Gaussian interferometric power

The interferometric power of a bipartite quantum state quantifies the precision, measured by quantum Fisher information, that such a state enables for the estimation of a parameter embedded in a unitary dynamics applied to one subsystem only, in the worst-case scenario where a full knowledge of the generator of the dynamics is not available a priori. For finite-dimensional systems, this quantity was proven to be a faithful measure of quantum correlations beyond entanglement. Here we extend the notion of interferometric power to the technologically relevant setting of optical interferometry with continuous-variable probes. By restricting to Gaussian local dynamics, we obtain a closed formula for the interferometric power of all two-mode Gaussian states. We identify separable and entangled Gaussian states which maximize the interferometric power at fixed mean photon number of the probes and discuss the associated metrological scaling. At fixed entanglement of the probes, highly thermalized states can guarantee considerably larger precision than pure two-mode squeezed states.

Figure 1

Black-box optical interferometry.

Figure 2

Gaussian IP versus mean photon number of the probing mode $A$ for ${10}^5$ entangled (lighter) and separable (darker) Gaussian states. The standard quantum limit ${\mathcal{P}}_G^A={\overline{n}}_A$ (solid line) and the Heisenberg limit ${\mathcal{P}}_G^A={\overline{n}}_A({\overline{n}}_A+1)$ (dashed line) are indicated.

Figure 3

Gaussian IP normalized by the mean photon number of mode $A$, plotted versus the logarithmic negativity $E_{\mathcal{N}}$ for ${10}^5$ entangled Gaussian states. The dashed line accommodates pure states. See text for details of the other boundary curves.