Characterizing the Generalized Einstein-Podolsky-Rosen State and Extensions

TL;DR

The paper analyzes continuous-variable entanglement using the generalized EPR state, deriving a closed-form GEM $\mathcal{E}^2=2 \frac{(\sigma-\Omega)^2}{\sigma^2+\Omega^2}$ and mapping SPDC biphotons to this form with $\sigma=\sqrt{\frac{L\lambda_p}{6\pi}}$ and $\Omega=2\sigma_p$, resulting in $\mathcal{E}^2=2 \frac{(\sqrt{L\lambda_p/(24\pi\sigma_p^2)}-1)^2}{L\lambda_p/(24\pi\sigma_p^2)+1}$; it compares GEM to the Schmidt number $K=\frac14(\frac{\sigma}{\Omega}+\frac{\Omega}{\sigma})^2$ and provides an experimentally accessible method to determine entanglement via $\mathcal{E}^2=2\left(1-\frac{\sigma_{(1|2)}}{\sigma_1}\right)$ tied to the cross-spectral density $W$, enabling width measurements through inversion interferometry. The work also extends the framework to non-Gaussian entanglement, presenting a non-Gaussian state with GEM $\mathcal{E}^2=2 - \frac{\sigma\Omega(3\Omega^4+2\Omega^2\sigma^2+3\sigma^4)}{(\sigma^2+\Omega^2)^3}$ and showing that entanglement can be stronger than in Gaussian cases, highlighting the broader applicability to quantum information tasks with continuous variables.

Abstract

In their seminal paper, Einstein Podolsky and Rosen (EPR) had introduced a momentum entangled state for two particles. That state, referred to as the EPR state, has been widely used in studies on entangled particles with continuous degrees of freedom. Later that state was generalized to a form that allows varying degree of entanglement, known as the generalized EPR state. In a suitable limit it reduces to the EPR state. The generalized EPR state is theoretically analyzed here and its entanglement quantified in terms of a recently introduced generalized entanglement measure. This state can also be applied to entangled photons produced from spontaneous parametric down conversion (SPDC). The present analysis is then used in quantifying the entanglement of photons produced from the SPDC process, in terms of certain experimental parameters. A comparison is also made with the Schmidt number, which is normally used as an entanglement measure in such situations. A procedure for experimentally determining the entanglement of SPDC photons has also been described. Furthermore, an additional state exhibiting non-Gaussian entanglement has been examined, and its entanglement has been quantified.

Figures (7)

• Figure 1: Schematic diagram showing the generation of entangled particles. The generated particles travel in opposite direction along z-axis. We consider their entanglement in the transverse direction.
• Figure 2: The generalized entanglement measure plotted as percent entanglement $\mathcal{E}^2\times 100/2$, against $\sigma/\Omega$.
• Figure 3: The generalized entanglement measure plotted as percent entanglement $\mathcal{E}^2\times 100/2$, against $\sigma$ and $\Omega$.
• Figure 4: Schematic diagram showing the generation of SPDC photons. A photon from a laser falls on a nonlinear crystal, resulting in the emission of two photons which are entangled. The two photons are typically separated via a polarization beam splitter.
• Figure 5: Variation of entanglement of two SPDC photons for $\sqrt{\frac{L\lambda_p}{6\pi}} \approx 0.01~\text{mm}$, as a functions of pump beam width $2\sigma_p$.