Multiparameter Gaussian quantum metrology

We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cramér-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.

Figure 1

An instance of multiparameter Gaussian quantum metrology. The initial state ${\widehat{\rho }}_0$ is a two-mode displaced squeezed state which passes through an interferometric setup before a joint measurement is made. One mode undergoes a phase transformation of $\varphi /2$ and the other of $-\varphi /2$, while both modes are affected by a phase covariant Gaussian channel ${\mathrm{\Lambda }}_{x,y}$ with noise parameters $x$ and $y$. We determine optimal strategies for the estimation of the three parameters $\{\varphi ,x,y\}$.

Figure 2

Results obtained from the metrological scheme outlined in Fig. 1. Left column: Analysis at low energy, $\overline{n}=0.005$. Right column: Analysis at higher energy, $\overline{n}=5$. Top: Optimal proportion of input energy to put in the displacement, $p_{\mathrm{opt}}$, comparing individual and simultaneous estimation schemes. The unmeshed (green online) plane marks $p_{\mathrm{opt}}=1/2$, above which more energy should be used for displacement than squeezing. Middle: Minimal achievable error ${\mathrm{\Delta }}_{\mathrm{opt}}$ for the two strategies. Bottom: Performance ratio $\mathcal{R}={\mathrm{\Delta }}_{\mathrm{opt}}^{\mathrm{ind}}/{\mathrm{\Delta }}_{\mathrm{opt}}^{\mathrm{sim}}$. The solid (red online) line marks $\mathcal{R}=1$; when $\mathcal{R}>1$, the simultaneous estimation scheme outperforms the individual one. All the presented results are independent of the value of the unknown phase $\varphi$. All the quantities plotted are dimensionless.

Figure 3

Comparing optimizing the input state for each parameter independently (“ind, ind”), to optimizing for the combined error (“ind, com”), as in Fig. 2. Left column: Analysis at low energy, $\overline{n}=0.005$. Right column: Analysis at higher energy, $\overline{n}=5$. Top: Optimal proportion of input energy to put in the displacement, $p_{\mathrm{opt}}$, comparing individual estimations of each parameter. The unmeshed (green online) plane marks $p_{\mathrm{opt}}=1/2$, above which more energy should be used for displacement than squeezing. Bottom: Minimal achievable error ${\mathrm{\Delta }}_{\mathrm{opt}}$ for the introduced independent estimation, in comparison with the two strategies discussed in Sec. 4. All the presented results are independent of the value of the unknown phase $\varphi$. All the quantities plotted are dimensionless.