The resource theory of informational nonequilibrium in thermodynamics
We review recent work on the foundations of thermodynamics in the light of quantum information theory. We adopt a resource-theoretic perspective, wherein thermodynamics is formulated as a theory of what agents can achieve under a particular restriction, namely, that the only state preparations and transformations that they can implement for free are those that are thermal at some fixed temperature. States that are out of thermal equilibrium are the resources. We consider the special case of this theory wherein all systems have trivial Hamiltonians (that is, all of their energy levels are degenerate). In this case, the only free operations are those that add noise to the system (or implement a reversible evolution) and the only nonequilibrium states are states of informational nonequilibrium, that is, states that deviate from the maximally mixed state. The degree of this deviation we call the state’s nonuniformity; it is the resource of interest here, the fuel that is consumed, for instance, in an erasure operation. We consider the different types of state conversion: exact and approximate, single-shot and asymptotic, catalytic and noncatalytic. In each case, we present the necessary and sufficient conditions for the conversion to be possible for any pair of states, emphasizing a geometrical representation of the conditions in terms of Lorenz curves. We also review the problem of quantifying the nonuniformity of a state, in particular through the use of generalized entropies, and that of quantifying the gap between the nonuniformity one must expend to achieve a single-shot state preparation or state conversion and the nonuniformity one can extract in the reverse operation. Quantum state conversion problems in this resource theory can be shown to be always reducible to their classical counterparts, so that there are no inherently quantum-mechanical features arising in such problems. This body of work also demonstrates that the standard formulation of the second law of thermodynamics is inadequate as a criterion for deciding whether or not a given state transition is possible.
- I Introduction
- II Preliminaries
- III Quasi-order of states under noisy operations
IV Nonuniformity monotones
- IV.1 Nonuniformity monotones built from convex functions on the reals
- IV.2 Nonuniformity monotones arising from the geometry of the Lorenz curve
- IV.3 Nonuniformity monotones from other Schur-convex functions
V Exact state conversion
- V.1 A standard form of nonuniformity: sharp states
- V.2 Nonuniformity of formation and distillable nonuniformity
- V.3 Nonuniformity cost and yield of state conversion
- V.4 Catalysis
- V.5 Inadequacy of the second law as a criterion for state conversion
- VI Approximate state conversion
- VII Conclusions
- VIII Acknowledgements
- A Proofs
- B Strong noisy-trumping and Rényi nonuniformities
- C Approximate distillation
A resource theory is defined by identifying a set of experimental operations that are considered free, that is, a set which can be used without limit, while the rest are considered expensive and are thus treated as resources Coecke et al. (2014). A quantum resource theory is defined by identifying a set of free operations within the set of all quantum operations. For instance, the restriction to local quantum operations and classical communication defines the resource theory of entanglement Horodecki et al. (2009), the restriction to symmetric quantum operations defines the resource theory of asymmetry Gour and Spekkens (2008); Marvian and Spekkens (2013); Bartlett et al. (2007), and the restriction to stabilizer operations defines a theory of quantum computational resources Veitch et al. (2014).
The resource theory that is the subject of this article is defined by the following set of free quantum operations:
one can implement any unitary on a system,
one can prepare any system in the uniform quantum state (i.e., the completely mixed state),
one can take a partial trace over any subsystem of a system.
In the most general such operation, an ancilla prepared in the uniform state is adjoined to the system of interest, they are coupled unitarily, and finally one takes a partial trace over some subsystem. These have been called noisy operations in Horodecki et al. (2003a, b), where this resource theory was first studied111They have also been called exactly factorizable maps by Shor Shor (2010); see also Haagerup and Musat (2011). If one comes to possess a system that is not in the uniform state, then because such a state cannot be prepared for free, it is a resource. We call such states nonuniform, and we call the aspects of a state that are relevant to the resource theory the state’s nonuniformity properties.
Nonuniform states are also resources insofar as they can be useful for simulating operations that are outside the noisy class: if a certain process taking a given initial state to a desired final state is not achievable via noisy operations, the process can always be rendered possible by adding an appropriate nonuniform state to the given initial state and suffering the consumption (partial or complete) of this added resource. An example of a process that would normally be disallowed under noisy operations, but becomes possible by supplying (and using up) an added nonuniformity resource, is an erasure operation. Erasure is “informational work”, and therefore one can understand the nonuniformity of a state as the potential for doing informational work.
The resource theory of nonuniformity is interesting primarily because of the light that it sheds on another, more intricate and more relevant, resource theory: the one where the free operations are those that are thermal relative to some temperature , and the resources are states that are not thermal at this temperature, or athermal states Janzing et al. (2000); Horodecki and Oppenheim (2013); Brandão et al. (2015, 2013); Skrzypczyk et al. (2013, 2014).222The resource-theoretic approach to thermodynamics is cognate with the earlier work of Lieb and Yngvason Lieb and Yngvason (1999), where classical thermodynamics is formulated in terms of interconvertibility relations between equilibrium states.
The thermal operations relative to a background temperature are defined by the following capabilities: One can prepare any subsystem in its thermal state at temperature , one can implement any unitary that commutes with the total Hamiltonian333It is presumed that the couplings between subsystems can be entirely controlled by the experimenter, so that in the absence of any such intervention, the overall Hamiltonian is just a sum of the free Hamiltonians for each subsystem., and one can trace over the subsystems. It follows that the resource theory of nonuniformity is simply a special case of the resource theory of athermality (for any temperature) where all systems of interest have trivial Hamiltonians (i.e., all their energy levels are degenerate).
To achieve a proper conceptual understanding of thermodynamics, it is critical to disentangle those aspects of the theory that are due to considerations of energetics and those that are due to considerations of information theory. Indeed, there is now a large literature on the role of information theory in thermodynamics, centered around such topics as Maxwell’s demon Maruyama et al. (2009), Szilard’s engine Szilard (1929), the thermodynamic reversibility of computation and Landauer’s principle Bennett (1973, 1982); Landauer (1961), and the use of maximum entropy principles in statistical mechanics Jaynes (1957a, b). (See also Leff and Rex (2003).) By focussing on the resource theory of athermality for the special case of systems that are energy-degenerate, we can begin to understand what aspects of thermodynamics are purely informational. In essence, we are here studying the particular type of thermal nonequilibrium corresponding to purely informational nonequilibrium. The term ‘nonuniformity’ can be understood as a shorthand for ‘informational nonequilibrium’.
