Computation of the smooth max-mutual information via semidefinite programming

We present an iterative algorithm based on semidefinite programming (SDP) for computing the quantum smooth max-mutual information $I_{max}^{\epsilon }\left({\rho }_{\mathrm{AB}}\right)$ 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.