Quantum steerability: Characterization, quantification, superactivation, and unbounded amplification

Abstract

Quantum steering, also called Einstein-Podolsky-Rosen steering, is the intriguing phenomenon associated with the ability of spatially separated observers to steer-by means of local measurementsthe set of conditional quantum states accessible by a distant party. In the light of quantum information, all steerable quantum states are known to be resources for quantum information processing tasks. Here, via a quantity dubbed steering fraction, we derive a simple, but general criterion that allows one to identify quantum states that can exhibit quantum steering (without having to optimize over the measurements performed by each party), thus making an important step towards the characterization of steerable quantum states. The criterion, in turn, also provides upper bounds on the largest steering-inequality violation achievable by arbitrary finite-dimensional maximally entangled states. For the quantification of steerability, we prove that a strengthened version of the steering fraction is a convex steering monotone and demonstrate how it is related to two other steering monotones, namely, steerable weight and steering robustness. Using these tools, we further demonstrate the superactivation of steerability for a well-known family of entangled quantum states, i.e., we show how the steerability of certain entangled, but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state. In particular, our approach allows one to explicitly construct a steering inequality to manifest this phenomenon. Finally, we prove that there exist examples of quantum states (including some which are unsteerable under projective measurements) whose steering-inequality violation can be arbitrarily amplified by allowing joint measurements on as little as three copies of the same state. For completeness, we also demonstrate how the largest steering-inequality violation can be used to bound the largest Bell-inequality violation and derive, analogously, a simple sufficient condition for Bell-nonlocality from the latter.

Quantum fidelity measures for mixed states

“…For example, by using the fidelity bounds, it was proved [94] that any entangled twoqubit (mixed) state is useful for quantum teleportation if arbitrary local operations assisted by classical communications (including arbitrary local filtering operations) are allowed prior to the teleportation experiment. Similarly, a connection has been shown, between no-cloning teleportation and the requirement of a steerable resource [95], between the fully-entangled fraction and a steerable [52] resource, as well as a Bellnonlocal [96] resource, see, e.g., [97] and references therein.…”

Resource Preservability

“…This property forbids any possibility to activate the Rpreservability. This is, however, not true due to the existence of activation properties of certain resources [66][67][68][69][70]. More precisely, in Appendix A we show that in some Rtheories, one can construct a free operation T ∈ O N R such that T ⊗k / ∈ O N R for some k > 0.…”

Superactivation of quantum steering

“…While completing this manuscript, we became aware of the work of [46] who independently derived a sufficient condition for steerability in terms of the fully entangled fraction and demonstrated the super-activation of steering for the isotropic states. We now reproduce a lemma first presented in Ref.…”