The Fidelity of Recovery is Multiplicative

TL;DR

This paper generalizes the fidelity of recovery measure in quantum information, proving its multiplicative property using semi-definite programming, which simplifies existing proofs of bounds on quantum correlations.

Contribution

It introduces a generalized fidelity of recovery measure and demonstrates its multiplicativity, simplifying the proof of a key lower bound on quantum mutual information.

Findings

The generalized FoR is multiplicative.

The proof uses semi-definite programming duality.

Simplifies previous operational proofs.

Abstract

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e. the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandao et al. [Phys. Rev. Lett. 115(5):050501, 2015] of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.