Exact mean and variance of the squared Hellinger distance for random density matrices

• URL: http://arxiv.org/abs/2409.14560v1
• Date: Sun, 22 Sep 2024 18:56:49 GMT
• Title: Exact mean and variance of the squared Hellinger distance for random density matrices
• Authors: Vinay Kumar, Kaushik Vasan, Santosh Kumar,
• Abstract summary: The Hellinger distance between quantum states is a significant measure in quantum information theory. We propose an approximation for the corresponding probability density function based on the gamma distribution.
• Score: 11.495104812547021
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is also easier to compute than the Bures distance, another measure that shares these properties. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.

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