arXiv subjects

A. K. Rajagopal

Publications and source records attributed to A. K. Rajagopal.

At least 73 records · Page 4Linked to original sources

arXiv2003

Validity of the second law in nonextensive quantum thermodynamics

The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.

cond-mat.stat-mech↗

arXiv2003

Statistical mechanical foundations of power-law distributions

The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic paaroaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -- maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements on each of these methods are briefly pointed out: in the first approach, the function space and the counting algorithm while in the second, "additivity" property of the entropy with respect to the composition of statistically independent systems. In the past few decades, a large number of systems, which are not necessarily in thermodynamic equilibrium (such as glasses, for example), have been found to display power-law distributions, which are not describable by the above-mentioned methods. In this paper, parallel to all the inquiries underlying the BG program described above are given in a brief form. In particular, in the probabilistic derivations, one employs a different function space and one gives up "additivity" in the variational scheme with a different form for the entropy. The requirement of stability makes the entropy choice to be that proposed by Tsallis. From this a generalized thermodynamic description of the system in a quasi-equilibrium state is derived. A brief account of a unified consistent formalism associated with systems obeying power-law distributions precursor to the exponential form associated with thermodynamic equilibrium of systems is presented here.

cond-mat.stat-mech↗

arXiv2003

Heat and Entropy in nonextensive thermodynamics

The concepts of quantity of heat and work are deduced in the non-extensive statistical mechanics context, following steps in parallel to those employed in the extensive statistical mechanics.

cond-mat.stat-mech↗

arXiv2003

Revivals and entanglement from initially entangled mixed states of a damped Jaynes-Cummings model

An exact density matrix of a phase-damped Jaynes - Cummings model (JCM) with entangled Bell-like initial states formed from a model two-state atom and sets of adjacent photon number states of a single mode radiation field is presented. The entanglement of the initial states and the subsequent time evolution is assured by finding a positive lower bound on the concurrence of local 2x2 projections of the full 2xinfinity JCM density matrix. It is found that the time evolution of the lower bound of the concurrence systematically captures the corresponding collapse and revival features in atomic inversion, relative entropies of atomic and radiation, mutual entropy, and quantum deficit. The atom and radiation subsystems exhibit alternating sets of collapses and revivals in a complementary fashion due to the initially mixed states of the atom and radiation employed here. This is in contrast with the result obtained when the initial state of the dissipationless system is a factored pure state of atom and radiation, where the atomic and radiation entropies are necessarily the same. The magnitudes of the entanglement lower bound and the atomic and radiation revivals become larger as both magnitude and phase of the Bell-like initial state contribution increases. The time evolution of the entropy difference of the total system and that of the radiation subsystem exhibits negative regions called "supercorrelated" states which do not appear in the atomic subsystem. Entangled initial states are found to enhance this supercorrelated feature. Finally, the effect of phase damping is to randomize both the subsystems for asymptotically long times .

quant-ph↗

arXiv2002

Aspects of the Second Law of Thermodynamics from Quantum Statistical Mechanics to Quantum Information Theory

The Kullback-Leibler inequality is a way of comparing any two density matrices. A technique to set up the density matrix for a physical system is to use the maximum entropy principle, given the entropy as a functional of the density matrix, subject to known constraints. In conjunction with the master equation for the density matrix, these two ingredients allow us to formulate the second law of thermodynamics in its widest possible setting. Thus problems arising in both quantum statistical mechanics and quantum information can be handled. Aspects of thermodynamic concepts such as the Carnot cycle will be discussed. A model is examined to elucidate the role of entanglement in the Landauer erasure problem.

