- Arul Lakshminarayan

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# Arul Lakshminarayan

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Arul Lakshminarayan

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Arul Lakshminarayan

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Lakshminarayan, A.

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- PublicationOut-of-time-ordered correlators and the Loschmidt echo in the quantum kicked top: How low can we go?(01-07-2021)
;Pg, Sreeram; Show more The out-of-time-ordered correlators (OTOCs) and the Loschmidt echo are two measures that are now widely being explored to characterize sensitivity to perturbations and information scrambling in complex quantum systems. Studying few qubits systems collectively modeled as a kicked top, we solve exactly the three- and four-qubit cases, giving analytical results for the OTOC and the Loschmidt echo. While we may not expect such few-body systems to display semiclassical features, we find that there are clear signatures of the exponential growth of OTOC even in systems with as low as four qubits in appropriate regimes, paving way for possible experimental measurements. We explain qualitatively how classical phase space structures like fixed points and periodic orbits have an influence on these quantities and how our results compare to the large-spin kicked top model. Finally we point to a peculiar case at the border of quantum-classical correspondence which is solvable for any number of qubits and yet has signatures of exponential sensitivity in a rudimentary form.Show more - PublicationOut-of-time-order correlators of nonlocal block-spin and random observables in integrable and nonintegrable spin chains(01-06-2022)
;Shukla, Rohit Kumar; Mishra, Sunil KumarShow more Out-of-time-order correlators (OTOC) in the Ising Floquet system, which can be both integrable and nonintegrable, are studied. Instead of localized spin observables, we study contiguous symmetric blocks of spins or random operators localized on these blocks as observables. We find only power-law growth of OTOC in both integrable and nonintegrable regimes. In the nonintegrable regime, beyond the scrambling time, there is an exponential saturation of the OTOC to values consistent with random matrix theory. This motivates the use of "prescrambled"random block operators as observables. A pure exponential saturation of OTOC in both integrable and nonintegrable system is observed, without a scrambling phase. Averaging over random observables from the Gaussian unitary ensemble, the OTOC is found to be exactly same as the operator entanglement entropy, whose exponential saturation has been observed in previous studies of such spin chains.Show more - PublicationScrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest timescale(15-03-2020)
;Prakash, RaviShow more Out-of-time-order correlators (OTOCs), vigorously being explored as a measure of quantum chaos and information scrambling, is studied here in the natural and simplest multiparticle context of bipartite systems. We show that two strongly chaotic and weakly interacting subsystems display two distinct regimes in the growth of OTOCs. The first is dominated by intrasubsystem scrambling, when an exponential growth with a positive Lyapunov exponent is observed until the Ehrenfest time. This regime is essentially independent of the interaction, while the second one is an interaction dominated exponential approach to saturation that is universal and described by a random matrix model. This simple random matrix model of weakly interacting strongly chaotic bipartite systems, previously employed for studying entanglement and spectral transitions, is approximately analytically solvable for its OTOC. The example of two coupled kicked rotors is used to demonstrate the different regimes, and the extent to which the random matrix model is applicable. We remark that the second regime implies an emergent invariance of the OTOC under local unitary transformations suggesting that the rate of relaxation is connected to the entangling power of the interaction. That the two regimes correspond to delocalization in the subsystems followed by intersubsystem mixing is seen via the participation ratio in phase space. We also point out that the second, universal, regime alone exists when the observables are in a sense locally prescrambled.Show more - PublicationFrom dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy(01-12-2021)
;Aravinda, S. ;Rather, Suhail AhmadShow more Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are “as random as a coin toss.” Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power of the basic two-particle unitary building block, , of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries and such that the building block is leads to an identification of the average mixing rate as being determined predominantly by the entangling power . Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.Show more - PublicationThirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem(25-02-2022)
;Rather, Suhail Ahmad ;Burchardt, Adam ;Bruzda, Wojciech ;Rajchel-MieldzioÄ‡, Grzegorz; Zyczkowski, KarolShow more The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))6, which saturates the Singleton bound and allows one to encode a six-level state into a triplet of such states.Show more - PublicationConstruction and Local Equivalence of Dual-Unitary Operators: From Dynamical Maps to Quantum Combinatorial Designs(01-10-2022)
