  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.
