf{\Omega }}$ can be transformed into another resource W. Formally,

$$V\to W\iff \exists f\in \mathcal{T}\mathrm{s}.\mathrm{t}.f\left(V\right)\subseteq W.$$

This pre-order combines the action of the transformation and the natural partial order on sets: operationally, this means that forgetting information (going to a less specific description) is always allowed in the resource theory. For example, in a resource theory of unital maps on qubits, we would find that we can always reach the specification of an $\epsilon -$neighbourhood of the maximally mixed state: $\left\{\right|0\rangle \langle 0\left|\right\}\to {\mathcal{B}}^{\epsilon }({\mathbb{1}}_2/2),$ since there is a transformation $f\in \mathcal{T}$ that achieves $f\left(\left\{\right|0\rangle \langle 0\left|\right\}\right)=\{{\mathbb{1}}_2/2\}$ and the maximally mixed state is a more specific description than an $\epsilon$-ball around it, $\{{\mathbb{1}}_2/2\}\subseteq {\mathcal{B}}^{\epsilon }({\mathbb{1}}_2/2).$ (Note: Unital maps are all TPCP maps that preserve the identity, $\mathcal{E}\left(\mathbb{1}\right)=\mathbb{1}$. See Section 3.3 for more details on the resource theory of unital maps and on how our results apply to this theory.)

3.2. Insights

Let us now introduce the main insights that this approach brings into the subject of quantifying resources. In Section 2, we saw that both currencies and the target are sets of resources. In our formalism, this means that they are sets of specifications, $\mathcal{C},\mathcal{S}\subseteq S^{\mathsf{\Omega }}$. In the following we explore some examples. Formal definitions, results and proofs can be found in Appendix A.

3.2.1. Rough Currencies and Single-Shot Statements

Specifications are particularly useful when a theory has a known currency ${\mathcal{C}}_{\mathrm{ideal}}$ that cannot be implemented experimentally—it may rely on idealizations like pure or perfectly uncorrelated states, for example. The experimenter may instead have access to coarser specifications of resources, for example $\epsilon$-neighbourhoods of the currency states ${\mathcal{C}}_{\mathrm{real}}={\left\{{\mathcal{B}}^{\epsilon }\left(\rho \right)\right\}}_{\rho \in {\mathcal{C}}_{\mathrm{ideal}}}$. If ${\mathcal{C}}_{\mathrm{real}}$ is ordered (up to equivalences), it automatically forms a currency for some target space. Otherwise (for example, if two elements overlap too much), we may remove or replace some of the elements with other accessible resources until we obtain an ordered set. Naturally, we would not expect ${\mathcal{C}}_{\mathrm{real}}$ to be as powerful a currency as ${\mathcal{C}}_{\mathrm{ideal}}$: it might not reach the entire state space (e.g., because we do not have access to pure states), or it might offer a coarser quantification (because we removed some elements). Nevertheless, we can always identify a maximal target set of specifications for which ${\mathcal{C}}_{\mathrm{real}}$ is universal. In this case the target would include the set of $\epsilon$-neighbourhoods of all states reached by ${\mathcal{C}}_{\mathrm{ideal}}$ (if the theory is stable, e.g., linear, under these neighbourhoods, as explained in [1]). Once we find the appropriate target for ${\mathcal{C}}_{\mathrm{real}}$, we may use all the tools of currencies, like cost, yield and checking for fairness, which apply to transformations between specifications, not only between states.

We can also address questions about single-shot transformations, of the sort ‘what is the cost of reaching a final state, if we allow for a small error tolerance?’ The agent encodes that error tolerance in an operational notion of closeness on the state space (like the trace distance), and uses it to build specifications of $\epsilon$-balls, $V^{\epsilon }\supseteq V$ [1], Section V. It follows that $Cost{\left(V\right)}_{\mathcal{C}}\ge Cost{\left(V^{\epsilon }\right)}_{\mathcal{C}}$ and $Yield{\left(V\right)}_{\mathcal{C}}\ge Yield{\left(V^{\epsilon }\right)}_{\mathcal{C}}$. The exact result will depend on the theory [13,42]; in the upcoming example of unital maps, the cost is characterized by smooth entropy measures. Similarly, with a Stage II currency we can talk about the balance of transitions between two resources $V^{\epsilon }$ and $W^{\epsilon }$. The same reasoning applies to any currency and target made of arbitrary specifications of resources more generally than just for well-behaved $\epsilon$-neighbourhoods.

