Double Markovity for quantum systems
Masahito Hayashi, Jinpei Zhao
TL;DR
This work addresses the challenge of extending the classical subadditivity--doubling--rotation (SDR) paradigm to quantum information by introducing two quantum analogues of double Markovity. It proves a tripartite result equating simultaneous quantum Markov conditions $A-B-C$ and $A-C-B$ to the existence of compatible projective measurements on $B$ and $C$ that yield a common label $J$ with $A-J-(B,C)$, and a conditional four-party result under full support showing $A-(B,D)-C$ and $A-(C,D)-B$ are equivalent to $A-D-(B,C)$. These theorems rely on a quantum Markov decomposition framework and, for the second result, a minimal direct-sum decomposition with a uniqueness property under full support. Together, they remove a key bottleneck for transferring SDR-type equality-case analyses to quantum settings, enabling sharper entropy- and Gaussian-extremality results in quantum information theory.
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.