Spectral statistics for the difference of two Wishart matrices

Abstract

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.

arXiv comment23 pages, 5 figures, Mathematica codes included as ancillary files (Accepted for publication in J. Phys. A: Math. Theor.)

References in corpus (55)

• Induced measures in the space of mixed quantum states
• Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
• Biorthogonal ensembles
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Statistical properties of random density matrices
• Spectrum of the Product of Independent Random Gaussian Matrices
• Recent exact and asymptotic results for products of independent random matrices
• The Wasteland of Random Supergravities
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Singular value correlation functions for products of Wishart random matrices
• Singular value statistics of matrix products with truncated unitary matrices
• Universal microscopic correlation functions for products of independent Ginibre matrices
• A note on biorthogonal ensembles
• Eigenvalue statistics for product complex Wishart matrices
• Asymptotics of random density matrices
• Derivation of determinantal structures for random matrix ensembles in a new way
• On the distinguishability of random quantum states
• Universal microscopic correlation functions for products of truncated unitary matrices
• Distinguishability of generic quantum states
• Heavy Tails in Calabi-Yau Moduli Spaces
• Eigenvalue Distributions of Reduced Density Matrices
• Exact Relation between Singular Value and Eigenvalue Statistics
• Products of Random Matrices from Polynomial Ensembles
• Entanglement in random pure states: Spectral density and average von Neumann entropy
• Lyapunov exponents for products of complex Gaussian random matrices
• Determinantal point processes in the plane from products of random matrices
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Degenerate Gauss hypergeometric functions
• Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for product of two Gaussian matrices
• Free products of large random matrices - a short review of recent developments
• Probability of all eigenvalues real for products of standard Gaussian matrices
• Recursion scheme for the largest $\beta$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport
• Eigenvalue statistics for the sum of two complex Wishart matrices
• The difference between two random mixed quantum states: exact and asymptotic spectral analysis
• Hard edge limit of the product of two strongly coupled random matrices
• Dropping the independence: singular values for products of two coupled random matrices
• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• Smallest eigenvalue density for regular or fixed-trace complex Wishart-Laguerre ensemble and entanglement in coupled kicked tops
• Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance
• The Scale of Inflation in the Landscape
• Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices
• On the number of real eigenvalues of products of random matrices and an application to quantum entanglement
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• Polynomial Ensembles and Pólya Frequency Functions
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Eigenvector statistics of the product of Ginibre matrices
• On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Kac-Rice fixed point analysis for single- and multi-layered complex systems
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Schwinger-Dyson and loop equations for a product of square Ginibre random matrices
• On the Exact Distribution of Mutual Information of Two-user MIMO MAC Based on Quotient Distribution of Wishart Matrices
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Time-inhomogeneous random Markov chains

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