In addition, many results in athermality theory can be inferred from results in nonuniformity theory. As such, developing the latter can help to answer questions in the former. Another motivation for focussing on the case of trivial Hamiltonians is that in building up one’s understanding of a field, it is always useful to start with the simplest case.
In this article, we focus on determining necessary and sufficient conditions for it to be possible to convert one state to another under noisy operations—exactly and approximately, single-shot and asymptotically, with and without a catalyst—and on finding measures of nonuniformity.
We demonstrate that the answers to these sorts of questions in the quantum case can be inferred from their answers in the classical resource theory of nonuniformity, which is defined in precise analogy with the quantum theory, as follows. A classical system is represented by a physical state space, or equivalently by a random variable, the valuations of which correspond to the physical states (i.e. the points in that space). The states of such a system that are of interest in the resource theory (and which are the analogues of the quantum states) are statistical states, that is, probability distributions over the physical state space. The free operations are:
one can implement any permutation on a system’s physical state space,
one can prepare any system in the uniform statistical state (i.e. the uniform probability distribution),
one can marginalize over any subsystem of a system.
The fact that state conversion problems in the quantum resource theory of nonuniformity reduce to their classical counterparts shows that there is nothing inherently quantum in any problem of state conversion in this theory.
This article seeks to present what is currently known about the resource theory of nonuniformity in a systematic and pedagogical fashion. In particular, we characterize the equivalence classes of states under noisy operations by Lorenz curves, thereby providing a geometric perspective on the problem of defining measures of nonuniformity and state conversion problems, a perspective that makes the proofs of certain results more straightforward and intuitive.
We draw on various sources.
The resource-theoretic approach to thermodynamics has seen a great deal of activity in the past few years, primarily by researchers in the field of quantum information Janzing et al. (2000); Brandão et al. (2013); Horodecki and Oppenheim (2013); Brandão et al. (2015); Faist et al. (2012); Dahlsten et al. (2009); Egloff et al. (2012); Linden et al. (2010); Skrzypczyk et al. (2013, 2014); Del Rio et al. (2011); Åberg (2013); Allahverdyan et al. (2004). Many of the results we present are rederivations of results from these works, or special cases thereof. In this sense, our article provides a review of some of this literature.
Insofar as such work (and this article) does not confine its attention to macroscopic thermodynamic variables, but rather focusses on the quantum state of the microscopic degrees of freedom, it is perhaps best described as a resource-theoretic approach to statistical mechanics. Because it also does not confine its attention to the family of states that are thermal for some temperature, it concerns nonequilibrium statistical mechanics.444It is important to distinguish between two notions of nonequilibrium here. In the resource theory of athermality, there is always a background temperature defining which states are free, and any state that is not thermal at temperature is considered a resource. In this context, if a state is thermal for a temperature differing from , it is natural to describe it as being out of equilbrium. Conventionally, however, we use the term equilibrium statistical mechanics to refer to situations wherein every system is in a thermal state, but where different systems may be at different temperatures. In this context, nonequilibrium means not thermal relative to any temperature. Finally, because it seeks to understand not just what occurs on average but what happens in a single shot, in particular, in the worst case, it is an application to statistical mechanics of the ideas of single-shot information theory.555And insofar as the thermodynamic limit typically involves such averaging procedures, it is concerned with thermodynamic questions outside of this limit.
Because of the reduction of quantum questions about state conversion to their classical counterparts, there are a number of works on classical statistical mechanics that can be brought to bear on our problem. In particular, our analysis is informed by the work of Ernst Ruch and his collaborators, who adopted a kind of resource-theoretic perspective on state conversion problems already in the 1970s Ruch (1975a); Ruch and Lesche (1978); Ruch (1975b); Ruch et al. (1978); Ruch and Mead (1976).
We also make heavy use of the mathematical literature on majorization theory Hardy et al. (1952); Bhatia (1997); Horn and Johnson (2012); Joe (1990), in particular, the canonical text on the subject by Marshall, Olkin and Arnold Marshall et al. (2010), which we shall refer to throughout the article as simply MOA.
Finally, in various places (in particular in our discussion of measures of nonuniformity) we make use of work concerning the zoo of generalized entropies Gorban et al. (2010); van Erven and Harremoës (2010); Renner (2008); Renner and Wolf (2004); Datta (2009).
We now review the structure of the article.
Section II covers preliminary material. In particular, we clarify the definition of noisy quantum operations, nonuniformity monotones and conversion witnesses, and we explain why state conversion problems in the quantum resource theory of nonuniformity can be reduced to their classical counterparts.
In Section III, we characterize the quasi-order of states under noisy operations. We begin by determining the equivalence classes of states under noisy operations. We show that these can be associated with a mathematical object known as the Lorenz curve of the state. We then present the necessary and sufficient conditions for one state to be deterministically converted to another by noisy operations in terms of the Lorenz curves of the states.
In Section IV, we discuss various techniques for constructing nonuniformity monotones. We show that various generalized entropies and relative entropies lead to useful monotones, some of which have operational interpretations. We also show how certain monotones can be inferred from the geometry of the Lorenz curve.
We consider various kinds of exact state conversion in Section V. We begin by defining a standard form of the resource of nonuniformity in terms of a one-parameter family of states which we term the sharp states. In terms of these sharp states, we can determine the amount of nonuniformity one requires to form a given state, as well as the amount of nonuniformity that one can distill from that state. These are shown to be related to nonuniformity monotones based on the Rényi entropies of order 0 and order . We also consider the amount of nonuniformity that must be paid to achieve a given state conversion or which can be recovered in addition to achieving the state conversion. We apply these results to determine some simple sufficient conditions for state conversion. Next, we consider exact state conversion in the presence of a catalyst, that is, a nonuniform state which can be used in the conversion process but which must be returned intact at the end of the protocol. In particular, we note the presence of nontrivial catalysis in our resource theory, but also the uselessness of sharp states as catalysts. We review the necessary and sufficient conditions for a state conversion to be possible by catalysis. Finally, we discuss the inadequacy of the standard formulation of the second law of thermodynamics (nondecrease of entropy) as a criterion for deciding whether a state conversion is possible, with or without a catalyst, and we discuss the criteria that actually do the job.
