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

Abstract

We derive steerability criteria applicable for both finite and infinite dimensional quantum systems using covariance matrices of local observables. We show that these criteria are useful to detect a wide range of entangled states particularly in high dimensional systems and that the Gaussian steering criteria for general M × N -modes of continuous variables are obtained as a special case. Extending from the approach of entanglement detection via covariance matrices, our criteria are based on the local uncertainty principles incorporating the asymmetric nature of steering scenario. Specifically, we apply the formulation to the case of local orthogonal observables and obtain some useful criteria that can be straightforwardly computable, and testable in experiment, with no need for numerical optimization.In quantum world, there exist some strong correlations that cannot be described in classical ways providing thereby a crucial basis for applications, e.g. in quantum information processing. Among different forms of quantum correlations, the most well studied are quantum entanglement [1] and nonlocality [2]. Nonlocality is the strongest correlation that does not admit any local realistic models [3], in which the joint probability for the outcomes a and b of local measurements A and B, respectively, are explained bywhere a hidden-variable λ is chosen according to the distribution p λ . On the other hand, quantum entanglement is the correlation distinguished from classical correlation within the framework of quantum mechanics. That is, if a quantum state shows correlation that cannot be explained by the formwhere the superscript Q refers to the restriction to quantum statistics only, it is called quantum entangled.Recently, an intermediate form of correlation between quantum entanglement and nonlocality was rigorously defined in [4]-quantum steering-and it has attracted a great deal of interest during the past decade. The concept of quantum steering envisions a situation where Alice performs a local measurement on her system, which makes it possible to steer Bob's local state depending on her choice of measurement setting [5,6]. 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,where Alice's statistics P λ (a|A) is unrestricted while Bob' statistics P Q λ (b|B) obeys quantum principles. P LHS is obviously a subset of P LHV as seen from its construction, which makes EPR steering more accessible in experiment than nonlocality [9]. There have been other remarkable works on quantum steering including its connection to measurement incompatibility [10] and the phenomenon of one-way steering [11,12], etc.. However, we need to have a more comprehensive set of steering criteria readily testable...

Quantum steering of Gaussian states via non-Gaussian measurements

“…In this Article we demonstrate that there exist two-mode Gaussian states for which Gaussian measurements cannot manifest steering, but non-Gaussian measurements can. For this purpose, we employ a formulation of steering criterion based on local uncertainty relations involving non-Gaussian measurements 47 . We study steerability of mixed Gaussian states, specifically, Gaussian states under a lossy and an amplifying channel that represent typical noisy environments.…”

Uncertainty Relations for Coarse–Grained Measurements: An Overview

“…Since 2007, EPR-steering has been understood to make up part of a hierarchy of quantum correlations, situated between entanglement [ 81 , 82 ] and Bell non-locality [ 83 ]. In addition to methods utilizing variance-based URs [ 84 ], entropic URs, such as those in Section 2.2 , can be used to identify EPR-steering [ 85 , 86 ] and to quantify high-dimensional entanglement [ 87 , 88 ]. Some of these URs can be used to test security in continuous variable quantum cryptography [ 89 , 90 ], and it has been shown that violation of entropic EPR-steering criteria are directly related to the secret key rate in one-sided device independent cryptography [ 91 ].…”

Steering Witness and Steering Criterion of Gaussian States

“…For example, Cavalcanti and James in [ 21 ] obtained the experimental criterion of EPR steering from entropy uncertainty relations. Ji et al in [ 22 ] obtained steerability criteria by using covariance matrices of local observables, which are applicable for both finite- and infinite-dimensional quantum systems. Wittmann et al in [ 23 ] gave EPR steering inequalities with three Pauli measurements; and then, as a generalization of the Pauli matrices, Marciniak et al in [ 24 ] found EPR steering inequalities with mutually unbiased bases.…”