arXiv subjects

Vaibhav Madhok

Publications and source records attributed to Vaibhav Madhok.

36 records · Page 2Linked to original sources

arXiv2018

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

Many-body states that are invariant under particle relabelling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining a new random matrix ensemble, we find the average linear entropy and von Neumann entropy which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multi-qubit system we find that entanglement, mutual information and TMI all increase for large subsystems across the Ehrenfest or log-time and saturate at the random state values if there is global chaos. During this time the out-of-time order correlators (OTOC) evolve exponentially implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.

quant-ph↗

arXiv2018

Quantum chaos, thermalization and tunneling in an exactly solvable few body system

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important - more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to non-integrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the 3 and 4 qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons based experiment realizing the 3-qubit case, we find agreement for short times, including a peculiar step-like behaviour in correlations of some states. In the case of 4-qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.

quant-ph↗

arXiv2018

Evolutionary dynamics from deterministic microscopic ecological processes: A toy model for evolutionary processes

The central goal of a dynamical theory of evolution is to abstract the mean evolutionary trajectory in the trait space by considering ecological processes at the level of the individual. In this work, we develop such a theory for a new class of deterministic individual based models describing individual births and deaths, which captures the essential features of standard stochastic individual-based models and become identical with the latter under maximal competition. The key motivation is to derive the canonical equation of adaptive dynamics from this microscopic ecological model, which can be regarded as a "toy model" for evolution, in a simple way and give it an intuitive geometric interpretation. Another goal is to study evolution and sympatric speciation under "maximal" competition. We show that these models, in the deterministic limit of adaptive dynamics, lead to the same equations that describe the unraveling of the mean evolutionary trajectory as those obtained from the standard stochastic models. We further study conditions under which these models lead to evolutionary branching and find them to be similar with those obtained from the standard stochastic models. We find that though deterministic models result in a strong competition that leads to a speed up in the temporal dynamics of a population cloud in the phenotypic space as well as an increase in the rate of generation of biodiversity, it does not seem to result in an absolute increase in biodiversity as far as total number of species are concerned. Hence, the "toy model" essentially captures all the features of the standard stochastic model. Interestingly, the notion of a fitness function does not explicitly enter in our derivation of the canonical equation, thereby advocating a mechanistic view of evolution.

q-bio.PE↗

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

Equations of Evolutionary Dynamics in High Dimensions

We study quasi-species and closely related evolutionary dynamics like the replicator-mutator equation in high dimensions. In particular, we show that under certain conditions the fitness of almost all quasi-species becomes independent of mutational probabilities and the initial frequency distributions of the sequences in high dimensional sequence spaces. This result is the consequence of the concentration of measure on a high dimensional hypersphere and its extension to Lipschitz functions known %knows as the Levy's Lemma. Therefore, evolutionary dynamics almost always yields the same value for fitness of the quasi-species, independent of the mutational process and initial conditions, and is quite robust to mutational changes and fluctuations in initial conditions. Our results naturally extend to any Lipschitz function whose input parameters are the frequencies of individual constituents of the quasi-species. This suggests that the functional capabilities of high dimensional quasi-species are robust to fluctuations in the mutational probabilities and initial conditions. We discuss the consequences of our study for the replicator-mutator equation.

q-bio.PE↗

arXiv2016

Quasi-Species in High Dimensional Spaces

We show that, under certain assumptions, the fitness of almost all quasi-species becomes independent of mutational probabilities and the initial frequency distributions of the sequences in high dimensional sequence spaces. This result is the consequence of the concentration of measure on a high dimensional hypersphere and its extension to Lipschitz functions knows as the Levy's Lemma. Therefore, evolutionary dynamics almost always yields the same value for fitness of the quasi-species, independent of the mutational process and initial conditions, and is quite robust to mutational changes and fluctuations in initial conditions. Our results naturally extend to any Lipschitz function whose input parameters are the frequencies of individual constituents of the quasi-species. This suggests that the functional capabilities of high dimensional quasi-species are robust to fluctuations in the mutational probabilities and initial conditions.

