Microcanonical thermodynamics in general physical theories

Microcanonical thermodynamics studies the operations that can be performed on systems with well-defined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: (i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, (ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and (iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.

The most basic instance of thermodynamics is for systems with definite energy. There, the natural choice of free state is the microcanonical state,i.e. the uniform mixture of all states with the same energy. In situations where the experimenter lacks control over the preparation of the system, it is natural to expect that the system's state will fluctuate randomly from one experiment to the next, so that the overall statistics is described by the microcanonical state. The choice of free operations is less obvious. The three main choices considered in the literature on quantum thermodynamics are:

States, transformations, and measurements
OPTs describe the experiments that can be performed on a given set of systems by a given set of physical processes. The framework is based on a primitive notion of composition, whereby every pair of physical systems A and B can be combined into a composite system, denoted by Ä A B. Physical processes can be connected in sequence or in parallel to build circuits, such as In this example, A, ¢  A , A , B, and ¢ B are systems, ρ is a bipartite state, , ¢ and  are transformations, a and b are effects. Note that inputs are on the left and outputs are on the right.
For generic systems A and B, we denote by • St( ) A the set of states of system A, • Eff( ) A the set of effects on A, • Transf( ) A, B the set of transformations from A to B, and by Transf( ) A the set of transformations from A to A, • (or , for short) the sequential composition of two transformations  and , with the input of  matching the output of , •  A the identity transformation on system A, represented by the plain wire •   Ä the parallel composition (or tensor product) of the transformations  and .
Among the list of valid physical systems, every OPT includes the trivial system I, corresponding to the degrees of freedom ignored by theory. The trivial system acts as a unit for the composition of systems: for every system A, one has Ä = Ä = I A A I A. States (resp. effects) are transformations with the trivial system as input (resp. output). Circuits with no external wires, like the circuit in equation (1), are called scalars. We will often use the notation r ( | ) a to denote the scalar ð2Þ and of the notation  r ( | | ) a to denote the scalar In the OPT framework, the scalar r ( | ) a is identified with a real number in the interval [ ] 0, 1 , interpreted as the probability of a joint occurrence of the state ρ and the effect a in an experiment consisting of a state preparation (containing ρ as one of the possible states), followed by a measurement (containing a as one of the possible effects).
The fact that scalars are real numbers induces a notion of sum for transformations, so that the sets St( ) A ,

Transf(
) A, B , and Eff( ) A become spanning sets of real vector spaces. By dimension of the state space St( ) A , we mean the dimension of the vector space spanned by the states of A.

Tests and channels
In general, a physical process can be non-deterministic, i.e. it can result into a set of alternative transformations, heralded by different outcomes, which can (at least in principle) be accessed by an experimenter. General nondeterministic processes are described by tests: a test from A to B is a collection of transformations , where X is the set of outcomes. If A (resp. B) is the trivial system, the test is called a preparation-test (resp. observation-test). If the set of outcomes X contains a single element, we say that the test is deterministic, because one, and only one transformation can take place. We will denote the sets of deterministic states, transformations, and effects as DetSt( ) A , DetTransf( ) A, B , and DetEff( ) A respectively. We refer to deterministic transformations as channels.
A transformation  from A to B is called reversible if there exists another transformation  -1 from B to A such that    = -1 A and   = -1 B . It is not hard to see that reversible transformations are deterministic, i.e. they are 'channels'. If there exists a reversible transformation transforming A into B, we say that A and B are operationally equivalent, denoted by  A B. Physically, this means that every experiment performed on system A can be (at least in principle) converted into an experiment on system B, and vice versa. The composition of systems is required to be symmetric, meaning that Ä Ä  A B B A. Physically, this means that for every pair of systems A and B there exists a reversible transformation that swaps A with B.

Pure transformations
The notion of pure transformation plays centre stage in our work. Intuitively, pure transformations represent the most fine-grained processes allowed by the theory. To make this intuition precise, we need a few definitions.
The first definition is coarse-graining-the operation of joining two or more outcomes of a test into a single outcome: the test We say that the test Pure transformations are the most refined transformations: We denote the sets of pure transformations, pure states, and pure effects as PurTransf( ) A, B , PurSt( ) A , and PurEff( ) A respectively. As usual, non-pure states are called mixed.

Purification
Another key notion in our paper is the notion of purification [57,59]. Consider a bipartite system Ä A Bin the state r AB . The state of system A alone is obtained by discarding system B-that is, by applying a channel that transforms system B into the trivial system. Discarding operations are represented by deterministic effects, i.e. deterministic transformations with trivial output. In quantum theory, every system has one and only one deterministic effect, corresponding to the partial trace on the Hilbert space of the system.
Given a deterministic effect or, in formula, AB . When r AB is pure and equation (5) is satisfied for some deterministic effect e, we say that r AB is a purification of r A and we call B the purifying system [57,59].

Definition 2. A purification
PurSt Y Î Ä ( ) A B is essentially unique [59] if for every pure state for some reversible transformation  .
In a completely general theory, there may be different ways to discard a system, corresponding to different deterministic effects. The deterministic effect is unique in causal theories, that is, theories where no signal can be sent from the future to the past [57].

Finiteness, closure, and convexity
In this paper we will make three standing assumptions. The first assumption is that our OPT describes finite systems, i.e. systems for which the state space is finite-dimensional. Operationally, this means that the state of each system is uniquely determined by the statistics of a finite number of finite-outcome measurements.
Our second assumption is that the space of transformations (and the spaces of states and effects, in particular) is closed under limits. Physically, this expresses the fact that a limit of operational procedures is itself an operational procedure, whereby the target transformation can be implemented with arbitrary accuracy. Mathematically, a sequence of transformations  The third standing assumption made throughout the paper is that the space of transformations Transf( ) A, B is convex for every A and B. Mathematically, this means that one has the implication Physically, this means that the experimenter can perform arbitrary randomised operations. Note that convexity is a natural assumption in every non-deterministic theory: provided that some experiment yields random outcomes, one can always repeat that experiment many times and approximate every probability distribution [57]. Then, the closure assumption guarantees that the limit probability distribution is also achievable within the theory. Thanks to this fact, the experimenter can perform arbitrary randomised tests.

