Fisher information and kinetic energy functionals: A dequantization approach

We strengthen the connection between information theory and quantum-mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed. The dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term. The purely quantum term, which results from the quantum fluctuations, is essentially identical to the Fisher information. The classical term is complementary to the Fisher information and, in this sense, it plays a role analogous to that of the Shannon entropy. We demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals.     

Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

Wigner-Yanase skew information vs. quantum Fisher information

Among concepts describing the information contents of quantum mechanical density operators, both the Wigner-Yanase skew information and the quantum Fisher information defined via symmetric logarithmic derivatives are natural generalizations of the classical Fisher information. We will establish a relationship between these two fundamental quantities and show that they are comparable.

     

Direct approach to quantum extensions of Fisher information

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.      

Separability and entanglement in tripartite states

While classical correlations can be freely distributed among many systems, this is not true for entanglement and quantum correlations. If a quantum system S^a is entangled with another quantum system S^b, then its entanglement with any third quantum system S^c cannot be arbitrary. This is the celebrated monogamy of entanglement. Implicit in this general statement is the plausible belief that only entanglement between the systems S^a and S^b constrains the entanglement between S^a and the third system S^c. We demonstrate that even classical correlations between S^a and S^b may impose surprisingly stringent restrictions on the possible entanglement between S^a and S^c. In particular, perfect bipartite classical correlations and full entanglement cannot coexist in any tripartite state. An intuitive explanation of this monogamy of hybrid classical and quantum correlations might be that the system S^a has a correlating capability, which cannot be used to establish any entanglement with a third system (but can still be used to establish classical correlations) if it is exhausted when correlated with S^b (in either a classical or quantum fashion). This may be interpreted as an alternate version of monogamy.     

Electric dipole in a magnetic field: Bound states without classical turning points

We show that both rigid and nonrigid dipoles can be trapped by an uniform external magnetic field in classical mechanics. The trapped states of the dipole present a nontrivial example of classical bound states embedded in a continuum (BSEC) that can be treated as analogues of quantum BSECs. For example, the classical motion of the dipole is confined to a finite region in space although there are no classical turning points. We also examine the quantum motion of the dipole in a magnetic field and show that for the most natural choices of the parameters, no quantum BSEC solutions exist. The possibilities of experimental investigations of BSECs are discussed. Deceased. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 189–205, November, 1998.     

Using the wigner function for quantum transport in device simulation

The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions.     

Uncertainty principle and quantum Fisher information

A family of inequalities, related to the uncertainty principle, has been recently proved by S. Luo, Z. Zhang, Q. Zhang, H. Kosaki, K. Yanagi, S. Furuichi and K. Kuriyama. We show that the inequalities have a geometric interpretation in terms of quantum Fisher information. Using this formulation one may naturally ask if this family of inequalities can be further extendend, for example to the RLD quantum Fisher information. We show that this is impossible by producing a family of counterexamples.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

Entanglement,fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

A characterization of extended monotone metrics

Classical information geometry has emerged from the study of geometrical aspect of the statistical estimation. Cencov characterized the Fisher metric as a canonical metric on probability simplexes from a statistical point of view, and Campbell extended the characterization of the Fisher metric from probability simplexes to positive cone . In quantum information geometry, quantum states which are represented by positive Hermitian matrices with trace one are regarded as an extension of probability distributions. A quantum version of the Fisher metric is introduced, and is called a monotone metric. Petz characterized the monotone metrics on the space of all quantum states in terms of operator monotone functions. A purpose of the present paper is to extend a characterization of monotone metrics from the space of all states to the space of all positive Hermitian matrices on finite dimensional Hilbert space. This characterization corresponds quantum modification of Campbell’s work.     

Quantumness of quantum ensembles

Quantum ensembles, as generalizations of quantum states, are a universal instrument for describing the physical or informational status in measurement theory and communication theory because of the ubiquitous presence of incomplete information and the necessity of encoding classical messages in quantum states. The interrelations between the constituent states of a quantum ensemble can display more or less quantum characteristics when the involved quantum states do not commute because no single classical basis diagonalizes all these states. This contrasts sharply with the situation of a single quantum state, which is always diagonalizable. To quantify these quantum characteristics and, in particular, to more clearly understand the possibilities of secure data transmission in quantum cryptography, based on certain prototypical quantum ensembles, we introduce some figures of merit quantifying the quantumness of a quantum ensemble, review some existing quantities that are interpretable as measures of quantumness, and investigate their fundamental properties such as subadditivity and concavity. Comparing these measures, we find that different measures can yield different quantumness orderings for quantum ensembles. This reveals the elusive and complex nature of quantum ensembles and shows that no unique measure can describe all the fundamental and subtle properties of quantumness.     

Normalization in Lie algebras via mould calculus and applications

We establish Écalle’s mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré–Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.     

Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

We show that in contrast to a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. We consider an approximation based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the second-order term. To escape technical difficulties related to the infinite dimensionality of the phase space for quantum mechanics, we consider finite-dimensional quantum mechanics. On one hand, this is a simple example with high pedagogical value. On the other hand, quantum information operates in a finite-dimensional state space. Therefore, our investigation can be considered a construction of a classical statistical model for quantum information. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 278–291, August, 2007.     

On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling

We discuss the local asymptotic behavior of the likelihood function associated with all the four characterizing parameters (α,β,δ,μ) of the Meixner Lévy process under high-frequency sampling scheme. We derive the optimal rate of convergence for each parameter and the Fisher information matrix in a closed form. The skewness parameter β exhibits a slower rate alone, relative to the other three parameters free of sampling rate. An unusual aspect is that the Fisher information matrix is constantly singular for full joint estimation of the four parameters. This is a particular phenomenon in the regular high-frequency sampling setting and is of essentially different nature from low-frequency sampling. As soon as either α or δ is fixed, the Fisher information matrix becomes diagonal, implying that the corresponding maximum likelihood estimators are asymptotically orthogonal.     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

Quantum and classical correlations in high-temperature dynamics of two coupled large spins

We study the dynamical correlations between two coupled spins depending on time and the value of the spin quantum numbers. In the high-temperature approximation, we obtain analytic expressions for the mutual information and the quantum and classical parts of correlations. We consider both orthogonal and nonorthogonal measurements in the basis of spin coherent states. We show that at small times, the quantum part of correlations becomes much less than the classical part as the spin quantum numbers increase, while the situation is quite different at times equal to half the quantum period.