Figure 1

Schematic illustration of an interferometric setup for the pure ECS. Two input states ($|{\mathrm{CSS}}_{\alpha }⟩$ and $|\alpha ⟩$) are applied to the first BS and become the ECS. After a phase shifter $U(\varphi )$ in a mode, the parity measurement is performed at the measurement stage.Reuse & Permissions

Figure 2

The optimal phase estimations for NOON, bat, and ECSs with no particle loss are depicted in black solid, blue dashed, and red dotted lines ($⟨n⟩=N/2={\mathcal{N}}_{\alpha }^2·|\alpha {|}^2$). Curves for NOON and bat states are shown as continuous for comparison, but are clearly only defined at the appropriate integers $N$ according to Eq. (8). For small $N$, $\delta {\varphi }_N$ is significantly bigger than $\delta {\varphi }_C$ while $\delta {\varphi }_N\approx \delta {\varphi }_C$ for large $N$. The crossover between ${\delta }_N$ and ${\delta }_B$ at $N=2$ indicates that the NOON and bat states are identical. The green solid and red long dashed lines show the phase estimation of the state given by Eq. (6) in Ref. 20 and $\Delta {\varphi }_{\mathrm{PM}}$ above Eq. (10), respectively.Reuse & Permissions

Figure 3

The graphs show the phase uncertainty with respect to particle loss ($T$, transmission rate of the BSs) for four states ($N=4$ and $\alpha =2$). The legend of Fig. 2 is used here while the dash-dotted green line indicates uncorrelated states [10]. As shown in the magnified inset, the ECS curve starts from ${\delta }_{C_2}\approx 0.205$ at $T=1$ and follows the NOON curve at $T\approx 1$.Reuse & Permissions