On time evolution of non-isolated physical systems

Conditions sufficient for a minimal quantum dynamical semigroup (QDS) to be conservative are proved for the class of problems in quantum optics under the assumption that the self-adjoint Hamiltonian of the QDS is a finite degree polynomial in the creation and annihilation operators. The degree of the Hamiltonian may be greater than the degree of the completely positive part of the generator of the QDS. The conservativity (or the unital property) of a minimal QDS implies the uniqueness of the solution of the corresponding master Markov equation, i.e., in the unital case, the formal generator determines the QDS uniquely; moreover, in the Heisenberg representation, the QDS preserves the unit observable, and in the Schrodinger representation, it preserves the trace of the initial state. The analogs of the conservativity condition for classical Markov evolution equations (such as the heat and the Kolmogorov--Feller equations) are known as nonexplosion conditions or conditions excluding the escape of trajectories to infinity.

Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.

This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.

Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm—utilizing a decomposition of nonunitary operators approach—that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.

Using the expression of the fidelity for the most general Gaussian quantum states, the quantum fidelity is studied for the states of a harmonic oscillator interacting with an environment, in particular with a thermal bath. The time evolution of the considered system is described in the framework of the theory of open systems based on quantum dynamical semigroups. By taking a correlated squeezed Gaussian state as initial state, we calculate the quantum fidelity for both undisplaced and displaced states. The time evolution of the quantum fidelity is analyzed depending on the squeezing and correlation parameters characterizing the initial Gaussian state and on the dissipation constant and temperature of the thermal bath.

This chapter describes certain fundamental differences between classical and quantum mechanics, their different postulates, the role of the observer, what is meant by local and non-local interactions, causality and determinism, and the role of force, energy, and momentum. A short introduction to the purely quantum mechanical aspect of superposition, measurement, and entanglement is provided to mentally prepare the reader for the chapters ahead.

The space of pure states of any physical system,classical or quantum, is identified as a Poisson spacewith a transition probability. These two structures areconnected through unitarity. Classical and quantum mechanics are each characterized by asimple axiom on the transition probability p. Unitaritythen determines the Poisson bracket of quantum mechanicsup to a multiplicative constant (identified with Planck's constant).

Sufficient conditions for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.

Bayesian approach to inverse time-dependent quantum mechanics

Using an analytical model potential which contains resonant and bound states, we show that the decay of the resonances can be simulated by Lindblad operators. For that purpose, the unitary time evolution of an initial Gaussian wave packet in the model potential is compared with the non-unitary time evolution, obtained by solving the Lindblad equation, of the same wave packet in a potential which coincides with the model potential in the region of interest but does not contain resonances. In the latter case, dissipative effects are accounted for by Lindblad operators which lead to phenomenological friction and diffusion constants in the equations of motion. We suggest how those constants can be determined in a non-heuristic way, being directly connected to the width of the resonance in the model potential which we calculate using the complex rotation method.

Because dissipative processes imply irreversibility and, therefore, a preferred direction in time, it is generally thought that quantum dynamical semigroups are the basic tools to introduce rigorously the dissipation in quantum mechanics. The most general form of the generators of such semigroups was given by Lindblad1. This formalism has been studied for the case of damped harmonic oscillators2,3 and applied to various physical phenomena, for instance, to the damping of collective modes in deep inelastic collisions in nuclear physics4. According to the axiomatic theory of Lindblad1, the usual von Neumann-Liouville equation ruling the time evolution of closed quantum systems is replaced in the case of open systems by the following time-homogeneous quantum mechanical Markovian master equation for the density operator ρ:KeywordsHarmonic OscillatorDensity OperatorWigner FunctionCollective ModeOpen Quantum SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper explores the consequences of the orthodox resolution of the measurement problem for the axiomatic base of non-relativistic elementary quantum mechanics. It is argued that the standard resolution of the measurement problem generates a paradox whose dissolution may be achieved through an enrichment of the axiomatic foundations of quantum mechanics. These results are also linked to some recent creative proposals by Nancy Cartwright concerning the nature of the so-called “reduction of the wave packet”.

A one-parameter model for the spread of Avian Influenza A/H5N1

Recent inquiries into optical communication have raised questions as to the validity of classical detection and estimation theory for weak light fields. Helstrom [1] proposed that the axioms of quantum mechanics be incorporated into a quantum approach to optical estimation and detection. In this paper, we discuss two important results, the quantum equivalent of the minimum-mean-square-error (MMSE) estimator and the quantum Cramér-Rao bound for estimation of the parameters of an electromagnetic field. The first result, a new one, is applied to linear modulation. We show that homodyning is the optimal demodulation scheme in that case. Parallels to the classical MMSE estimator are drawn. The Cramér-Rao bound, first derived in the quantum ease by Helstrom [1], is applied to specific estimation problems. Details are left to the references, but interesting results are presented.

The axioms of quantum mechanics