Piotr Ługiewicz

From quantum to quantum via decoherence

Various physical effects resulting from decoherence are discussed in the algebraic framework. In particular, it is shown that the environment may induce not only classical properties like superselection rules, pointer states or even classical behavior of the quantum system, but, what is more, it also allows the transition from statistical description of infinite quantum systems to quantum mechanics of systems with a finite number of degrees of freedom. It is shown that such transition holds for the quantum spin system in the thermodynamic limit interacting with the phonon field.

Decoherence in infinite quantum systems

Environmentally induced classical properties in quantum spin systems in the thermodynamic limit are discussed. In particular, it is shown that such a system, subjected to a specific interaction with another quantum system, may be described for all practical purposes as a classical dynamical system. This provides an alternative approach to the classical ? 0 limit for the emergence of reversible classical evolution from dissipative quantum dynamics.

Quantum dynamical semigroups for finite and infinite Bose systems

A new class of quasifree quantum Markov semigroups on C*-algebras of canonical commutation relations is introduced and discussed. Two applications to decoherence in the Heisenberg representation are given. In the first one the dynamical semigroup which leads to the appearance of decoherence induced superselection rules corresponding to the boundary conditions of a quantum particle in a finite interval is considered. The second example analyzes the possibility of the transition from infinite systems to systems with a finite number of degrees of freedom.

Qutrit geometric discord

Abstract Properties of the trace norm geometric discord of the system of two qutrits are studied. The geometric discord of qutrit Bell states, Werner states and bound entangled states is computed.

Measurement-induced qudit geometric discord

We study measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d = 3 systems) are here strictly proved for arbitrary d. The present study supports the observations, coming also from other approaches, that systems with show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d = 2 and d = 3. Qubit systems are exceptionally simple. The underlying geometry of state spaces and related Lie groups is responsible for this.

NONLINEAR MARKOV SEMIGROUPS ON C*-ALGEBRAS

A notion of a nonlinear quantum dynamical semigroup is introduced and discussed. Some sufficient conditions, expressed solely in terms of the duality map, in order that a multivalued mapping on a C*-algebra generates the nonlinear Markov semigroup are proposed.

Ergodic properties of diffusion-type quantum dynamical semigroups

We develop a framework for investigation of asymptotic behavior of quantum Markov semigroups on C*-algebras associated with noncommutative elliptic generators. An exponential rate of convergence toward the projection onto the fixed point subalgebra has been established and, in a particular case of the semigroup selecting rotationally invariant states of a three-dimensional quantum system, the time factor of such a convergence has been estimated.

Theory of the Decoherence Effect in Finite and Infinite Open Quantum Systems Using the Algebraic Approach

Quantum mechanics is the greatest revision of our conception of the character of the physical world since Newton. Consequently, David Hilbert was very interested in quantum mechanics. He and John von Neumann discussed it frequently during von Neumann’s residence in Gottingen. He published in 1932 his book Mathematical Foundations of Quantum Mechanics. In Hilbert’s opinion it was the first exposition of quantum mechanics in a mathematically rigorous way. The pioneers of quantum mechanics, Heisenberg and Dirac, neither had use for rigorous mathematics nor much interest in it. Conceptually, quantum theory as developed by Bohr and Heisenberg is based on the positivism of Mach as it describes only observable quantities. It first emerged as a result of experimental data in the form of statistical observations of quantum noise, the basic concept of quantum probability.