Violation of Bell's inequalities with preamplified homodyne detection

We show that the use of probabilistic noiseless amplification in entangled coherent state-based schemes for the test of quantum nonlocality provides substantial advantages. The threshold amplitude to falsify a Bell-CHSH nonlocality test, in fact, is significantly reduced when amplification is embedded into the test itself. Such a beneficial effect holds also in the presence of detection inefficiency. Our study helps in affirming noiseless amplification as a valuable tool for coherent information processing and the generation of strongly nonclassical states of bosonic systems.

Figure 1

Scheme for the violation of the CHSH inequality with amplified entangled coherent state. We show the source of ECS states, the local oscillator (LO) needed for homodyne measurements, and the decomposition of the local unitary transformations ${\widehat{U}}_j$ given in terms of the rotations $\widehat{R}\left({\theta }_j\right)$ and local amplification ${\widehat{G}}_j(j=A,B)$. We also show the symbols for beam splitters and homodyners.

Figure 2

Bell-CHSH function $\mathcal{B}(\tilde{\alpha },\Theta )$, optimized over the set of rotation angles $\Theta$, plotted against $\alpha$ for $g=1$ (solid red line), $g=2$ (blue dashed line), and $g=3$ (purple dot-dashed line). The light straight line marks the local realistic bound.

Figure 3

Bell function (optimized numerically over the set of rotation angles $\Theta$) plotted against the coherent-state amplitude $\alpha$ for $g=1.0$ (red curve) and $g=1.4$ (blue curve). Inset: Same as in the main panel but for $g=1.0$ (red curve) and $g=1.1$ (blue curve) and $\alpha \in [0,2]$. Even small increases in the gain factor result in noticeable reductions of the threshold for the violation of the Bell-CHSH inequality.

Figure 4

Numerically optimized Bell's function plotted against the amplitude of the coherent states with detection inefficiencies. The black point indicates the value of $\alpha$ for which the violation occurs with perfect detectors ($\eta =1$) and no amplification. Setting $\eta =0.9$ we obtained the purple curve for $g=1.0$ and the green curve for $g=1.4$.

Figure 5

Numerically optimized Bell's function plotted against the amplitude of the coherent states with effective rotations for $g=1.0$ (purple curve) and $g=1.3$ (green curve). The black point represents the value of $\alpha$ for which the violation occurs with ideal rotations and no amplification.