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

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.

Figure 1

Scheme to generate an arbitrary TMST state [52]. An EPR source prepares a two-mode squeezed vacuum state with squeezing parameter $s$, Eq. (14). One mode (right pointing blue arrow) of the state freely propagates towards Bob while the other mode (left pointing red arrow) propagates through a phase-insensitive Gaussian channel towards Alice. The channel can be decomposed [53, 54] into a quantum-limited attenuator realized by a beam splitter with intensity transmissivity $\eta$ followed by a quantum-limited amplifier implemented by a two-mode squeezer with a squeezing parameter $r$. The state shared by Alice and Bob is then a TMST state ${\widehat{\rho }}_{AB}^{\text{TMST}}(s,\eta ,r)$ with covariance matrix given by Eq. (18). See text for further details.

Figure 2

Plot of the left-hand side $M_{{\widehat{\rho }}_{AB}^{\text{EPR}}}^{\left(j\right)}\left(s\right)$ [Eq. (31)] of the squeezing criterion [Eq. (9)] using pseudospin measurements of type $j$, with $j=\text{i}$ (solid green) and $j=\text{ii}$ (dashed magenta), for a two-mode squeezed vacuum state ${\widehat{\rho }}_{AB}^{\text{EPR}}\left(s\right)$ [Eq. (14)] as a function of the squeezing parameter $s$. All the quantities plotted are dimensionless.

Figure 3

Comparison of EPR steering criteria for the lossy state ${\widehat{\rho }}_{AB}^{\text{TMST}}(s,\eta ,0)$ as a function of the initial squeezing $s$ and the attenuator transmissivity $\eta$ (see Fig. 1). $A\to B$ steering is detected by each criterion in the parameter region above the corresponding threshold curve. The considered criteria are steerability by Gaussian measurements, Eq. (32) [10] (thin solid black); nonlinear steering criterion with type-i pseudospin measurements, Eq. (33) [36] (dashed blue); local uncertainty relation criterion with two-level orthogonal observables, Eq. (35) [37] (dashed red); moment criterion, Eq. (9) [42], evaluated in this paper with type-i pseudospin measurements [38] (thick solid green); moment criterion, Eq. (9) [42], evaluated in this paper with type-ii pseudospin measurements [40] (dot-dashed magenta). In the shaded (yellow) region of parameters, steering is identified only by our criterion based on type-i pseudospin observables. All the quantities plotted are dimensionless.

Figure 4

EPR steering of TMST states ${\widehat{\rho }}_{AB}^{\text{TMST}}(s,\eta ,r)$. (a) Plot of the threshold for the transmissivity $\eta$ to detect steering as a function of the amplifier squeezing $r$, for different initial squeezing $s$ (from bottom to top: $s=0.4,0.7,1.0$). (b) Plot of the threshold for $\eta$ to detect steering as a function of $s$ (solid lines), for different $r$ (from bottom to top: $r=0.1,0.2,0.4$). Steering is detected in the regions above the corresponding curves. In both panels, solid lines indicate thresholds for the moment criterion Eq. (9) using type-i pseudospin measurements, while dashed lines indicate thresholds for the criterion Eq. (8) using Gaussian measurements. All the quantities plotted are dimensionless.

Figure 5

Threshold probability $p_{\mathrm{steer}}\equiv p_{\mathrm{steer}}^{\left(\text{i}\right)}$ [Eq. (41)] characterizing the EPR steering of the CV Werner state (37) as a function of the squeezing parameter $s$ and the thermal noise parameter $u$. According to inequality (9), the state is steerable by type-i pseudospin measurements if $p>p_{\mathrm{steer}}^{\left(\text{i}\right)}$. All the quantities plotted are dimensionless.

Figure 6

Threshold probabilities characterizing the EPR steering of the CV Werner state (37) with $u=s$. The state is steerable by type-i pseudospin measurements when $p>p_{\mathrm{steer}}^{\left(\text{i}\right)}$ [Eq. (41)] (solid blue line), by type-ii pseudospin measurements when $p>p_{\mathrm{steer}}^{\left(\text{ii}\right)}$ [Eq. (43)] (dashed green line), and by Gaussian measurements when $p>p_{\mathrm{steer}}^{\left(\text{G}\right)}$ (dotted red line) [Eq. (45)]. All the quantities plotted are dimensionless.