In Section VI, we shift our attention to approximate state conversion. We begin by formalizing the notion of approximate state conversion in terms of a state being mapped to one that is -close to another relative to some contractive metric on the state space, and discuss subtleties of the quantum to classical reduction. We then review the notion of smoothed entropies. We reconsider the problem of determining the nonuniformity of formation and the distillable nonuniformity when the state conversion is allowed to be approximate, and we show that the answers are provided by the smoothed versions of the entropies that characterize exact conversion. We then use these results to provide a simple proof of the rate of asymptotic conversion between states, that is, the rate at which one state can be converted approximately to another in the limit of arbitrarily many copies. The asymptotic results are in turn applied to determine the nonuniformity cost and yield of approximate state conversion when one has arbitrarily many copies to convert. Finally, we consider various notions of approximate state conversion in the presence of a catalyst.
In Section VII, we discuss a few open problems and highlight some of the overarching conclusions of the article.
This article provides a review and synthesis of many known results, but along the way we also present a number of novel results, which we now summarize.
We demonstrate that all exact state conversion problems in the quantum resource theory of nonuniformity reduce to the corresponding problems in the classical resource theory, and that all approximate state conversion problems can also be reduced to their classical counterparts as long as one makes a judicious choice of the metric over states by which one judges the degree of approximation. We generalize many known results on state conversion and on majorization to the case where the input system and the output system are of different dimensions. In particular, we describe a general scheme for building nonuniformity monotones from Schur-convex functions in this case. We introduce the notion of a state conversion witness: a function of a pair of states in terms of which one can specify a necessary condition for the possibility of conversion from one state to the other, or a necessary condition for the impossibility of such a conversion, or both. We demonstrate how such witnesses are the appropriate tool for summarizing some known results, and are as such more versatile than monotones. We introduce a class of states, which we call the sharp states, that serve as a gold standard form of nonuniformity and that naturally generalize the notion of ‘ pure bits’ to the case that is not an integer. We use sharp states to quantify the nonuniformity of formation (or of distilation) of an arbitrary state and the nonuniformity cost (or yield) of an arbitrary state conversion. In the case of approximate conversion, we prove exact expressions for the nonuniformity of formation and of distillation for a class of judiciously-chosen metrics over the states. For exact conversion using a catalyst, we note the uselessness of sharp states as catalysts and its implication that having access to an ideal measurement cannot catalyze any state conversion. Finally, we present a new and simplified proof of the rate of asymptotic conversion between nonuniform states.
ii.1 Free operations, monotones, and witnesses
We begin by formalizing the definition of a noisy quantum operation. A quantum operation is a completely positive trace-preserving linear map , where is the set of linear bounded operators acting on . In this paper, all Hilbert spaces are assumed to be finite-dimensional.
Definition 1 (Noisy quantum operations)
A noisy quantum operation is one that admits of the following decomposition.666For a fixed pair of Hilbert spaces , the set of maps from to of the form (1) is not in general topologically closed Shor (2010). Thus, strictly speaking, we define noisy quantum operations as those linear maps that can be arbitrarily well approximated in operator norm by maps of the form (1). This reflects the intuition that there is no physical difference between exact and arbitrarily accurate implementation of any map. Consequently, the set of noisy quantum operations becomes topologically closed. This allows us, in the following, to prove noisiness of quantum operations by showing how they can be approximated by maps of the form (1). Our noisy quantum operations are what Shor (2010) calls strongly factorizable maps. There must exist a finite-dimensional ancilla space and a unitary on such that, for all input states ,
where is the space complementary to in the total Hilbert space, that is, , and where .
Although at first glance, “degree of purity” might appear to be a good name for what we are calling “nonuniformity”, at second glance one recognizes that it is not because standard usage of the term “purity” takes a pure state of a 2-level system and a pure state of a 3-level to have equal degrees of purity, while the latter is a stronger resource than the former in the resource theory we are considering. The fact that these two pure states are not equivalent under noisy operations is counterintuitive for many quantum information theorists because it contrasts with the situation in entanglement theory, the resource theory with which they are most familiar. The difference arises because the free operations in entanglement theory (local operations and classical communication) allow one to prepare pure product states for free, thereby allowing a given state to be embedded into an arbitrarily large space, whereas noisy operations do not allow one to prepare pure states for free. More generally, two states with identical structure on their support have different amounts of resourcefulness depending on the dimension of the space in which the state is embedded. 777This inequivalence can also be made slightly more intuitive by noting that it holds also in the resource theory of athermality and accounts for the fact that a Szilard engine at temperature extracts different amounts of work from the two states: if it operates on a 2-level system that is prepared in a pure initial state, it can extract an expected work of , whereas if it operates on a 3-level system that is prepared in a pure initial state, it can extract . The resourcefulness of a state under noisy operations is therefore not determined by its proximity to a pure state, but rather its distance from the uniform state, and therefore the term “nonuniformity” is more appropriate than the term “purity” as a description of the resource.
An important question in any resource theory is how to quantify the resource. A resource monotone is a function from states to the real numbers such that if by the free operations, then . For the set of operations given by noisy operations, we refer to the resource monotones as nonuniformity monotones:
Definition 2 (Quantum nonuniformity monotone)
A function , mapping density operators to real numbers, is a nonuniformity monotone if for any two states and (possibly of different dimensions), by noisy operations implies that .
Any measure of nonuniformity must at least be a nonuniformity monotone. When quantifying the relative resourcefulness of different states, there is a strong compulsion to think that the researcher’s task is to find the “one true measure to rule them all”. This tendency stems from an implicit assumption that any property worth quantifying must necessarily be totally ordered. In fact, it is more common to find that resource states only form a quasi-order, and no individual scalar measure can capture a quasi-order. So the search for the one true measure is in vain. The “total-order fallacy” must be resisted.