Theories of systems with constraints
The language of general probabilistic theories is largely interpretation-independent. As such, it has the flexibility to model very different physical scenarios, or even to model different fragments of the same physical theory. In this paper we use the framework to model scenarios of microcanonical thermodynamics, where the systems under consideration have a well-defined energy. More generally, the microcanonical approach can be applied to systems with additional constraints-e.g. to systems of particles confined in a given volume, or constrained to have a fixed value of the angular momentum. In these scenarios, the microcanonical state is interpreted as the state of 'minimum information' compatible with the constraints. In this section we outline how the OPT framework can be used to describe physical systems subject to constraints.

Constrained systems in quantum theory
Before delving into general theories, it may help to analyse the example of quantum theory. Let us consider first the case of a quantum system S constrained to a fixed value of the energy. The constraint is implemented by specifying the system's Hamiltonian H S and by restricting the allowed states to (mixtures of) eigenstates of the Hamiltonian for a fixed eigenvalue, say E. The quantum states compatible with the constraint are the density matrices ρ satisfying the condition where P E is the projector on the eigenspace of H S with eigenvalue E. For example, the system S could be an electron in a hydrogen atom, in the absence of external fields. In general, the basis states of the electron are labelled as ñ |n l m m , , , s , where n l m , , , and m s are the principal, orbital, magnetic, and spin quantum number respectively. The electron may be constrained to the lowest energy shell, corresponding to n=1 and = = l m 0. In this case, the allowed states are contained in a twodimensional subspace, spanned by the 'spin-up' and 'spin-down' , Under this restriction, the electron's spin can be regarded as an effective qubit.
Constraints other than energy preservation can be treated in a similar way. We consider constraints of the form where  is a linear map on the state space of the system. This form is suggested by equation (10), where the linear S ,  S being the identity channel on system S. Given a set of constraints  { } i , one can define an effective system, whose states are the density matrices ρ satisfying the conditions  r = ( ) 0 i for every i. The constrained quantum system can be denoted as 1 are the linear maps representing the constraints. The physical transformations of the effective system are the physical transformations of S that send every input state satisfying the constraint into an output state satisfying the constraints. For the energy constraint (10), this means that the transformations of the effective systems should be energy-preserving.
A familiar example of effective system is the polarisation of a single photon. At the fundamental level, the single photon is just an excitation of the electromagnetic field, e.g. corresponding to the wave vector k. One can regard the single photon as an effective system by restricting the attention to the two-dimensional space spanned by the states ñ | H k, , 1 and ñ | V k, , 1 , corresponding to vertical and horizontal polarisation, respectively. In this case, we can see two constraints working together: a constraint on the wave vector and a constraint on the energy of the field. Note that the effective description in terms of single photons is accurate only as long as the dynamics of the field is confined into the 'single-photon subspace with wave vector k'.
So far, we have defined effective systems at the single-system level. An important question is how to define the composition of effective systems. Consider two effective systems 1 , where S A and S B are the original, unconstrained systems. A natural way to define the effective composite system Ä A Bis to select the states of the unconstrained composite system Ä S S A B that satisfy both constraints-i.e. to select the density matrices ρ such that When the effective systems A and B result from an energy constraint, the effective system Ä A Bdescribes a system consisting of two parts, each of which with its own, well-defined energy. In this case, the constraints (12) can be summarised in a single equation, namely where E A and E B are the energies of the two local systems, and P E A and Q E B are the projectors on the corresponding eigenspaces. One might be tempted to define the composite system Ä A Bin a different way, without imposing that each individual part has a definite energy. Indeed, one could imagine that, when the two systems S A and S B are brought into contact, they start exchanging energy, with the only constraint that the total energy has to remain constant. The resulting states would be density matrices that satisfy the (generally) weaker condition r r The reason why we do not make such a choice can be illustrated with a simple example. Suppose that S A and S B are two spatial modes of the electromagnetic field, with wave vectors k A and k B , respectively. Systems A and B could be single photons, i.e. effective systems corresponding to states of the field in the first excited level. Now, if the energy of the two modes is the same, an energy-preserving evolution could transform the initial state States of this kind cannot be interpreted as states of two single photons. Note that, instead, the constraint (12) correctly identifies the correct set of states-including, among others, entangled states such as the Bell state Motivated by this and by similar examples, we reserve the notation Ä A Bfor effective systems defined by the constraint (12). Other effective systems, like the system defined by the constraint (14), can be treated in our framework, but will be regarded as different from the product system Ä A B.

Constrained systems in general theories
The construction outlined in the quantum case can be easily extended to arbitrary physical theories. A constraint for system S can be defined as an element  of the real vector space Transf  ( ) S spanned by the physical transformations in Transf( ) S . The constraint is satisfied by the states ρ such that  1 , we define the composite system Ä A Bto be the effective system This definition is consistent with interpretation of system Ä A Bas a composite system made of two, independently addressable parts A and B. For example, local measurements on one side of a bipartite state of Ä A Binduce states of the correct system (either A or B) on the other side.

Effective theories
Given a theory and a set of constraints composed as in equation (19), one can build a new effective theory, which consists only of effective systems. For example, one can build an effective theory where every system has definite energy, and where every composite systems consist of subsystems with definite energy. For a given system A in such a theory, all the states in St( ) A have-by fiat-the same energy. Likewise, all the transformations in Transf( ) A will be-by fiat-energy-preserving. For every pair of systems A and B, the composite system Ä A B consists of two parts, each of which with its own, well-defined energy. The joint transformations in Transf Ä ( ) A B will be interpreted as operations that preserve the energy of the first part and the energy of the second part.
One benefit of the effective picture is that one does not need to specify the constraints-in principle, every linear constraint can fit into the framework. In this way, we can circumvent the thorny issue of defining the notion of Hamiltonian in general probabilistic theories (see [62]): in the effective description, we can simply regard each effective system as a system with trivial Hamiltonian, which assigns the same energy to all states of the system.

The microcanonical framework
In this section we build a microcanonical framework for general physical theories. We will adopt the effective description, wherein every system is interpreted as the result of a constraint-typically, but not essentially, a constraint on the energy.

