Ultra-precise phase estimation without mode entanglement.

We explore optical quantum engineering of phase-parameterized continuous-variable (CV) probe states to exploit the nonclassical light to solve the problem of precise phase estimation. The optical interferometer consists of a single beam splitter (BS) with tunable transmittance and reflectance, and two single-mode squeezed vacuum states (SMSVs). The reference SMSV state is mixed with a weakly squeezed state carrying an unknown phase at the beam splitter to form an output hybrid entangled state. Then, in the measurement mode, the number of photons is measured to generate the target CV state parameterized by the unknown phase. By using a non-unitary encoding of the probe state with an unknown phase, we propose a sub-Heisenberg metrology protocol, in which the quantum Cramer-Rao (QCR) boundary is saturated by the intensity measurement. The advantage of quantum engineering of CV probe states for ultra-precise phase estimation of an unknown phase is due solely to the nonclassical photonic properties of the measurement-induced CV states of definite parity, and the protocol execution is independent of the mode entanglement.