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.

arXiv commentPublished version

References in corpus (37)

• Level-Spacing Distributions and the Airy Kernel
• Average Entropy of a Subsystem
• A Random Matrix Approach to Cross-Correlations in Financial Data
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Induced measures in the space of mixed quantum states
• Random Matrix Theory and Chiral Symmetry in QCD
• Level-Spacing Distributions and the Bessel Kernel
• Level-Spacing Distributions and the Airy Kernel
• Testing statistical bounds on entanglement using quantum chaos
• Statistical properties of random density matrices
• Random quantum correlations and density operator distributions
• Dynamical aspects of quantum entanglement for weakly coupled kicked tops
• Statistical distribution of quantum entanglement for a random bipartite state
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Distribution of the k-th smallest Dirac operator eigenvalue
• Entanglement production in coupled chaotic systems : Case of the Kicked Tops
• Phase transitions of bipartite entanglement
• Smallest Dirac Eigenvalue Distribution from Random Matrix Theory
• Random Bures mixed states and the distribution of their purity
• Random pure states: quantifying bipartite entanglement beyond the linear statistics
• Entanglement of random vectors
• Quantum ergodicity and entanglement in kicked coupled-tops
• Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
• Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
• Eigenvalue distributions for some correlated complex sample covariance matrices
• Entanglement in random pure states: Spectral density and average von Neumann entropy
• Beyond universality in random matrix theory
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Distribution of the Smallest Eigenvalue in the Correlated Wishart Model
• Entanglement and localization transitions in eigenstates of interacting chaotic systems
• Universal scaling of spectral fluctuation transitions for interacting chaotic systems
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge
• Entangled random pure states with orthogonal symmetry: exact results
• Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model

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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