3.2.2. Local Resources and Currencies

In our approach, we start from a global theory and model local resources as specifications in a global space. Doing this allows us to go beyond the tensor product to combine local resources. For example, given a Hilbert space ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, a marginal state ${\rho }_A$ is a compact description of the set of all global states compatible with that marginal,

$${\widehat{\rho }}_A:=\{{\sigma }_{AB}:{Tr}_B\left({\sigma }_{AB}\right)={\rho }_A\}.$$

A general way to compose local resources is to combine these specifications, for example

$${\widehat{\rho }}_A\cap {\widehat{\tau }}_B=\{{\sigma }_{AB}:{\sigma }_A={\rho }_A\wedge {\sigma }_B={\tau }_B\}.$$

This way, we can treat genuinely local knowledge without imposing additional non-local assumptions—for example, we do not assume that the local states in the product state ${\rho }_A\otimes {\tau }_B\in {\widehat{\rho }}_A\cap {\widehat{\tau }}_B$. If we do have some additional knowledge about the strength of correlations, we can include it in the specification. For example, the knowledge that the mutual information between the two subsystems is at most $\epsilon$ can be expressed as a specification

$$I{(A:B)}_{\le \epsilon }:=\{{\sigma }_{AB}:I{(A:B)}_{\sigma }\le \epsilon \},$$

and our global specification then becomes

$${\widehat{\rho }}_A\cap {\widehat{\tau }}_B\cap I{(A:B)}_{\le \epsilon }.$$

Often a currency $\mathcal{C}$ and a target $\mathcal{S}$ consist of local resources lying in different subsystems. Consider for example the case of LOCC, where a standard currency consists of different numbers of copies of Bell pairs. These can be stored in a wallet system shared by two agents Alice and Bob: we may think of a decomposition of the global Hilbert space as

$${\mathcal{H}}_{\mathrm{global}}=\underset{\mathrm{wallet}}{\underbrace{\left({⨂}_{i=1}^N(A_i\otimes B_i)\right)}}\otimes \underset{\mathrm{target}}{\underbrace{A^{\prime }\otimes {\tilde{B}}^{\prime }}},$$

where each $A_i$ and $B_i$ is a qubit. The currency is made of specifications of a certain number n of copies of Bell pairs in the wallet, $\mathcal{C}={\left\{{\mathsf{\Psi }}^n\right\}}_{n=1}^N,$ with ${\mathsf{\Psi }}^n=\{{\sigma }_{\mathrm{global}}:{\sigma }_{A_1{B}_1\cdots A_n{B}_n}=|\psi \rangle \langle \psi {|}^{\otimes n}\},$ and $|\psi \rangle =\left(\right|00\rangle +|11\rangle )/\sqrt{2}$. Note that the currency is naturally ordered as ${\mathsf{\Psi }}^n\to {\mathsf{\Psi }}^m$ for $n>m$ (as ${\mathsf{\Psi }}^n\subset {\mathsf{\Psi }}^m$, that is we can always go from more to less Bell pairs by forgetting some—in other words, by discarding or tracing them out). Teleportation implies that this currency is universal for a target as large as the wallet, $log(max(|A^{\prime }|,|B^{\prime }\left|\right))\le N$. In our language it means that we can pick the target $\mathcal{S}$ to be any set of specifications that are local in $A^{\prime }\otimes B^{\prime }$, for example $\mathcal{S}={\left\{\widehat{{\rho }_{A^{\prime }{B}^{\prime }}}\right\}}_{\rho }$, with $\widehat{{\rho }_{A^{\prime }{B}^{\prime }}}=\{{\sigma }_{\mathrm{global}}:{\sigma }_{A^{\prime }{B}^{\prime }}={\rho }_{A^{\prime }{B}^{\prime }}\}$ (Appendix B). The cost and yield of target resources are the usual entanglement of formation and distillation, respectively [13].