q-bio.PE↗

arXiv2016

Efficient Simulations of Individual Based Models for Adaptive Dynamics and the Canonical Equation

We propose a faster algorithm for individual based simulations for adaptive dynamics based on a simple modification to the standard Gillespie Algorithm for simulating stochastic birth-death processes. We provide an analytical explanation that shows that simulations based on the modified algorithm, in the deterministic limit, lead to the same equations of adaptive dynamics as well as same conditions for evolutionary branching as those obtained from the standard Gillespie algorithm. Based on this algorithm, we provide an intuitive and simple interpretation of the canonical equation of adaptive dynamics. With the help of examples we compare the performance of this algorithm to the standard Gillespie algorithm and demonstrate its efficiency. We also study an example using this algorithm to study evolutionary dynamics in a multi-dimensional phenotypic space and study the question of predictability of evolution.

q-bio.PE↗

arXiv2015

Comment on "Entanglement and chaos in the kicked top"

We comment on the investigation of the connection between chaos and dynamically generated entanglement in Phys. Rev. E, \textbf{83}, 016207, (2011). While, in the referred paper, the authors give an explicit example of a state initially localized in the regular region and still has entanglement properties similar to the states localized in the chaotic region, a few clarifications related to previous works are in order. Firstly, it is crucial to point out that such a behavior can occur for states initially localized near the border between the chaotic region and regular island, which seems to be the case in the example provided in the referred paper. We comment on the time evolution of such states and the extent to which these can be regarded as having "regular dynamics". Secondly, the degree to which entanglement is correlated with the chaos in the system is better understood when we analyze the same initial state and increase the chaoticity in the system gradually. We also discuss in what capacity entanglement can be regarded as a signature of chaos in such studies.

quant-ph↗

arXiv2015

Individual-Based models for adaptive diversification in high-dimensional phenotype spaces

Most theories of evolutionary diversification are based on equilibrium assumptions: they are either based on optimality arguments involving static fitness landscapes, or they assume that populations first evolve to an equilibrium state before diversification occurs, as exemplified by the concept of evolutionary branching points in adaptive dynamics theory. Recent results indicate that adaptive dynamics may often not converge to equilibrium points and instead generate complicated trajectories if evolution takes place in high-dimensional phenotype spaces. Even though some analytical results on diversification in complex phenotype spaces are available, to study this problem in general we need to reconstruct individual-based models from the adaptive dynamics generating the non-equilibrium dynamics. Here we first provide a method to construct individual-based models such that they faithfully reproduce the given adaptive dynamics attractor without diversification. We then show that a propensity to diversify can by introduced by adding Gaussian competition terms that generate frequency dependence while still preserving the same adaptive dynamics. For sufficiently strong competition, the disruptive selection generated by frequency-dependence overcomes the directional evolution along the selection gradient and leads to diversification in phenotypic directions that are orthogonal to the selection gradient.

q-bio.PE↗

arXiv2015

Characterizing and Quantifying Quantum Chaos with Quantum Tomography

We explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. The tomographic record consists of a time series of expectation values of a Hermitian operator evolving under application of the Floquet operator of a quantum map that possesses (or lacks) time reversal symmetry. We find that the rate of information gain, and hence the fidelity of quantum state reconstruction, depends on the symmetry class of the quantum map involved. Moreover, we find an increase in information gain and hence higher reconstruction fidelities when the Floquet maps employed increase in chaoticity. We make predictions for the information gain and show that these results are well described by random matrix theory in the fully chaotic regime. We derive analytical expressions for bounds on information gain using random matrix theory for different class of maps and show that these bounds are realized by fully chaotic quantum systems.

quant-ph↗

arXiv2014

Chaos in high-dimensional dynamical systems

For general dissipative dynamical systems we study what fraction of solutions exhibit chaotic behavior depending on the dimensionality $d$ of the phase space. We find that a system of $d$ globally coupled ODE's with quadratic and cubic non-linearities with random coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from $\sim 10^{-5} - 10^{-4}$ for $d=3$ to essentially one for $d\sim 50$. In the limit of large $d$, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity but does not depend on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling and for the probability of chaos.

