arXiv subjects

Arul Lakshminarayan

Publications and source records attributed to Arul Lakshminarayan.

At least 55 records · Page 3Linked to original sources

arXiv2018

Quantum correlations as probes of chaos and ergodicity

Long-time average behavior of quantum correlations in a multi-qubit system, collectively modeled as a kicked top, is addressed here. The behavior of dynamical generation of quantum correlations such as entanglement, discord, concurrence, as previously studied, and Bell correlation function and tangle, as identified in this study, is a function of initially localized coherent states. Their long-time average reproduces coarse-grained classical phase space structures of the kicked top which contrast, often starkly, chaotic and regular regions. Apart from providing numerical evidence of such correspondence in the semiclassical regime of a large number of qubits, we use data from a recent transmons based experiment to explore this in the deep quantum regime of a 3-qubit kicked top. The degree to which quantum correlations can be regarded as a quantum signature of chaos, and in what ways the various correlation measures are similar or distinct are discussed.

quant-ph↗

arXiv2017

Persistent entanglement in a class of eigenstates of quantum Heisenberg spin glasses

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 ran- dom 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.

quant-ph↗

arXiv2016

Can local dynamics enhance entangling power?

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 applications of any entangling operator interlaced with random local gates, we prove that both investigated quantities approach their asymptotic values in a simple exponential form. 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.

quant-ph↗

arXiv2016

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely realignment and partial transpose. This bipartite quantity, denoted here as $R_{12}$, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three qubit states via their two-qubit reduced density matrices. Upper and lower bounds on concurrence and negativity of two-qubit density matrices for all ranks are given in terms of $R_{12}$. Ansatz states satisfying these bounds are given and verified using various numerical methods. In rank-2 case it is shown that the states satisfying the lower bound on $R_{12}$ {\it vs} concurrence define a class of three qubit states that maximizes the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure $R_{12}$ is conjectured, via numerical sampling, to be always larger than the concurrence and negativity. In particular this is shown to be true for the physically interesting case of $X$ states.

quant-ph↗

arXiv2016

Entanglement and localization transitions in eigenstates of interacting chaotic systems

The entanglement and localization in eigenstates of strongly chaotic subsystems are studied as a function of their interaction strength. Excellent measures for this purpose are the von-Neumann entropy, Havrda-Charv{\' a}t-Tsallis entropies, and the averaged inverse participation ratio. All the entropies are shown to follow a remarkably simple exponential form, which describes a universal and rapid transition to nearly maximal entanglement for increasing interaction strength. An unexpectedly exact relationship between the subsystem averaged inverse participation ratio and purity is derived that infers the transition in the localization as well.

quant-ph↗

arXiv2015

Local entanglement structure across a many-body localization transition

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.

cond-mat.str-el↗

arXiv2015

Universal scaling of spectral fluctuation transitions for interacting chaotic systems

The spectral properties of interacting strongly chaotic systems are investigated for growing interaction strength. A very sensitive transition from Poisson statistics to that of random matrix theory is found. We introduce a new random matrix ensemble modeling this dynamical symmetry breaking transition which turns out to be universal and depends on a single scaling parameter only. Coupled kicked rotors, a dynamical systems paradigm for such transitions, are compared with this ensemble and excellent agreement is found for the nearest-neighbor-spacing distribution. It turns out that this transition is described quite accurately using perturbation theory.

nlin.CD↗

arXiv2015

Real eigenvalues of non-Gaussian random matrices and their products

We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the number of matrices in the product increases. Here we present numerical evidence that this phenomenon is robust with respect to the probability distribution of matrix elements, and is therefore a general property that merits detailed investigation. Since the elements of the product matrix are no longer distributed as those of the single matrix nor they remain independent random variables, we study the role of these two factors in detail. We study numerically the properties of the Hadamard (or Schur) product of matrices and also the product of matrices whose entries are independent but have the same marginal distribution as that of normal products of matrices, and find that under repeated multiplication, the probability of all eigenvalues to be real increases in both cases, but saturates to a constant below unity showing that the correlations amongst the matrix elements are responsible for the approach to one. To investigate the role of the non-normal nature of the probability distributions, we present a thorough analytical treatment of the $2 \times 2$ single matrix for several standard distributions. Within the class of smooth distributions with zero mean and finite variance, our results indicate that the Gaussian distribution has the maximum probability of real eigenvalues, but the Cauchy distribution characterised by infinite variance is found to have a larger probability of real eigenvalues than the normal. We also find that for the two-dimensional single matrices, the probability of real eigenvalues lies in the range [5/8,7/8].

math-ph↗

arXiv2015

Records in the classical and quantum standard map

Record statistics is the study of how new highs or lows are created and sustained in any dynamical process. The study of the highest or lowest records constitute the study of extreme values. This paper represents an exploration of record statistics for certain aspects of the classical and quantum standard map. For instance the momentum square or energy records is shown to behave like that of records in random walks when the classical standard map is in a regime of hard chaos. However different power laws is observed for the mixed phase space regimes. The presence of accelerator modes are well-known to create anomalous diffusion and we notice here that the record statistics is very sensitive to their presence. We also discuss records in random vectors and use it to analyze the {\it quantum} standard map via records in their eigenfunction intensities, reviewing some recent results along the way.

