Quantum umlaut information

arXiv (Cornell University)

Abstract

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states $ρ_{AB}$ as a reversed variant of the quantum mutual information: $U(A;B)_ρ= \min_{σ_B} D(ρ_A\otimes σ_B\|ρ_{AB})$ in terms of the quantum relative entropy $D$. As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel -- as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.

Authors 6

1. Filippo Girardi
2. Aadil Oufkir Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Germany

3. Bartosz Regula

   RIKEN · RIKEN Center for Quantum Computing · RIKEN Pioneering Research Institute

   Affiliation as printed

   Mathematical Quantum Information RIKEN Hakubi Research Team , RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC) , Wako , Saitama 351-0198 , Japan

4. Marco Tomamichel

   Centre for Quantum Technologies · National University of Singapore

   Affiliation as printed

   Centre for Quantum Technologies , National University of Singapore , Singapore

   Department of Electrical and Computer Engineering , National University of Singapore , Singapore

5. Mario Berta Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Germany

6. Ludovico Lami