Boosting linear-optical Bell measurement success probability with predetection squeezing and imperfect photon-number-resolving detectors

Linear-optical realizations of Bell state measurement (BSM) on two single-photon qubits succeed with probability $p_s$ no higher than 0.5. However, predetection quadrature squeezing, i.e., quantum noise limited phase sensitive amplification, in the usual linear-optical BSM circuit, can yield $p_s\approx 0.643$. The ability to achieve $p_s>0.5$ has been found to be critical in resource-efficient realizations of linear-optical quantum computing and all-photonic quantum repeaters. Yet, the aforesaid value of $p_s>0.5$ is not known to be the maximum achievable using squeezing, thereby leaving it open whether close-to-$100\%$ efficient BSM might be achievable using squeezing as a resource. In this paper, we report insights on why squeezing-enhanced BSM achieves $p_s>0.5$. Using this, we show that the previously reported $p_s\approx 0.643$ at single-mode squeezing strength $r=0.6585$—for unambiguous state discrimination (USD) of all four Bell states—is an experimentally unachievable point result, which drops to $p_s\approx 0.59$ with the slightest change in $r$. We, however, show that squeezing-induced boosting of $p_s$ with USD operation is still possible over a continuous range of $r$, with an experimentally achievable maximum occurring at $r=0.5774$, achieving $p_s\approx 0.596$. Finally, deviating from USD operation, we explore a trade space between $p_s$, the probability with which the BSM circuit declares a “success,” versus the probability of error $p_e$, the probability of an input Bell state being erroneously identified given the circuit declares a success. Since quantum error correction could correct for some $p_e>0$, this tradeoff may enable better quantum repeater designs by potentially increasing the entanglement generation rates with $p_s$ exceeding what is possible with traditionally studied USD operation of BSMs.

Figure 1

The optical circuit that identifies a randomly input Bell state $|\zeta \rangle \in \left\{|{\psi }_{+}\rangle ,|{\psi }_{-}\rangle ,|{\varphi }_{+}\rangle ,|{\varphi }_{-}\rangle \right\}$ with success probability $p_s=0.5$. With probability $1-p_s=0.5$, it produces an erasure (“I don't know”) outcome. This is an example of unambiguous state discrimination (USD).

Figure 2

An optical circuit which can identify unambiguously a randomly input Bell state $|\zeta \rangle \in \left\{|{\psi }_{+}\rangle ,|{\psi }_{-}\rangle ,|{\varphi }_{+}\rangle ,|{\varphi }_{-}\rangle \right\}$ with $p_s>50\%$. $S\left(\xi \right)$ represents a single-mode squeezer (e.g., an OPA), where $\xi =\frac{1}{2}{e}^{i\varphi }\tanh r$ is the degree of squeezing.

Figure 3

Bell measurement success probability ($p_s$) vs squeezing intensity ($r$) for a squeezed Bell measurement circuit.

Figure 4

Bell measurement success probability ($p_s$) vs squeezing intensity ($r$) for a predetection-squeezed BSM circuit, and with PNR detectors with maximum number resolution $n_{\max }\in \left\{1,2,...,12\right\}$. The case of $n_{\max }=\infty$ from Fig. 3 is also shown for comparison (top black curve).

Figure 5

Bell measurement success probability ($p_s$) vs squeezing amplitude ($r$), with respect to error probability ($p_e$), for a predetection-squeezed BSM circuit, and with PNR detectors restricted to a maximum number resolution $n_{\max }=7$. The black dots represent the results of USD measurements with $p_e=0$. Error probability ($p_e$) is restricted to [0%,10%], [0%,1%], [0%,0.1%], and [0%,0.01%] in the top, second from top, second from bottom, and bottom plots, respectively.

Figure 6

Bell measurement success probability ($p_s$) vs measurement confidence ($\alpha$), with respect to error probability ($p_e$), for a squeezed Bell measurement circuit, measured with PNR detectors restricted a PNR of $n_{\text{max}}=7$. The black dots represent the results of USD measurements with $p_e=0$.

Figure 7

Maximum Bell measurement success probability ($p_s$) vs measurement confidence ($\alpha$) for a squeezed Bell measurement circuit, shown for PNR detectors with PNR limited to $n_{\text{max}}\in [1,12]$. Each curve terminates at the maximum possible PSD $p_s$ which can be obtained for a given $n_{\text{max}}$.

Figure 8

Predetection quadrature squeezing Bell measurement setup with detector loss represented as predetection beam splitters.

Figure 9

Maximum Bell measurement success probability ($p_s$) vs measurement confidence ($\alpha$) for a squeezed Bell measurement circuit, shown for lossy PNR detectors with PNR limited to $n_{\text{max}}=[1,9]$ with efficiency $\eta \in [0.90,1]$. The black lines represent the success probabilities for lossless detectors shown in Fig. 7.

Figure 10

Maximum Bell measurement success probability ($p_s$) vs measurement confidence ($\alpha$) for a squeezed Bell measurement circuit, shown for lossy PNR detectors with PNR limited to $n_{\text{max}}=[7,9]$ with efficiency $\eta \in [0.90,1]$.