Structure of states which satisfy strong subadditivity of quantum entropy with equality
Patrick Hayden, Richard Jozsa, Denes Petz, Andreas Winter
TL;DR
This work provides a complete characterization of quantum states that saturate the strong subadditivity inequality for von Neumann entropy. By combining Petz’s equality condition for monotonicity of relative entropy with Koashi–Imoto’s structure theorem, the authors show that SSA equality occurs exactly for short quantum Markov chains: there exists a decomposition ${\cal H}_B=\bigoplus_j({\cal H}_{b^L_j}\otimes{\cal H}_{b^R_j})$ such that $\rho_{ABC}=\bigoplus_j q_j\,\rho_{A b^L_j}\otimes\rho_{b^R_j C}$, yielding $I(A:C|B)=0$ and conditional independence of $A$ and $C$ given $B$. The result integrates a recovery map viewpoint and provides explicit structure, from which familiar entropic conditions for quantum error correction and Holevo bound saturation emerge as corollaries. The work also clarifies the connection between SSA equality and quantum Markov properties, offering a concrete framework for when quantum operations preserve correlations. Open questions include extensions to infinite dimensions and approximate equality scenarios.
Abstract
We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so-called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.