The principle of equal a priori probabilities
The starting point of the microcanonical approach is the principle of equal a priori probabilities, stating that one should assign the same probability to all the microstates of the system compatible with a given macrostate. In our language, the 'microstates' are the deterministic pure states, representing those preparations of the system that are both deterministic and maximally fine-grained. The 'macrostate' is specified by a constraint, such as the constraint of fixed energy. The principle of equal a priori probabilities states that the system should be described by a uniform mixture of all deterministic pure states satisfying the constraint. For example, the microcanonical state of a (finite-dimensional) quantum system at energy E is described by the density matrix where S E is the manifold of pure states in the eigenspace of the system's Hamiltonian corresponding to the eigenvalue E, and y ( ) p d E is the uniform probability distribution over S E . In the effective picture, the microcanonical state is nothing but the maximally mixed state where y d is the uniform probability distribution over the pure states of the system. A traditional problem in the foundations of statistical mechanics is to determine the conditions under which the principle of equal a priori probabilities holds. Here we will not delve into this problem, which involves a great deal of detail about the physics of the system and of its dynamics. Instead, we will focus on the general conditions that must be satisfied in order to formulate the principle of equal a priori probabilities in physical theories other than classical and quantum mechanics.
In general probabilistic theories, the key problem is to define what we mean by 'equal a priori probabilities'. In quantum mechanics, there is a canonical choice: the unitarily invariant probability distribution on the pure states of the system. The obvious extension to general theories is to consider the probability distributions that are invariant under all reversible transformations. The problem is, however, that there may be more than one invariant probability distribution. This point is illustrated in the following example: Example 1. Consider a toy theory where the space of the deterministic states of one of the systems is a half-disk in the two-dimensional plane, as in figure 1(a). For this system, the pure states are the states on the half-circle (in green in the figure), and can be parameterised with a polar angle θ between 0 and π. Now, the reversible transformations send deterministic states into deterministic states and, therefore, must be symmetry transformations of the state space. For the half-disk, the only symmetry transformations are the identity transformation and the reflection around the symmetry axis (in black in the figure). Hence, every probability distribution that assigns the same probability distribution to the points θ and p q is guaranteed to be invariant under reversible transformations. This means that the notion of 'equal a priori probabilities' is not uniquely defined. The situation would be different if the state space of the system were a full disk, as illustrated in figure 1(b). In this case, every rotation of the disk could be (at least in principle) a reversible transformation of the system. The invariant probability distribution would be unique, and given by the probability density q = p ( ) p 1 2 . Note that in the above examples we only specified the state space and the transformations of a single system, without giving the full-blown OPT. It is easy to see that such a theory does indeed exist. In general, one can always build a 'minimal OPT' that includes a given system with a given state space, a given set of transformations, and a given set of measurements. The construction was shown in [81, example 2]. In the minimal OPT, the composite systems are made of many copies of the given system, and their allowed states, transformations, and operations are (mixtures of) product states, product transformations, and product operations.
The above example shows that there exist probabilistic theories where the notion of 'equal a priori probabilities' on pure states is not uniquely defined. In order to formulate the principle of equal a priori probabilities, we put forward the following requirement: Requirement 1. For every (finite) system there exists a unique invariant probability distribution on the deterministic pure states.
This requirement is far from trivial. In fact, it is equivalent to an important property, independently considered in the literature on the axiomatisation of quantum theory [54,[82][83][84]: Theorem 1. For every finite system A, the following are equivalent: 1. There exists a unique invariant probability distribution on the deterministic pure states of system A.
2. Every deterministic pure state of system A can be obtained from every other deterministic pure state of the same system through a reversible transformation.
Proof. The idea of the proof is that the set of deterministic pure states of system A can be decomposed into a disjoint union of orbits generated by the group of reversible transformations. More formally, for every two (deterministic) pure states α and a¢, one can define the equivalence relation for some reversible transformation  . In this way, the set of deterministic pure states is partitioned into equivalence classes, known as homogeneous spaces. Moreover, each homogeneous space is a closed set, because the group of reversible transformations is closed [57] and has a finite-dimensional representation on the state space of system A. Now, one can define an invariant probability distribution for every equivalence class. Indeed, it is enough to define the invariant measure on the pure states induced by the invariant measure on the group of reversible transformations. Hence, the condition that there is only one invariant probability distribution implies that there must be only one equivalence class. In other words, every two pure states are connected by a reversible transformation.
Conversely, if there is only one equivalence class for the relation~r ev there is only one invariant probability distribution. This is because the normalised invariant measure on a homogeneous space is uniquely defined. + The mutual convertibility of pure states under reversible transformations was introduced by Hardy [54] as an axiom for the derivation of quantum theory and has been assumed, either directly or indirectly, in all the recent derivations inspired by quantum information theory (see [68,70,82,83] and the contributed volume [60] for an overview). Theorem 1 provides one more motivation for the convertibility of pure states, identified as the necessary condition for the formulation of the principle of equal a priori probabilities.

The microcanonical state
Every theory satisfying requirement 1 has a canonical notion of 'uniform distribution over the pure states of the system'. We can then apply the principle of equal a priori probabilities and define the microcanonical state as the uniform mixture the invariant probability distribution over the deterministic pure states of system A. The convexity of the state space guarantees that the microcanonical state is indeed a state. Moreover, since the state space is finite-dimensional, it is possible to replace the integral in equation (22) with a finite sum. This means that the microcanonical state can (in principle) be generated by picking deterministic pure states at random from a finite set.
The microcanonical state has two important properties, proved in appendix A: 1. it is invariant under arbitrary reversible dynamics of the effective system; 2. it can be generated from every other deterministic pure state of the effective system through a random reversible dynamics.
Property 1 expresses the fact that the microcanonical state is an equilibrium state, in the sense that it does not evolve under any of the reversible dynamics compatible with the constraints. Note that the notion of equilibrium here is different from the notion of thermal equilibrium, which refers to interactions with an external bath. Instead of thermal equilibrium, we consider here a dynamical equilibrium, consisting in the fact that the probability assignments made by the microcanonical state are stable under all possible evolutions of the system. Property 2 refers to the fact that the system can-at least in principle-be brought to equilibrium. Physically, we can imagine a situation where the experimenter has no control on the system's preparation, but has control on the system's dynamics through some classical control fields. In this picture, property 2 guarantees that the experimenter can prepare the microcanonical state by drawing at random the parameters of her control fields. Further along this line, one can also imagine scenarios where the randomisation occurs naturally as a result of fluctuations of the fields. Property 2 is important from the resource-theoretic approach, where the microcanonical state is often regarded as free, or 'easy to prepare'.

