Relative entropy for von Neumann subalgebras

Abstract

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

arXiv commentMinor changes. Added reference and examples; 32 pages. Comments are welcome!

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Cited by in corpus (8)

• Complete entropic inequalities for quantum Markov chains
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Approximate tensorization of the relative entropy for noncommuting conditional expectations
• Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Approximate Unitary $k$-Designs from Shallow, Low-Communication Circuits
• Quantum teleportation in the commuting operator framework
• Resource-Dependent Complexity of Quantum Channels