Investigating Einstein-Podolsky-Rosen steering of continuous-variable bipartite states by non-Gaussian pseudospin measurements

Abstract

Einstein-Podolsky-Rosen (EPR) steering is an asymmetric form of correlations which is intermediate between quantum entanglement and Bell nonlocality, and can be exploited as a resource for quantum communication with one untrusted party. In particular, steering of continuous-variable Gaussian states has been extensively studied theoretically and experimentally, as a fundamental manifestation of the EPR paradox. While most of these studies focused on quadrature measurements for steering detection, two recent works revealed that there exist Gaussian states which are only steerable by suitable non-Gaussian measurements. In this paper we perform a systematic investigation of EPR steering of bipartite Gaussian states by pseudospin measurements, complementing and extending previous findings. We first derive the density-matrix elements of two-mode squeezed thermal Gaussian states in the Fock basis, which may be of independent interest. We then use such a representation to investigate steering of these states as detected by a simple nonlinear criterion, based on second moments of the correlation matrix constructed from pseudospin operators. This analysis reveals previously unexplored regimes where non-Gaussian measurements are shown to be more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. We further consider an alternative set of pseudospin observables, whose expectation value can be expressed more compactly in terms of Wigner functions for all two-mode Gaussian states. However, according to the adopted criterion, these observables are found to be always less sensitive than conventional Gaussian observables for steering detection. Finally, we investigate continuous-variable Werner states, which are non-Gaussian mixtures of Gaussian states, and find that pseudospin measurements are always more effective than Gaussian ones to reveal their steerability. Our results provide useful insights on the role of non-Gaussian measurements in characterizing quantum correlations of Gaussian and non-Gaussian states of continuous-variable quantum systems.

Homodyne detection of non-Gaussian quantum steering

“…In panel (a) we show how Reid's criterion fails to witness any quantum steering in this case. In panel (b) we witness steering through In panel (a) we show the observations arising from the application of Reid's criterion S hom R (A) (28). The legend is the same as in panel (b).…”

“…5 we show the steering witness, as measured in the two directions. In panel (a) we use Reid's criterion (28), which leads to strongly asymmetric results, as no EPR steering from the mode where the photon was subtracted is observed. Yet, remarkably, in panel (b) the vio-lation of the metrological inequality, not only witnesses steering from the photon subtracted mode, but actually leads to a larger value for the steering witness.…”

“…One possible approach relies on conditional quantum state tomography and semidefinite programming [27]. Alternatively, many protocols are based on second order correlations [18], and for non-Gaussian states these protocols require non-Gaussian measurements [28]. The latter is twofold undesired: First, it is appealing to rely strictly on Gaussian continuous-variable (CV) measurements, such as homodyne detection.…”

Steering Witness and Steering Criterion of Gaussian States

“…Recent works also revealed that there exist Gaussian states which are only steerable by suitable non-GPOVMs. In [ 25 ], the authors considered pseudospin measurements instead of GPOVMs for any two-mode Gaussian states and found that these observables are always less sensitive than conventional Gaussian observables for steering detection. Note that GPOVMs are accessible in laboratory by means of homodyne detections and Gaussian transformations.…”

“…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. For continuous-variable systems, the authors in [ 25 ] performed a systematic investigation of EPR steering for bipartite Gaussian states by pseudospin measurements. Kogias and Adesso [ 26 ] gave a measure of EPR steering for two-mode continuous variable states.…”