Optimal unambiguous comparison of two unknown squeezed vacua

We propose a scheme for the unambiguous state comparison (USC) of two unknown squeezed vacuum states of the electromagnetic field. Our setup is based on linear optical elements and photon-number detectors, and it achieves optimal USC in an ideal case of unit quantum efficiency. In realistic conditions, i.e., for nonunit quantum efficiency of photodetectors, we evaluate the probability of getting an ambiguous result as well as the reliability of the scheme, thus showing its robustness in comparison to previous proposals.

Figure 1

(a) Schematic diagram of a setup for USC of the squeezed vacuum states $S(\xi )|0\rangle$ and $S(\zeta )|0\rangle$. (b) Scheme leading to the same output state $|{\Psi }_{\mathrm{out}}(\xi ,\zeta )\rangle \rangle$ as in (a) when the two squeezing phases are unknown but equal, i.e., $\arg (\xi )=\arg (\zeta )$. We defined $r_{\pm }=(\xi \pm \zeta )/2$. See the text for comments and details.

Figure 2

Conditional probability of revealing the difference of two squeezed vacuum states $|{\psi }_1\rangle =S(r)|0\rangle$ and $|{\psi }_2\rangle =S(s)|0\rangle$ in the ideal case ($\eta =1$) as a function of ${\delta }_{-}=|r-s|$. Solid lines, from top to bottom, correspond to the optimal squeezed-states comparator (blue) and the universal comparator (red line). The dashed line is the upper bound on the probability in the case of only two possible squeezings. See Sec. 4c for details.

Figure 3

$p_{\eta }(0|r,r)$ (solid lines) and $p_{\eta }(D|r,r)$ (dot-dashed lines) as functions of $r$ for different values of the efficiency $\eta$; from top to bottom (solid) and from bottom to top (dot-dashed): $\eta =0.999$ (red), $0.99$ (green), $0.90$ (blue), $0.50$ (magenta).

Figure 4

Top: Reliability $R_D(\eta ;r,s)$ as a function of ${\delta }_{-}$ for fixed ${\delta }_{+}=1.0$ and different values of the efficiency. Bottom: Reliability $R_D(\eta ;r,s)$ as a function of ${\delta }_{+}$ for difference ${\delta }_{-}=0.2$ and different values of the efficiency. In both plots, from top to bottom: $\eta =0.999$ (red), $0.99$ (green), $0.90$ (blue), $0.50$ (magenta).