The difference between two random mixed quantum states: exact and asymptotic spectral analysis

Abstract

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson's theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.

34 pages, 7 figures. Paper has been restructured with first section presenting all the results, with proofs in subsequent sections

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• Spectral statistics for the difference of two Wishart matrices
• Derivative principles for invariant ensembles
• Duistermaat-Heckman measure and the mixture of quantum states
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
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• On a Matrix Ensemble for Arbitrary Complex Quantum Systems
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