Possibility of state cloning is analyzed in two types of generalizations of quantum mechanics with nonlinear evolution. It is first shown that nonlinear Hamiltonian quantum mechanics does not admit cloning without the cloning machine. It is then demo
nstrated that the addition of the cloning machine, treated as a quantum or as a classical system, makes the cloning possible by nonlinear Hamiltonian evolution. However, a special type of quantum-classical theory, known as the mean-field Hamiltonian hybrid mechanics, does not admit cloning by natural evolution. The latter represents an example of a theory where it appears to be possible to communicate between two quantum systems at super-luminal speed, but at the same time it is impossible to clone quantum pure states.
Lagrangian formulation of quantum mechanical Schrodinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system res
ults in a well defined second order equation on a real vector space. The Klein-Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrodinger equation.
We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and t
he quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfilment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfilment of the quasi-independence approximation and certain forms of synchronization between the individual units.
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 indica
te if the quantum subsystem does or does not have additional conserved observables.
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 an
d 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.
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 the classical descriptions of the respective parts of t
he 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.
We point out the crucial difference between the relative and absolute phase observables treated in our contribution cite{1} and in the Comment by Hall and Pegg cite{HP} respectively. The main contribution of our work is to show that the quantum expec
tation of the relative phase is highly discontinuous function of the frequency and to point out interesting dependence of the phase on the number-theoretic nature of the frequencies.
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 distingu
ishable 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.
The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations comprised of $N$ stochastic excitable units each is performed by studying an approximate system, obtained by replacing eac
h population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the inter-ensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical inter-population couplings.
Constrained Hamiltonian description of the classical limit is utilized in order to derive consistent dynamical equations for hybrid quantum-classical systems. Starting with a compound quantum system in the Hamiltonian formulation conditions for class
ical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The presented formalism suggests that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.