cond-mat.stat-mech↗

arXiv2002

Separability and correlations in composite states based on entropy methods

This work is an enquiry into the circumstances under which entropy methods can give an answer to the questions of both quantum separability and classical correlations of a composite state. Several entropy functionals are employed to examine the entanglement and correlation properties guided by the corresponding calculations of concurrence. It is shown that the entropy difference between that of the composite and its marginal density matrices may be of arbitrary sign except under special circumstances when conditional probability can be defined appropriately. This ambiguity is a consequence of the fact that the overlap matrix elements of the eigenstates of the composite density matrix with those of its marginal density matrices also play important roles in the definitions of probabilities and the associated entropies, along with their respective eigenvalues. The general results are illustrated using pure and mixed state density matrices of two-qubit systems. Two classes of density matrices are found for which the conditional probability can defined: (1) density matrices with commuting decompositions and (2) those which are decohered in the representation where the density matrices of the marginals are diagonal. The first class of states encompass those whose separability is currently understood as due to particular symmetries of the states. The second are a new class of states which are expected to be useful for understanding separability. Examples of entropy functionals of these decohered states including the crucial isospectral case are discussed.

quant-ph↗

arXiv2001

Robust and Fragile Entanglement of Three Qubits: Relation to Permutation Symmetry

Entanglement properties of a basic set of eight entangled three particle pure states possessing certain permutation symmetries are studied. They fall into four sets of two entangled states, differing in their patterns of robustness to entanglement when one of the states is lost. These features are related to the permutation symmetries of the spin states of the three particles and their corresponding marginal two-particle states. It is interesting to note that the eight entangled three-qubit states discussed here are eigenstates of a three-spin Heisenberg Hamiltonian.

quant-ph↗

arXiv2001

Dissipative quantum theory: Implications for quantum entanglement

Three inter-related topics are discussed here. One, the Lindblad dynamics of quantum dissipative systems; two, quantum entanglement in composite systems and its quantification based on the Tsallis entropy; and three, robustness of entanglement under dissipation. After a brief review of the Lindblad theory of quantum dissipative systems and the idea of quantum entanglement in composite quantum systems illustrated by describing the three particle systems, the behavior of entanglement under the influence of dissipative processes is discussed. These issues are of importance in the discussion of quantum nanometric systems of current research.

quant-ph↗

arXiv2001

Theory of ferromagnetism in (A,Mn)B semiconductors

A brief review of theory of ferromagnetism of dilute magnetic semiconductors of the form (A,Mn)B based on the double exchange model is first given. A systematic investigation of the phenomena extending the current theory is outlined. We begin with an investigation of the regions of instability of the nonmagnetic towards the ferromagnetic state of a system of Mn-atoms doped in AB-type semiconductor. A self-consistent many-body theory of the ferromagnetic state is then developed, going beyond the mean field approaches by including fluctuations of the Mn-spins and the itinerant hole-gas. A functional theory suitable for computation of system properties such as Curie temperature as a function of hole and the Mn-concentration, spin-current, etc. is formulated.

cond-mat.str-el↗

arXiv2000

Nonuniqueness of Canonical Ensemble Theory arising from Microcanonical Basis

Given physical systems, counting rule for their statistical mechanical descriptions need not be unique, in general. It is shown that this nonuniqueness leads to the existence of various canonical ensemble theories which equally arise from the definite microcanonical basis. Thus, the Gibbs theorem for canonical ensemble theory is not universal, and the maximum entropy principle is to be appropriately modefied for each physical context.

quant-ph↗

arXiv2000

Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half spaces

The Levy-type distributions are derived using the principle of maximum Tsallis nonextensive entropy both in the full and half spaces. The rates of convergence to the exact Levy stable distributions are determined by taking the N-fold convolutions of these distributions. The marked difference between the problems in the full and half spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Levy indices. An important result emerging from the present analysis is deduced if interpreted in terms of random walks, implying the dependence of the asymptotic long-time behaviors of the walks on the ranges of the Levy indices if N is identified with the total time of the walks.

cond-mat.stat-mech↗

arXiv2000

Macroscopic thermodynamics of equilibrium characterized by power-law canonical distributions

Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.

cond-mat.stat-mech↗

arXiv2000

Decoherence, Correlation, and Entanglement in a Pair of Coupled Quantum Dissipative Oscillators

