Abstract The entropy principle asserts that spontaneous processes drive systems toward states of higher disorder. Operational origin of such randomness is a question of utmost importance. The seminal Birkhoff theorem (also known as Birkhoff–von Neumann theorem) identifies doubly stochastic transformations as convex mixtures of reversible transformations, thereby providing an epistemic account of disorder. Quantum theory, however, does not satisfy this equivalence in general. In this work, we investigate this question within the framework of Generalized Probabilistic Theories (GPTs). We prove a violation of the Birkhoff theorem in the square-bit model, a canonical non-quantum GPT, and further show that this violation persists asymptotically under tensor products of any finite, yet arbitrarily large, number of copies of even-polygon models. This provides evidence that the failure of the Birkhoff theorem is an intrinsic operational feature of a wide class of GPTs, rather than a peculiarity of quantum theory. Furthermore, we show that, beyond quantum theory, the preorder over states induced by mixtures of reversible transformations can differ from that induced by doubly stochastic transformations, leading to inequivalent notions of state convertibility and distinct resource theories of purity. Beyond its foundational significance, we show that the violation of the Birkhoff theorem admits a concrete operational interpretation: in the square-bit theory, it serves as a genuine resource for enhancing the performance of Random Access Codes. Finally, we discuss the construction of entropy functions in GPTs that remain consistent with thermodynamic principles, in the spirit of von Neumann’s formulation.
Abstract The entropy principle asserts that spontaneous processes drive systems toward states of higher disorder. Operational origin of such randomness is a question of utmost importance. The seminal Birkhoff theorem (also known as Birkhoff–von Neumann theorem) identifies doubly stochastic transformations as convex mixtures of reversible transformations, thereby providing an epistemic account of disorder. Quantum theory, however, does not satisfy this equivalence in general. In this work, we investigate this question within the framework of Generalized Probabilistic Theories (GPTs). We prove a violation of the Birkhoff theorem in the square-bit model, a canonical non-quantum GPT, and further show that this violation persists asymptotically under tensor products of any finite, yet arbitrarily large, number of copies of even-polygon models. This provides evidence that the failure of the Birkhoff theorem is an intrinsic operational feature of a wide class of GPTs, rather than a peculiarity of quantum theory. Furthermore, we show that, beyond quantum theory, the preorder over states induced by mixtures of reversible transformations can differ from that induced by doubly stochastic transformations, leading to inequivalent notions of state convertibility and distinct resource theories of purity. Beyond its foundational significance, we show that the violation of the Birkhoff theorem admits a concrete operational interpretation: in the square-bit theory, it serves as a genuine resource for enhancing the performance of Random Access Codes. Finally, we discuss the construction of entropy functions in GPTs that remain consistent with thermodynamic principles, in the spirit of von Neumann’s formulation.
In accordance with the entropy principle of thermodynamics, under spontaneous evolutions, physical systems always evolve towards states with equal or greater randomness. But, where does this randomness originate? Renowned Birkhoff-von Neumann theorem, often referred to as Birkhoff theorem, identifies source of this randomness to be the stochastic application of reversible operations on the system under study, thereby ensuring its epistemic origin. Analogue of this theorem is known to fail in the quantum case. Here, we extend this investigation beyond quantum mechanics to a broader class of operational theories described within the framework of general probabilistic theories (GPTs). In this generalized framework, we establish Birkhoff-violation as the prevalent trait; in fact the asymptotic variant of the theorem gets violated. We then demonstrate that Birkhoff-violation in GPTs can lead to consequences that are atypical to quantum theory. For instance, we report manifestation of Birkhoff-violation in a communication task, which otherwise is not observed in quantum world. We also show that, unlike the quantum case, in other operational theories the state transformation criteria can be distinct under mixtures of reversible transformations and doubly stochastic evolutions, leading to different resource theories of purity. Despite these exotic implications, we analyze how to define a coherent notion of entropy in this generalized framework, while upholding alignment with von Neumann's thought experiment.
Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.
Generalized probabilistic theories (GPTs) provide a unified framework for describing probabilistic physical theories, encompassing classical and quantum theories as well as hypothetical theories beyond quantum mechanics. Since most GPTs are highly unrealistic and far removed from known physical theories, it is important to constrain them by physically reasonable principles. One of the most important such principles is consistency with thermodynamics, which has been extensively studied through toy models involving semipermeable membranes (SPMs) implementing measurements. On the other hand, information thermodynamics, which plays a central role in understanding the relationship between measurement and thermodynamics in classical and quantum theory, has remained largely undeveloped in GPTs. In this work, we construct information thermodynamics in GPTs and provide a unified framework for analyzing the relationship between measurement, feedback, information erasure, and the second law of thermodynamics. We also formulate a general framework for SPM models and analyze the thermodynamic cost of measurements implemented by SPMs. As a result, we show that no work can be extracted in contradiction with the second law as long as the measurement processes are consistent with entropy nondecrease, and derive sufficient conditions for this property for several entropy definitions proposed in GPTs. Moreover, by considering measurement processes violating these conditions, we construct explicit GPT systems realizing isothermal SPM cycles from which positive work can be extracted. These results demonstrate that violations of the second law can arise from the lack of fundamental entropy properties or discrepancies between entropy definitions, and provide a unified and model-independent foundation for understanding the relationship between thermodynamics and measurement in GPTs.
Quantum magic squares are generalizations of doubly stochastic matrices to arbitrary, possibly non-commutative algebras, more specifically C∗-algebras. Birkhoff-von Neumann Theorem states that every doubly stochastic matrix is a convex combination of permutation matrices. One inevitable question is whether a quantum magic square is a convex combinations of quantum permutation matrices. It has been proven by Cuevas, Drescher, and Netzer that the generalization of the Birkhoff-von Neumann Theorem in the quantum setting is not true in general. A quantum magic square is a convex combination of Arveson extreme points but not necessarily a convex combination of quantum permutation matrices. Additionally, the relationships between different subsets of quantum magic squares, including their inclusion properties, have been studied in this context.
We investigate antidistinguishability of states within the framework of general probabilistic theories (GPTs). We formulate antidistinguishability, strong and equal antidistinguishability as refined notions that imposed additional constraint on the measurement effects. We establish general results relating these notions of antidistinguishability and derive an upper bound on the cardinality of equally antidistinguishable sets in terms of the affine dimension of the state space. We then study antidistinguishability in polygonal theories, obtaining conditions for antidistinguishability of a set of states. In consequence, we show that the set of all pure states in a polygon model is antidistinguishable. Additionally, we identify broad families of strongly and equally antidistinguishable states. Finally, using Random Exclusion Codes, whose success probability is governed by the antidistinguishability of different sets of encoding states, we probe the nonclassicality of polygon theories. We find that certain polygon models can outperform the optimal quantum value, while their optimal performance converges to the quantum limit in the large-polygon limit.
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.