Resource theories provide a framework for thermodynamics beyond the thermodynamic limit. For systems with definite energy,the key resource is purity. In this paper we formulate three alternative resource theories of purity in the framework of general probabilistic theories (GPTs). Each resource theory corresponds to a different choice of free operations, including i)random reversible operations, ii)noisy operations, generated by reversible interactions with ancillas in the microcanonical state, and iii)unital operations, which preserve the microcanonical state.We determine the requirements the underlying GPT must satisfy in order to support well-behaved resource theories with the above sets as free operations. Then, we focus on a special class of physical theories, called sharp theories with purification, which are appealing for an information-theoretic foundation of thermodynamics. For every sharp theory with purification we show that the convertibility of states implies the validity of a majorisation condition. Moreover, we characterise exactly when majorisation is sufficient for convertibility via random reversible operations. Under this condition, we prove that i) random reversible, noisy, and unital operations obey inclusion relations like in quantum theory, ii) despite being different, the three sets of operations are equivalent in terms of convertibility, iii) there exists a one-to-one correspondence between measures of purity and Schur-convex functions, and iv) purity resources are dual to entanglement resources.
Resource theories provide a framework for thermodynamics beyond the thermodynamic limit. For systems with definite energy,the key resource is purity. In this paper we formulate three alternative resource theories of purity in the framework of general probabilistic theories (GPTs). Each resource theory corresponds to a different choice of free operations, including i)random reversible operations, ii)noisy operations, generated by reversible interactions with ancillas in the microcanonical state, and iii)unital operations, which preserve the microcanonical state.We determine the requirements the underlying GPT must satisfy in order to support well-behaved resource theories with the above sets as free operations. Then, we focus on a special class of physical theories, called sharp theories with purification, which are appealing for an information-theoretic foundation of thermodynamics. For every sharp theory with purification we show that the convertibility of states implies the validity of a majorisation condition. Moreover, we characterise exactly when majorisation is sufficient for convertibility via random reversible operations. Under this condition, we prove that i) random reversible, noisy, and unital operations obey inclusion relations like in quantum theory, ii) despite being different, the three sets of operations are equivalent in terms of convertibility, iii) there exists a one-to-one correspondence between measures of purity and Schur-convex functions, and iv) purity resources are dual to entanglement resources.
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.
We study the informational underpinnings of thermodynamics and statistical mechanics, using an abstract framework, general probabilistic theories, capable of describing arbitrary physical theories. This allows one to abstract the informational content of a theory from the concrete details of its formalism. In this framework, we extend the treatment of microcanonical thermodynamics, namely the thermodynamics of systems with a well-defined energy, beyond the known cases of classical and quantum theory, formulating two necessary requirements for a well-defined thermodynamics. We adopt the recent approach of resource theories, where one studies the transitions between states that can be accomplished with a restricted set of physical operations. We formulate three different resource theories, differing in the choice of the restricted set of physical operations. To bridge the gap between the objective dynamics of particles and the subjective world of probabilities, one of the core issues in the foundations of statistical mechanics, we propose four information-theoretic axioms. They are satisfied by quantum theory and more exotic alternatives, including a suitable extension of classical theory where classical systems interact with each other creating entangled states. The axioms identify a class of theories where every mixed state can be modelled as the reduced state of a pure entangled state. In these theories it is possible to introduce well-behaved notions of majorisation, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. The three resource theories define the same notion of resource if and only if, on top of the four axioms, the dynamics of the underlying theory satisfy a condition called "unrestricted reversibility". Under this condition we derive a duality between microcanonical thermodynamics and pure bipartite entanglement.
Entanglement is one of the most striking features of quantum mechanics, and yet it is not specifically quantum.More specific to quantum mechanics is the connection between entanglement and thermodynamics, which leads to an identification between entropies and measures of pure state entanglement.Here we search for the roots of this connection, investigating the relation between entanglement and thermodynamics in the framework of general probabilistic theories.We first address the question whether an entangled state can be transformed into another by means of local operations and classical communication.Under two operational requirements, we prove a general version of the Lo-Popescu theorem, which lies at the foundations of the theory of pure-state entanglement.We then consider a resource theory of purity where free operations are random reversible transformations, modelling the scenario where an agent has limited control over the dynamics of a closed system.Our key result is a duality between the resource theory of entanglement and the resource theory of purity, valid for every physical theory where all processes arise from pure states and reversible interactions at the fundamental level.As an application of the main result, we establish a one-to-one correspondence between entropies and measures of pure bipartite entanglement.The correspondence is then used to define entanglement measures in the general probabilistic framework.Finally, we show a duality between the task of information erasure and the task of entanglement generation, whereby the existence of entropy sinks (systems that can absorb arbitrary amounts of information) becomes equivalent to the existence of entanglement sources (correlated systems from which arbitrary amounts of entanglement can be extracted).
This is an extended abstract based on the preprint arXiv:1608.04459. We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms 'Causality, Purity Preservation, Pure Sharpness, and Purification' identify purity as a fundamental ingredient for every sensible theory of thermodynamics. Indeed, in physical theories satisfying these axioms, called sharp theories with purification, every mixed state can be modelled as the marginal of a pure entangled state, and every unsharp measurement can be modelled as a sharp measurement on a composite system. We show that these theories support a well-behaved notion of entropy and of Gibbs states, by which one can derive Landauer's principle. We show that in sharp theories with purification some bipartite states can have negative conditional entropy, and we construct an operational protocol exploiting this feature to overcome Landauer's principle
In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This elementary structure plays an ubiquitous role in quantum mechanics, quantum information theory, and quantum statistical mechanics, where it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalized. The first axiom is Causality, which ensures that the marginal of a bipartite state is well defined. Then, Purity Preservation states that the set of pure transformations is closed under composition. The third axiom is Purification, which allows to assign a pure state to the composition of a system with its environment. Finally, we introduce the axiom of Pure Sharpness, stating that for every system there exists at least one pure effect occurring with unit probability on some state. For theories satisfying our four axioms, we show a constructive algorithm for diagonalizing every given state. The diagonalization result allows us to formulate a majorization criterion that captures the convertibility of states in the operational resource theory of purity, where random reversible transformations are regarded as free operations.
We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory where classical systems can be entangled with other, non-classical systems. Theories satisfying our axioms support well-behaved notions of majorization, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. We conjecture that every theory admitting a sensible thermodynamics must be extendable to a sharp theory with purification.