"Squashed Entanglement" - An Additive Entanglement Measure
Matthias Christandl, Andreas Winter
TL;DR
This work introduces squashed entanglement, a new entanglement monotone defined as the infimum of half the quantum conditional mutual information over all tripartite extensions of a bipartite state. It proves that $E_{sq}$ is LOCC monotone, convex, additive on tensor products, and generally superadditive, while providing upper and lower bounds relative to entanglement cost and distillable entanglement; for pure states it reduces to the standard entanglement measure. The authors also draw an analogy with classical intrinsic information to motivate the construction and discuss how $E_{sq}$ bounds both $E_D$ and $E_F$, offering a promising axis for resolving longstanding additivity questions. Continuity remains an open challenge, with partial results and a proposed quantum Fannes-type inequality that would establish full continuity. The work outlines future directions, including deeper connections to PPT-related measures and broader implications in quantum information theory.
Abstract
In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call "squashed entanglement": it is a lower bound on entanglement of formation and an upper bound on distillable entanglement. Furthermore, it is convex, additive on tensor products, and superadditive in general. Continuity in the state is the only property of our entanglement measure which we cannot provide a proof for. We present some evidence, however, that our quantity has this property, the strongest indication being a conjectured Fannes type inequality for the conditional von Neumann entropy. This inequality is proved in the classical case.