Quantification of Gaussian Quantum Steering

Abstract

Einstein-Podolsky-Rosen steering incarnates a useful nonclassical correlation which sits between entanglement and Bell nonlocality. While a number of qualitative steering criteria exist, very little has been achieved for what concerns quantifying steerability. We introduce a computable measure of steering for arbitrary bipartite Gaussian states of continuous variable systems. For two-mode Gaussian states, the measure reduces to a form of coherent information, which is proven never to exceed entanglement, and to reduce to it on pure states. We provide an operational connection between our measure and the key rate in one-sided device-independent quantum key distribution. We further prove that Peres' conjecture holds in its stronger form within the fully Gaussian regime: namely, steering bound entangled Gaussian states by Gaussian measurements is impossible. Steering is the quantum mechanical phenomenon that allows one party, Alice, to change (i.e., to "steer") the state of a distant party, Bob, by exploiting their shared entanglement. This phenomenon, fascinatingly discussed by Schrödinger [8,9], was already noted by Einstein, Podolksy, and Rosen (EPR) in their famous 1935 paper [12], and is at the heart of the so-called EPR paradox [13]. There it was argued that steering implied an unacceptable "action at a distance," which led EPR to claim the incompleteness of quantum theory. The EPR expectations for local realism were mostly extinguished by Bell's theorems [14,15], which showed that no locally causal theory can reproduce all the correlations observed in nature [16]. The first experimental criterion for the demonstration of the EPR paradox, i.e., for the detection of quantum steering, was later proposed by Reid [17], but it was not until 2007 that the particular type of nonlocality captured by the concept of steering [8,9,12] was in fact formalized [10,18].From a quantum information perspective [10], steering corresponds to the task of verifiable entanglement distribution by an untrusted party. If Alice and Bob share a state which is steerable in one way, say from Alice to Bob, then Alice is able to convince Bob (who does not trust Alice) that their shared state is entangled, by performing local measurements and classical communication [10]. Notice that steering, unlike entanglement, is an asymmetric property: a quantum state may be steerable from Alice to Bob, but not vice versa. On the operational side, it has been recently realized that steering provides security in one-sided device-independent quantum key distribution (QKD) [19], where the measurement apparatus of one party only is untrusted. These protocols are less demanding than totally device-independent ones, for which Bell nonlocality is known to be necessary [3]. Experimentally, at variance with the case of Bell tests, a demonstration of steering free of detection and locality loopholes is in reach [19][20][21][22], which makes one-sided device-independent QKD appealing for current technology and quantum steering a practically useful concept. EPR ste...

Generation of Quantum Correlations in Bipartite Gaussian Open Quantum Systems

“…It has been shown in Refs. [35,36] that the steerability A → B is present if and only if the following relation does not hold: A measure has been proposed of how much a state with covariance matrix σ is A → B steerable with Gaussian measurements, by quantifying the amount by which the condition (14) is violated, as follows [37]: G A→B (σ) = max{0, − ln 2ν B }, where ν B is the symplectic eigenvalue of the Schur complement of A in covariance matrix σ. This quantity vanishes if and only if the state σ is not steerable by Gaussian measurements and is invariant under local symplectic transformations.…”

“…The general quantity proposed in Refs. [37,38] has a particularly simple form when the steered party has one mode:…”

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

“…We illustrate the usefulness of our methods by detecting steerability of some higher-dimensional states, for which few criteria are known so far. Moreover, we show that our method leads to the Gaussian steering criteria for general M × N -modes of continuous variables (CVs) as a special case [4,16,17].…”

“…We illustrate the usefulness of our methods by detecting steerability of some higher-dimensional states, for which few criteria are known so far. Moreover, we show that our method leads to the Gaussian steering criteria for general M × N -modes of continuous variables (CVs) as a special case [4,16,17].Non-steerability-Let us begin with the notion of nonsteerablity. Assume that two separate observers, Alice and Bob, share a bipartite quantum state ρ AB on H = H A ⊗ H B , where d A (d B ) is the dimension of H A (H B ).…”

Genuine multipartite nonlocality of permutationally invariant Gaussian states

“…Very recently, displaced parity and pseudospin observables have been considered to detect steerability of bipartite Gaussian states [59,60], and proven useful to reveal a larger set of steerable states than what can be characterized by using Gaussian measurements alone [55,61]. Identifying the boundaries of the sets of steerable or nonlocal Gaussian states (in bipartite as well as multipartite continuous variable systems), and the maximum allowed violations of corresponding inequalities when acting with specific classes of feasible measurements, would be helpful to identify optimal resources for fully or partially device-independent quantum communication using continuous variable systems.…”