Generalized Stein's lemma and asymptotic equipartition property for subalgebra entropies

Summary · qwen2.5:32b

The article presents a proof for the validity of the generalized quantum Stein's lemma when the second hypothesis is the state space of any finite-dimensional subalgebra, achieved through a strong asymptotic equipartition property for smooth subalgebra entropies. This finding links resource convertibility to quantum hypothesis testing and demonstrates that the relative entropy of a subalgebra represents the asymptotic dilution cost under suitable operations, offering insights into connecting different quantum resources based on subalgebras. Notably, this work provides an alternative proof using distinct operator-algebraic techniques compared to recent independent resolutions by Hayashi-Yamasaki and Lami.

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arXiv:2401.03090v3 Announce Type: replace-cross Abstract: The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.