Quantum mutual information redistribution in a pure tripartite state by a number partitioning algorithm

Abstract

Quantum information distribution in a tripartite state plays a fundamental role in quantum information processes. Here we investigate how a bipartite unitary transformation ${U}_{AB}$ redistributes the quantum mutual information with the third party $C$ in a tripartite pure state ${|\ensuremath{\psi}\ensuremath{\rangle}}_{ABC}$ in a ${d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B}\ifmmode\times\else\texttimes\fi{}{d}_{C}$-dimensional Hilbert space. In particular, we focus on finding out the optimal unitary transformation ${U}_{AB}^{*}$ that maximizes the quantum mutual entropy between party $A$ and party $C, I(A:C)=S({\ensuremath{\rho}}_{A})\ensuremath{-}S({\ensuremath{\rho}}_{B})+S({\ensuremath{\rho}}_{C})$. We show that the mutual entropy $I(A:C)$ is upper bounded by $2S({\ensuremath{\rho}}_{C})$ derived from the Araki-Lieb inequality. This upper bound can be realized via an optimal unitary transformation for any pure state with the rank ${r}_{C}$ of ${\ensuremath{\rho}}_{C}$ satisfying ${r}_{C}\ensuremath{\le}{d}_{A}$. For a generic pure state with ${r}_{C}>{d}_{A}$, the upper bound cannot be realized by any bipartite unitary transformation. To maximize the mutual entropy in the later case, we propose a fast numerical algorithm to produce an approximate optimal unitary transformation, where our optimization is transformed into a modified number partition problem. The validness of our algorithm is confirmed by its comparison with the results from the Adam algorithm for parameterized unitary transformations. Our approximate algorithm thus provides a practical protocol to implement redistribution of quantum mutual information for a tripartite quantum state with high dimensions.