Revealing entanglement through local features of phase-space distributions

Abstract

We formulate an infinite hierarchy of continuous-variable separability criteria in terms of quasiprobability distributions and their derivatives evaluated at individual points in phase space. Our approach is equivalent to the Peres--Horodecki criterion and sheds light on how distillable entanglement manifests in the phase-space picture. We demonstrate that already the lowest-order variant constitutes a powerful method for detecting the elusive non-Gaussian entanglement of relevant state families. Further, we devise a simple measurement scheme that relies solely on passive linear transformations and coherent ancillas. By strategically probing specific phase-space regions, our method offers clear advantages over existing techniques that rely on access to the full phase-space distributions.

Figures (8)

• Figure 1: Negative regions of the minor $M_2(\alpha,\beta;1)$ for $N=3,4,5$ from left to right, with $\alpha,\beta$ being real. Entanglement is witnessed over entire regions rather than at isolated points in phase space, making the witness robust.
• Figure 2: Separability criterion $M_2(\alpha_\textrm{opt},\beta_\textrm{opt};\sigma)\overset{\text{sep.}}{\geq} 0$ for various $N$ as a function of the $\sigma$-parameter, where $\alpha_\textrm{opt},\beta_\textrm{opt}$ are the coordinates for the global minimum of the Husimi-based criterion, $M_2(\alpha,\beta;1)$. The Husimi-based criterion is optimal, with increasing $N$ requiring larger $\sigma$ for flagging entanglement.
• Figure 3: Separability criterion $M_2(\alpha_\textrm{opt},\beta_\textrm{opt};1)\overset{\text{sep.}}{\geq} 0$ for lossy NOON states \ref{['eq:LossyNOONStates']} with loss parameter $(1-\tau)$. Losses impact larger values of $N \gtrsim 6$, whilst entanglement is certified even for high loss-levels when $N \lesssim 3$.
• Figure 4: Negative regions of the Husimi-based criterion $M_2(\alpha,\alpha;1)$ for the pure odd cat state $\boldsymbol{\rho}_\textrm{cat}(\gamma,\gamma,0)$. We evaluate the minor at $\arg(\alpha)=\arg(\gamma)$ (top, linear scale) and $\arg(\alpha)=\arg(\gamma)+\pi/2$ (bottom, log scale). The latter enables efficient entanglement detection for arbitrary coherent amplitudes $\lvert\gamma\rvert$.
• Figure 5: $M_2(\mathrm{i}\mkern1mu\gamma,\mathrm{i}\mkern1mu\gamma;1)$ for odd cat states $\boldsymbol{\rho}_\textrm{cat}(\gamma,\gamma,p)$ for varying dephasing parameter $p$. Entanglement is certified for all $p<1$.