arXiv subjects

Arul Lakshminarayan

Publications and source records attributed to Arul Lakshminarayan.

At least 73 records · Page 4Linked to original sources

arXiv2011

Entanglement optimizing mixtures of two-qubit states

Entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states. When two or three pure real states are mixed it is shown that 28.5% and 5.12% of the cases respectively, are optimal. Conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived. For four or more pure states it is shown that there are no such sets of real states. The implications of these on superposition of two or more dimerized states is discussed. A corollary of these results also show in how many cases rebit concurrence can be the same as that of qubit concurrence.

quant-ph↗

arXiv2011

Localised zero-energy modes in the Kitaev model with vacancy-disorder

We study the effects of vacancy disorder on the Kitaev model defined on a hexagonal lattice. We show that the vacancy disorder induces a zero-mode that is localized at the defect site. We derive analytical forms for these localized wave functions in both the gapped and gapless phases of the Kitaev model. We conjecture that the vacancy disorder can be utilized as a probe of the quantum phase transition (from the gapped to gapless phases) in this model. The behavior of the Inverse Participation Ratio (IPR) in the gapless phase and across the transition is also studied numerically. Comments are made about the behavior of site-site entanglement in the single particle states for the case of a single vacancy.

cond-mat.str-el↗

arXiv2011

On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems

The Kolmogorov-Sinai (K-S) entropy is a central measure of complexity and chaos. Its calculation for many-body systems is an interesting and important challenge. In this paper, the evaluation is formulated by considering $N$-dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem. This approach makes explicit a duality relation that is exactly analogous to one found in a generalized Anderson tight-binding model, and leads to a formally exact expression for the finite-time K-S entropy. Within this formalism there is a hierarchy of approximations, the final one being a diagonal approximation that only makes use of instantaneous Hessians of the potential to find the K-S entropy. By way of a non-trivial illustration, the K-S entropy of $N$ identically coupled kicked rotors (standard maps) is investigated. The validity of the various approximations with kicking strength, particle number, and time are elucidated. An analytic formula for the K-S entropy within the diagonal approximation is derived and its range of validity is also explored.

nlin.CD↗

arXiv2010

Fluctuations in classical sum rules

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 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.

nlin.CD↗

arXiv2008

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

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 {\em strongly correlated} random variables for all values of N (and not just for large N).

cond-mat.stat-mech↗

arXiv2007

Extreme statistics of complex random and quantum chaotic states

An exact analytical description of extreme intensity statistics in complex random states is derived. These states have the statistical properties of the Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory. Although the components are correlated by the normalization constraint, it is still possible to derive compact formulae for all values of the dimensionality N. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-N formulae.

quant-ph↗

arXiv2007

The modular multiplication operator and the quantized bakers maps

The modular multiplication operator, a central subroutine in Shor's factoring algorithm, is shown to be a coherent superposition of two quantum bakers maps when the multiplier is 2. The classical limit of the maps being completely chaotic, it is shown that there exist perturbations that push the modular multiplication operator into regimes of generic quantum chaos with spectral fluctuations that are those of random matrices. For the initial state of relevance to Shor's algorithm we study fidelity decay due to phase and bit-flip errors in a single qubit and show exponential decay with shoulders at multiples or half-multiples of the order. A simple model is used to gain some understanding of this behavior.

quant-ph↗

arXiv2007

Classical bifurcations and entanglement in smooth Hamiltonian system

We study entanglement in two coupled quartic oscillators. It is shown that the entanglement, as measured by the von Neumann entropy, increases with the classical chaos parameter for generic chaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects a class of quantum eigenstates, called the channel localized states. For these states, the entanglement is a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near a anti-pitchfork bifurcation. We place these results in the context of the close connections that may exist between entanglement measures and conventional measures of localization that have been much studied in quantum chaos and elsewhere. We also point to an interesting near-degeneracy that arises in the spectrum of reduced density matrices of certain states as an interplay of localization and symmetry.