cond-mat.dis-nn↗

arXiv2013

Multiparty quantum communication using multiqubit entanglement and teleportation

We propose a 2N qubit entangled channel that can be used to teleport N qubits in a network to a single receiver. We describe the construction of this channel and explicitly demonstrate how the protocol works. The protocol is different from teleportation using $N$ Bell pairs shared between $N$ senders and a single receiver, as it ensures that all parties have to participate in order for the teleportation to be successful. This can be advantageous in various scenarios and we discuss potential application of this protocol to voting.

quant-ph↗

arXiv2013

Signatures of chaos in the dynamics of quantum discord

We identify signatures of chaos in the dynamics of discord in a multiqubit system collectively modelled as a quantum kicked top. The evolution of discord between any two qubits is quasiperiodic in regular regions, while in chaotic regions, the quasiperiodicity is lost. As the initial wave function is varied from the regular regions to the chaotic sea, a contour plot of the time averaged discord remarkably reproduces the structures of the classical stroboscopic map. We also find surprisingly opposite behaviour of two-qubit discord versus two-qubit entanglement. Our calculations provide the first evidence of signatures of chaos in dynamically generated discord.

quant-ph↗

arXiv2013

Information gain in tomography - A quantum signature of chaos

We find quantum signatures of classical chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and hence higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory.

quant-ph↗

arXiv2012

Quantum Correlations, Chaos and Information

Quantum chaos is the study of quantum systems whose classical description is chaotic. How does chaos manifest itself in the quantum world? In this spirit, we study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section - it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudo-random states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. Continuing on our journey to find the footprints of chaos in the quantum world, we explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. We find an increase in the rate of information gain and hence higher fidelities in the process when the Floquet maps employed increase in chaoticity. We make predictions for the information gain using random matrix theory in the fully chaotic regime and show a remarkable agreement between the two. The last part of this thesis is devoted to the study of the nature of quantum correlations themselves. We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information.

quant-ph↗

arXiv2012

Quantum discord as a resource in quantum communication

As quantum technologies move from the issues of principle to those of practice, it is important to understand the limitations on attaining tangible quantum advantages. In the realm of quantum communication, quantum discord captures the damaging effects of a decoherent environment. This is a consequence of quantum discord quantifying the advantage of quantum coherence in quantum communication. This establishes quantum discord as a resource for quantum communication processes. We discuss this progress, which derives a quantitative relation between the yield of the fully quantum Slepian-Wolf protocol in the presence of noise and the quantum discord of the state involved. The significance of quantum discord in noisy versions of teleportation, super-dense coding, entanglement distillation and quantum state merging are discussed. These results lead to open questions regarding the tradeoff between quantum entanglement and discord in choosing the optimal quantum states for attaining palpable quantum advantages in noisy quantum protocols.

quant-ph↗

arXiv2011

Role of quantum discord in quantum communication

Positivity of quantum discord is shown to be equivalent to the strong sub additivity of the Von-Nuemann entropy. This leads to a connection between the mother protocol of quantum information theory [A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465, 2537, (2009)] and quantum discord. We exploit this to show that quantum discord is a measure coherence in the performance of the mother protocol. Since the mother protocol is a unification of an important class of problems (those that are bipartite, unidirectional and memoryless), we show quantum discord to be a measure of coherence in these protocols. Our work generalizes an earlier operational interpretation of quantum discord provided in terms of quantum state merging [V. Madhok and A. Datta, Phys. Rev. A 83, 032323, (2011)].

quant-ph↗

arXiv2010

Interpreting quantum discord through quantum state merging

We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information. Our interpretation has an intuitive explanation based on the strong subadditivity of von Neumann entropy. We use our result to provide operational interpretations of other quantities like the local purity and quantum deficit. Finally, we discuss in brief some instances where our interpretation is valid in the single copy scenario.

quant-ph↗