Proposal for a Loophole-Free Bell Test Using Homodyne Detection

We propose a feasible optical setup allowing for a loophole-free Bell test with efficient homodyne detection. A non-Gaussian entangled state is generated from a two-mode squeezed vacuum by subtracting a single photon from each mode, using beam splitters and standard low-efficiency single-photon detectors. A Bell violation exceeding 1% is achievable with 6 dB squeezed light and a homodyne efficiency around 95%. A detailed feasibility analysis, based upon the recent experimental generation of single-mode non-Gaussian states, suggests that this method opens a promising avenue towards a complete experimental Bell test.

Figure 1

Proposed experimental setup. The source (controlled by Sophie) is based on a master laser beam, which is converted into second harmonic in a nonlinear crystal (SHG). After spectral filtering (F), the second harmonic beam pumps an OPA which generates two-mode squeezed vacuum in modes $A$ and $B$. Single photons are conditionally subtracted from modes $A$ and $B$ with the use of the beam splitters ${\mathrm{B}\mathrm{S}}_A$ and ${\mathrm{B}\mathrm{S}}_B$ and single-photon detectors ${\mathrm{P}\mathrm{D}}_A$ and ${\mathrm{P}\mathrm{D}}_B$. Alice (Bob) measures a quadrature of mode $A$ ($B$) using a balanced homodyne detector that consists of a balanced beam splitter ${\mathrm{B}\mathrm{S}}_3$ (${\mathrm{B}\mathrm{S}}_4$) and a pair of highly efficient photodiodes. The local oscillators ${\mathrm{L}\mathrm{O}}_A$ and ${\mathrm{L}\mathrm{O}}_B$ are extracted from the laser beam by means of two additional beam splitters ${\mathrm{B}\mathrm{S}}_1$ and ${\mathrm{B}\mathrm{S}}_2$. The random switching of the relative phase $\theta$ ($\varphi$) between ${\mathrm{L}\mathrm{O}}_A$ and $A$ (${\mathrm{L}\mathrm{O}}_B$ and $B$) can be performed using fast electro-optical modulators.

Figure 2

Violation of Bell-CHSH inequality with the conditionally prepared non-Gaussian state. (a) One-dimensional cut of the Wigner function of the conditionally generated non-Gaussian two-mode state (with $\lambda =0.5$, $T=0.95$, and $\eta =30\%$) along the line $x^B=-x^A$, $p^A=p^B=0$. Notice the regions where $W$ is negative. (b) Bell parameter $S$ as a function of the squeezing $\lambda$ of the initial two-mode squeezed vacuum. The curves are plotted for perfect detectors ($\eta ={\eta }_{\mathrm{B}\mathrm{H}\mathrm{D}}=100\%$) with $T=0.9$ (solid line), $T=0.95$ (dashed line), and $T=0.99$ (dot-dashed line). The open circles mark the points where $T\lambda =0.57$. (c) Bell parameter $S$ as a function of the efficiency $\eta$ of the single-photon detectors, for $\lambda T=0.57$ and ${\eta }_{\mathrm{B}\mathrm{H}\mathrm{D}}=100\%$. (d) Bell parameter $S$ as a function of the efficiency ${\eta }_{\mathrm{B}\mathrm{H}\mathrm{D}}$ of the balanced homodyne detectors, for $\lambda T=0.57$ and $\eta =30\%$.