cond-mat.stat-mech↗

arXiv2014

Protocol using kicked Ising dynamics for generating states with maximal multipartite entanglement

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 till the blocks are maximally entangled at which stage an unravelling begins at the same rate till the blocks are unentangled. The model is a variant of the transverse field Ising model and can be implemented with CNOT and single qubit gates. The inter qubit 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.

quant-ph↗

arXiv2014

Diagonal unitary entangling gates and contradiagonal quantum states

Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $ρ=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $ρ$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.

quant-ph↗

arXiv2013

On the number of real eigenvalues of products of random matrices and an application to quantum entanglement

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 2 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 ($1/\sqrt{2}$) for a single matrix, it is shown numerically that the probability that {\it 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 is also presented, where an exponential approach of the expected number to the dimension of the matrix seems to hold.

math-ph↗

arXiv2013

Quenching and generation of random states in a kicked Ising model

The kicked Ising model with both a pulsed transverse and a continuous longitudinal field is studied numerically. Starting from a large transverse field and a state that is nearly an eigenstate, the pulsed transverse field is quenched with a simultaneous enhancement of the longitudinal field. The generation of multipartite entanglement is observed along with a phenomenon akin to quantum resonance when the entanglement does not evolve for certain values of the pulse duration. Away from the resonance, the longitudinal field can drive the entanglement to near maximum values that is shown to agree well with those of random states. Further evidence is presented that the time evolved states obtained do have some statistical properties of such random states. For contrast the case when the fields have a steady value is also discussed.

cond-mat.stat-mech↗

arXiv2013

Using Partial Transpose and Realignment to generate Local Unitary Invariants

Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson {\it et. al.}, Phys. Rev. A {\bf 83}, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher dimensional states are briefly discussed.

quant-ph↗

arXiv2013

Record statistics in random vectors and quantum chaos

The record statistics of complex random states are analytically calculated, and shown that the probability of a record intensity is a Bernoulli process. The correlation due to normalization leads to a probability distribution of the records that is non-universal but tends to the Gumbel distribution asymptotically. The quantum standard map is used to study these statistics for the effect of correlations apart from normalization. It is seen 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 random states.

cond-mat.stat-mech↗

arXiv2012

Entanglement signatures for the dimerization transition in the Majumdar-Ghosh model

The transition from a gapless liquid to a gapped dimerized ground state that occurs in the frustrated antiferromagnetic Majumdar-Ghosh (or $J_1-J_2$ Heisenberg) model is revisited from the point of view of entanglement. We study the evolution of entanglement spectra, a "projected subspace" block entropy, and concurrence in the Schmidt vectors through the transition. The standard tool of Schmidt decomposition along with the existence of the unique MG point where the ground states are degenerate and known exactly, suggests the projection into two orthogonal subspaces that is useful even away from this point. Of these, one is a dominant five dimensional subspace containing the complete state at the MG point and the other contributes marginally, albeit with increasing weight as the number of spins is increased. We find that the marginally contributing subspace has a minimum von Neumann entropy in the vicinity of the dimerization transition. Entanglement content between pairs of spins in the Schmidt vectors, studied via concurrence, shows that those belonging to the dominant five dimensional subspace display a clear progress towards dimerization, with the concurrence vanishing on odd/even sublattices, again in the vicinity of the dimerization, and maximizing in the even/odd sublattices at the MG point. In contrast, study of the Schmidt vectors in the marginally contributing subspace, as well as in the projection of the ground state in this space, display pair concurrence which decrease on both the sublattices as the MG point is approached. The robustness of these observations indicate their possible usefulness in the study of models that have similar transitions, and have hitherto been difficult to study using standard entanglement signatures.

cond-mat.str-el↗

arXiv2012

Entanglement between two subsystems, the Wigner semicircle and extreme value statistics

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $ρ_{12}^{T_2}$. The density of states of $ρ_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $ρ_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.

quant-ph↗

arXiv2011

Entanglement transitions in random definite particle states

Entanglement within qubits are studied for the subspace of definite particle states or definite number of up spins. A transition from an algebraic decay of entanglement within two qubits with the total number $N$ of qubits, to an exponential one when the number of particles is increased from two to three is studied in detail. In particular the probability that the concurrence is non-zero is calculated using statistical methods and shown to agree with numerical simulations. Further entanglement within a block of $m$ qubits is studied using the log-negativity measure which indicates that a transition from algebraic to exponential decay occurs when the number of particles exceeds $m$. Several algebraic exponents for the decay of the log-negativity are analytically calculated. The transition is shown to be possibly connected with the changes in the density of states of the reduced density matrix, which has a divergence at the zero eigenvalue when the entanglement decays algebraically.

quant-ph↗