Necessary and Sufficient Quantum Information Characterization of Einstein-Podolsky-Rosen Steering

Abstract

Steering is the entanglement-based quantum effect that embodies the "spooky action at a distance" disliked by Einstein and scrutinized by Einstein, Podolsky, and Rosen. Here we provide a necessary and sufficient characterization of steering, based on a quantum information processing task: the discrimination of branches in a quantum evolution, which we dub subchannel discrimination. We prove that, for any bipartite steerable state, there are instances of the quantum subchannel discrimination problem for which this state allows a correct discrimination with strictly higher probability than in absence of entanglement, even when measurements are restricted to local measurements aided by one-way communication. On the other hand, unsteerable states are useless in such conditions, even when entangled. We also prove that the above steering advantage can be exactly quantified in terms of the steering robustness, which is a natural measure of the steerability exhibited by the state. Not all entangled states are steerable, and not all steerable states exhibit nonlocality [4,5], but states that exhibit steering allow for the verification of their entanglement in a semi-device independent way: there is no need to trust the devices used by the steering party, and the ability to determine the conditional states of the steered party is sufficient [4,5,9]. In general, besides its foundational interest, steering is interesting in practice in bipartite tasks, like quantum key distribution (QKD) [10], where it is convenient and/or appropriate to trust the devices of one of two parties, but not necessarily of the other party. For example, by exploiting steering it is possible to obtain key rates unachievable in a full device-independent approach [11], but still assuming less about the devices than in a standard QKD approach [12]. For these reasons, steering has attracted a lot of interest in recent times, both theoretically and experimentally [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], mostly directed to the verification of steering. Nonetheless, an answer to the question "What is steering useful for?" that applies to states that exhibit steering can arguably be considered limited [9,12]. Furthermore, the quantification of steering has just started to be addressed [24].In this Letter we fully characterize and quantify steering in an operational way that nicely matches the asymmetric features of steering, and that breaks new ground in the investigation of the usefulness of steering. We prove that every steerable state is a resource in a quantum information task that we dub subchannel discrimination, in a practically relevant scenario where measurements can only be performed locally.Subchannel discrimination is the identification of which branch of a quantum evolution a quantum system undergoes (see Fig. 1). It is well known that entanglement between a probe and an ancilla can help in discriminating different channels [31][32][33][34][35][36][37][38][39][40][41]. In [42] it was proven that actually every ...

Measures and applications of quantum correlations

“…All of these forms of non-classical correlations can enable classically impossible tasks. For instance, device-independent quantum cryptography requires nonlocality [5], entanglement-assisted subchannel discrimination depends on steering [14], while quantum teleportation and dense coding exploit plain entanglement [3]. In this review we shall focus on the lower end of the spectrum, i.e., quantum correlations (QCs) beyond entanglement.…”

Sufficient criterion for guaranteeing that a two-qubit state is unsteerable

“…It was demonstrated experimentally, see e.g. [7], and finds application in quantum information processing [8][9][10].…”

Steering criteria via covariance matrices of local observables in arbitrary-dimensional quantum systems

“…The probability of Alice getting the output a for measurement A k is given by T r ρ a|k = λ P (λ)P (a|A k , λ), and the set of un-normalized states ρ a|k is referred to as an assemblage [8]. If the assemblage for a given state ρ AB is written in the form of Eq.…”

“…This notion is practically relevant when Bob wants to confirm quantum correlation although he cannot trust Alice or her devices at all [4], leading to some applications, e.g. one-sided device-independent cryptography [7] and subchannel discrimination [8]. In view of joint probability distribution, steering is the quantum correlation that can rule out the local hidden state (LHS) models,…”