Nonetheless, resource monotones still have an important role to play. A set of such monotones can capture the quasi-order, in which case they are called a complete set of monotones. Also, if we define an operational task that requires the resource, it is then a well-defined question which resource monotone accurately quantifies the degree of success achievable in the task (according to some figure of merit) for a given state. The states become totally ordered relative to the task.
In this article, we describe some methods for generating a large number of nontrivial nonuniformity monotones, and we describe operational interpretations for some of these. We also identify some complete sets of nonuniformity monotones.
Besides quantifying nonuniformity, we will be primarily interested in problems of state conversion. A useful tool for characterizing such problems is the concept of a state conversion witness.
Definition 3 (Quantum state conversion witness)
Let be a real-valued function on pairs of quantum states, and (possibly of different dimensions). is said to be a go witness if implies that under the free quantum operations. is said to be a no-go witness if implies that under the free quantum operations. Finally, is said to be a complete witness if it is both a go witness and a no-go witness.
No-go witnesses were previously introduced under the name of relative monotones Sanders and Gour (2009).
Any resource monotone defines a no-go witness by because implies , which implies that . It follows that also defines a resource witness – that is, a function such that implies that is a resource – by , where is any free state. A complete set of monotones defines a complete witness by .
ii.2 Reducing quantum state conversion problems to their classical counterparts
To define the classical resource theory of nonuniformity, we must characterize the set of noisy classical operations. Just as we have confined our attention to finite-dimensional Hilbert spaces in the quantum case, we confine our attention to discrete variables in the classical case (that is, finite information-carrying capacity in both cases). Let be the discrete physical state space of the input system and be the discrete physical state space of the output system. Let be the simplex of probability distributions over , and let be the smallest linear vector space in which can be embedded. Clearly, every probability distribution on a sample space can be represented by a vector in . We call each such probability distribution a state. When we say that states are normalized, we refer to the fact that their components sum to total probability one.
The classical analogue of a quantum operation that is completely positive and trace-preserving is a classical operation which preserves positivity and normalization, hence taking probability distributions to probability distributions. For finite dimension, this is represented by a stochastic matrix, that is, a matrix whose entries are real, are nonnegative, and satisfy .
We can now provide a definition of the free operations.
Definition 4 (Noisy classical operations)
A noisy classical operation is a positivity-preserving and normalization-preserving map that admits of the following decomposition.888Analogously to the definition of noisy quantum operations, the set of classical operations of the form (2) between fixed vector spaces and is not topologically closed. This follows from the simple observation that input vectors with all rational entries are mapped to output vectors with all rational entries. Thus, as in the quantum case, we define noisy classical operations from to as those linear maps that can be arbitrarily well approximated in operator norm by maps of the form (2). There must exist an ancilla system with a discrete physical state space and a permutation on with an induced representation on such that, for all input states ,
where is the normalized uniform distribution on and is the physical state space complementary to , that is, .
We note an important difference between the structure of the set of noisy quantum operations and the set of noisy classical operations.
Recall that a unital operation is one for which , where and are the identity operators on and , respectively Mendl and Wolf (2009).
Noisy quantum operations form a strict subset of the unital operations, and, in the case of equal dimension of input and output space, a strict superset of the mixtures of unitaries.
The proof is described in Appendix A.
To compare with the classical case, we must specify the classical analogues of each of the classes of operations appearing in Lemma 5.
The classical analogue of a unital quantum operation is a stochastic matrix that takes the uniform distribution on the input system to the uniform distribution on the output system, hence where () is the dimension of the input (output) vector space. We will call such a stochastic matrix uniform-preserving. Note that, if the input and output spaces are of equal dimension, a uniform-preserving stochastic matrix satisfies , in which case it is said to be doubly-stochastic. Finally, the classical analogue of a mixture of unitaries is a mixture of permutations of the physical state space.
The structure of the set of noisy classical operations is much more straightforward than the quantum one.
The set of noisy classical operations coincides with the set of uniform-preserving stochastic matrices and, in the case of equal dimension of input and output spaces (where the uniform-preserving stochastic matrices are the doubly-stochastic matrices), it coincides with the set of mixtures of permutations.
Here also, we relegate the proof to Appendix A. Note that the fact that every doubly-stochastic matrix can be written as a convex combination of permutations is known as Birkhoff’s theorem Birkhoff (1946).
Because all unitaries are free operations in the quantum resource theory of nonuniformity, the only feature of a quantum state that is relevant for its nonuniformity properties is the vector of its eigenvalues (where an eigenvalue appears multiple times in the vector if it is degenerate and we explicitly include any zero eigenvalues). For any state , we denote the vector of eigenvalues, listed in non-increasing order, by . Therefore, the condition under which we can deterministically convert to by noisy operations must be expressible in terms of and .
We can now state the reduction of the quantum state conversion problem to the corresponding classical problem, leaving the proof to Appendix A.
There exists a noisy quantum operation that achieves the quantum state conversion if and only if there is a noisy classical operation that achieves the classical state conversion .
Lemma 7 implies that all questions about exact state interconversion in the quantum resource theory of nonuniformity are answered by the classical theory, so studying the latter is sufficient.999Note, however, that one does find a separation between the quantum and the classical theories if one assumes an additional restriction on the free operations, namely, that the systems to which one has access are correlated with others to which one has no access. The reason is that a mixture of entangled states cannot be transformed by a local unitary to a mixture of product states. Later on (in Lemma 55), we will prove an analogous statement for approximate conversion.
In particular, Lemma 7 implies a reduction of quantum nonuniformity monotones and quantum state conversion witnesses to their classical counterparts, which are defined analogously to the quantum notions, Definitions 2 and 3, but with quantum states replaced by probability distributions, and noisy quantum operations replaced by noisy classical operations.
Consider , a real-valued function over probability distributions, and , a real-valued function over quantum states, such that , where is the vector of eigenvalues of . Then is a quantum nonuniformity monotone if and only if is a classical nonuniformity monotone.