Composition of microcanonical states
At the level of single systems, requirement 1 guarantees the existence of a microcanonical state. But how does the microcanonical state behave under the composition of systems? Traditionally, this question is not addressed in textbook presentations, where the microcanonical state is associated with isolated systems, i.e. systems that do not interact with other systems. From the operational point of view, however, it is natural to consider scenarios where the experimenter has more than one system at her disposal.
Composition is especially important in the context of resource theories, where it is natural to ask how resources interact when combined together. To illustrate this point, it is useful to consider the quantum resource theory of noisy operations [46][47][48]. There, the microcanonical states are treated as free. Since the experimenter can generate the microcanonical states c A and c B at no cost, then she can generate the product state c c Ä A B at no cost too. If we insist that the microcanonical states are the only free states in the resource theory of noisy operations, the product state c c Ä A B must be the microcanonical state of the composite system Ä Equation (23) is consistent with the intuitive interpretation of the microcanonical state as 'the state of minimum information compatibly with the constraints'. Indeed, equation (23) amounts to saying that, if one has minimum information on the parts of a system, then one has minimum information about the whole. This is indeed the case in quantum theory, where the product of two maximally mixed states is maximally mixed. Recall that here we are dealing with effective systems, which exist only as long the corresponding constraints are enforced. For energy constraints, the composite of two effective systems A and B is defined as a system consisting of two parts, each constrained to a specific value of the energy. Consistently with this interpretation, the microcanonical state of system Ä A Bis the 'maximally mixed state' in the manifold of quantum states with fixed local energies, as defined in equation (13).
Following the example of quantum mechanics, we require that minimum information about the parts imply minimum information about the whole: Requirement 2. The microcanonical state of a composite system is the product of the microcanonical states of its components. In formula: for every pair of effective systems A and B.
We call equation (24) the condition of informational equilibrium. Note that, again, here we are not referring to thermal equilibrium between the two subsystems. This is clear from the fact that we do not allow an energy flow between the two systems A and B. Instead, we allow a flow of information, implemented by the joint dynamics of the composite system Ä A B. It is natural to ask which physical principles guarantee the condition of informational equilibrium. One such principle is local tomography [54,55,57], namely the requirement that the state of multipartite systems be determined by the joint statistics of local measurements. However, local tomography is not necessary for informational equilibrium. For example, quantum theory on real Hilbert spaces violates local tomography, but still satisfies the condition of informational equilibrium. In this paper, we will not assume local tomography in our set of physical principles. Nevertheless, our principles will guarantee the validity of the condition of informational equilibrium.

Microcanonical theories
We are now ready to extend the microcanonical framework from quantum and classical theory to general physical theories.
Definition 3. An operational-probabilistic theory, interpreted as a theory of effective systems, is microcanonical if requirements 1 and 2 are satisfied.
Physically, a microcanonical theory is a theory where (i) every system has a well-defined notion of uniform mixture of all pure states, and (ii) uniform mixtures are stable under parallel composition of systems. Microcanonical theories provide the foundation for the definition of three important resource theories, analysed in the following sections.

Three resource theories
In this section we study three different notions of state convertibility in microcanonical theories. We adopt the resource-theoretic framework of [20,21], where one fixes a set of free operations, closed under sequential and parallel composition. A basic question in the resource-theoretic framework is whether a given state ρ can be transformed into another state σ by means of free operations. When this is possible, ρ is regarded as 'more resourceful' than σ, denoted as F  r s, where F is the set of free operations. Mathematically, the relation F  is a preorder on the states. In the following we define three resource theories and their corresponding preorders.

The random reversible (RaRe) resource theory
Our first resource theory is based on the notion of random reversible channel [67]: is a probability distribution and, for every i,  i is a reversible channel on system A.
Physically, RaRe channels are the operations that can be implemented with limited control over the reversible dynamics of the system. Mathematically, it is immediate to check that RaRe channels have all the properties required of free operations: the identity channel is RaRe, the sequential composition of two RaRe channels is a RaRe channel, and so is the parallel composition. We call the resulting resource theory the RaRe resource theory, and we denote by RaRe  the corresponding preorder. Note that the RaRe resource theory can be formulated in every OPT, even in OPTs that do not satisfy requirements 1 and 2. Such generality, however, comes at a price: the RaRe resource theory has no free states. This is because states are operations with trivial input, while the only free operations in the RaRe theory are transformations where the input and the output coincide.
Despite not having free states, the RaRe resource theory can have minimally resourceful states, defined as follows Definition 5. In a resource theory with free operations F, a state ρ is minimally resourceful if the condition F  r s implies s r = .
In the RaRe resource theory, minimally resourceful states are easy to characterise: , for some RaRe channel . Since ρ is invariant, it must satisfy the relation r r = . Hence, s r = . + For theories satisfying requirement 1, proposition 1 implies that the microcanonical state is minimally resourceful: indeed, we know that the microcanonical state is invariant under reversible transformations.

The noisy resource theory
While the RaRe resource theory can be defined in every OPT, we now discuss a second resource theory that can only be defined in physical theories satisfying requirements 1 and 2. In this resource theory, free operations are generated by letting the system interact with ancillas in the microcanonical state. These operations, usually called 'noisy' [46][47][48], are defined as follows: where E and ¢ E are suitable systems such that Ä ¢Ä ¢  A E A E ,  is a reversible transformation, and e is a deterministic effect, representing a possible way to discard system ¢ E .
Note that here we only allow reversible transformations, instead of mixtures of reversible transformations. In principle, one could consider arbitrary RaRe channels, as in the previous subsection. The main reason why we stick to reversible transformations (without randomisation) is that we want to be consistent with the existing literature [46][47][48]. Note also that definition 6 is interesting per se, because it does not rely on the availability of external sources of randomness: instead, all the randomness is accounted for in the preparation of the microcanonical state in the right-hand side of equation (25).
Definition 6 has a slightly unpleasant aspect: the set of basic noisy operations is generally not closed. In quantum theory, for example, there exist counterexamples where the limit of a sequence of basic noisy operations is not a basic noisy operation [51]. It is then convenient to take the closure of the set of basic noisy operations: Definition 7. A channel  is a noisy operation if it is the limit of a sequence of basic noisy operations  { } n .
The set of noisy operations satisfies all the requirements for being a set of free operations: the identity is a noisy operation, and the parallel and sequential composition of two noisy operations are noisy operations, thanks to the condition of informational equilibrium (24). The resource theory where the set of free operations is the set of noisy operations will be called the noisy resource theory. The corresponding preorder on states will be denoted by Noisy  .

