Optimal entangled coherent states in lossy quantum-enhanced metrology

We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decreases with more loss in the interferometry. Under the constraint of the input mean photon number, we obtain that the optimal entangled coherent state is more robust than a separable coherent state, even in a high photon loss rate. The optimal entangled coherent state preserves quantum advantage over the standard interferometric limit of the separable coherent state. We also show that the corresponding optimal measurement requires correlation measurement bases.

Figure 1

Entangled coherent state for quantum phase estimation in the presence of photon loss. $R$ is a photon loss rate. $\varphi$ represents a phase shifter, $exp\left(i\varphi \widehat{n}\right)$. The input entangled coherent state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ is prepared after a $50:50$ beam splitter.

Figure 2

Degree of entanglement for the entangled coherent state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$, as a function of $|\alpha -\beta |$.

Figure 3

Economical point of quantum phase estimation using the state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ without photon loss. Given a value of $\alpha$, we find the optimal value of $\beta$ achieving the maximum value of $F_Q/\langle {\widehat{n}}_a\rangle$, the quantum Fisher information over the input mean photon number in mode $a$. At $\alpha <0.4$, ${\beta }_{\text{opt}}=-\alpha$, which were not shown here. The optimal value of $\beta$ is numerically obtained under the constraint of $-\alpha \le \beta <\alpha$.

Figure 4

Photon loss in both arms: Economical point of quantum phase estimation using the state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ with photon loss rate $R$. Given a value of $\alpha$, we find the optimal value of $\beta$ achieving the maximum value of $F_Q/\langle {\widehat{n}}_a\rangle$, the quantum Fisher information over the input mean photon number in mode $a$, depending on the photon loss rate $R$. (a) $\alpha =0.6$, (b) $\alpha =1$, (c) $\alpha =1.8$, and (d) $\alpha =3$. We consider the constraint of $-\alpha \le \beta <\alpha$.

Figure 5

Photon loss in both arms: Comparison between the state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ and a separable coherent state ${|\alpha \rangle }_a{|\alpha \rangle }_b$ for the quantum Fisher information with $N_{\text{av}}=\langle {\widehat{n}}_a\rangle$, in the constraint of input mean photon number. $R$ is the photon loss rate. Black dashed line, ${|\alpha \rangle }_a{|\alpha \rangle }_b$; red dotted curve, ${|\alpha \rangle }_a{|0\rangle }_b+{|0\rangle }_a{|\alpha \rangle }_b$; blue solid curve, ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$. (a) $\beta =-0.2\alpha$, (b) $\beta =0.3\alpha$, (c) $\beta =0.5\alpha$, and (d) $\beta =0.7\alpha$.

Figure 6

Photon loss in one arm: Economical point of quantum phase estimation using the state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ with photon loss rate $R$. Given a value of $\alpha$, we find the optimal value of $\beta$ achieving the maximum value of $F_Q/\langle {\widehat{n}}_a\rangle$, the quantum Fisher information over the input mean photon number in mode $a$, depending on the photon loss rate $R$. (a) $\alpha =0.6$, (b) $\alpha =1$, (c) $\alpha =1.8$, and (d) $\alpha =3$. We consider the constraint of $-\alpha \le \beta <\alpha$.

Figure 7

Photon loss in one arm: Comparison between the state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ and a separable coherent state ${|\alpha \rangle }_a{|\alpha \rangle }_b$ for the quantum Fisher information with $N_{\text{av}}=\langle {\widehat{n}}_a\rangle$, in the constraint of input mean photon number. $R$ is the photon loss rate. Black dashed line, ${|\alpha \rangle }_a{|\alpha \rangle }_b$; red dotted curve, ${|\alpha \rangle }_a{|0\rangle }_b+{|0\rangle }_a{|\alpha \rangle }_b$; blue solid curve, ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$. (a) $\beta =-0.1\alpha$, (b) $\beta =0.2\alpha$, (c) $\beta =0.4\alpha$, and (d) $\beta =0.5\alpha$.

Figure 8

Photon loss in both arms: Output entanglement of the optimal entangled coherent state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ as a function of photon loss rate $R$ for fixed values of (a) $\alpha =1$, (b) $\alpha =1.8$, (c) $\alpha =3$, and (d) $\alpha =5$. Green dashed curve, $\beta =-0.2\alpha$; red dotted curve, $\beta =0$; purple dot-dashed curve, $\beta =0.5\alpha$; blue solid curve, $\beta =0.7\alpha$.

Figure 9

Photon loss in one arm: Output entanglement of the optimal entangled coherent state ${|\alpha \rangle }_a{|\beta \rangle }_b+{|\beta \rangle }_a{|\alpha \rangle }_b$ as a function of photon loss rate $R$ for fixed values of (a) $\alpha =1$, (b) $\alpha =1.8$, (c) $\alpha =3$, and (d) $\alpha =5$. Green dashed curve, $\beta =-0.1\alpha$; red dotted curve, $\beta =0$; purple dot-dashed curve, $\beta =0.4\alpha$; blue solid curve, $\beta =0.5\alpha$.

Figure 10

Economical point as a function $R,\alpha ,$ and $\beta$. Given a value of $\alpha$, we find the optimal value of $\beta$ in the range of $-\alpha \le \beta <\alpha$ to maximize $F_Q/\langle {\widehat{n}}_a\rangle$. At $R\to 0$, the optimal value of $\beta$ goes to $-\alpha$ for small $\alpha$ and it becomes zero for large $\alpha$.

Figure 11

Photon loss in both arms: Comparison between the state ${|\alpha \rangle }_a{|\beta \rangle }_b-{|\beta \rangle }_a{|\alpha \rangle }_b$ and a separable coherent state ${|\alpha \rangle }_a{|\alpha \rangle }_b$ for the quantum Fisher information with $N_{\text{av}}=\langle {\widehat{n}}_a\rangle$, in the constraint of input mean photon number. $R$ is the photon loss rate. Black dashed line, ${|\alpha \rangle }_a{|0\rangle }_b$; red dotted curve, ${|\alpha \rangle }_a{|0\rangle }_b-{|0\rangle }_a{|\alpha \rangle }_b$; blue solid curve, ${|\alpha \rangle }_a{|\beta \rangle }_b-{|\beta \rangle }_a{|\alpha \rangle }_b$. (a) $\beta =-\alpha$, (b) $\beta =0.3\alpha$, (c) $\beta =-\alpha$, and (d) $\beta =0.3\alpha$.

Figure 12

Photon loss in one arm: Comparison between the state ${|\alpha \rangle }_a{|\beta \rangle }_b-{|\beta \rangle }_a{|\alpha \rangle }_b$ and a separable coherent state ${|\alpha \rangle }_a{|\alpha \rangle }_b$ for the quantum Fisher information with $N_{\text{av}}=\langle {\widehat{n}}_a\rangle$, in the constraint of input mean photon number. $R$ is the photon loss rate. Black dashed line, ${|\alpha \rangle }_a{|0\rangle }_b$; red dotted curve, ${|\alpha \rangle }_a{|0\rangle }_b-{|0\rangle }_a{|\alpha \rangle }_b$; blue solid curve, ${|\alpha \rangle }_a{|\beta \rangle }_b-{|\beta \rangle }_a{|\alpha \rangle }_b$. (a) $\beta =-\alpha$, (b) $\beta =0.3\alpha$, (c) $\beta =0.6\alpha$, and (d) $\beta =0.4\alpha$.