Relating the Bures measure to the Cauchy two-matrix model

Abstract

The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding joint probability density function of its eigenvalues was identified. Moreover a relation with the Cauchy two-matrix model was discovered but never thoroughly investigated, leaving open in particular the following question: How are the kernels of the Pfaffian point process of the Bures random matrix ensemble related to the ones of the determinantal point process of the Cauchy two-matrix model and moreover, how can it be possible that a Pfaffian point process derives from a determinantal point process? We give a very explicit answer to this question. The aim of our work has a quite practical origin since the calculation of the level statistics of the Bures ensemble is highly mathematically involved while we know the statistics of the Cauchy two-matrix ensemble. Therefore we solve the whole level statistics of a density operator drawn from the Bures prior.

39 pages; new version contains an additional chapter about the hard edge scaling limit

References in corpus (15)

• The Cauchy two-matrix model
• Unifying approach to the quantification of bipartite correlations by Bures distance
• Eigenvalue statistics for product complex Wishart matrices
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Differential equations for singular values of products of Ginibre random matrices
• Raney distributions and random matrix theory
• Preservation of the geometric quantum discord in noisy environments
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)
• Discord of response
• Quantifying the nonlinearity of a quantum oscillator
• Spectral density of generalized Wishart matrices and free multiplicative convolution
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• Thermal effects on the sudden changes and freezing of correlations between remote atoms in cavity QED network
• Quantum Correlations in Cavity QED Networks
• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments

Cited by in corpus (24)

• Singular value statistics of matrix products with truncated unitary matrices
• Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Bures-Hall Ensemble: Spectral Densities and Average Entropies
• The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy
• Exact variance of von Neumann entanglement entropy over the Bures-Hall measure
• Dropping the independence: singular values for products of two coupled random matrices
• Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble
• Average capacity of quantum entanglement
• BKP hierarchy and Pfaffian point process
• GUE-chGUE Transition preserving Chirality at finite Matrix Size
• Entropy fluctuation formulas of fermionic Gaussian states
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Matrix model for the total descendant potential of a simple singularity of type $D$
• Large n limit for the product of two coupled random matrices
• Integrable structure of products of finite complex Ginibre random matrices
• Approximation of Splines in Wasserstein Spaces
• Moments of quantum purity and biorthogonal polynomial recurrence
• Volume-law entanglement entropy of typical pure quantum states
• Average mutual information for random fermionic Gaussian quantum states