Optimal quantum estimation of loss in bosonic channels

We address the estimation of the loss parameter of a bosonic channel probed by Gaussian signals. We derive the ultimate quantum bound on precision and show that no improvement may be obtained by having access to the environment degrees of freedom. We found that, for small losses, the variance of the optimal estimator is proportional to the loss parameter itself, a result that represents a qualitative improvement over the shot noise limit. An observable based on the symmetric logarithmic derivative is derived, which attains the ultimate bound and may be implemented using Gaussian operations and photon counting.Comment: 4 pages, 2 figures, replaced with published versio

Optimal phase measurements with pure Gaussian states

We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisenberg limit for large samples. This proves that this limit can be attained with local projective Von Neuman measurements.Comment: 9 pages. Revised version: two new sections added, revised conclusions. Corrected prose. Corrected reference

Information geometry of Gaussian channels

We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated from distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a byproduct, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications: It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulae for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states, and provide the optimal observables for the estimation of the channel parameters (e.g. bath couplings, squeezing, and temperature).Comment: 19 pages, 4 figure