  Quantum metrology exploits quantum correlations to make precise measurements
with limited particle numbers. By utilizing inter- and intra- mode correlations
in an optical interferometer, we find a state that combines entanglement and
squeezing to give a 7-fold enhancement in the quantum Fisher information (QFI)
-- a metric related to the precision -- over the shot noise limit, for low
photon numbers. Motivated by practicality we then look at the squeezed
cat-state, which has recently been made experimentally, and shows further
precision gains over the shot noise limit and a 3-fold improvement in the QFI
over the optimal Gaussian state. We present a conceptually simple measurement
scheme that saturates the QFI, and we demonstrate a robustness to loss for
small photon numbers. The squeezed cat-state can therefore give a significant
precision enhancement in optical quantum metrology in practical and realistic
conditions.
