Gaussian systems for quantum-enhanced multiple phase estimation

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 Cramér-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.

Figure 1

A $d+1$ mode interferometer with the most general pure Gaussian input, produced from a general pure and separable Gaussian state followed by a passive element. The resultant state undergoes the phase shifts to be estimated.

Figure 2

A two-mode interferometer with the most general pure Gaussian input, produced from a general pure and separable Gaussian state followed by a passive element. The resultant state undergoes the phase shift to be estimated. This is repeated $d$ times, i.e., for the number of phases to be estimated or it can be viewed as $d$ parallel, individual estimations. The total energy for simultaneous and individual estimation is the same.

Figure 3

$R$ as a function of number of phases and squeezing.

Figure 4

$R$ for large number of phases as a function of realistic squeezing. The lower limit is approached quickly for experimentally feasible squeezing.