The unital resource theory
In the third resource theory, the set of free operations includes all the operations that transform microcanonical states into microcanonical states. The rationale for considering these transformations, called unital channels, is their generality: if we insist that the microcanonical states are the only free states, unital channels are the most general transformations that send free states into free states. In other words, they are the most general operations that do not create resources out of free states Mathematically, the unital channels are defined as follows: Unital channels are the operational generalisation of doubly stochastic matrices in classical probability theory [49,50,66].
The set of unital channels enjoys all the properties required of a set of free operations: the identity is a unital channel, and thanks to the condition of informational equilibrium, the sequential and parallel composition of unital channels is a unital channel. The resource theory where free operations are unital channels will be called the unital resource theory. The corresponding preorder on states will be denoted by Unital  . In classical probability theory, the inclusion is actually an equality, as a consequence of Birkhoff's theorem [66,85]. Remarkably, in quantum theory there exist unital channels that are not random unitary, meaning that the inclusion (26) is generally strict. The simplest example is due to Landau and Streater [49]: for a quantum particle of spin j, they defined the map

Containment relations
where J J J , , x y z are the three components of the spin operator. It is easy to see that the map  j is trace-preserving and identity-preserving-that is, it is a unital channel. On the other hand, Landau and Streater showed that the map  j cannot be decomposed as a mixture of unitary channels unless = j 1 2 [49]. We have seen that all RaRe channels are unital. Noisy operations are also unital, as shown by the following Proposition 2. Every noisy operation is unital.
Proof. Suppose that  is a basic noisy operation, decomposed as in equation (25). Then, one has The inclusion is strict in quantum theory, where Haagerup and Musat have found examples of unital channels that cannot be realised as noisy operations [52]. It remains to understand the relation between RaRe channels and noisy operations. In quantum theory, the set of noisy operations (strictly) contains the set of RaRe channels as a proper subset [51]. In a generic theory, however, this containment relation may not hold. As a counterexample, consider the variant of quantum theory where only local operations are allowed: in this case, RaRe channels are not contained in the set of noisy operations, because all the interactions are trivial.
The inclusions (26) and (29) are the most general result one can derive from the definitions alone. To go further, we need to introduce axioms. In the next sections, we will introduce a set of axioms that imply deeper relations between the RaRe, noisy, and unital resource theories. In addition, the axioms will imply a connection with the mathematical theory of majorisation and with the resource theory of entanglement.

Four axioms
In this section we review the four axioms used in this paper. These axioms-causality, purity preservation, pure sharpness, and purification-define a special class of theories, which we call sharp theories with purification.

Sharp theories with purification
Sharp theories with purification are defined by the following four axioms. The first axiom-causality-states that no signal can be sent from the future to the past: Axiom 1 (Causality [57,59,61,68]). The probability that a transformation occurs in a test is independent of the settings of tests performed on the output.
The second axiom-purity preservation-states that no information can leak to the environment when two pure transformations are composed: Axiom 2 (Purity preservation [86]). Sequential and parallel compositions of pure transformations yield pure transformations.
The third axiom-pure sharpness-guarantees that every system possesses at least one elementary property, in the sense of Piron [87]: Axiom 3 (Pure sharpness [88]). For every system there exists at least one pure effect occurring with unit probability on some state. Quantum theory, both on complex and real Hilbert spaces, satisfies purification. Remarkably, even classical theory can be regarded as a sub-theory of a larger physical theory where purification is satisfied [62].

Definition 9. An OPT is a sharp theory with purification if it satisfies axioms 1-4.
In the rest of the section we will outline the main kinematic properties of sharp theories with purification.

Well-defined marginal states
By definition, sharp theories with purification satisfy causality, which in turn is equivalent to the requirement that, for every system A, there exists a unique deterministic effect Eff Î ( ) u A A (or simply u, when no ambiguity can arise) [57]. The uniqueness of the deterministic effect implies that the marginals of a bipartite state are uniquely defined. For a bipartite state St r Î Ä ( ) A B , we will denote the marginal on system A as ð30Þ in analogy with the notation used in quantum theory.
In a causal theory, it is immediate to see that a state ρ can be prepared deterministically if and only if it is normalised, namely

Diagonalisation
In sharp theories with purification, one can prove that every state can be diagonalised, that is, decomposed as a random mixture of perfectly distinguishable pure states. It follows from the axioms that the eigenvalues are uniquely defined by the state (see [62] for the proof). The uniqueness of the spectrum is a non-trivial consequence of the axioms: notably, [89,90] exhibited examples of theories (other than sharp theories with purification) where states can be diagonalised, but the same state can have two different diagonalisations with two different spectra.

State-effect duality
Sharp theories with purification exhibit a duality between normalised pure states and normalised pure effects-a normalised effect being an effect a such that r = ( | ) a 1 for some state. Denoting the set of normalised pure effects by PurEff ( ) A 1 , the duality reads as follows:

Proposition 3 ([88]
). There is a bijective correspondence between normalised pure states and normalised pure effects.
such that a a = ( | ) †

1.
Physically, the meaning of the duality is that every pure state can be certified by a (unique) pure effect, which occurs with unit probability only on that particular state. The duality between pure states and pure effects can be lifted to a duality between maximal sets of perfectly distinguishable pure states and perfectly distinguishing observation tests, defined as follows: A maximal set of perfectly distinguishable pure states will be called pure maximal set for short. With this notation, the duality reads As a consequence, the product of two pure maximal sets is a pure maximal set for the composite system: It is possible to show that all pure maximal sets in a given system have the same cardinality [62]. For a generic system A, we will denote the cardinality of the maximal sets by d A . We will refer to d A as the dimension of system A. We stress that the dimension d A should not be confused with the dimension of the normalised state space St ( ) A 1 : in quantum theory, the dimension d A is the dimension of the system's Hilbert space, while the dimension of the space of density matrices isd 1 A 2 . Proposition 5 shows that the dimension of a composite system is the product of the dimensions of the components, namely for every pair of systems A and B. This property has been dubbed information locality by Hardy [70,71].