nlin.CD↗

arXiv2007

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands

Some statistical properties of finite-time stability exponents in the standard map can be estimated analytically. The mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants. Whereas the mean carries information about the strength of the interaction, and only indirect information about dynamical correlations, the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength. In particular, the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time-scalings. The average of the stability matrix' inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter. The usual accelerator modes and the small islands created through double saddle node bifurcations, which come halfway between the positions in interaction strength of the usual accelerator modes, are clearly visible in the variance, whose time scaling is capable of detecting the presence of islands as small as 0.01% of the phase space. We study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions. We also discuss the nature of this local approximation in some detail.

nlin.CD↗

arXiv2007

Entanglement, avoided crossings and quantum chaos in an Ising model with a tilted magnetic field

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.

quant-ph↗

arXiv2006

The Fourier transform of the Hadamard transform: Multifractals, Sequences and Quantum Chaos

We introduce a class of functions that limit to multifractal measures and which arise when one takes the Fourier transform of the Hadamard transform. This introduces generalizations of the Fourier transform of the well-studied and ubiquitous Thue-Morse sequence, and introduces also generalizations to other intriguing sequences. We show their relevance to quantum chaos, by displaying quantum eigenfunctions of the quantum bakers map that are approximated well by such measures, thereby extending our recent work where we pointed to the existence of ``Thue-Morse'' states.

nlin.CD↗

arXiv2006

Quantum chaos in the spectrum of operators used in Shor's algorithm

We provide compelling evidence for the presence of quantum chaos in the unitary part of 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 CUE 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 bakers map.

quant-ph↗

arXiv2006

Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map

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 for many states of the quantum baker's map. These new 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.

nlin.CD↗

arXiv2005

Entanglement measures in quantum and classical chaos

Entanglement is a Hilbert-space based measure of nonseparability of states that leads to unique quantum possibilities such as teleportation. It has been at the center of intense activity in the area of quantum information theory and computation. In this paper we discuss the implications of quantum chaos on entanglement, showing how chaos can lead to large entanglement that is universal and describable using random matrix theory. We also indicate how this measure can be used in the Hilbert space formulation of classical mechanics. This leads us to consider purely Hilbert-space based measures of classical chaos, rather than the usual phase-space based ones such as the Lyapunov exponents, and can possibly lead to understanding of partial differential equations and nonintegrable classical field theories.

nlin.CD↗

arXiv2005

Shuffling cards, factoring numbers, and the quantum baker's map

It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of this operator the eigenfunctions of the quantum baker's map are compressed by factors of around five or more. We show explicitly its connection to an operator that is closely related to the usual quantum baker's map. This permutation operator has interesting connections to the art of shuffling cards as well as to the quantum factoring algorithm of Shor via the quantum order finding one. Hence we point out that this well-known quantum algorithm makes crucial use of a quantum chaotic operator, or at least one that is close to the quantization of the left-shift, a closeness that we also explore quantitatively.

nlin.CD↗

arXiv2005

Multifractal eigenstates of quantum chaos and the Thue-Morse sequence

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 quasi-periodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to 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.

nlin.CD↗

arXiv2005

Multipartite Entanglement in a One-Dimensional Time Dependent Ising Model

We study multipartite entanglement measures for a one-dimensional Ising chain that is capable of showing both integrable and nonintegrable behaviour. 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 shows that large multipartite entanglement is accompanied by diminishing two-body correlations, and that time averaged multipartite entanglement measures can be enhanced in nonintegrable systems.

quant-ph↗

arXiv2005

Entangling power of quantum chaotic evolutions via operator entanglement

We study operator entanglement of the quantum chaotic evolutions. This study shows that properties of the operator entanglement production are qualitatively similar to the properties reported in literature about the pure state entanglement production. This similarity establishes that the operator entanglement quantifies {\it intrinsic} entangling power of an operator. The term `intrinsic' suggests that this measure is independent of any specific choice of initial states.

quant-ph↗