# Computation of the smooth max-mutual information via semidefinite programming

## TL;DR

This paper introduces a new algorithm using semidefinite programming to compute a quantum information measure called smooth max-mutual information.

## Contribution

A novel semidefinite programming method for computing quantum smooth max-mutual information with proven strong duality.

## Key findings

- The algorithm is accurate under a rank condition for marginal states in the smoothing environment.
- The method provides an upper bound when the rank condition is not satisfied.
- The approach extends SDP-based techniques for quantum information processing tasks.

## Abstract

We present an iterative algorithm based on semidefinite programming (SDP) for computing the quantum smooth max-mutual information \documentclass[12pt]{minimal}
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\begin{document}$$I^\varepsilon _{\max }(\rho _{AB})$$\end{document}Imaxε(ρAB) of bipartite quantum states in any dimension. The algorithm is accurate if a rank condition for marginal states within the smoothing environment is satisfied and provides an upper bound otherwise. Central to our method is a novel SDP, for which we establish primal and dual formulations and prove strong duality. With the direct application of bounding the one-shot distillable key of a quantum state, this contribution extends SDP-based techniques in quantum information theory. Thereby it improves the capabilities to compute or estimate information measures with application to various quantum information processing tasks.