Arul Lakshminarayan

We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter Λ. The behaviors apply equally well to few- and many-body systems, e.g., interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems, as long as their subsystems are quantum chaotic and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading-order behaviors depend on Λ. The marginal case of the 1/2 moment, which is related to the distance to the closest maximally entangled state, is an exception having a ΛlnΛ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charvát-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e., the entanglement spectrum, show a transition from power laws and Lévy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

Classical sum rules arise in a wide variety of physical contexts. Asymptotic expressions have been derived for many of these sum rules in the limit of long orbital period (or large action). Although sum-rule convergence may well be exponentially rapid for chaotic systems in a global phase-space sense with time, individual contributions to the sums may fluctuate with a width which diverges in time. Our interest is in the global convergence of sum rules as well as their local fluctuations. It turns out that a simple version of a lazy baker map gives an ideal system in which classical sum rules, their corrections, and their fluctuations can be worked out analytically. This is worked out in detail for the Hannay-Ozorio sum rule. In this particular case the rate of convergence of the sum rule is found to be governed by the Pollicott-Ruelle resonances, and both local and global boundaries for which the sum rule may converge are given. In addition, the width of the fluctuations is considered and worked out analytically, and it is shown to have an interesting dependence on the location of the region over which the sum rule is applied. It is also found that as the region of application is decreased in size the fluctuations grow. This suggests a way of controlling the length scale of the fluctuations by considering a time dependent phase-space volume, which for the lazy baker map decreases exponentially rapidly with time. © 2010 The American Physical Society.

Local entanglement between pairs of spins, as measured by concurrence, is investigated in a disordered spin model that displays a transition from an ergodic to a many-body localized phase in excited states. It is shown that the concurrence vanishes in the ergodic phase and becomes nonzero and increases in the many-body localized phase. This happen to be correlated with the transition in the spectral statistics from Wigner to Poissonian distribution. A scaling form is found to exist in the second derivative of the concurrence with the disorder strength. It also displays a critical value for the localization transition that is close to what is known in the literature from other measures. An exponential decay of concurrence with distance between spins is observed in the localized phase. Nearest neighbor spin concurrence in this phase is also found to be strongly correlated with the disorder configuration of on-site fields: nearly similar fields implying larger entanglement.

Out-of-time-order correlators being explored as a measure of quantum chaos, are studied here in a coupled bipartite system. Each of the subsystems can be chaotic or regular and lead to very different out-of-time-order correlators growths both before and after the scrambling or the Ehrenfest time. We present preliminary results then on weakly coupled subsystems which have very different Lyapunov exponents. We also review the case when both the subsystems are strongly chaotic when a random matrix model can be pressed into service to derive an exponential relaxation to saturation.

We present a solvable model of iterating cluster state protocols that lead to entanglement production, between contiguous blocks, of 1 ebit per iteration. This continues until the blocks are maximally entangled, at which stage an unravelling begins at the same rate until the blocks are unentangled. The model is a variant of the transverse-field Ising model and can be implemented with controlled-not and single-qubit gates. The interqubit entanglement as measured by the concurrence is shown to be zero for periodic chain realizations, while for open boundaries there are very specific instances at which these can develop. Thus we introduce a class of simply produced states with very large multipartite entanglement content of potential use in measurement-based quantum computing.

It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate typicality and study its properties. Analyzing multiple, say n, applications of any entangling operator, U, interlaced with random local gates we prove that both investigated quantities approach their asymptotic values in a simple exponential form. These values coincide with the averages for random nonlocal operators on the full composite space and could be significantly larger than that of Un. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.

The probability that there are k real eigenvalues for an n-dimensional real random matrix is known. Here, we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of two real two-dimensional random matrices has real eigenvalues to an issue of optimal quantum entanglement, this is fully analytically solved. It is shown that in π/4 fraction of such products the eigenvalues are real. Being greater than the corresponding known probability () for a single matrix, it is shown numerically that the probability that all eigenvalues of a product of random matrices are real tends to unity as the number of matrices in the product increases indefinitely. Some other numerical explorations, including the expected number of real eigenvalues, are also presented, where an exponential approach of the expected number to the dimension of the matrix seems to hold. © 2013 IOP Publishing Ltd.

The record statistics of the quantum standard map is shown to capture the classical transition to chaos. It is shown that in the mixed phase-space regime the number of intensity records is a power law in the dimensionality of the state as opposed to the logarithmic growth for the random states. The exponent of this power law is close to 0.5 at the critical value of the chaos parameter, K ≃ 0.98 of the standard map, an exponent that is also obtained for random walks. These findings are based on the record statistics of complex, normalized random states for which we have shown that the probability of a record intensity is a Bernoulli process. © EPLA, 2013.

The eigenstates of a quantum spin glass Hamiltonian with long-range interaction are examined from the point of view of localisation and entanglement. In particular, low particle sectors are examined and an anomalous family of eigenstates is found that is more delocalised but also has larger inter-spin entanglement. These are then identified as particle-added eigenstates from the one-particle sector. This motivates the introduction and the study of random promoted two-particle states, and it is shown that they may have large delocalisation such as generic random states and scale exactly like them. However, the entanglement as measured by two-spin concurrence displays different scaling with the total number of spins. This shows how for different classes of complex quantum states entanglement can be qualitatively different even if localisation measures such as participation ratio are not.

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important - more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to nonintegrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the three- and four-qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons-based experiment realizing the three-qubit case, we find agreement for short times, including a peculiar steplike behavior in correlations of some states. In the case of four qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.