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Mutual Information Bounded by Fisher Information

Wojciech Górecki, Xi Lu, Chiara Macchiavello, Lorenzo Maccone

Abstract

We derive a general upper bound to mutual information in terms of the Fisher information. The bound may be further used to derive a lower bound for the Bayesian quadratic cost. These two provide alternatives to other inequalities in the literature (e.g.~the van Trees inequality) that are useful also for cases where the latter ones give trivial bounds. We then generalize them to the quantum case, where they bound the Holevo information in terms of the quantum Fisher information. We illustrate the usefulness of our bounds with a case study in quantum phase estimation. Here, they allow us to adapt to mutual information (useful for global strategies where the prior plays an important role) the known and highly nontrivial bounds for the Fisher information in the presence of noise. The results are also useful in the context of quantum communication, both for continuous and discrete alphabets.

Mutual Information Bounded by Fisher Information

Abstract

We derive a general upper bound to mutual information in terms of the Fisher information. The bound may be further used to derive a lower bound for the Bayesian quadratic cost. These two provide alternatives to other inequalities in the literature (e.g.~the van Trees inequality) that are useful also for cases where the latter ones give trivial bounds. We then generalize them to the quantum case, where they bound the Holevo information in terms of the quantum Fisher information. We illustrate the usefulness of our bounds with a case study in quantum phase estimation. Here, they allow us to adapt to mutual information (useful for global strategies where the prior plays an important role) the known and highly nontrivial bounds for the Fisher information in the presence of noise. The results are also useful in the context of quantum communication, both for continuous and discrete alphabets.
Paper Structure (5 sections, 6 theorems, 51 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 51 equations, 2 figures.

Table of Contents

  1. Derivation of inequalities (\ref{['eq:maxq']}) and (\ref{['eq:cauchy']})
  2. Obtaining \ref{['eq:xilu']}
  3. Applying the bound for Gaussian distribution
  4. Mutual information in occurrence of noise
  5. Holevo information bounded by Quantum Fisher Information -- proof of \ref{['thm:qmi_bound']}

Key Result

Theorem 1

If the parameter is guaranteed to lie inside of a finite-size set, namely $\textrm{supp}\,p(\phi)\subseteq [a,b]$, then

Figures (2)

  • Figure 1: The bound for the mutual information $I(X,\Phi)$ in terms of the Fisher information $F(\phi)$ allows for the transfer of the results obtained for local estimation to global estimation. Here the parameter $\phi$ is considered as realization of a random variable $\Phi$.
  • Figure 2: The mutual information bound given by \ref{['eq:xilu']} with maximum Fisher information, for the dephasing channel and amplitude damping channel. The two dashed lines help to compare the bound with HS and SQL.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2