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Higher-order spacing ratios are investigated analytically using a Wigner-like surmise for Gaussian ensembles of random matrices. For a kth order spacing ratio (r^{(k)},k>1) the matrix of dimension 2k+1 is considered. A universal scaling relation for this ratio, known from earlier numerical studies, is proved in the asymptotic limits of r^{(k)}â0 and r^{(k)}â∞.
RESUMO
Quantum chaotic kicked top model is implemented experimentally in a two-qubit system comprising of a pair of spin-1/2 nuclei using nuclear magnetic resonance techniques. The essential nonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were realized using radio-frequency pulses. After a variable number of kicks, quantum state tomography was employed to reconstruct the single-qubit reduced density matrices, using which we could extract von Neumann entropies and Husimi distributions. These measures enabled the study of correspondence with classical phase space as well as probing distinct features of quantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top.
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Quantum correlations reflect the quantumness of a system and are useful resources for quantum information and computational processes. Measures of quantum correlations do not have a classical analog and yet are influenced by classical dynamics. In this work, by modeling the quantum kicked top as a multiqubit system, the effect of classical bifurcations on measures of quantum correlations such as the quantum discord, geometric discord, and Meyer and Wallach Q measure is studied. The quantum correlation measures change rapidly in the vicinity of a classical bifurcation point. If the classical system is largely chaotic, time averages of the correlation measures are in good agreement with the values obtained by considering the appropriate random matrix ensembles. The quantum correlations scale with the total spin of the system, representing its semiclassical limit. In the vicinity of trivial fixed points of the kicked top, the scaling function decays as a power law. In the chaotic limit, for large total spin, quantum correlations saturate to a constant, which we obtain analytically, based on random matrix theory, for the Q measure. We also suggest that it can have experimental consequences.
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The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as A and B, is computed analytically using a Coulomb gas method. It is shown that this probability, for large N, goes as exp[-ßN^{2}Φ(ζ)], where the parameter ß is the Dyson index of the ensemble, ζ is the large deviation parameter, while the rate function Φ(ζ) is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems 1 and 2, obtained by further partitioning the subsystem A, using the properties of the density matrix's partial transpose ρ_{12}^{Γ}. The density of states of ρ_{12}^{Γ} is found to be close to the Wigner's semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter ζ. Log negativity is used to quantify the entanglement between subsystems 1 and 2. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.