3.2.3. Independence without Composition

We promised that specifications allow us to define independence between a currency and a target without the need to talk of subsystems or any traditional notion of composition. At heart, independence means that we can change one quantity (the currency) without affecting the other (the target), as discussed in Section 2. While this trivially holds in the neat case of a tensor product structure between states and subsystems, the notion is more general.

The first requirement for independence was that all states of the currency can coexist with all states of the target—that is, if we combine the knowledge C about the currency and the knowledge V about the target, the resulting specification $C\cap V$ contains at least one global state compatible with this knowledge. If on the other hand $C\cap V=\varnothing$, this tells us that the two specifications contained contradictory knowledge—which could happen for example if currency and target were defined on the same degree of freedom, so we could not have both at once. Therefore, the first condition for Stage II currencies is compatibility: for all currency and target resources $C\in \mathcal{C}$ and $V\in \mathcal{S}$, we should have $C\cap V\ne \varnothing$. We do not impose any extra subsystem structure on $\mathcal{C}$ and $\mathcal{S}$, which are otherwise just sets of specifications, nor do we require a formal operation of composition of ‘local’ resources.

The second key idea is that we can manipulate the currency without disturbing the target. In our language, this means that if we can transform $C\to C^{\prime }$, then we can also act on the combined knowledge of currency and target as $C\cap V\to C^{\prime }\cap V$, for all target resources $V\in \mathcal{S}$. Naturally, the Bell pairs from the previous example form a Stage II currency (Proposition A7). Another example of a Stage II currency is work in quantum thermodynamics (Proposition A8). More generally, these conditions can be satisfied even when correlations between currency and target cannot be ruled out—in quantum theory, they do not need to be in a tensor product. Indeed, local knowledge such as $\widehat{{\rho }_A}\cap \widehat{{\sigma }_B}$ is sufficient to ensure independence, and therefore to define the balance of transformations ([1], Section IV).

3.3. Example: Unital Maps

Let us look at the particular example of a resource theory in which the allowed operations are given by unital, completely positive trace preserving maps on quantum states [43]. On the level of state transformations and for classical systems, this resource theory also coincides with the resource theory of noisy operations [15,42,43,44,45] (for arbitrary quantum states, the actual set of unital maps is a superset of the maps achievable by noisy operations). As such, the resource theory of unital maps characterizes a range of physical situations from an agent processing information in a noisy environment to thermodynamics on degenerate energy eigenstates. On the level of quantum state transformations, the resource theory of unital maps is well understood: the pre-order on quantum states is given by majorization [15,42,45,46,47,48,49,50,51,52,53,54,55,56,57], and we can quantify resources through smooth entropies [40,42,58,59,60,61,62]. For simplicity, we restrict our analysis to the case of same input and output dimensions.

However, for specifications the situation is not so clear: how can we assign value to general resources $V\in S^{\mathsf{\Omega }}$? Are there simple necessary and sufficient conditions for resource transformations $V\to W$? Solving these questions would allow us to characterize the pre-order for realistic descriptions of resources, including approximations around quantum states and specifications of a few selected properties of the system. Furthermore, it would give us insight into how to define entropy of specifications, study how much work is needed to erase the information encoded in a specification (analogous to Landauer’s principle [42,63,64]), and how to draw a link between microscopic and macroscopic thermodynamics, as outlined in [1]. Finally, the question of when a specification can be transformed into another bears resemblance to the question of the work cost of a general process [42] as well as to the question of when a set of states can be transformed into another set [65,66,67,68].

Here, we introduce three alternative currencies for the resource theory of unital maps on specifications. First, we define a Stage I currency and compute the cost and yield of specifications, which already give us necessary and sufficient conditions for resource transformations. Then we give a similar Stage II currency, and discuss examples of alternative currencies that make use of specifications more explicitly. Finally, we study smooth transformations between $\epsilon$-neighbourhoods of states. Our choice of currency is inspired by [40], where entropies for quantum states were derived from similar considerations—our results in this section could hence be interpreted as a step towards understanding entropies for specifications.

