Smallest eigenvalue density for regular or fixed-trace complex Wishart-Laguerre ensemble and entanglement in coupled kicked tops

Abstract

The statistical behaviour of the smallest eigenvalue has important implications for systems which can be modeled using a Wishart-Laguerre ensemble, the regular one or the fixed trace one. For example, the density of the smallest eigenvalue of the Wishart-Laguerre ensemble plays a crucial role in characterizing multiple channel telecommunication systems. Similarly, in the quantum entanglement problem, the smallest eigenvalue of the fixed trace ensemble carries information regarding the nature of entanglement. For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme suggested by Edelman to directly obtain the exact expression for the smallest eigenvalue density. In the case of complex Wishart-Laguerre matrices, for finding exact and explicit expressions for the smallest eigenvalue density, existing results based on determinants become impractical when the determinants involve large-size matrices. In this work, we derive a recurrence scheme for the complex case which is analogous to that of Edelman's for the real case. This is used to obtain exact results for the smallest eigenvalue density for both the regular, and the fixed trace complex Wishart-Laguerre ensembles. We validate our analytical results using Monte Carlo simulations. We also study scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of coupled kicked tops.

Published version

References in corpus (8)

• Statistical distribution of quantum entanglement for a random bipartite state
• Entanglement production in coupled chaotic systems : Case of the Kicked Tops
• Entanglement of random vectors
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
• Eigenvalue distributions for some correlated complex sample covariance matrices
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model

Cited by in corpus (12)

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• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum
• Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance
• Entanglement in coupled kicked tops with chaotic dynamics
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Spectral statistics for the difference of two Wishart matrices
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• Volume of the set of LOCC-convertible quantum states
• Probing the randomness of ergodic states: extreme-value statistics in the ergodic and many-body-localized phases