Consider , a real-valued function over pairs of probability distributions, and , a real-valued function over pairs of quantum states, such that . Then is a witness for state conversion under noisy quantum operations if and only if is a witness for state conversion under classsical noisy operations.
It may seem surprising that the quantum problem of state conversion reduces to its classical counterpart even though the strict inclusions of the mixtures of unitaries within the noisy operations, and the noisy operations within the unital operations (Lemma 5), have no classical counterparts (Lemma 6). The solution to this puzzle is that, for the purposes of state conversion, the three classes of quantum operations have the same power.
Let and . Then, the following propositions are equivalent:
by a noisy operation
by a unital operation.
If and are of equal dimension, then (i) and (ii) are also equivalent to
by a mixture of unitaries.
See Appendix A for the proof. For the case of and of equal dimension, Uhlmann’s Theorem Uhlmann (1970) implies that by a mixture of unitaries if and only if the spectrum of majorizes that of ; see Nielsen and Vidal (2001) for a discussion. 101010The implication from an operation being unital to majorization of the final state’s spectrum by the initial state’s has also been noted in Chefles (2002).
There are questions in the resource theory that do not concern state conversion. For instance, one may ask about the possibility of simulating an operation that is outside the free set, given access to some resource state. For such questions, the quantum solution does not necessarily reduce to the classical one. We do not consider such problems in this article. From this point onwards, therefore, we can restrict our attention to the classical resource theory.
Iii Quasi-order of states under noisy operations
iii.1 Equivalence classes under noisy operations
The first step in understanding state conversion within the classical resource theory of nonuniformity is to determine when two states are equivalent relative to noisy operations, by which it is meant that they can be reversibly interconverted, one to the other, deterministically, by noisy operations. In this case, we say that the states have precisely the same nonuniformity properties.
Definition 11 (Exact state conversion)
We write if there exists a noisy classical operation such that .
Definition 12 (Noisy-equivalence of states)
Two states, and , are said to be noisy-equivalent if they are reversibly interconvertible, that is, if and .
Since noisy classical operations can have an input of dimension and an output of dimension , state conversion is a map from a vector on to a vector on . As such, it is useful to introduce the set
Any finite-dimensional probability distribution is a vector in for some , and therefore a member of the set . For a distribution , we will denote by the integer for which .
The simplest case to consider is when and are of equal dimension. In this case, if and are reversibly interconvertible by noisy operations, there is a permutation that takes one to the other. The proof is as follows. Suppose the doubly stochastic matrix taking to is denoted and the one taking to is denoted . We can write (respectively ) as a mixture of permutations (respectively ):
where and all . Thus
Consider the convex set consisting of all convex combinations of permutations of . The state itself is an extremal point of this convex set. Thus, for all , we must have , and so is the same state (call it ) for all . But , so is a permutation of .
If we let denote the vector having the same components as but permuted such that they are in descending order, the condition for noisy-equivalence can be expressed as .
Note that if and are of equal dimension and noisy-equivalent, and if is the marginal of a correlated state on a larger system, and similarly for , then because we can get from to by a permutation, the conversion can be achieved while preserving all correlations with other systems.
The more interesting case is where and have unequal dimension: First note that one can reversibly interconvert and for any uniform state on an arbitrary ancilla. We get from to simply by injecting an ancilla in the uniform state, which is allowed under noisy operations, and we get from to simply by discarding the ancilla, which is also allowed under noisy operations.
It follows that one can reversibly interconvert and by noisy operations if and only if one can reversibly interconvert and , where and are uniform states for any arbitrary pair of ancillas. In particular, this is true if and only if one can reversibly interconvert and for ancillas having dimensions and such that , so that and are of equal dimension. As shown above, in this case, the interconversion is possible if and only if there is a permutation that takes to . Given that uniform states are invariant under permutations, this condition is equivalent to
This condition, therefore, is necessary and sufficient for reversible interconvertibility of and under noisy operations.
In the resource theory of nonuniformity, the only properties of a state that are relevant to determining its value as a resource are its nonuniformity properties, that is, the features of the state that determine its noisy-equivalence class. It is therefore useful to replace the state by a mathematical object that represents only the nonuniformity properties of .
Given that is reversibly interconvertible with , this mathematical object must be invariant under adding and removing ancillas in the uniform state. If we plot histograms of and , we see that the envelope of each is a step function and that the steps’ relative heights are equal. An example is given in Fig. 1. We can make these step functions strictly equal by rescaling them in an appropriate way. The rescaling is chosen such that the area of each bar of the histogram remains the same, while the range of the function becomes . Specifically, we define the function , with range , such that the th step extends over the range and has height . Equivalently,
where denotes the integer floor of . Again, an example is provided in Fig. 1. We call the uniform-rescaled histogram of .
Clearly, adding or removing a uniform state (of arbitrary dimension) leaves the uniform-rescaled histogram invariant, , as illustrated in Fig. 1. Furthermore, if two states of the same dimension are equal up to a permutation, their uniform-rescaled histograms are equal. In particular, it follows that if and only if . But so we can conclude that holds if and only if . We have shown, therefore, that the uniform-rescaled histogram of a state is a mathematical object that characterizes the noisy-equivalence class of that state.
Now consider the integral of from to as a function of ,
Clearly, the curve traced by extends between (0,0) and (1,1) regardless of the state . It contains all the information contained in the rescaled histogram . Indeed, one can recover the latter by taking the derivative of . As such, it is another mathematical object that characterizes the noisy-equivalence class. Examples of this curve for various states are provided in Fig. 1. For the uniform state , this curve is simply the diagonal line extending between (0,0) and (1,1).
There is another way of defining this curve which is worth noting. First, define for as the sum of the largest components of ,
and define (Note that for a normalized probability distribution). is sometimes called the Ky Fan -norm of Horn and Johnson (2012); Bhatia (1997). Then we can characterize as the linear interpolation of the points
is called the Lorenz curve of Lorenz (1905), (MOA, Sec. 1.A).111111Actually, the Lorenz curve of is conventionally taken to be the linear interpolation of , where is the sum of the smallest components of Lorenz (1905), or equivalently, it is taken to be the integral . But the conventional definition is just the inversion about the line extending from (0,0) to (1,1) of our definition, and so the two curves have precisely the same information content. We here choose to adopt the opposite of the usual convention because in this way the majorization relation between states, , coincides with the inequality relating the height of the Lorenz curves of those states, (see Eq. (9) below). Also, our convention coincides with the one adopted in Horodecki and Oppenheim (2013). It will be seen to be one of the primary tools for characterizing the resource theory of nonuniformity.
