Nikola Burić

Hamiltonian quantum dynamics with separability constraints

Abstract Schroedinger equation on a Hilbert space H , represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space P H . Separable states of a bipartite quantum system form a special submanifold of P H . We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.

Statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems

General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.

Emergence of classical behavior from the quantum spin

Classical Hamiltonian system of a point moving on a sphere of fixed radius is shown to emerge from the constrained evolution of quantum spin. The constrained quantum evolution corresponds to an appropriate coarse-graining of the quantum states into equivalence classes, and forces the equivalence classes to evolve as single units representing the classical states. The coarse-grained quantum spin with the constrained evolution in the limit of the large spin becomes indistinguishable from the classical system.

Relations Between Different Notions of Degrees of Freedom of a Quantum System and Its Classical Model

There are at least three different notions of degrees of freedom (DF) that are important in comparison of quantum and classical dynamical systems. One is related to the type of dynamical equations and inequivalent initial conditions, the other to the structure of the system and the third to the properties of dynamical orbits. In this paper, definitions and comparison in classical and quantum systems of the tree types of DF are formulated and discussed. In particular, we concentrate on comparison of the number of the so called dynamical DF in a quantum system and its classical model. The comparison involves analyzes of relations between integrability of the classical model, dynamical symmetry and separability of the quantum and the corresponding classical systems and dynamical generation of appropriately defined quantumness. The analyzes is conducted using illustrative typical systems. A conjecture summarizing the observed relation between generation of quantumness by the quantum dynamics and dynamical properties of the classical model is formulated.

Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

Representation of classical dynamics by unitary transformations has been used to develop unified description of hybrid classical-quantum systems with particular type of interaction, and to formulate abstract systems interpolating between classical and quantum ones. We solved the problem of unitary description of two interpolating systems with general potential interaction. The general solution is used to show that with arbitrary potential interaction between the two interpolating systems the evolution of the so called unobservable variables is decoupled from that of the observable ones if and only if the interpolation parameters in the two interpolating systems are equal.

Dynamical time versus system time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system time operator, i.e., for the free one-dimensional particle, is obtained.

Phase space theory of quantum–classical systems with nonlinear and stochastic dynamics

Abstract A novel theory of hybrid quantum–classical systems is developed, utilizing the mathematical framework of constrained dynamical systems on the quantum–classical phase space. Both, the quantum and classical descriptions of the respective parts of the hybrid system are treated as fundamental. Therefore, the description of the quantum–classical interaction has to be postulated, and includes the effects of neglected degrees of freedom. Dynamical law of the theory is given in terms of nonlinear stochastic differential equations with Hamiltonian and gradient terms. The theory provides a successful dynamical description of the collapse during quantum measurement.

Constrained quantum dynamics and coarse-grained description of a quantum system of nonlinear oscillators

Constrained Hamiltonian dynamics is exploited to provide the mathematical framework of a coarse-grained description of the quantum system of nonlinear interacting oscillators. The coarse graining is treated as an equivalence relation on the set of quantum states resulting in the emergence of classical phase space. The equivalence relation imposes constraints on the Hamiltonian dynamics of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of the corresponding classical system on a sufficiently large scale.

Positive-operator-valued measures in the Hamiltonian formulation of quantum mechanics

In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.

Orbits of hybrid systems as qualitative indicators of quantum dynamics

Abstract Hamiltonian theory of hybrid quantum–classical systems is used to study dynamics of the classical subsystem coupled to different types of quantum systems. It is shown that the qualitative properties of orbits of the classical subsystem clearly indicate if the quantum subsystem does or does not have additional conserved observables.