Belavkin–Staszewski Quantum Markov Chains

Year: 2025

Type: article

Abstract: It is well known that the conditional mutual information of a quantum state is zero if, and only if, the quantum state is a quantum Markov chain. Replacing the Umegaki relative entropy in the definition of the conditional mutual information by the Belavkin–Staszewski (BS) relative entropy, we obtain the BS-conditional mutual information, and we call the states with zero BS-conditional mutual information Belavkin–Staszewski quantum Markov chains. In this article, we establish a correspondence which relates quantum Markov chains and BS-quantum Markov chains. This correspondence allows us to find a recovery map for the BS-entropy in the spirit of the Petz recovery map. Furthermore, we show that, over the set of BS-quantum Markov chains, this correspondence constitutes an entanglement-breaking map. Moreover, we prove a structural decomposition of the Belavkin–Staszewski quantum Markov chains and also study states for which the BS-conditional mutual information is only approximately zero. We subsequently extend the aforementioned correspondence, structural decomposition, and recovery map to arbitrary pairs of states and quantum channels. As an application of the correspondence, we find a family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.

Source: Annales Henri Poincaré

DOI: 10.1007/s00023-025-01644-1

Authors: Andreas Bluhm, Ángela Capel, Pablo Costa Rico, Anna Jenčová

Institutions: Institut polytechnique de Grenoble, Centre National de la Recherche Scientifique, Université Grenoble Alpes, University of Cambridge, University of Tübingen, Munich Center for Quantum Science and Technology, Technical University of Munich, Slovak Academy of Sciences, Mathematical Institute of the Slovak Academy of Sciences

Language: English

Topic: Quantum Information and Cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche, Bundesministerium für Bildung und Forschung, Ministerstvo školstva, vedy, výskumu a športu Slovenskej republiky, Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg, Agentúra na Podporu Výskumu a Vývoja, Technische Universität München, Vedecká Grantová Agentúra MŠVVaŠ SR a SAV, Universidad de Granada, QuantERA, Horizon 2020 Framework Programme, Horizon 2020

OpenAlex ID: W7105517651