Arul Lakshminarayan

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving many states of the quantum baker's map. These transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry. © 2006 IOP Publishing Ltd.

Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems. © Indian Academy of Sciences.

The entanglement dynamics of spin chains is investigated using Heisenberg-XY spin Hamiltonian dynamics. The various measures of two-qubit entanglement are calculated analytically in the time-evolved state starting from initial states with no entanglement and exactly one pair of maximally entangled qubits. The localizable entanglement between a pair of qubits at the end of chain captures the essential features of entanglement transport across the chain, and it displays the difference between an initial state with no entanglement and an initial state with one pair of maximally-entangled qubits. © 2005 Elsevier B.V. All rights reserved.

We analyze certain eigenstates of the quantum baker's map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasiperiodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to the Thue-Morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal. © 2005 The American Physical Society.

The recurrence of fidelity in nearly integrable systems was discussed. It was found that in accordance with the correspondence principle, recurrence was observed when the underlying classical dynamics was well approximated by the harmonic oscillator. The analysis showed that the classical-quantum correspondence of fidelity was very short for states initially in the rotational Kolmogorov-Arnold-Moser region.

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. © 2005 The American Physical Society.

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. © 2007 The American Physical Society.

We provide compelling evidence for the presence of quantum chaos in the unitary part of the operator usually employed in Shor’s factoring algorithm. In particular we analyze the spectrum of this part after proper desymmetrization and show that the fluctuations of the eigenangles as well as the distribution of the eigenvector components follow the circular unitary ensemble of random matrices, of relevance to quantized chaotic systems that violate time-reversal symmetry. However, as the algorithm tracks the evolution of a single state, it is possible to employ other operators; in particular, it is possible that the generic quantum chaos found above becomes of a nongeneric kind such as is found in the quantum cat maps and in toy models of the quantum baker’s map. © 2006 The American Physical Society.

We derive a number of local identities involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2iπ/s), where s is any integer. Third, we systematize the local identities by deriving four local 'master identities' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobi elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.

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). © 2008 Springer Science+Business Media, LLC.