A pair of coupled quantum dissipative oscillators, serving as a model for a nanosystem, is here described by the Lindblad equation. Its dynamic evolution is shown to exhibit the features of decoherence (spatial extent of quantum behavior), correlation (spatial scale over which the system localizes to its physical dimensions), and entanglement (a special quantum feature making its appearance first in such bipartite systems) as a function of the coupling constants of the Lindblad equation. One interesting feature emerging out of this calculation is that the entanglement may exhibit revivals in time. An initially entangled state need not remain so for all time and may exhibit regions of nonentanglement. Interpreting the parameters of the Lindblad theory as environmental features in certain experimental situations, this model calculation gives us clues to possible control of decoherence, correlation, and entanglement. We indicate possible interpretation of the Lindblad parameters as control parameters in more general contexts of some recent experiments.

quant-ph↗

arXiv2000

Control of Decoherence and Correlation in Single Quantum Dissipative Oscillator Systems

A single quantum dissipative oscillator described by the Lindblad equation serves as a model for a nanosystem. This model is solved exactly by using the ambiguity function. The solution shows the features of decoherence (spatial extent of quantum behavior), correlation (spatial scale over which the system localizes to its physical dimensions), and mixing (mixed- state spatial correlation). A new relation between these length scales is obtained here. By varying the parameters contained in the Lindblad equation, it is shown that decoherence and correlation can be controlled. We indicate possible interpretation of the Lindblad parameters in the context of experiments using engineered reservoirs.

quant-ph↗

arXiv2000

Two Qubits in the Dirac Representation

A general two qubit system expressed in terms of the complete set of unit and fifteen traceless, Hermitian Dirac matrices, is shown to exhibit novel features of this system. The well-known physical interpretations associated with the relativistic Dirac equation involving the symmetry operations of time-reversal T, charge conjugation C, parity P, and their products are reinterpreted here by examining their action on the basic Bell states. The transformation properties of the Bell basis states under these symmetry operations also reveal that C is the only operator that does not mix the Bell states whereas all others do. In a similar fashion, expressing the various logic gates introduced in the subject of quantum computers in terms of the Dirac matrices shows for example, that the NOT gate is related to the product of time-reversal and parity operators.

quant-ph↗

arXiv2000

Levy distribution in half space based on nonextensive statistical mechanics

Probability distributions defined on the half space are known to be quite different from those in the full space. Here, a nonextensive entropic treatment is presented for the half space in an analytic and self-consistent way. In this development, the ordinary first moment of the random variable X is divergent in contrast to the case of the full space. A general (nu)-th moment of X is considered as a constraint in the principle of maximum Tsallis entropy. The infinite divisibility of the distribution with an arbitrary (nu) larger than zero and convergence of its N-fold convolution to the exact Levy-stable distribution is discussed in detail. A feature of this derivation is that the Levy index is related to both the values of (nu) and the index of nonextensivity.

cond-mat.stat-mech↗

arXiv2000

Statistical Mechanical Foundations for Systems with Nonexponential Distributions

Traditionally the exponential canonical distributions of Gibbsian statistical mechanics are given theoretical justification in at least four different ways: steepest descent method, counting method, Khinchin's method based on te central limit theorem, and maximum entropy principle of Jaynes. Equally ubiquitous power-law canonical distributions are shown to be given similar justification by appropriately adopting these formulations.

cond-mat.stat-mech↗

arXiv2000

Towards Nonadditive Quantum Information Theory

A definition of the nonadditive (nonextensive) conditional entropy indexed by q is presented. Based on the composition law in terms of it, the Shannon-Khinchin axioms are generalized and the uniqueness theorem is established for the Tsallis entropy. The nonadditive conditional entropy, when considered in the quantum context, is always positive for separable states but takes negative values for entangled states, indicating its utility for characterizing entanglement. A criterion deduced from it for separability of the density matrix is examined in detail by using a bipartite spin-half system. It is found that the strongest criterion for separability obtained by Peres using an algebraic method is recovered in the present information-theoretic approach.

quant-ph↗