It is worth emphasizing that we can also infer that noisy equivalence implies equality of Lorenz curves directly from the condition that . It suffices to note that, via the definition of Lorenz curves in terms of Ky-Fan -norms, this condition is equivalent to and then to note that the Lorenz curve is invariant under adding and removing ancillas in a uniform state, .
To summarize, we have shown that
Proposition 13 (Conditions for noisy equivalence)
A pair of states and are noisy-equivalent if and only if the following equivalent conditions holds
their uniform-rescaled histograms are equal,
their Lorenz curves are equal,
The uniform-rescaled histogram of and the Lorenz curve of both capture all and only the nonuniformity properties of . It follows that for any notion of state conversion we wish to study (exact or approximate; single-shot, multi-copy or asymptotic; catalytic or noncatalytic), the necessary and sufficient conditions under which one state can be converted to another can always be expressed in terms of either of these objects. Furthermore, any nonuniformity monotone or state conversion witness must be expressible entirely in terms of them as well.
The application of these mathematical objects to thermodynamics was first recognized in Ruch (1975a); Ruch and Lesche (1978); Ruch (1975b); Ruch et al. (1978); Ruch and Mead (1976). What we have called the “uniform-rescaled histogram” was discussed in Ruch et al. (1978) under the title of the “density diagram”. More recently, in Egloff et al. (2012), these old tools have begun to be used again in the context of an information-theoretic approach to thermodynamics. In this article, an operation of “Gibbs-rescaling” is introduced which is akin to our use of the uniform-rescaled histogram in place of the distribution itself (although with a different scaling convention for the horizontal axis). The analogue of the Lorenz curve for an athermal state, that is, the generalization of Lorenz curves to a system with a nontrivial Hamiltonian, has recently been studied in Horodecki and Oppenheim (2013) and applied in Brandão et al. (2015).
With this characterization of the noisy equivalence class in hand, we can clarify a point that was made in the introduction, namely, that the nonuniformity properties of a state depend on the dimension of the space in which the state is embedded. As already noted, this is because embedding a state in a higher-dimensional space—that is, padding the state with extra zeros—is not a noisy operation. In terms of the uniform-rescaled histogram, padding a state with extra zeros corresponds to squeezing the entire histogram of the state to the left and leaving only zeros on the right side, which obviously results in a different histogram. In the Lorenz curve picture, it corresponds to squeezing the Lorenz curve to the left and adding a horizontal segment at value 1 on the right end, again resulting in something that is obviously distinct from the original Lorenz curve.
iii.2 Deterministic interconversion of nonuniform states
We now turn to a consideration of the necessary and sufficient conditions on a pair of states and such that there exists a deterministic noisy operation taking to . We are here asking about one-way state conversion, i.e., there need not be any deterministic noisy operation taking to . We begin by presenting the general result.
iii.2.1 The result
(Conditions for deterministic conversion) if and only if
the uniform-rescaled histogram of , , can be taken to that of , , by moving probability density only towards the right (i.e. from lower to higher values of ).
the Lorenz curve of is everywhere greater than or equal to the Lorenz curve of :
When these conditions hold, we say that noisy-majorizes .
The rest of this section provides the proof of Proposition 14. We start with the case in which and have equal dimension.
Recall the definition of the majorization relation (Definition A.1 of MOA).
Definition 15 (Majorization)
Letting and be normalized probability vectors with equal dimension , we say that majorizes and write if
Because and are normalized probability distributions, .
The connection with noisy classical operations is made through the following famous result Hardy et al. (1952):
Lemma 16 (Hardy, Littlewood, Polya)
if and only if there is a doubly stochastic matrix such that .
Given that the set of noisy classical operations with equal input and output dimensions are represented by the set of doubly-stochastic matrices (Lemma 6), we immediately obtain that, for and of equal dimensions, if and only if .
An equivalent means of expressing the condition of majorization is in terms of Lorenz curves. Noting that the expressions in the inequalities of Eq. (10) are just the Ky Fan norms and recalling that the Lorenz curve is the linear interpolation of points , we see that in the case of and of equal dimension, we have that if and only if the Lorenz curve of is nowhere less than the Lorenz curve of :
This proves (ii) of Proposition 14 for the case of states of equal dimension.
When and are states of unequal dimension, the condition for deterministic conversion is not simply majorization; this is why we use the term noisy-majorization to describe the condition in the general case. It is determined using the same trick that was deployed in the characterization of the noisy equivalence classes. It suffices to note that if and only if there exist uniform states and such that (because adding and removing uniform states are noisy operations), and that if we choose and such that , then and are states of equal dimension. If we define as the least common multiple of and , , then it suffices to choose and , in which case and have dimension .
Recalling that we have established Condition (ii) of Proposition 14 for states of equal dimension, and the fact that the Lorenz curve of for a uniform state is equal to the Lorenz curve of , it follows that if and only if the Lorenz curve of is nowhere below the Lorenz curve of . This proves Condition (ii) of Proposition 14 for all states.
Finally, recalling that the Lorenz curve is the cumulative integral of the uniform-rescaled histogram, any motion of density rightward in the uniform-rescaled histogram corresponds to a decrease of the height of the Lorenz curve over some subset of its domain, while motion of density leftward corresponds to an increase of height. Condition (ii) of Proposition 14, therefore, implies (i).
This concludes the proof of Proposition 14.
