Abstract
Gaussian states are of increasing interest in the estimation of physical parameters because they are easy to prepare and manipulate in experiments. In this article, we derive formulae for the optimal estimation of parameters using two- and multi-mode Gaussian states. As an application of our result, we derive the optimal Gaussian probe states for the estimation of the parameter characterizing a one-mode squeezing channel.
Highlights
One of the main aims of quantum metrology is to find the ultimate precision bound on the estimation of a physical parameter encoded in a quantum state
In the previous section we derived an exact expression for two-mode Gaussian states, in recent work [14] a general formula for the quantum Fisher information was derived as a limit of a particular infinite series
We have derived an exact formula for the quantum Fisher information of an arbitrary two-mode Gaussian state
Summary
One of the main aims of quantum metrology is to find the ultimate precision bound on the estimation of a physical parameter encoded in a quantum state. Calculating the quantum Fisher information gives us an idea of how well we can estimate the parameter when only a fixed amount of measurements are available. Gao and Lee derived an exact formula [16] for the quantum Fisher information for the multi-mode Gaussian states in terms of the inverse of certain tensor products, elegantly generalizing the previous results, with some possible drawbacks, especially in the necessity of inverting relatively large matrices. In the case when the Williamson decomposition of the covariance matrix is known, the infinite series can be evaluated This gives a general and exact formula for the quantum Fisher information of any multi-mode Gaussian state in terms of its decomposition. In appendix F we provide a table of frequently used notation that are constant throughout the paper and appear repeatedly
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