Double Markovity for quantum systems

TL;DR

This paper develops quantum analogues of double Markovity, enabling the extension of classical SDR techniques to quantum information theory by characterizing Markov conditions in tripartite and four-party quantum states.

Contribution

It establishes quantum conditions for double Markovity, removing a key obstacle in applying classical SDR methods to quantum systems.

Findings

Characterization of quantum Markov conditions A-B-C and A-C-B.

Equivalence of certain four-party quantum Markov conditions.

Removal of a bottleneck in extending SDR techniques to quantum systems.

Abstract

The subadditivity-doubling-rotation (SDR) technique is a powerful route to Gaussian optimality in classical information theory and relies on strict subadditivity and its equality-case analysis, where double Markovity is a standard tool. We establish quantum analogues of double Markovity. For tripartite states, we characterize the simultaneous Markov conditions A-B-C and A-C-B via compatible projective measurements on B and C that induce a common classical label J yielding A-J-(BC). For strictly positive four-party states, we show that A-(BD)-C and A-(CD)-B hold if and only if A-D-(BC) holds. These results remove a key bottleneck in extending SDR-type arguments to quantum systems.