3.3.1. Stage I Currency

Let the global state space $\mathsf{\Omega }$ of the theory be all quantum states on a d-dimensional Hilbert space. Specifications $V\in S^{\mathsf{\Omega }}$ are then sets of such states. We can define a universal currency $\mathcal{C}={\left\{C^k\right\}}_{k=1}^d\cup \left\{\mathsf{\Omega }\right\}$ as the set of uniform states of different ranks,

$$C^k=\left\{\frac{1}{k}\sum _{i=1}^k|i\rangle \langle i|\right\}=\left\{\frac{{\Pi }^k}{k}\right\},$$

where ${\Pi }^k$ denotes the projector onto the first k eigenstates of a fixed basis. We take the value function

$$Val\left(C^k\right)=log\left(d\right)-logk,$$

such that states with lower rank are more valuable. (Note that $Val\left(\mathsf{\Omega }\right)=Val\left(C^d\right)=0$, since the two are interconvertible under unital operations.) We prove that $\mathcal{C}$ indeed forms a currency for the global target $\mathcal{S}=S^{\mathsf{\Omega }}$ in Proposition A9. The cost of a specification V in terms of $\mathcal{C}$ is then given by

$$\begin{array}{cc}\hfill Cost{\left(V\right)}_{\mathcal{C}}& =logd-\underset{\rho \in V}{sup}log\lfloor 2^{H_{\min }\left(\rho \right)}\rfloor \hfill \end{array}$$

where $H_{min}$ is the min-entropy [41,60,62] defined as $H_{min}\left(\rho \right)=-log{\lambda }_{\max }\left(\rho \right)$, with ${\lambda }_{\max }\left(\rho \right)$ being the largest eigenvalue of $\rho$, and $\lfloor ·\rfloor$ the nearest integer (from below) to the enclosed expression (Proposition A10). This result can be seen as an analogue to the lower bound for entropy of quantum states $\rho$ found in [40], which obtained exactly $H_{\min }\left(\rho \right)$ as a monotone. Here, we see that more generally there is an optimization over the states in the specification.

Note that as we evaluate the cost, we need to round down ${\lambda }_{\max }^{-1}\left(\rho \right)$, because there is a limit to how well we can approximate ${\lambda }_{\max }$ by means of a rational $\frac{1}{n}$. For example, in a two-level system, our currency would only contain elements of two different values: 1 (pure state), and 0 (fully mixed state). Then, the cost of a single-qubit specification is either 0 (for all specifications containing the fully mixed state) or 1 (for all others), but nothing in between. This is a finite-size effect that is not critical for large systems; it nevertheless tells us that our choice of currency, while universal, is not very fine-grained. An alternative currency for single qubits could be for example ${\mathcal{C}}^{\prime }={\left\{C_p\right\}}_{p=0}^{1/2}\cup \left\{\mathsf{\Omega }\right\}$, with $C_p=\left\{p\right|0\rangle \langle 0|+(1-p)|1\rangle \langle 1\left|\right\}$, which would recover $Cost{\left(V\right)}_{{\mathcal{C}}^{\prime }}=logd-{sup}_{\rho \in V}{H}_{min}\left(\rho \right)$. Note: In Ref. [40], this issue was circumvented by extending the state space from density matrices to continuous step functions. On the level of state transformations, the theory of unital maps (or noisy operations) boils down to a simple majorization condition, which can easily be extended to such step functions. However, here, we would like to be a little more careful, especially since we deal with specifications V in general, for which the pre-order structure is more complicated and not fully characterized by majorization.