Microcanonical thermodynamics in sharp theories with purification
Here we show that sharp theories with purification satisfy our requirements 1 and 2 for the construction of the microcanonical framework. Moreover, we will show that sharp theories with purification exhibit a simple inclusion relation between RaRe and noisy operations.

The microcanonical state
We start by showing that every sharp theory with purification satisfies requirement 1, which enables the formulation of the principle of equal a priori probabilities. Thanks to theorem 1, we only need to show that every two pure states of the same system are connected by a reversible transformation. This fact is an immediate consequence of purification: Proposition 6 ([57]). For every theory satisfying purification, for every system A in the theory, and for every pair of deterministic pure states a and a¢ of system A, there exists a reversible transformation  such that  a a ¢ = .
Proposition 6 guarantees that for every system A there exists a unique probability distribution y ( ) p d A , which is invariant under all reversible dynamics. In turn, the probability distribution y ( ) p d A can be used to define the microcanonical state c A .
In sharp theories with purification, the microcanonical state enjoys a remarkable property: the state can be decomposed into a uniform mixture of perfectly distinguishable pure states.
is any orthonormal basis for the eigenspace of the Hamiltonian with eigenvalue E. It is worth noting that equation (36) is often chosen as the definition of the microcanonical state in quantum statistical mechanics. Proposition 7 shows that a similar definition is possible in every sharp theory with purification. One may be tempted to use equation (34) to define the microcanonical state in arbitrary physical theories. However, the fact that the state is independent of the choice of maximal set is not guaranteed to hold in every theory. For this reason, we prefer to define the microcanonical state as the uniform mixture of all pure states with a given energy, rather than the uniform mixture of a particular maximal set of pure states. Physically, the uniform mixture of all pure states represents the result of fully uncontrolled, but energy conserving fluctuations in the experimental setup. From a subjective point of view, the uniform mixture represents the complete lack of knowledge besides the knowledge of the value of the energy: not even the 'energy eigenbasis' is assumed to be known.

The condition of informational equilibrium
We have seen that sharp theories with purification satisfy requirement 1-the uniqueness of the uniform distribution over the pure states. We now show that requirement 2-the condition of informational equilibrium -is satisfied too. Proof. Pick two pure maximal sets for A and B, say In summary, sharp theories with purification satisfy our two requirements for the general microcanonical framework. In the following, we will show that sharp theories with purification also guarantee an important inclusion relation between the set of RaRe channels and the set of noisy operations.

Inclusion of RaRe into noisy
In sharp theories with purification, one can establish an inclusion between RaRe channels and noisy operations. To obtain this result, we first restrict our attention to rational RaRe channels, i.e. RaRe channels of the form   = å p i i i where each p i is a rational number. With this definition, we have the following lemma: In every sharp theory with purification, every rational RaRe channel is a basic noisy operation.
In quantum theory, this statement is quite immediate, as pointed out in [48]: where Tr anc is the partial trace on the n-dimensional system used as ancilla, and U is the control-unitary gate 1 being a list of unitary gates, n 1 of which are equal to U 1 , n 2 equal to U 2 , and so on. The situation is in general more complicated in sharp theories with purification. The reason is that the simple construction of equations (38) and (39) cannot be reproduced. The analogue of the control-unitary U is a control-reversible transformation, which performs a reversible transformation on the target system depending on the state of a control system. However, later in the paper we will show that not every sharp theory with purification admits control-reversible transformations. In fact, we will show that the existence of controlreversible transformation is equivalent to a non-trivial property of the dynamics, which we will call 'unrestricted reversibility'.
The non-trivial content of lemma 1 is that the inclusion RationalRaRe Noisy Í holds in every sharp theory with purification, without the need of assuming unrestricted reversibility. To prove such a result we need a new construction, more elaborate than the simple construction of equations (38) and (39). The details can be found in appendix B. Now, since rational RaRe channels are dense in the set of RaRe channels, and since the set of noisy operations is closed (see definition 7), we obtain the following theorem: Theorem 3. In every sharp theory with purification, RaRe channels are noisy operations.
The inclusion of RaRe channels in the set of noisy operations is generally strict: for example, in quantum theory there exist noisy operations that are not RaRe channels [51]. In summary, we have the inclusions

State convertibility and majorisation
In this section we investigate the convertibility of states in the RaRe, noisy, and unital resource theories. The main result is that, in every sharp theory with purification, an input state can be converted into an output state by a unital channel if and only if the vector of the eigenvalues of the output state is majorised by the vector of the eigenvalues of the input state. Since the set of unital channels contains the sets of noisy operations, and RaRe channels, our results establishes majorisation as a necessary condition for convertibility under noisy operations and RaRe channels. Later in the paper, we will determine the physical condition under which majorisation is also sufficient.

State convertibility
In sharp theories with purification, the inclusions (40)  valid for every pair of states ρ and σ of the same system. Note that the unital relation Unital  is the weakest, i.e. the easiest to satisfy. In the following we will provide a necessary and sufficient condition for the unital preorder.

Unital channels and doubly stochastic matrices
In a broad sense, unital channels are the generalisation of doubly stochastic matrices. In sharp theories with purification, there is also a more explicit connection: form an observation-test and that  is a channel, and therefore  = u u [57]. On the other hand, one has having used proposition 7 and the fact that unital channels leave χ invariant. In conclusion, equations (43) and (44) show that the matrix D is doubly stochastic. + Vice versa, every doubly stochastic matrix defines a unital channel: the third equality following from the definition of doubly stochastic matrix, and the fourth equality following from the diagonalisation of the state χ (proposition 7). + Lemmas 2 and 3 establish a direct connection between unital channels and doubly stochastic matrices. Using this connection, in the following we establish a relation between the unital resource theory and the theory of majorisation.

Majorisation criterion for state convertibility under unital channels
Here we show that the ability to convert states in the unital resource theory is completely determined by a suitable majorisation criterion. Let us start by recalling the definition of majorisation [66]: x y k d x y , a n d . Majorisation can be equivalently characterised in terms of doubly stochastic matrices: one has  x y if and only if = D y x, where D is a doubly stochastic matrix [66,91].
In every sharp theory with purification, majorisation is a necessary and sufficient condition for convertibility under unital channels: Theorem 4. Let ρ and σ be normalised states, and let p and q be the vectors of their eigenvalues, respectively. The state ρ can be converted into the state σ by a unital channel if and only if p majorises q. In formula: The proof is provided in appendix C. Note that since RaRe channels and noisy operations are special cases of unital channels, majorisation is a necessary condition for convertibility in the RaRe and noisy resource theories.

