Asymptotic Birkhoff-Violation in Operational Theories: Thermodynamic Implications and Information Processing

TL;DR

This paper explores how Birkhoff-violation in general probabilistic theories affects thermodynamics and information processing, revealing phenomena absent in quantum mechanics and proposing a generalized entropy concept.

Contribution

It extends Birkhoff's theorem analysis beyond quantum mechanics within GPTs, demonstrating asymptotic violations and their implications for thermodynamics and resource theories.

Findings

Birkhoff-violation is prevalent in GPTs and asymptotic violations occur.

Birkhoff-violation leads to unique communication advantages not seen in quantum theory.

Different resource theories of purity emerge due to distinct state transformation criteria.

Abstract

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.