  Quantum fidelity is a measure to quantify the closeness of 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 for practical purposes. We find here the optimal POVMs for quantum
fidelity between multi-mode Gaussian states in a closed analytical form. Our
general finding is specified for selected single-mode Gaussian states of
particular interest and we identify three types of optimal measurements: a
number-resolving detection, a projection on the eigenbasis of operator
$\hat{x}\hat{p}+\hat{p}\hat{x}$, and a quadrature detection, each of which is
applied 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, enabling one to easily find the optimal
measurements for displacement, phase, squeezing, and loss parameter estimations
using Gaussian states.
