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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msubsup><mml:mi>I</mml:mi> <mml:mo>max</mml:mo> <mml:mi>ε</mml:mi></mml:msubsup> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>ρ</mml:mi> <mml:mrow><mml:mi>AB</mml:mi></mml:mrow> </mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> 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.