Gaussian systems for quantum enhanced multiple phase estimation

TL;DR

This paper demonstrates that Gaussian states in a multimode interferometer can achieve quantum-enhanced simultaneous estimation of multiple phases, with a fundamental limit of a twofold precision improvement without requiring entanglement.

Contribution

It derives the quantum Fisher information matrix for Gaussian inputs in multimode phase estimation and shows the bound can be saturated analytically, revealing a fundamental Gaussian state limitation.

Findings

Simultaneous estimation offers up to twice the precision of individual estimation.

Gaussian states do not require entanglement for quantum enhancement in this context.

The quantum Cramér-Rao bound can be analytically calculated and saturated.

Abstract

For a fixed average energy, the simultaneous estimation of multiple phases can provide a better total precision than estimating them individually. We show this for a multimode interferometer with a phase in each mode, using Gaussian inputs and passive elements, by calculating the covariance matrix. The quantum Cram\'{e}r-Rao bound provides a lower bound to the covariance matrix via the quantum Fisher information matrix, whose elements we derive to be the covariances of the photon numbers across the modes. We prove that this bound can be saturated. In spite of the Gaussian nature of the problem, the calculation of non-Gaussian integrals is required, which we accomplish analytically. We find our simultaneous strategy to yield no more than a factor-of-2 improvement in total precision, possibly because of a fundamental performance limitation of Gaussian states. Our work shows that no modal entanglement is necessary for simultaneous quantum-enhanced estimation of multiple phases.