;Rather, Suhail Ahmad ;Aravinda, S.Show more While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization of dual-unitary operators themselves are only partially understood. A nonlinear map on the space of unitary operators has been proposed in Phys. Rev. Lett. 125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries. Here, we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries. A subset of dual-unitary operators having maximum entangling power are 2-unitary operators or perfect tensors and these are equivalent to four-party absolutely maximally entangled states. It is known that they only exist if the local dimension is larger than d=2. We use the nonlinear map, and introduce stochastic variants of it, to construct explicit examples of new dual and 2-unitary operators. A necessary criterion for their local unitary equivalence to distinguish classes is also introduced and used to display various concrete results and a conjecture in d=3. It is known that orthogonal Latin squares provide a "classical combinatorial design"for constructing permutations that are 2-unitary. We extend the underlying design from classical to genuine quantum ones for general dual-unitary operators and give an example of what might be the smallest-sized genuinely quantum design of a 2-unitary in d=4.Show more - PublicationEntanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength(23-10-2020)
;Jonnadula, Bhargavi; ;Å»yczkowski, KarolShow more Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power ep and a complementary quantity, the gate typicality gt. We characterize the boundaries of the set K2 representing all two-qubit gates projected onto the plane (ep,gt) showing that the fractional powers of the swap operator form a parabolic boundary of K2, while the other bounds are formed by two straight lines. In this way, a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of swap persists in the set KN for gates of higher dimensions (N>2). Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating "thermalization"in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.Show more - PublicationQuantum coherence controls the nature of equilibration and thermalization in coupled chaotic systems(01-02-2023)
;Pulikkottil, Jethin J.; ;Srivastava, Shashi C.L. ;Kieler, Maximilian F.I. ;BÃ¤cker, ArndTomsovic, StevenShow more A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interaction with a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax toward an equilibrium. It is found that quantum coherence of the initial product states in the energy eigenbasis of the subsystems - quantified by the off-diagonal elements of the subsystem density matrices - can be viewed as a resource for equilibration and thermalization as manifested by the entanglement generated. Results are given for four distinct perturbation strength regimes, the ultraweak, weak, intermediate, and strong regimes. For each, three types of tensor product states are considered for the initial state: uniform superpositions, random superpositions, and individual subsystem eigenstates. A universal timescale is identified involving the interaction strength parameter. In particular, maximally coherent initial product states (a form of uniform superpositions) thermalize under time evolution for any perturbation strength in spite of the fact that in the ultraweak perturbative regime the underlying eigenstates of the system have a tensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity as the interaction vanishes. Moreover, it is shown that in the ultraweak regime the initial entanglement growth of the system whose initial states are maximally coherent is quadratic-in-time, in contrast to the widely observed linear behavior.Show more - PublicationCreating Ensembles of Dual Unitary and Maximally Entangling Quantum Evolutions(14-08-2020)
;Rather, Suhail Ahmad ;Aravinda, S.Show more Maximally entangled bipartite unitary operators or gates find various applications from quantum information to many-body physics wherein they are building blocks of minimal models of quantum chaos. In the latter case, they are referred to as "dual unitaries."Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called "2-unitaries"and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists in any local dimension, which results in the formation of such special classes of unitary operators. We outline an iterative protocol, a nonlinear map on the space of unitary operators, that creates ensembles whose members are arbitrarily close to being dual unitaries. For qutrits and ququads we find that a slightly modified protocol yields a plethora of 2-unitaries.Show more - Publication9 Ã— 4 = 6 Ã— 6: Understanding the Quantum Solution to Euler's Problem of 36 Officers(01-01-2023)
;Zyczkowski, K. ;Bruzda, W. ;Rajchel-MieldzioÄ‡, G. ;Burchardt, A. ;Ahmad Rather, S.Show more The famous combinatorial problem of Euler concerns an arrangement of 36 officers from six different regiments in a 6×6 square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to superpositions of quantum states. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each quantum officer is represented by a superposition of at most 4 classical states. The entire quantum combinatorial design involves 9 Bell bases in nine complementary 4-dimensional subspaces.Show more