As for the yield of specifications in terms of $\mathcal{C}$, let us first look at the single-qubit example. Suppose that we start from the specification $V=\left\{\right|0\rangle \langle 0|,|1\rangle \langle 1\left|\right\}$. While for any individual pure state the yield is 1 since we can always generate the pure currency state from it, the same is not true for the specification V. The reason is that there is no single protocol that achieves the pure currency state for both states $|0\rangle \langle 0|$ and $|1\rangle \langle 1|$, so that it could be applied without knowing which one is actually the case. Instead, in the case of V one can only generate the fully mixed currency state, which can be done with a single process that works for both cases (the discard-and-prepare map $\mathcal{E}\left(\rho \right)=\mathbb{1}/2$, which is unital). The yield of V is hence 0—the same as the yield for a general convex mixture of the two pure states in V.

In general, we find that the yield of a specification in terms of $\mathcal{C}$ is given by

$$\begin{array}{cc}\hfill Yield{\left(V\right)}_{\mathcal{C}}& =logd-\underset{\rho \in V^{\mathbb{P}}}{max}{H}_0\left(\rho \right),\hfill \end{array}$$

where $V^{\mathbb{P}}$ denotes the convex hull of V, which consists of all the convex (probabilistic) mixtures of the states in V ([1], Section VI) and $H_0\left(\rho \right)$ is the order zero quantum Rényi entropy defined as $H_0\left(\rho \right)=logrank\left(\rho \right)$ [41,60,62]. This is proven in Proposition A11.

3.3.2. Stage II Currency

We can now analyse an independent currency that lives in a different subsystem to the target specifications. Consider a global state space of density matrices on two systems $W\otimes S$, where W will be our wallet and system S the target, with dimensions $d_W\ge d_S$. More precisely, we can show that the set $\mathcal{C}={\left\{C^k\right\}}_{k=1}^{d_W}\cup \left\{\mathsf{\Omega }\right\}$, with

$$C^k=\left\{{\sigma }_{WS}:{Tr}_S\left({\sigma }_{WS}\right)=\frac{{\Pi }^k}{k}\right\}$$

forms a currency for the target $\mathcal{S}$ defined through

$$V\in \mathcal{S}\iff V=\{{\sigma }_{WS}:{Tr}_C\left({\sigma }_{WS}\right)\in V_S\}$$

for some set $V_S$ of density matrices on system S (Proposition A12). This currency furthermore satisfies independence according to Definition A5, even if most elements of the currency are not in a tensor product with target resources.

We can easily see that the currency is not fair, because of the finite-size effects discussed previously, but in the limit of large wallet dimension $d_W$, we can show that

$$Cost{\left(V\right)}_{\mathcal{C}}\approx log{d}_S-\underset{{\rho }_S\in V_S}{sup}log{H}_{min}\left({\rho }_S\right),$$

to an arbitrarily good approximation (Theorem A5). This issue has also been discussed in [40], where it was noted that access to large ancilla systems can help approximate the step functions arbitrarily well.

3.3.3. Alternative Currency

As we saw, there can be many options for a currency within a resource theory, and the ultimate choice is up to the user (for example, whether they will be treating small or large systems, or whether they can easily distinguish states that are close). We would like to finalize this example with a currency that makes more explicit use of specifications to formalize lack of knowledge. The idea is to model an agent who knows only that the system could be in any mixture of k pure states— weaker than knowing that the system is in a uniform mixture of those k states. Consider ${\mathcal{C}}_2={\left\{C_2^k\right\}}_k\cup \left\{\mathsf{\Omega }\right\}$, with

$$C_2^k={\left(\bigcup _{1\le i\le k}\left\{\right|i\rangle \langle i\left|\right\}\right)}^{\mathbb{P}},$$

where instead of taking maximally mixed states of different ranks like $\mathcal{C}$ before, we take the convex hull of differently sized sets of orthogonal pure states on the currency system.

In comparison to our original currency $\mathcal{C}$, note that for each k, $C^k\subseteq {\mathcal{C}}_2^k$. The set ${\mathcal{C}}_2$ forms again a currency for the target $S^{\mathsf{\Omega }}$, and is in fact equivalent to the currency $\mathcal{C}$ before since, under unital operations, $C^k\to C_2^k$ and $C_2^k\to C^k$ for all k. In particular, we can take the usual value function $Val\left(C_2^k\right)=logd-logk$ and obtain the same expressions for cost and yield. The convex hull in the expression for $C_2^k$ is crucial—without it, we would still get a currency but with different results for the cost and yield of resources (Proposition A13).