The application of majorization theory to state conversion in thermodynamics was studied extensively in Ruch (1975a); Ruch and Lesche (1978); Ruch (1975b); Ruch et al. (1978); Ruch and Mead (1976). The quantum information community became familiar with majorization due to its role in the resource theory of entanglement Nielsen (1999); Nielsen and Vidal (2001). The problem of state conversion in thermodynamics was first considered from a quantum information perspective in Janzing et al. (2000), where some necessary conditions on state conversion were derived. The necessary and sufficient conditions for state conversion under thermal operations were first determined in Horodecki and Oppenheim (2013), where the relation was called thermo-majorization. The results described in this section are the specialization of the thermo-majorization relation to the case of a trivial Hamiltonian.
iii.2.2 Some consequences
The order over states induced by deterministic conversion is not a total order but a quasi-order. We call it the noisy quasi-order. One easily generates pairs of states that are not noisy-ordered relative to one another by simply drawing a pair of valid Lorenz curves where one is not everywhere above the other.121212The order is a quasi-order (also known as a pre-order) rather than a partial order, because we can have and for . This occurs whenever is a nontrivial permutation of or requires addition or removal of a uniform state. While the states form a quasi-order, the noisy-equivalence classes of states form a partial order.
Note that one can easily recover the condition for noisy-equivalence (Proposition 13) from the condition for noisy-majorization (Proposition 14) by recognizing that reversible interconvertibility requires noisy-majorization in both directions.
Another simple corollary of the deterministic-conversion result concerns the relation between a state of a composite and its marginal state. Suppose is a state of a composite system and is the marginal state on , that is, , where . is noisy-majorized by , that is, for all .
For noisy-equivalence of and , we require that which implies that . It follows that marginalization is reversible only if the marginalized system is uncorrelated with the rest and is in a uniform state.
Proposition 14 also implies that the height of the Lorenz curve at a given value of in the region is a nonuniformity monotone, , and the set of such heights, , form a complete set of nonuniformity monotones. Although this is an infinite set, if and are both of finite dimension (the only case we consider in this article), one can decide the convertability question by looking at a finite number of monotones. The following is the pertinent result.
For and of finite dimension, noisy-majorizes if and only if at the points for all . In other words, it suffices to consider the monotones
The proof is simply that is linear between the distinguished values of . Given that is concave, if at these points, then at all .
This result can be rephrased in terms of state-conversion witnesses as follows.
Each of the functions
for is a no-go witness for . That is, if then it is not the case that .
is a complete witness for the state conversion. That is, if and only if .
In Section V.5, we will discuss how this result implies the inadequacy of the standard formulation of the second law of thermodynamics.
In the previous section, we determined the conditions under which it is possible to transform to by a noisy operation, but the proof was not constructive. In this section, we describe a practical implementation of the appropriate noisy operation. We begin with states of equal dimensions.
First of all, we recall the definition of majorization in terms of a sequence of T-transforms. A T-transform is a doubly-stochastic matrix that is nontrivial on a single block. For the block corresponding to levels and , we denote the T-transform by . The only permutations that act only on levels and are the identity, denoted , and the permutation that swaps and , which we denote by . Therefore, by Birkhoff’s Theorem Birkhoff (1946), the most general form of is
where . It follows that if and are -dimensional vectors and , then
Equivalently, defining and , we have
We see that if (), then the weights of the pair of levels and become equal, and if (), then these weights become closer to equal (assuming they were unequal to begin with). In the context of income inequality, a T-transform is called a “Robin Hood transfer” Arnold (1987).
Lemma 19 (Muirhead, Hardy, Littlewood, Polya)
For states of equal dimensions, if and only if there is a finite sequence of T-transforms taking to . The number of steps required is at most , where is the dimension.
Proof. We follow the proof in (MOA, Lemma B.1, p. 32). Let denote the state after the th step of the sequence of T-transforms, so that and . Consider the step that takes to . Let denote the largest index such that the weight for that index is strictly larger for than it is for , that is, (“ex” denotes “excess”). Let denote the smallest index such that the weight for that index is strictly smaller for than it is for , that is, (“df” denotes “deficient”). By definition, we must have . This is illustrated in Fig. 3.
We consider a protocol where, in the step in the sequence, one transfers the maximum weight possible from level to level while ensuring that one still has for and for (so that, in particular, still majorizes ). This maximum weight, denoted , cannot exceed the difference of weights in level nor the difference of weights in level , so
The resulting transformation is
Because the transfer tends to make the weights of the two levels closer to equal, it is clearly a T-transform. Specifically, it is the transform , where . After the step of the sequence, either the index is reduced by 1 or the index is increased by 1 (or both), so that at the next step, one is transferring weight between a different pair of levels. Clearly, if majorizes , then a sequence of such transformations can take to in a finite number of steps.
Finally, we show that at most steps are required. Suppose denotes the number of levels wherein and differ in value. Because the number of differences is reduced by 1 at every step of the sequence, and the last step takes two differences to no differences, it follows that one requires steps. , however, is at most .
This covers the case of states of equal dimension. Next, we consider how to implement a noisy operation that achieves deterministic conversion of states of unequal dimensions. It follows from Lemma 19 that we can achieve the transformation by a sequence of T-transforms, with the number of steps in the sequence being at most where . Fig. 3 depicts the set of T-transforms that maps to (where and by the protocol described in the proof of Lemma 19. Note that we must divide the -axis into bins of size to depict how the sequence of T-transforms acts on the uniform-rescaled histograms. Note also that this figure provides an intuitive proof of Proposition 14 (i). Finally, note how the T-transforms act on the Lorenz curves. Each T-transform acts on the pair of bins for which the difference between the slope differential of the Lorenz curves in the first bin and the slope differential of the Lorenz curves in the second bin is largest. After a particular T-transform is complete, the slope differential in one of the two bins becomes zero, and consequently, the Lorenz curves have the same slope in that bin. By this process, the Lorenz curve of the initial state approaches that of the target state.
Iv Nonuniformity monotones
In this section, we discuss the properties and method of construction of functions which serve as monotones under classical noisy operations. A reader interested in getting a quick introduction to nonuniformity monotones (without too much technical detail) will find the tables in this section useful: Table 1 lists some monotones derived from convex functions of one real variable, outlining the steps of the derivation; Table 2 lists some monotones which can be derived from the geometry of the Lorenz curve; and Table 3 lists the operational significance of some of the monotones.
