Testing Bell inequalities with photon-subtracted Gaussian states

Recently, photon-subtracted Gaussian states (PSGSs), which well approximate superpositions of coherent states (SCSs) with small amplitudes, were generated by several experimental groups. We compare the ideal SCSs and the PSGSs for experimental tests of certain types of Bell inequalities. In particular, we analyze the effects of the key experimental components used to generate PSGSs: mixedness of the Gaussian states, limited transmittivity of the beam splitter, and an avalanche photodetector that cannot resolve photon numbers. As a result of this analysis, the degrees of mixedness and the beam splitter transmittivity that can be allowed for successful tests of Bell inequalities are revealed.

Figure 1

Maximized fidelity $F$ between an ideal odd SCS of amplitude $\alpha$ and a pure photon-subtracted Gaussian state.

Figure 2

(a) Optimized Bell-CHSH function $B={\mid B_{\mathrm{CHSH}}\mid }_{\max }$ for a PSGS (solid curve) and a SCS (dashed curve), each of them divided at a 50:50 beam splitter, with photon number parity measurements and displacement operations. (b) The optimized Bell-CH function $B={\mid B_{\mathrm{CH}}\mid }_{\max }$ for a PSGS (solid curve) and a SCS (dashed curve), each of them divided at a 50:50 beam splitter, with photon measurements and displacement operations.

Figure 3

(a) Numerically optimized Bell-CHSH function $B={\mid B_{\mathrm{CHSH}}\mid }_{\max }$ for a PSGS using a pure Gaussian state of $r=0.3$ ($2.61\mathrm{dB}$, solid curve) and mixed Gaussian states (dashed curve). The mixed Gaussian state corresponds to $-2.56\mathrm{dB}$ below and $2.65\mathrm{dB}$ above the vacuum level (the two dimensional variance $A\times B$ is about 1.02). The horizontal axis represents the transmittivity $T$ of the beam splitter used to generate the PSGS. (b) Success probability of the pure (solid curve) and mixed (dashed curve) cases shown in (a).

Figure 4

(a) Numerically optimized Bell-CH function $B={\mid B_{\mathrm{CH}}\mid }_{\max }$ for a PSGS using a pure Gaussian state of $r=0.3$ ($2.61\mathrm{dB}$, solid curve) and mixed Gaussian states. The mixed state case represented by the dashed curve corresponds to $-2.52\mathrm{dB}$ below and $2.69\mathrm{dB}$ above the vacuum level (the variance $A\times B$ is about 1.04). The case represented by the dotted curve corresponds to $-2.43\mathrm{dB}$ below and $2.78\mathrm{dB}$ above the vacuum level (the variance $A\times B$ is about 1.08). (b) Numerically optimized Bell-CH function for a PSGS using an experimentally feasible Gaussian state. The squeezing degree is $-3.57\mathrm{dB}$ below and $4.26\mathrm{dB}$ above the vacuum level. The two-dimensional variance $A\times B$ in this case is about 1.17. (c) Success probability $P_s$ of the case shown in (b).

Figure 5

(a) Numerically optimized Bell-CHSH function $B={\mid B_{\mathrm{CHSH}}\mid }_{\max }$ for a PSGS using a pure Gaussian state of $r=0.3$ with a beam splitter of transmittivity $T$ and a realistic detector of efficiency $ϵ$. The quantum efficiency of the detector is considered to be $ϵ=1$ (solid line), 0.8 (dashed line), and 0.6 (dotted line). (b) Success probability $P_s$ for the cases shown in (a).

Figure 6

(a) Numerically optimized Bell-CH function $B={\mid B_{\mathrm{CH}}\mid }_{\max }$ for a PSGS using a pure Gaussian state of $r=0.3$ with a beam splitter of transmmittivity $T$ and a realistic detector of efficiency $ϵ$. The quantum efficiency of the detector is considered to be $ϵ=1$ (solid line), 0.8 (dashed line), and 0.6 (dotted line). (b) Numerically optimized Bell-CH function $B$ against the detection efficiency $ϵ$ for $T=0.95$ (solid line) and 0.98 (dashed line).

Figure 7

Numerically optimized Bell-CH function $B={\mid B_{\mathrm{CH}}\mid }_{\max }$ for a PSGS using a pure Gaussian state of $r=0.3$ against the modal purity factor $P_m$ with a beam splitter of transmittivity $T=0.99$ (solid line) and 0.95 (dashed line). The detection efficiency was supposed to be $ϵ=0.6$.