Figure 1

Setup for quantum metrology with MZI. BS1, BS2: 50:50 beam splitters. There are losses for light beams in each path; ${\eta }_1,{\eta }_2$ represent transmissivity of light beams. $\varphi$ is a phase shifter which takes the amount of phase shift $\pm \varphi$ to creation and annihilation operators of the incident light mode.Reuse & Permissions

Figure 2

(Left) Estimation precision $\delta \varphi$ vs average photon number $n$. Lowest (dashed black) line: modified HL. Uppermost (dashed black) line: SQL. Thick solid green line and thin solid blue line: quantum limits $\frac{1}{\sqrt{F_Q}}$ for two-arm-equal-loss (${\eta }_1={\eta }_2=0.8$) and one-arm-loss (${\eta }_1=0.8,{\eta }_2=1$) models, respectively. Dot-dashed red line: [12]. (Right) Estimation precision $\delta \varphi$ vs transmissivity $\eta$. Lower horizontal dashed black line: modified HL. Upper horizontal dashed black line: SQL. Thick solid green line and thin solid blue line: quantum limits $\frac{1}{\sqrt{F_Q}}$ for two-arm-equal-loss and one-arm-loss models, respectively. Thick dashed green line and thin dashed blue line: classical limits [3] $\frac{1}{\sqrt{n\eta }}$ for two-arm-equal-loss and $\frac{1+\sqrt{\eta }}{2\sqrt{n\eta }}$ for one-arm-loss models, respectively. Dot-dashed red line: [12].Reuse & Permissions

Figure 3

(Left) Optimal measurement point ${\varphi }_o$ vs average photon number $n$ and transmissivity $\eta$. (Right) Phase estimation error $\Delta \varphi$ vs average photon number $n$ for one-arm losses ($\eta =99\%$). Purple (uppermost) line: classical limit [3]. Blue (lowest) line: quantum limit of exact QFI. Green (middle) line: optimal phase measurement error via parity detection.Reuse & Permissions

Figure 4

Repeated-experiment phase error $\delta \varphi$ vs average photon number $n$ for one-arm losses [$\eta =99\%,98\%,97\%,96\%$ from (a) to (d), $N=200$ fixed]. Red horizontal line: classical limit [3]. Blue (lowest) line: quantum limit of exact QFI. Green (middle) line: optimal phase measurement error via parity detection.Reuse & Permissions