Generalized Quantum Stein's Lemma and Second Law of Quantum Resource Theories

CQT Talk

Speaker: Masahito Hayashi, The Chinese University of Hong Kong, Shenzhen
Abstract: The second law of thermodynamics is the cornerstone of physics, characterizing the convertibility between thermodynamic states through a single function, entropy. Given the universal applicability of thermodynamics, a fundamental question in quantum information theory is whether an analogous second law can be formulated to characterize the convertibility of resources for quantum information processing by such a single function. In 2008, a promising formulation was proposed, linking resource convertibility to the optimal performance of a variant of the quantum version of hypothesis testing. Central to this formulation was the generalized quantum Stein's lemma, which aimed to characterize this optimal performance by a measure of quantum resources, the regularized relative entropy of resource. If proven valid, the generalized quantum Stein's lemma would lead to the second law for quantum resources, with the regularized relative entropy of resource taking the role of entropy in thermodynamics. However, in 2023, a logical gap was found in the original proof of this lemma, casting doubt on the possibility of such a formulation of the second law. In this work, we address this problem by developing alternative techniques and successfully proving the generalized quantum Stein's lemma. Based on our proof, we reestablish and extend the formulation of quantum resource theories with the second law, applicable to both static and dynamical resources. These results resolve the fundamental problem of bridging the analogy between thermodynamics and quantum information theory.
This is a joint work with Hayata Yamasaki, and the full paper is available from https://arxiv.org/abs/2408.02722