Quantum entanglement in phase space

Abstract

While commonly used entanglement criteria for continuous variable systems are based on quadrature measurements, here we study entanglement detection from measurements of the Wigner function. These are routinely performed in platforms such as trapped ions and circuit QED, where homodyne measurements are difficult to be implemented. We provide complementary criteria which we show to be tight for a variety of experimentally relevant Gaussian and non-Gaussian states. Our results show novel approaches to detect entanglement in continuous variable systems and shed light on interesting connections between known criteria and the Wigner function.

Figures (12)

• Figure 1: Full information of a bipartite continuous-variable system is contained in a joint Wigner function $W_{AB}(x_A,p_A,x_B,p_B)$ defined in a four-dimensional phase space. Revealing entanglement from measurements of this function is in general a difficult task. To overcome this problem, we present entanglement criteria based on measurements on a two-dimensional slice $W_{AB}(x,p,x',p')$, where $x'$ and $p'$ are linear functions of $x,p$, representing a coordinate transformation.
• Figure 2: Comparison between criteria \ref{['crit1']}, \ref{['crit2']} and Simon's for TMST states. Since the states considered here are Gaussian, Simon's criterion is both necessary and sufficient SimonPRL2000. It identifies all states above the black solid line, $\eta = \tanh^2{(r)}$, as entangled (shaded region, independent of $s$). This bound coincides with the one given by criterion \ref{['crit1']}. On the other hand, criterion \ref{['crit2']} detects entanglement only for states above the dashed lines, each for a different $s$ value.
• Figure 3: Entanglement detection for dephased cat states Eq. \ref{['eq:rhocat']}. (a) Slice of the joint Wigner function defined by $W_{AB}(x,p,x,-p)$, for $\gamma=2$ and $\epsilon=1$. Yellow dashed circles indicate the smallest region $R$ needed to violate criterion \ref{['crit2']}. (b) Amount of violation for criterion \ref{['crit2']} with $\theta=\pi/4$, i.e. $\iint_{-\infty}^{\infty}|W_{AB}(x,p,x,-p)|\text{d} x \text{d} p-1/(4\pi)$, and for PPT criterion, i.e. $|\lambda_{\min}(\rho^{T_B})|$, as a function of $\epsilon$ and for three values of $\gamma$. Entanglement can be revealed by both criteria whenever $\gamma,\epsilon>0$.
• Figure 4: Comparison of the entanglement detection capabilities of different criteria, i.e. the criteria (\ref{['crit1']},\ref{['crit2']},\ref{['crit3']}), Duan-Simon criterion SimonPRL2000DuanPRL, Hillery-Zubairy criterion HilleryPRL2006, QFI criterion ManuelPRA2016 and PPT criterion PeresPRL1996SeparabilityHorodeckiPLA1996, for TMST, Werner and dephased cat states.
• Figure 5: Comparison between EPR steering criterion $\mathcal{M}$ and entanglement criterion \ref{['crit1']} and Simon's for TMST states. Both criterion \ref{['crit1']} and Simon's provide a tight bound for entanglement (black), which is the line $\eta=\tanh^2{(r)}$. $\mathcal{M}$ reveals EPR steering only for the states above red, yellow and green lines, which correspond to different squeezing parameter $s$.