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

Abstract

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.

Bures–Hall ensemble: spectral densities and average entropies

“…which upon performing the inverse Laplace transform yields C (F ) C = Γ[n(n + 2α + 1)/2] = Γ(α + γ + 1) = Γ(mn − n 2 /2), (C. 6) where γ = (n − 1)(n + 2α + 2)/2 as in (43). Plugging this back in (C.4) gives (38).…”

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

“…The opposite limit, Λ → ∞, is also likely to allow for analytic calculations. As the main object of interest has been the spectra of the reduced density matrices, the relevant random matrix ensemble is the fixed-trace Wishart ensemble [110] for which a variety of results are already known [56,103,105,[110][111][112]. Thus it would be interesting to connect the perturbative regimes with various power laws to this random matrix regime.…”

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

“…In this section, we compare the analytical results obtained in the preceding section with the mean square Hilbert-Schmidt distance obtained using random density matrices generated via coupled kicked tops. Coupled quantum kicked tops, inter alia, have been used extensively to study the bipartite entanglement and effect of chaos [22,[103][104][105][106][107][108][109]. In Ref.…”

“…The fixed density matrix is chosen as n −1 1 n , which represents the maximally mixed state. Before we proceed to calculate the average distance between the quantum states, to demonstrate that the algorithm does produce density matrices distributed according to the Hilbert-Schmidt measure, we compare the corresponding eigenvalue density with the random matrix prediction for β = 2 [19,22], viz.,…”