Fidelity of recovery, geometric squashed entanglement, and measurement recoverability

TL;DR

This paper introduces new quantum information measures based on the fidelity of recovery, including geometric squashed entanglement and measurement recoverability, and explores their properties and implications for quantum correlations.

Contribution

It defines and analyzes the geometric squashed entanglement and surprisal of measurement recoverability, establishing their properties and relation to quantum discord and entanglement.

Findings

Geometric squashed entanglement is a valid entanglement measure and a 1-LOCC monotone.

Surprisal of measurement recoverability characterizes quantum discord-like correlations.

States with near-zero discord are approximate fixed points of entanglement breaking channels.

Abstract

This paper defines the fidelity of recovery of a quantum state on systems $A$, $B$, and $C$ as a measure of how well one can recover the full state on all three systems if system $A$ is lost and a recovery operation is performed on system $C$ alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information $I(A;B|C)$, including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continuity. We then define a (pseudo) entanglement measure based on this quantity, which we call the geometric squashed entanglement. We prove that the geometric squashed entanglement is a 1-LOCC monotone, that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is invariant with respect to local isometries, subadditive, continuous, and normalized on maximally entangled states. We next define the surprisal of measurement recoverability, which is an information quantity in the spirit of quantum discord, characterizing how well one can recover a share of a bipartite state if it is measured. We prove that this discord-like quantity satisfies several properties, including non-negativity, faithfulness on classical-quantum states, invariance with respect to local isometries, a dimension bound, and normalization on maximally entangled states. This quantity combined with a recent breakthrough of Fawzi and Renner allows to characterize states with discord nearly equal to zero as being approximate fixed points of entanglement breaking channels. Finally, we discuss a multipartite fidelity of recovery and several of its properties.