Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras

Abstract:We establish duality relations for conditional entropies relative to subalgebras of arbitrary von Neumann algebras, including the Umegaki relative entropy as well as the sandwiched- and Petz-Rényi divergences. Even in finite dimensions, these results extend the familiar dualities for conditional entropies from tensor-product subsystems to general subalgebras, while also lifting the framework to the fully infinite-dimensional setting. As an application, we establish an infinite-dimensional extension of quantum Stein's lemma when the alternative hypothesis is the state space of a finite-index subalgebra, which gives an operational interpretation of the relative entropy to a subalgebra as the optimal asymptotic error exponent in composite quantum hypothesis testing-the first beyond finite dimensions.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Operator Algebras (math.OA)

arXiv:2610.00772 [quant-ph]

(or arXiv:2610.00772v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00772 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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