Fidelity of recovery, squashed entanglement, and measurement recoverability

Abstract

This paper defines the fidelity of recovery of a tripartite 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 (i.e., monotone non-increasing with respect to local operations and classical communication from Bob to Alice), 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 nonnegativity, 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 (equivalently, they are recoverable from the state of a measuring apparatus). Finally, we discuss a multipartite fidelity of recovery and several of its properties.

Measures and applications of quantum correlations

“…Since every entanglement-breaking channel can be written as a sequence of a measurement map followed by a preparation (recovery) map, as discussed in Section 2.3.5, such a facet of QCs can be quantitatively captured by how far the state ρ AB of the overall bipartite system is from being a fixed point of an entanglement-breaking channel [50], along the lines of Definition 6. Given a distance D δ , we can then define the recoverability type measures of QCs as Q…”

“…The recoverability approach to QCs, as outlined in Section 3.2.7, has been explored only quite recently; for this reason, there is only one measure which has been considered in explicit detail so far [50], as we now review.…”

“…It then turns out that a state ρ AB is classical-quantum if, and only if, it is left invariant by at least one entanglement-breaking channel acting on subsystem A [50], i.e.,…”

“…Instances of recoverability measures of QCs as defined in recent literature [50] are presented in Section 3.9.…”

“…Surprisal of measurement recoverability The "surprisal of measurement recoverability" Q R F A has been originally defined by resorting to the (negative logarithm of) Uhlmann fidelity [50], corresponding the one-sided definition in Eqs. (62) with D δ (ρ, σ) = − log F(ρ, σ), where the fidelity F(ρ, σ) is given by Eq.…”