Characterisation of unital monotones
The majorisation criterion determines whether a state is more resourceful than another. To be more quantitative, one can introduce monotones [19][20][21]-i.e. functions that are non-increasing under free operations: Definition 13. A monotone under the free operations F for system A is a function St   ( ) P: A satisfying the condition When F is the set of unital operations, we refer to P as unital monotones. In sharp theories with purification, unital monotones have an elegant mathematical characterisation: Proof. Theorem 4 shows that the convertibility of states under unital channels is fully captured by their eigenvalues. Consequently, a unital monotone will be a function only of the eigenvalues of a state: there exists a function This means that f is Schur-convex. Conversely, given a Schur-convex function f one can define a function P f on the state space, as r = ( ) ( ) P f p : f , p being the spectrum of ρ. This function is easily proved to be a unital monotone, thanks to theorem 4.

The counterexample of doubled quantum theory
We have seen that majorisation is a necessary and sufficient condition for state convertibility in the unital resource theory. Is majorisation sufficient also for convertibility in the RaRe resource theory? Now we show that the answer is negative by constructing a counterexample, which we call 'doubled quantum theory'.

Individual systems
Consider a theory where every non-trivial system is the direct sum of two identical quantum systems with Hilbert spaces  0 and  1 , respectively. Physically, we can think of the two Hilbert spaces as two superselection sectors. We associate each 'doubled quantum system' with a pair of isomorphic Hilbert spaces   ( ) , 0 1 , with    0 1 . We define the states of the doubled quantum system to be of the form where r 0 and r 1 are two density matrices in the two sectors, respectively, and p is a probability. Likewise, we define the effects to be all quantum effects of the form = Å e e e 0 1 , where e 0 and e 1 are two quantum effects in the two sectors. The allowed channels from the input system   ( ) , 1 , even though the two channels act in the same way on every input state of the form (52). Operationally, the difference between the channels  and  will become visible when the channels are applied locally on one part of a composite system.
The existence of different physical transformations that are indistinguishable at the single-system level is made possible by the fact that doubled quantum theory does not satisfy the local tomography axiom, as we show in appendix D.1. The inability to distinguish transformations at the single-system level is a fairly generic trait of theories where the local tomography axiom does not hold. Quantum theory on real Hilbert spaces also exhibit this trait [57,59] (but only for non-pure transformations [57]).

Composite systems
The peculiarity of doubled quantum theory is the way systems are composed. The product of two doubled quantum systems   ( ) , As an example, consider the composite system of two doubled qubits, corresponding to 2 . An example of state of the composite system is the pure state . This means that the coherence between the two summands in the state(55) is invisible at the single-system level.
From a physical point of view, doubled quantum theory can be thought of as ordinary quantum theory with a superselection rule on the total parity. Every system is split into two identical sectors of even and odd parity, respectively. When systems are composed, the sectors are grouped together based on the total parity, so that superpositions between subspaces with the same parity are allowed.

In doubled quantum theory, majorisation is not sufficient for convertibility under RaRe channels
In appendix D we summarise a few operational features of doubled quantum theory. In particular, we show that doubled quantum theory is a sharp theory with purification. Nevertheless, now we show that majorisation does not guarantee the convertibility of states under RaRe channels.
Permutability, strong symmetry, and reversible controllability are logically distinct requirements. For example, strong symmetry implies permutability, but the converse is not true in general, as shown by the example of the square bit [55] in figure 3 (see appendix E for more details).
Although different in general, permutability, strong symmetry, and reversible controllability become equivalent in sharp theories with purification: Proposition 10. In every sharp theory with purification, permutability, strong symmetry, and reversible controllability are equivalent requirements.
The proof is presented in appendix F. The fact that three desirable properties become equivalent under our axioms gives a further evidence that the axioms capture an important structure of physical theories.
Since permutability, strong symmetry, and reversible controllability are equivalent in the present context, we conflate them into a single notion: Definition 14. A sharp theory with purification has unrestricted reversibility if the theory satisfies permutability, or strong symmetry, or reversible controllability.

When the three resource theories of purity are equivalent
We now characterise exactly when the RaRe, noisy, and unital resource theories are equivalent in terms of state convertibility. Owing to the inclusions RaRe Noisy Unital Í Í , a sufficient condition for the equivalence is that the convertibility under unital channels implies the convertibility under RaRe channels. The characterisation is as follows: Theorem 5. In every sharp theory with purification, the following statements are equivalent: 1. the RaRe, noisy, and unital resource theories are equivalent in terms of state convertibility 2. the theory has unrestricted reversibility.
Proof. The implication  2 1was already proven in [88] To prove the implication  1 2, we show that condition 1 implies the validity of strong symmetry. Let a = { }  criterion guarantees that ρ can be converted into σ by a unital channel, and vice versa (theorem 4). Now, our hypothesis is that convertibility under unital channels implies convertibility under RaRe channels. The mutual convertibility of ρ and σ under RaRe channels implies that there exists a reversible transformation  such that  s r = [67,95]. Applying the effect a ¢ † 1 to both sides of the equality  s r = , we obtain Combining equation (63) with the equality r s = , we obtain the condition r s = 1 1 . Applying to r 1 and s 1 the same argument we used for ρ and σ, we obtain the equality a a = ¢ 2 2 . Iterating the procedured 1 times, we finally obtain the equality a Hence, every two maximal sets of perfectly distinguishable pure states are connected by a reversible transformation. + Theorem 5 gives necessary and sufficient conditions for the equivalence of the three resource theories of microcanonical thermodynamics. In addition, it provides a thermodynamic motivation for the condition of unrestricted reversibility.

