Versatile Gaussian probes for squeezing estimation

We consider an instance of “black-box” quantum metrology in the Gaussian framework, where we aim to estimate the amount of squeezing applied on an input probe, without previous knowledge on the phase of the applied squeezing. By taking the quantum Fisher information (QFI) as the figure of merit, we evaluate its average and variance with respect to this phase in order to identify probe states that yield good precision for many different squeezing directions. We first consider the case of single-mode Gaussian probes with the same energy, and find that pure squeezed states maximize the average quantum Fisher information (AvQFI) at the cost of a performance that oscillates strongly as the squeezing direction is changed. Although the variance can be brought to zero by correlating the probing system with a reference mode, the maximum AvQFI cannot be increased in the same way. A different scenario opens if one takes into account the effects of photon losses: coherent states represent the optimal single-mode choice when losses exceed a certain threshold and, moreover, correlated probes can now yield larger AvQFI values than all single-mode states, on top of having zero variance.

Figure 1

Sketch of a physical scenario in which a squeezing operation along an initially unknown direction is applied on the probing system. Stage (i): versatile single-mode or two-mode probes are prepared, without knowing the angle $\theta$ characterizing the estimation process in which they will be used. Stage (ii): the probing systems are sent to the squeezing device, and in each one-way trip they acquire a certain phase $\theta$. Stage (iii): upon recollection of the probes, the angle $\theta$ becomes known, and can be used to devise the best measurement allowed by quantum mechanics, from which an estimator $\widehat{\epsilon }$ is recovered. Note that the knowledge of $\theta$ can also be used to apply the unitary correction ${\mathcal{R}}_{-\theta }^{\left(A\right)}$ on the final state of the probes. Although this does not change the QFI, it allows us to write the total encoding unitary operation in the same form that appears in Eq. (23).

Figure 2

Noiseless AvQFI for single-mode Gaussian probes with respect to the photon number $n_A$. Blue dots: ${10}^5$ single-mode Gaussian state uniformly sampled with the method of Appendix pp5; red top solid line: pure undisplaced squeezed states; red bottom dashed line: undisplaced thermal states; black dot-dashed line: coherent states.

Figure 3

Range of QFI values that could be achieved for any fixed $n_A$ by varying $\theta$, for pure and undisplaced squeezed single-mode probes ${\rho }_A^{\left(\text{sq}\right)}$. The corresponding AvQFI is plotted as a black line for comparison.

Figure 4

AvQFI with respect to $\nu$ and ${\left|\xi \right|}^2/\left(2n_A\right)$ for $\epsilon =1,n_A=5$, and $\eta =0.95$. We uniformly sampled ${10}^5$ random Gaussian states with $\varphi =0$ and $\psi =\pi /2$, according to the method detailed in Appendix pp5.

Figure 5

Optimal value for the ratio ${\left|\xi \right|}^2/\left(2n_A\right)$, leading to the maximum AvQFI value for a fixed pair $(n_A,\eta )$. The plot is obtained by considering $\epsilon =1$.

Figure 6

Relative increase $\mathcal{I}$ in precision obtained by using two-mode squeezed vacuum probes rather than the optimal single-mode input state, numerically evaluated for ${10}^4$ pairs $(n_A,\eta )$ or $(N,\eta )$ when $\epsilon =1$. (a) Comparison for fixed $n_A$. (b) Comparison for fixed $N$.

Figure 7

Optimal value for the ratio ${\left|\xi \right|}^2/\left(2n_A\right)$, leading to the maximum AvQFI value for ${10}^4$ pairs $(n_A,\eta )\in [0,10]\times [0,1]$ and different values of $\epsilon$. (a) $\epsilon =0.5$. (b) $\epsilon =0.1$.

Figure 8

Relative increase $\mathcal{I}$ in precision for fixed $n_A$ obtained by using two-mode squeezed vacuum probes rather than the optimal single-mode input state, numerically evaluated for ${10}^4$ pairs $(n_A,\eta )\in [0,10]\times [0,1]$ and different values of $\epsilon$. (a) $\epsilon =0.5$. (b) $\epsilon =0.1$.

Figure 9

Relative increase $\mathcal{I}$ in precision for fixed $N$ obtained by using two-mode squeezed vacuum probes rather than the optimal single-mode input state, numerically evaluated for ${10}^4$ pairs $(N,\eta )\in [0,10]\times [0,1]$ and different values of $\epsilon$. (a) $\epsilon =0.5$. (b) $\epsilon =0.1$.