Variance of relative entropy measured across random quantum states
Variance of relative entropy over random states
Information Theory
Summary
Measuring how different two quantum states are is important in quantum computing and physics. The authors look beyond just the average difference to understand how much these differences can vary when picking random states. They calculate exact formulas for this variation in two common ways of generating random states. These results show that the average difference is usually a good representation because the fluctuations around it are small in the studied scenarios. Their work uses advanced math dealing with random matrices and eigenvalues.
What this means in practice
- •For quantum computing engineers: Improve error quantification by using variance information of state differences in simulations of quantum algorithms.
- •For quantum communication teams: Enhance reliability assessments for quantum state transmission by incorporating typical fluctuation measures in state distinguishability.
Authors
Linfeng Wei, Lu Wei
Abstract
Average relative entropy over major generic random states is recently studied. In this work, the investigation is extended by deriving the variances of relative entropy over the Hilbert--Schmidt and Bures--Hall ensembles, which characterize its fluctuations around the average value. Exact results for the variances of relative entropy between independent states drawn from the same ensemble and from two different ensembles are obtained. These results further indicate that the average relative entropy is typical in the considered limiting regimes. In addition to the matrix-level calculations similar to the average relative entropy, eigenvalue correlation functions of the ensembles are involved in the variance computations through the use of zonal polynomials.