The equivalence in a nutshell
The results of this section can be summed up in the following theorem: Theorem 6. In every sharp theory with purification and unrestricted reversibility, the following are equivalent Proof. The implications  1 2 and  2 3follow from the inclusions (40). The implication  3 4follows from theorem 4. The implication  4 1 follows from the equivalence between majorisation and unital convertibility, combined with theorem 5. + Theorem 6 tells us that the RaRe, noisy, and unital resource theories are all equivalent in terms of state convertibility. It is important to stress that the equivalence holds despite the fact that the three sets of operations are generally different.
An important consequence of the equivalence is that the RaRe, noisy, and unital resource theories have the same quantitative measures of resourcefulness: be a real-valued function on the state space of system A. If P is a monotone under one of the sets RaRe, Noisy and Unital, then it is a monotone under all the other sets.
Since the preorders RaRe  , Noisy  , and Unital  coincide, we can say that the RaRe, noisy, and unital resource theories define the same notion of resource, which one may call purity. Accordingly, we will talk about 'the resource theory of purity' without specifying the set of free operations.

The entanglement-thermodynamics duality
We conclude the paper by showing that sharp theories with purification and unrestricted reversibility exhibit a fundamental duality between the resource theory of purity and the resource theory of entanglement [67]. The entanglement-thermodynamics duality is a duality between two resource theories: the resource theory of purity (with RaRe, or noisy, or unital channels as free operations) and the resource theory of pure bipartite entanglement (with local operations and classical communication as free operations). The content of the duality is that a pure bipartite state is more entangled than another if and only if the marginal states of the latter are purer than the marginal states of the former. More formally, the duality can be stated as follows [67]: Our earlier work [67] showed that the entanglement-thermodynamics duality can be proved from four axioms: causality, purity preservation, purification, and local exchangeability-the latter defined as follows:  where SWAP is the channel that exchanges system A and system B.
Since causality, purity preservation, and purification are already assumed among our axioms, proving the entanglement-thermodynamics duality is reduced to proving the validity of local exchangeability. The proof is presented in appendix G, which backs the following claim: Theorem 7. Every sharp theory with purification and unrestricted reversibility satisfies the entanglementthermodynamics duality.
As a consequence of the duality, the purity monotones characterised in the previous subsection are in oneto-one correspondence with measures of pure bipartite entanglement. For example, Shannon-von Neumann entropy of the marginals of a pure bipartite state can be regarded as the entanglement entropy [96][97][98], an entropic measure of entanglement that is playing an increasingly important role in quantum field theory [99,100] and condensed matter [101].

Conclusions
In this work we developed a microcanonical framework for general physical theories. The framework is based on two requirements: the uniqueness of the invariant probability distribution over pure states, needed to define the microcanonical state, and the stability of the microcanonical state under composition. Under these requirements, we defined three resource theories, where free operations are random reversible channels, noisy operations, and unital channels, respectively. We explored the connections between these three sets of operations in a special class of physical theories, called sharp theories with purification, which enable a fundamentally reversible description of every process. In sharp theories with purification, the sets of random reversible channels is contained in the set of noisy operations, which in turn is contained in the set of unital channels. Convertibility under unital channels is equivalent to majorisation, which is a necessary condition for convertibility under the other sets of operations. Majorisation becomes a sufficient condition for convertibility under all sets of operations if and only if the dynamics allowed by the theory have a property, called unrestricted reversibility. In this case, one obtains the entanglement-thermodynamics duality, which connects the entanglement of pure bipartite states with the purity of their marginals.
Our results identify sharp theories with purification and unrestricted reversibility as the natural candidate for the information-theoretic foundation of microcanonical thermodynamics. At the same time, it is interesting to go beyond the microcanonical scenario, and develop a general probabilistic framework for the canonical ensemble. Some steps in this direction can be found in a companion paper [62], where we give an operational definition of the Gibbs state, and use it in an information-theoretic derivation of Landauer's principle. These results are only the surface of a deep operational structure, where thermodynamic and information-theoretic features are interwoven at the level of fundamental principles. Many interesting directions of research remain open, including, for example, an extension of the notion of thermomajorisation [25], a derivation of the monotonicity of the relative entropy [102], and a derivation of the 'second laws of thermodynamics' [26] from operational axioms.

Acknowledgments
We thank the Referees for insightful comments that stimulated a substantial extension of our original manuscript.
This work has been supported by the Foundational Questions Institute Proof. The group of reversible transformations on system A has a finite-dimensional representation on the state space of system A. This representation defines a group of finite-dimensional matrices, call it G A . Note that the group G A is compact, because it is closed and finite-dimensional. Hence, one can construct the invariant measure  d and define the transformation where  = { } x x n 1 are reversible transformations on A, chosen so that n 1 of the channels are equal to  1 , n 2 are equal to  2 , and so on. Since the theory satisfies Purification, the channel  has a reversible extension [57,58], meaning that one has ðB3Þ where C and ¢ C are suitable systems, γ is a suitable pure state, and  is a reversible transformation. Now, by construction we have . Finally, let us show that  is a channel. To this end, it is enough to show that  = u u [57]. This property is satisfied if and only if  c = ( | | ) u 1, because every state lies in some convex decomposition of χ [57]. By the condition of informational equilibrium and equation (B10), we have ðB11Þ so  is a channel. Since every pure channel on a fixed system (here Ä Ä A B A) is reversible [57],  is reversible. Hence, equation (B10) shows that  is a basic noisy operation, with environment = Ä E B A. Now, the D ij ʼs are the entries of a doubly stochastic matrix D (lemma 2). Hence, equation (C2) implies that p majorises q.
Conversely, suppose that  p q and let D be a doubly stochastic matrix such that = D q p. Define the measure-and-prepare channel where the purity of , , and Y¢ follows from purity preservation. Like in the proof of proposition 10, we can prove that  and  are in fact channels, so  = u u B A and  = u u A B . Hence Y¢ and SWAP Y have the same marginals. Then, the uniqueness of purification applied to both systems A and B (viewed as purifying systems of one another) implies that there exist two reversible transformations  A and  B such that ðG7Þ ðG8Þ Hence, we have shown that there exist two local pure channels    = : B and    = : A that reproduce the action of the swap channel on the state Y. + Note that local exchangeability is implemented in this setting by pure channels.
In passing, we also mention that the validity of local exchangeability implies that every state admits a symmetric purification, in the following sense: Definition 17. [67] Let ρ be a state of system A and let Ψ be a pure state of Ä A A. We say that Ψ is a symmetric purification of ρ if ðG9Þ and ðG10Þ With the above notation, we have the following Proposition 15. In every sharp theory with purification and unrestricted reversibility, every state of every finite system admits a symmetric purification.
The existence of a symmetric purification for every state is guaranteed by theorem 3 of [67].