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.Comment: 12 pages, 1 Tabl

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.Comment: 13 pages; This is a thoroughly revised version of the original preprint which has now appearted in print. It also corrects several errors and misstatements in the published version. The main conclusions of the paper however remain intac

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.Comment: 10 pages, 2 figure

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.Comment: 19 pages, no figures. Invited talk at Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Santa Fe, USA, November 6-9, 200

Generalization of the Lie-Trotter Product Formula for q-Exponential Operators

The Lie-Trotter formula $e^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N$ is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical chemistry, to many-body and thermostatistical calculations. We generalize it for the q-exponential function $e_q (x) = [1+ (1-q) x]^{(1/(1-q))}$ (with $e_1(x)=e^x$), and prove $e_q(\hat{A}+\hat{B}+(1-q) [\hat{A}\hat{B}+\hat{B}\hat{A}] /2) = \lim_{N\to \infty} {[e_{1-(1-q)N}(\hat{A}/N)] [e_{1-(1-q)N}(\hat{B}/N)]}^N$. This extended formula is expected to be similarly useful in the nonextensive situationsComment: 5 pages, no figure

Majorana representation of symmetric multiqubit states

As early as 1932, Majorana had proposed that a pure permutation symmetric state of N spin-${\frac{1}{2}}$ particles can be represented by N spinors, which correspond geometrically to N points on the Bloch sphere. Several decades after its conception, the Majorana representation has recently attracted a great deal of attention in connection with multiparticle entanglement. A novel use of this representation led to the classification of entanglement families of permutation symmetric qubits—based on the number of distinct spinors and their arrangement in constituting the multiqubit state. An elegant approach to explore how correlation information of the whole pure symmetric state gets imprinted in its parts is developed for specific entanglement classes of symmetric states. Moreover, an elegant and simplified method to evaluate geometric measure of entanglement in N-qubit states obeying exchange symmetry has been developed based on the distribution of the constituent Majorana spionors over the unit sphere. Multiparticle entanglement being a key resource in several quantum information processing tasks, its deeper understanding is essential. In this review, we present a detailed description of the Majorana representation of pure symmetric states and its applicability in investigating various aspects of multiparticle entanglement

Open-​system quantum dynamics with correlated initial states, not completely positive maps, and non-​Markovianity

Dynamical A and B maps have been employed extensively by Sudarshan and co-​workers to investigate open-​system evolution of quantum systems. A canonical structure of the A map is introduced here. It is shown that this canonical A map enables us to investigate whether the dynamics is completely pos. (CP) or not completely pos. (NCP) in an elegant way and, hence, it subsumes the basic results on open-​system dynamics. Identifying memory effects in open-​system evolution is gaining increasing importance recently and, here, a criterion of non-​Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization-​though not related directly-​to the fidelity difference criterion proposed recently. Three typical examples of open-​system evolution of a qubit, prepd. initially in a correlated state with another qubit (environment)​, and evolving jointly under a specific unitary dynamics-​which corresponds to a NCP dynamical map-​are investigated by employing both the relative entropy difference and fidelity difference tests of non-​Markovianity. The two-​qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-​Markovianity manifests itself in all three examples

Unsharp measurements and joint measurability

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice’s pair of non-commuting observables in the presence of Bob’s entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice–Bob state, implementing incompatible POVMs at Bob’s end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix – which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables – could be realized if and only if compatible POVMs are employed

Quantum Entangled Supercorrelated States in the Jaynes-Cummings Model

The regions of independent quantum states, maximally classically correlated states, and purely quantum entangled (supercorrelated) states described in a recent formulation of quantum information theory by Cerf and Adami are explored here numerically in the parameter space of the well-known exactly soluable Jaynes-Cummings model for equilibrium and nonequilibrium time-dependent ensembles.Comment: 12 pages, 3 figure

Nonadditive conditional entropy and its significance for local realism

Based on the form invariance of the structures given by Khinchin's axiomatic foundations of information theory and the pseudoadditivity of the Tsallis entropy indexed by q, the concept of conditional entropy is generalized to the case of nonadditive (nonextensive) composite systems. The proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context, indicating its utility for characterizing quantum entanglement. A criterion deduced from it for separability of density matrices for validity of local realism is examined in detail by employing a bipartite spin-1/2 system. It is found that the strongest criterion is obtained in the limit q going to infinity.Comment: 12 pages, 1 figur