3.3.4. Smooth Transformations

As a special case of our framework, we can model scenarios where we allow for some error probability in the output of a transformation, or where we need to make sure that our protocol is robust against errors in the initial resources. To illustrate this, we can look at resources that correspond to approximate quantum states, ${\mathcal{B}}^{\epsilon }\left(\rho \right)$, according to some metric like the trace distance or the purified distance based on the fidelity (more general approximations are discussed in [1]). In this case, we recover the smooth-min entropy [41,60,62] as a monotone, as

$$Cost{\left(B^{\epsilon }\left(\rho \right)\right)}_{\mathcal{C}}=logd-log\lfloor 2^{H_{min}^{\epsilon }\left(\rho \right)}\rfloor .$$

On the other hand, smoothing the input does not always give us very meaningful results for this choice of currency, as for example $Yield{\left(B^{\epsilon }\left(\rho \right)\right)}_{\mathcal{C}}=0$ for all $\epsilon >0$. The usual bound for the balance of resource transformations in terms of cost and yield (which works by selling the initial resource and buying the final one) is not helpful when the yield is zero: for example, the most efficient way to go from $B^{\epsilon }\left(\rho \right)$ to $B^{{\epsilon }^{\prime }}\left(\rho \right)$ will never go through this currency. This effect occurs partly because the currency is relatively coarse-grained (it contains only flat states), and also because it consists of exact states rather than smooth specifications (unlike the previously discussed ${\mathcal{C}}_{\mathrm{real}}$). By smoothing the currency or adopting a version of yield that smooths over the output, we could arrive at more useful statements, involving for example the smooth max-entropy. First steps on smooth transformations can be found in Appendix B.

4. Discussion

While each resource theory has its own understanding of how to identify precious resources, there are common tools to address this question. In this work, we abstract from particular examples and identify the general concept of currencies that can be used to quantify the cost of resource transformations as well as the value of particular resources. Our approach allows us to go beyond commonly employed assumptions of perfect independence between the currency and the system of interest, exactly known quantum states, perfect and uncorrelated copies of states on individual subsystems, or any special form or properties of the currency states such as the scalability of equilibrium systems employed in [2,3,4,5]. As such, our results seem particularly relevant for thermodynamics of macroscopic systems or systems of limited control, where slight correlations cannot be ruled out or only a few properties of the systems of interest are known. Furthermore, our approach could prove powerful in the context of adversarial settings in cryptography, in which worst-case scenarios need to be assumed [69,70].

Using the formalism of specifications developed in [1], we can quantify the value of arbitrary descriptions of resources, paving the way to finding entropic quantities that characterize non-probabilistic states of knowledge.

As we saw in the example of the resource theory of unital maps, there are many possible choices of currency within a theory. Traditional choices such as copies of pure states or Bell pairs are useful to find monotones from asymptotic conversion rates [8,14]. On the other hand, in realistic settings with limited resources, these strict currencies may result in a big divergence between cost and yield of target resources, as happened for resources that were $\epsilon$-neighbourhoods of states. As a consequence of this gap, the resulting bounds on the balance of resource conversions become very loose, and do not give a very useful characterization of the pre-order of the resource statement. A commonly used way out is to always work in the regime of large systems, where these problems become less significant; however, we believe it is more natural to stay in the experimentally realistic setting and adapt our theoretical tools to meet the constraints in the lab. In order to find tighter and meaningful bounds, we might for example use finer currencies, or smooth currencies made of specifications (e.g., currencies that are themselves $\epsilon$-neighbourhoods of states). Alternatively, we might keep the currency but relax our definitions of cost, yield and balance to allow for smoothing on the input and output, corresponding to a small error probability.

