Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random Matrix Theory (RMT) modeling of composite quantum systems, investigated recently, entails an universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, that is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.PACS numbers : 03.65. Ud, 05.45.Mt Recently, entanglement has been discussed extensively due to its crucial role in quantum computation and quantum information theory [1]. Since a quantum computer is a many particle system, entanglement is inevitable and even desirable. Entanglement is important both at the hardware and software levels of a quantum computer, as the efficiency of all proposed quantum algorithms are based on it, hence its characterization as a quantum resource. The many particle nature of a quantum computer brings another phenomenon to the fore, that is chaos. Some studies have enquired whether chaos will help or hinder in the operation of a quantum computer [2]. At a more basic level several studies have explored the connections between quantum entanglement and classical chaos [3][4][5], two phenomena that are prima facie uniquely quantum and classical respectively. Such a connection between entanglement and chaos has been previously studied with the example of an N -atom Jaynes-Cummings model [3]. It was found that the entanglement rate is considerably enhanced if the initial wave packet was placed in a chaotic region. In another work of similar kind, the authors have related such rates to classical Lyapunov exponents with the help of a coupled kicked top model [4]. Recently, one of us studied entanglement in coupled standard maps [5] and found that entanglement increased with coupling strength, but after a certain magnitude of coupling strength corresponding to the emergence of complete classical chaos, the entanglement saturated. The saturation value depended on the Hilbert space dimensions and was less than its maximum possible value. This result implies that though there exists a maximum kinematical limit for entanglement, dynamically it is not possible to create it by using generic Hamiltonian evolutions on unentangled states. It should be emphasized that such bounds are statistical and are more unlikely to be violated the larger the Hilbert space dimension.Recent related work [6] calculates the mean entanglement of pure states for the case M = N by using a RMT model that allows specification of the joint probability distribution of the eigenvalues of the reduced density matrices (RDM). Below we calculate the entanglement from a eigenvalue distribution t...

We study the quantum entanglement caused by unitary operators that have classical limits that can range from the near integrable to the completely chaotic. Entanglement in the eigenstates and time-evolving arbitrary states is studied through the von Neumann entropy of the reduced density matrices. We demonstrate that classical chaos can lead to substantially enhanced entanglement. Conversely, entanglement provides a useful characterization of quantum states in higher-dimensional chaotic or complex systems. Information about eigenfunction localization is stored in a graded manner in the Schmidt vectors, and the principal Schmidt vectors can be scarred by the projections of classical periodic orbits onto subspaces. The eigenvalues of the reduced density matrices is sensitive to the degree of wave-function localization, and is roughly exponentially arranged. We also point out the analogy with decoherence, as reduced density matrices corresponding to subsystems of fully chaotic systems, are diagonally dominant.

Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have used the reduced Husimi function, the Husimi function of the reduced density matrix, to visualize the possible behaviors of a wave packet. We have studied a phase-space based measure of the complexity of a state and used random matrix theory (RMT) to model the strongly chaotic cases. Extensive numerical studies have been done for the entanglement production in coupled kicked tops corresponding to different underlying classical dynamics and different coupling strengths. An approximate formula, based on RMT, is derived for the entanglement production in coupled strongly chaotic systems. This formula, applicable for arbitrary coupling strengths and also valid for long time, complements and extends significantly recent perturbation theories for strongly chaotic weakly coupled systems.

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N strongly correlated random variables for all values of N (and not just for large N ).

We study multipartite entanglement measures for a one-dimensional Ising chain that is capable of showing both integrable and nonintegrable behavior. This model includes the kicked transverse Ising model, which we solve exactly using the Jordan-Wigner transform, as well as nonintegrable and mixing regimes. The cluster states arise as a special case and we show that while one measure of entanglement is large, another measure can be exponentially small, while symmetrizing these states with respect to up and down spins produces those with large entanglement content uniformly. We also calculate exactly some entanglement measures for the nontrivial but integrable case of the kicked transverse Ising model. In the nonintegrable case we begin on extensive numerical studies that show that large multipartite entanglement is accompanied by diminishing two-body correlations, and that time averaged multipartite entanglement measures can be enhanced in nonintegrable systems.

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 onsite fields: nearly similar fields implying larger entanglement.

Entanglement sharing among sites of one-particle states is considered using the measure of concurrence. These are the simplest in an hierarchy of number-specific states of many qubits and corresponds to "one-magnon" states of spins. We study the effects of onsite potentials that are both integrable and nonintegrable. In the integrable case we point to a metal-insulator transition that reflects on the way entanglement is shared. In the nonintegrable case the average entanglement content increases and saturates along with a transition to classical chaos. Such quantum chaotic states are shown to have universal concurrence distributions that are modified Bessel functions derivable within random matrix theory. Time-reversal breaking and time evolving states are shown to possess significantly higher entanglement sharing capacity that eigenstates of time-reversal symmetric systems. We use the ordinary Harper and kicked Harper Hamiltonians as model systems.

We study a one-dimensional Ising model with a magnetic field and show that tilting the field induces a transition to quantum chaos. We explore the stationary states of this Hamiltonian to show the intimate connection between entanglement and avoided crossings. In general entanglement gets exchanged between the states undergoing an avoided crossing with an overall enhancement of multipartite entanglement at the closest point of approach, simultaneously accompanied by diminishing two-body entanglement as measured by concurrence. We find that both for stationary as well as nonstationary states, nonintegrability leads to a destruction of two-body correlations and distributes entanglement more globally.