“Squashed entanglement”: An additive entanglement measure

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.

Fundamental rate-loss tradeoff for optical quantum key distribution

“…Maurer and Wolf 18 later introduced the intrinsic information I(X;YkZ) min{I(X; Y|Z 0 ): P X,Y,Z,Z 0 ¼ P X,Y,Z P Z 0 |Z }, and proved that this quantity optimized over all channel input distributions is a sharp upper bound on the secret key agreement capacity of p Y,Z|X . Leveraging strong parallels discovered between secrecy and quantum coherence [19][20][21] , Christandl and Winter 22 extended the intrinsic information quantity to the realm of quantum information theory. They defined the squashed entanglement E sq (A;B) r of a bipartite quantum state r AB and proved it to be an upper bound on the rate at which two parties can distil maximally entangled (Bell) states 0…”

“…We can interpret E sq (A;B) r as quantifying the minimum remnant quantum correlations between A and B after an adversary possessing the purifying system E performs a quantum operation on it with the intent of 'squashing down' the correlations that A and B share. It should also be noted that among the many entanglement measures, squashed entanglement is the only one known to satisfy all eight desirable properties that have arisen in the axiomatization of entanglement theory 22,[27][28][29] .…”

“…Note that, in the above formula, we can take a maximum rather than a supremum if the input space is finite-dimensional because, in this case, the input space is compact and the squashed entanglement measure is continuous 28 . In addition, we can restrict the optimization to be taken over pure bipartite states because of the convexity of squashed entanglement 22 . We now prove that E sq (N ) plays an operational role analogous to intrinsic information, that is, it upper bounds the secret key agreement capacity P 2 (N ).…”

Robustness of Quantum Markov Chains

“…Note that if there were a bound of the form ∆(ρ) ≤ f (I) log(d A d C ) -in particular not depending on the dimension of B -, then this would settle a question left open in [7]: namely, it would imply that the "squashed entanglement" E sq (ρ AB ) of a bipartite state ρ AB is zero if and only if the state is separable. (We are grateful to Paweł Horodecki for pointing this out to us.…”

Мера Сцепленности На Основе Наблюдаемых Корреляций

“…В настоящее время имеются разнообразные определения меры сцепленности, такие как сцепленность формиро-вания [5]- [8], дистиллируемая сцепленность [5], [7], [9], сцепленность относительной 454 ШУНЬ-ЛУН ЛО энтропии [10], отрицательность [11], сжатая сцепленность [12] и т.д. Прототипом всех этих понятий является сцепленность формирования, которая определяется для смешанного состояния ρ двух подсистем как…”