Computation of the Smooth Max-Mutual Information via Semidefinite Programming

TL;DR

We address the computation of the smooth max-mutual information $I^\varepsilon_{max}(\rho_{AB})$ for bipartite quantum states, a task that is generally hard due to bilinear smoothing constraints. The authors introduce an iterative SDP-based algorithm (a mountain-climbing style method) that alternates between solving two SDPs and, under a full-rank marginal constraint, converges to the exact value via a proven strong duality framework for a core SDP. They derive and analyze the primal and dual formulations of this core SDP, establishing strong duality and finite-step convergence, and show how the method extends existing SDP techniques for one-shot information measures. The resulting algorithm provides accurate estimates or upper bounds for $I^\varepsilon_{max}$, enabling improved bounds on one-shot distillable key and cryptographic capacities in noisy quantum environments.

Abstract

We present an iterative algorithm based on semidefinite programming (SDP) for computing the quantum smooth max-mutual information $I^\varepsilon_{\max}(ρ_{AB})$ 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.

Theorems & Definitions (8)

• Definition 1: Fidelity, sine distance
• Definition 2: Smoothing environment
• Definition 3: Max-relative entropy, smooth max-mutual information
• Definition 4: Semidefinite program
• Proposition 2
• proof
• Proposition 3
• proof