Optimal measurements for quantum fidelity between Gaussian states and its relevance to quantum metrology

Quantum fidelity is a measure to quantify the closeness between two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance both for fundamental interest and practical purposes. We find here the optimal POVMs for quantum fidelity between multimode Gaussian states in a closed analytical form. Our general finding is applied for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: an excitation-number-resolving detection, a projection onto the eigenbasis of operator $\widehat{x}\widehat{p}+\widehat{p}\widehat{x}$, and a quadrature variable detection, each of which corresponds to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation when two arbitrary states are infinitesimally close. It is applied for simplifying the derivations of quantum Fisher information and the associated optimal measurements, exemplified by displacement, phase, squeezing, and loss parameter estimation using Gaussian states.

Figure 1

Quantum fidelity between two states ${\widehat{\rho }}_0$ and ${\widehat{\rho }}_1$ can be measured by the minimal overlap between the probability distributions $p_0\left(x\right)$ and $p_1\left(x\right)$, where the measurement outcomes $x$ are obtained by an optimally chosen POVM $\left\{{\widehat{E}}_x\right\}$.

Figure 2

Classification of optimal measurements as a function of $r_0$ and ${\overline{n}}_0$ for a given ${\overline{n}}_1$. The regions where type (i) and type (ii) are optimal are divided by the the black curves, at which type (iii) is optimal, also including the intersection point when the Gibbs matrices of the states are identical.