Given that noisy operations may change the dimension of the space, every nonuniformity monotone is really a family of functions, , where is defined on a -dimensional space. We will denote by the function which reduces to on . Thus, ,
Where the dimensionality is clear from the context, we will omit the subscript in .
A monotone is said to be strict if only on level sets, that is, only when and are either noisy-equivalent (i.e., and ) or not ordered relative to one another (i.e., it is neither the case that nor that ).
Since appending ancillary systems in the uniform state is a reversible process under noisy operations, every nonuniformity monotone must satisfy , where is the uniform state of dimension .
Within a space of fixed dimension, the noisy quasi-order reduces to the majorization quasi-order, so any nonuniformity monotone , when restricted to states of a fixed dimension, must be a nonincreasing monotone with respect to the majorizaton quasi-order. The latter are known as Schur-convex functions and have been extensively studied. See, e.g., Chapter 3 of MOA.
Definition 20 (Schur-convexity)
A function mapping -dimensional probability distributions to the reals is Schur-convex iff for all -dimensional distributions and ,
As it turns out, for a function , the conditions that the restriction of to every -dimensional vector space be Schur-convex and that be invariant under the addition of a uniform state are necessary and sufficient for to be a nonuniformity monotone. We formalize this in the following.
A function is a nonuniformity monotone if and only if both of the following conditions hold:
For each , the restriction of to is Schur-convex.
For all , for all distributions ,
where is the -dimensional uniform distribution.
Proof. The forward implication is trivial, so our task is to show that the pair of conditions imply that is a nonuniformity monotone.
We begin by noting that for each , if the restriction of to is Schur-convex, then it follows by definition that if and are states of equal dimension, and majorizes , then . It remains to show that if and are of unequal dimension, and noisy-majorizes , then .
By assumption, , for any . Then, one can reason as follows. First, we can define states and that are of equal dimension, namely . Given that noisy-majorizes , and given that adding uniform states does not change the noisy-equivalence class of a state, it follows that noisy-majorizes . But since noisy-majorization between states of equal dimension is just majorization, it follows that majorizes . Then, by the Schur-convexity of the restriction of to , we conclude that . It follows that .
Note that if one takes an arbitrary family of Schur-convex functions, where , then the function which reduces to on need not be a nonuniformity monotone because it need not satisfy the requirement of invariance under adjoining a uniform state, Eq. (23). So the problem of defining nonuniformity monotones is not solved by merely finding families of Schur-convex functions. Nonetheless, the powerful characterization theorems for Schur-convex functions can be exploited to construct and characterize nonuniformity monotones. We will discuss these theorems in the following.
iv.1 Nonuniformity monotones built from convex functions on the reals
The following result is well-known; cf. (MOA, Proposition C.1).
For every convex function , the function defined by
is Schur-convex for each . That is, if and are of equal dimension , and if , then .
Proof. Recall that a function is convex if, for any pair of points ,
for all . If is twice differentiable, an equivalent definition of convexity is that for all .
Note also that the function is invariant under permutation of the components of . Hence it is sometimes described as a symmetric function.
The proof is then straightforward. Given that , it follows that for some doubly-stochastic matrix . By Birkhoff’s Theorem Birkhoff (1946), there exists a probability distribution and a set of permutations such that . Because is a sum of convex functions, it is also convex, and therefore, . But we noted above that is invariant under a reordering of the components of its argument, so that . It follows that .
Now that we have seen how to generate families of Schur-convex functions, the question arises of which of these families can yield a function that is invariant under adjoining uniform states.
iv.1.2 Schur-convexity relative to a distribution
Towards this end, we introduce the notion of majorization relative to a distribution ; see (MOA, Ch. 14, Sec. B, p. 585).
Within a space of a given dimension , consider those stochastic matrices that preserve a particular distribution . Call these the -preserving stochastic matrices.
Definition 23 (Majorization relative to )
A distribution is said to majorize another distribution relative to the distribution , denoted , if there exists a -preserving stochastic matrix such that .
Definition 24 (Schur-convexity relative to )
We say that a function is Schur-convex relative to if implies .
Intuitively, whereas a Schur-convex function quantifies the distance of some distribution to the uniform distribution , a Schur-convex function relative to quantifies the distance of to .
The following is a useful method of constructing Schur-convex functions relative to a distribution , from convex functions of one real variable.
Given a distribution with all , for every convex function , the function defined by
is Schur-convex relative to . That is, if and are of equal dimension , and if , then .
This is Proposition B.3 in Ch. 3 of MOA, proven in Veinott (1971). Functions of the form of Eq. (24) have also been proposed as a generalization of the notion of relative entropy in Csiszár (1967) (where it is called the -divergence of from ) and in Morimoto (1963), as discussed in Gorban et al. (2010).
Note that if we multiplied each by a factor in Eq. (24), we would still have a Schur-convex function relative to (because the function is also convex). Nonetheless, the case of has a special status. Suppose we define a fiducial distribution for every system and suppose that the fiducial distribution on a composite system is the product of the fiducial distributions on the components, that is, . In this case, we can prove the following.
For every convex function , the function defined by satisfies for any system .
Proof. It suffices to note that
where we have used the fact that .
It follows that any family of functions (one for every type of system), each of which is Schur-convex relative to the fiducial distribution on that system, can be used to construct resource monotones in a classical resource theory where those fiducial distributions are the free states.
In particular, in the context of the resource theory of athermality Brandão et al. (2013, 2015), where the free states are thermal states, we can construct athermality monotones from families of functions that are Schur-convex relative to the thermal state (note that the thermal state of a system depends not only on the dimension of that system, but also on its Hamiltonian).
iv.1.3 Schur-convexity relative to the uniform state
By specializing to the case where the fiducial distribution for a system of dimension is just the uniform distribution of that dimension, Lemma 26 provides a method of constructing a nonuniformity monotone from any convex function.
For every convex function , the function defined by
is a nonuniformity monotone.
Proof. We must show that the two conditions of Proposition 21 hold. It is given that , therefore it follows from Theorem 26 that for any , , and it follows from Theorem 25 that for any , the restriction of to a -dimensional space is Schur-convex relative to , hence Schur-convex.
Note that if a Schur-convex function can be expressed in the form