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On super additivity of Fisher information in fully Gaussian metrology

Javier Navarro, Simon Morelli, Mikel Sanz, Mohammad Mehboudi

Abstract

Famously, the quantum Fisher information -- the maximum Fisher information over all physical measurements -- is additive for independent copies of a system and the optimal measurement acts locally. We are left to wonder: does the same hold when the set of accessible measurements is constrained? Such constraints are necessary to account for realistic experimental restrictions. Here, we consider a fully Gaussian scenario focusing on only Gaussian measurements. We prove that the optimal Gaussian measurement protocol remains local, if the information is encoded in either the displacement or the covariance matrix. However, when the information is imprinted on both, this no longer holds true: we construct a simple global Gaussian measurement where the Fisher information becomes super additive. These results can improve parameter estimation tasks via feasible tools. Namely, in quantum optical platforms our proposed global operation requires only passive global operations and single mode Gaussian measurements. We demonstrate this in two examples where we estimate squeezing and losses. While in the former case there is a significant gap between the Fisher information of the optimal Gaussian measurement and the quantum Fisher information for a single copy, this gap can be reduced with joint Gaussian measurements and closed in the asymptotic limit of many copies.

On super additivity of Fisher information in fully Gaussian metrology

Abstract

Famously, the quantum Fisher information -- the maximum Fisher information over all physical measurements -- is additive for independent copies of a system and the optimal measurement acts locally. We are left to wonder: does the same hold when the set of accessible measurements is constrained? Such constraints are necessary to account for realistic experimental restrictions. Here, we consider a fully Gaussian scenario focusing on only Gaussian measurements. We prove that the optimal Gaussian measurement protocol remains local, if the information is encoded in either the displacement or the covariance matrix. However, when the information is imprinted on both, this no longer holds true: we construct a simple global Gaussian measurement where the Fisher information becomes super additive. These results can improve parameter estimation tasks via feasible tools. Namely, in quantum optical platforms our proposed global operation requires only passive global operations and single mode Gaussian measurements. We demonstrate this in two examples where we estimate squeezing and losses. While in the former case there is a significant gap between the Fisher information of the optimal Gaussian measurement and the quantum Fisher information for a single copy, this gap can be reduced with joint Gaussian measurements and closed in the asymptotic limit of many copies.
Paper Structure (7 sections, 54 equations, 2 figures)

This paper contains 7 sections, 54 equations, 2 figures.

Table of Contents

  1. Further details of the Result 1
  2. Information only on the displacement vector
  3. Information only on the covariance matrix
  4. Extension to isothermal states with higher number of modes
  5. More details on isothermal models and the proof of Eq. \ref{['eq:W_D_orthosymplectic']}
  6. Proof of Eq. \ref{['eq:W_diag_orthosym']}
  7. The maximum likelihood estimator for the squeezing estimation example

Figures (2)

  • Figure 1: Schematic of our proposed global Gaussian measurement that can demonstrate super additivity of the optimal Gaussian FI. The measurement strategy only require LON and local Homodyne/Heterodyne detection. In case of $m=2$ the LON reduces to a simple balanced beam splitter.
  • Figure 2: Monte Carlo simulation of a maximum likelihood estimation (MLE) strategy. (a) Shows how the MLE gets close to the true value with increasing data size $\nu$. (b) Shows how our proposed global strategy can improve local strategies and saturate the QCRB. Panels (c) and (d) show the relevance of the FI as a figure of merit; the MSE by using the MLE very quickly saturates the relevant CRB for two different scenarios with $m=2$ and $m=5$ respectively. The MSE is achieved by averaging the square error over $10^4$ different Monte Carlo simulations---thin gray lines---which shows the CRB is saturated with as few as $\nu=3$ repetitions. Here, we have set $\alpha = \sqrt{2}$, and true $r = 0.1$.

Theorems & Definitions (2)

  • proof
  • proof