Limit Distribution Theory for Quantum Divergences
- URL: http://arxiv.org/abs/2311.13694v2
- Date: Thu, 7 Dec 2023 09:45:22 GMT
- Title: Limit Distribution Theory for Quantum Divergences
- Authors: Sreejith Sreekumar and Mario Berta
- Abstract summary: We show that a limit distribution theory which characterizes the fluctuations of the estimation error is still premature.
As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting distribution is a normal, with its variance characterized in terms of the Pauli operators and states.
We utilize the knowledge of the aforementioned limit distribution to obtain performance guarantees for a multi-hypothesis testing problem.
- Score: 9.590806017527504
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Estimation of quantum relative entropy and its R\'{e}nyi generalizations is a
fundamental statistical task in quantum information theory, physics, and
beyond. While several estimators of these divergences have been proposed in the
literature along with their computational complexities explored, a limit
distribution theory which characterizes the asymptotic fluctuations of the
estimation error is still premature. As our main contribution, we characterize
these asymptotic distributions in terms of Fr\'{e}chet derivatives of
elementary operator-valued functions. We achieve this by leveraging an operator
version of Taylor's theorem and identifying the regularity conditions needed.
As an application of our results, we consider an estimator of quantum relative
entropy based on Pauli tomography of quantum states and show that the resulting
asymptotic distribution is a centered normal, with its variance characterized
in terms of the Pauli operators and states. We utilize the knowledge of the
aforementioned limit distribution to obtain asymptotic performance guarantees
for a multi-hypothesis testing problem.
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