4.1. Related Work

In abstracting from particular resource theories and identifying common tools to quantifying resources, our work is similar in spirit to [7,8,9]. However, while these works also explore the mathematical structure of resource theories, they study the case where resources have a clear local structure, equipped with a composition operation that ensures independence between currency and target (this is the case for symmetric monoidal categories [7,8] and in particular the tensor product of quantum states [9]). In other words, those works study Stage II currencies, with one direction of fairness (having less does not help) guaranteed. We see our approach as complementary to Fritz’s [8], in the sense that he explores in depth how much we can do once we have a definite notion of composition, while we ask the broader question of what independence and composition mean, and what we can still do when we cannot guarantee that local structure in the space of resources. Another difference is our interest in single-shot settings, compared to a stronger focus on conversion rates and asymptotic scenarios in [7,8,9].

We drew inspiration from many particular examples of resource theories, and especially from the abstract approach to thermodynamics in Lieb and Ynganvon’s work [2,3,4,5]. We generalize their ideas to arbitrary resource theories beyond thermodynamics. In addition, we weaken and clarify the assumptions needed to recover the monotones; in particular, we can avoid assuming any additional structure on the currency resources, such as the scalability requirement present in those works. Not surprisingly, our results resemble the quantities and considerations of [2,3,4,5,40], as well as those of approaches to quantum thermodynamics [17,45,58]. Our results, however, have a wider range of application.

Our work also bears resemblance to the article by Gallego et al. on defining work in quantum thermodynamics from operational principles [6]. Their paper studies how a set of axioms motivated by the second law of thermodynamics influences possible definitions of cost or balance of resource transformations. The setting of two players, Arthur and Merlin, in that paper relate very closely to our splitting of resources into currency and target. Furthermore, the definitions and properties of the resulting work function are similar to our statements about balance. For example, the total order on the currency resources that we demand seems to follow from the axioms in [6]—their approach can hence be seen as an interesting complement to ours. However, we go beyond the scope of their results not only since we consider general resource theories beyond quantum thermodynamics, but also because we do not limit ourselves to exact quantum states and tensor product composition. Furthermore, we discuss a range of properties of currencies and their implications, beyond seeking a unique notion of balance.

4.2. Directions

A natural and powerful application of our tools is quantum thermodynamics, where current resource-theoretical approaches oversimplify physical settings. For example, the popular framework of thermal operations assumes a weak coupling regime and models thermal baths as uncorrelated systems in perfect Gibbs states. This work provides the tools needed to move towards more realistic models for quantum thermodynamics. Our approach can be applied by the wider physics and information theory community—for example in cryptographic settings, where protocols must be robust against lack of knowledge and control of different agents.

In addition to looking at concrete applications, we can also further develop the approach presented here. For example, it would be key to generalize familiar monotones such as the relative entropy to the fixed point or free resources of the resource theory [9] to the case of specifications. Similarly, one could try to generalize known results from state to state transformations for particular resource theories, such as the conditions of majorization or thermo-majorization in noisy and thermal operations.

As we have seen for example in the case of approximate quantum states in the resource theory of unital maps (Section 3.3), there can be a large gap between the cost and the yield of resources. This is reminiscent of the phenomenon of bound entanglement in LOCC, when only a few or even no Bell pairs can be extracted from a state but one nonetheless requires a large number of Bell pairs to create it. Since our results show that the cost and yield of resources ultimately depend on the choice of currency, it would be interesting to see if similar statements could be made about bound entanglement in LOCC: if instead of Bell pairs one would consider a different set of reference states as a currency, would the phenomenon of bound entanglement always occur or could one eliminate the concept altogether?

Another direction of further research would be to look more closely at resource theories in which we can only find currencies for a limited target. A good example of this is the resource theory of thermal operations [14,16], in which work can be considered a currency only for states that are block-diagonal in the energy eigenbasis. Since the theory is time-translation symmetric, we cannot introduce coherence from purely incoherent resources [27,29,30]. This problem could perhaps be solved by introducing two or more different currencies that function together and may or not be traded against each other (similar to recent ideas on non-commuting conserved charges in thermodynamics [71,72,73])—for example, by introducing multiple wallets to store currencies like work and coherence.

Finally, it would be interesting to characterize the conditions under which currencies become fair, like the regime of large systems, and the connection to conversion rates.