Phase variance of squeezed vacuum states

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number $n$, as dictated by the Heisenberg limit, i.e., as $n^{-2}$, only for $N>4$. For $N⩽4$ this fundamental scaling breaks down and it becomes $n^{-N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling $n^{-2}$ is recovered for large samples.

Figure 1

(Color online) Log-log plot of the scaled phase variance $NV$ defined in Eq. (21) for $N$ copies of SVS, $1⩽N⩽9$ (top to bottom), as a function of the mean photon number $n$ (solid lines). For large $n$ the lines become steeper as $N$ increases, in agreement with Eq. (27). The slopes stabilize for $N⩾5$. For small $n$ the slopes are independent of $N$, as Eqs. (28, 36) show. The dotted line is the limiting curve $(N\to \infty )$ for both small and large $n$. It follows from the HL (36), as discussed in Sec. 5.

Figure 2

(Color online) Log-log plot of the rescaled phase variance $E^{2}V$ as a function of the total available energy $E=nN$ (in units of $\hslash \omega$). The thin solid lines correspond (from top to bottom) to $N=5,6,7$; the thick green line corresponds to $N=8$. The dashed lines correspond to $N⩾9$ in increasing order from bottom to top on the right side of the plot. For large $E$, variances scale as $E^{-2}$ (all lines have horizontal asymptotes) and $N=8$ clearly provides the smallest variance.