Belavkin-Staszewski Quantum Markov Chains

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 conditional expectations. As an application of the correspondence, we find the first family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.

References in corpus (13)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Quantum conditional mutual information and approximate Markov chains
• Monotonicity of quantum relative entropy revisited
• Universal recovery maps and approximate sufficiency of quantum relative entropy
• Different quantum f-divergences and the reversibility of quantum operations
• Sufficiency in quantum statistical inference
• Multivariate Trace Inequalities
• Quantum Approximate Markov Chains are Thermal
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Structure of sufficient quantum coarse-grainings
• A strengthened data processing inequality for the Belavkin-Staszewski relative entropy
• General Continuity Bounds for Quantum Relative Entropies
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures

Cited by in corpus (1)

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