Markov dilations and quantum detailed balance

We construct quantum stochastic processes whose multi-time correlation functions, with suitable time ordering, can be obtained from a quantum dynamical semigroup. We prove that such a process defines a stationary Markov dilation of the associated semigroup if and only if (up to technicalities) the semigroup satisfies the quantum detailed balance condition with respect to its stationary state.     

Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M_n(\mathbbC){M_n(\mathbb{C})} for all n ≥ 3, as well as an example of a one-parameter semigroup (T(t)) _t≥0 of Markov maps on M₄(\mathbbC){M_4(\mathbb{C})} such that T(t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality.     

The low density limit for anN-level system interacting with a free bose or fermi gas

It is proved that the reduced dynamics of anN-level system coupled to a free quantum gas converges to a quantum dynamical semigroup in the low density limit. The proof uses a perturbation series of the quantum BBGKY-hierarchy, and the analysis of this series is based on scattering theory. The limiting semigroup contains the full scattering cross section, but it does not depend on the statistics of the reservoir. The dynamics of the semigroup is discussed.     

Constructing the davies process of resonance fluorescence with quantum stochastic calculus

The starting point is a given semigroup of completely positive maps on the 2×2 matrices. This semigroup describes the irreversible evolution of a decaying two-level atom. By using the integral-sum kernel approach to quantum stochastic calculus, the two-level atom is coupled to an environment, which in this case will be interpreted as the electromagnetic field. The irreversible time evolution of the two-level atom then stems from the reversible time evolution of the atom and the field together. Mathematically speaking, a Markov dilation of the semigroup has been constructed. The next step is to drive the atom by a laser and to count the photons emitted into the field by the decaying two-level atom. For every possible sequence of photon counts, a map is constructed that gives the time evolution of the two-level atom implied by that sequence. The family of maps obtained in this way forms a so-called Davies process. In his book, Davies describes the structure of these processes, which brings us into the field of quantum trajectories. Within the model presented in this paper, the jump operators are calculated and the resulting counting process is briefly described.     

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra ? with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator of T _t , existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation of T _t is obtained through solving a canonical flow equation for maps on the right Fock module ?⊗Γ(L ²(ℝ₊,k ₀)), where k ₀ is some Hilbert space arising from a representation of ?^′. This gives rise to a *-homomorphism j _t of ?. Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration. Received: 15 June 1998/ Accepted: 4 March 1999     

Relativistic semigroup,the lorentz group,and tachyons. IV

The algebraic and topological properties of the relativistic semigroup are discussed. Its probability-theoretical features establish that the relativistic semigroup belongs to the type of complex Markov structures. From the functional point of view, the relativistic semigroup is a compact Lie semigroup which is contracting in partial spaces. Principles of measurability, observability, and stochasticity are formulated, and these lead to a space-time structure of complex Markov kind. Thus, a certain probability-theoretical gnosiology is also possible in the theory of relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–58, August, 1977.I thank D. D. Ivanenko for valuable advice and numerous discussions.     

On the generators of quantum dynamical semigroups

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB() is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.     

Quantum Graphical Models and Belief Propagation

Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersley–Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.     

Fixed Point Result for Environment-Induced Semigroups

Based on the environment-induced semigroup approach to the quantum measurement process, we show that a certain class of these semigroups, referred to as contractive uniformly k-Lipschitzian semigroups, exhibit a fixed point property. With regard to the quantum measurement problem, semigroups of this kind ensure decoherence and the selection of a single state from the family of pointer states. In fact, the common fixed point is the selected state.     

A Pseudo-cocycle for the Comultiplication on the Quantum <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">2</Emphasis>) Group

We show that the comultiplication on the quantum group SU _q (2) may be obtained from that on the quantum semigroup SU ₀(2) by twisting with a unitary 2-pseudo-cocycle. Work supported by the ARC Linkage International Fellowship LX0667294, and by the Korea Research Foundation Grant (KRF-2004-041-C00024).     

Coherent Population Trapping and Partial Decoherence in the Stochastic Limit

We use the stochastic limit technique to predict a new phenomenon concerning a two-level atom with degenerate ground state interacting with a quantum field. We show, that the field drives the state of the atom to a stationary state, which is non-unique, but depends on the initial state of the system through some conserved quantities. This non uniqueness follows from the degeneracy of the ground state of the atom, and when the ground subspace is two-dimensional, the family of stationary states will depend on a one-dimensional parameter. Only one of the stationary states in this family is a pure state and it coincides with the known trapped state. This means that by controlling the initial state (input) we can control the final state (output). The quantum Markov semigroup obtained in the limit admits an invariant pure state, but it is not true that all the extremal invariant states are pure. This is an interesting phenomenon also from mathematical point of view and its meaning will be discussed in a future paper. PACS numbers: 31.15.-p, 31.15.Gy, 32.80.Pj, 32.80.Qk     

Dividing Quantum Channels

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.     

Quantum mechanics of space and time

A formulation of relativistic quantum mechanics is presented independent of the theory of Hilbert space and also independent of the hypothesis of spacetime manifold. A hierarchy is established in the nondistributive lattice of physical ensembles, and it is shown that the projections relating different members of the hierarchy form a semigroup. It is shown how to develop a statistical theory based on the definition of a statistical operator. Involutions defined on the matrix representations of the semigroup are interpreted in terms ofCPT conjugations. The theory of particles of spin one-half and systems with higher spin is developed from first principles. Methods are also developed for defining energy, momentum, orbital angular momentum, and weighted spacetime coordinates without reference to a manifold.     

The Gamow vectors and the Schwinger effect

We introduce a ‘proper time’ formalism to study the instability of the vacuum in a uniform external electric field due to particle production. This formalism allows us to reduce a quantum field-theoretic problem to a quantum mechanical one in a higher dimension. The instability results from the inverted oscillator structure which appears in the Hamiltonian. We show that the ‘proper time’ unitary evolution splits into two semigroups. The semigroup associated with decaying Gamov vectors is related to the Feynman boundary conditions for the Green functions and the semigroup associated with growing Gamov vectors is related to the Dyson boundary conditions.     

‘Return to Equilibrium’ for Weakly Coupled Quantum Systems: A Simple Polymer Expansion

Recently, several authors studied small quantum systems weakly coupled to free boson or fermion fields at positive temperature. All the rigorous approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. We show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation aka Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation.     

Spectral decomposition of expanding probabilistic dynamical systems

We study probabilistic combinations of expanding dynamical systems, which we call expanding probabilistic dynamical systems, in one dimension. If the system is composed by exact endomorphisms we prove that the probabilistic dynamical system is an exact Markov semigroup, and we determine a generalized spectral decomposition of the associated Markov operator on densities for an example of the tent map coupled with the 2-Renyi map.     

Algebraic implications of composability of physical systems

From classical and quantum mechanics we abstract the concept of a two-product algebra. One of its products is left unspecified; the other is a Lie product and a derivation with respect to the first. From composition of physical systems we abstract the concept of composition classes of such two-product algebras, each class being a semigroup with a unit. We show that the requirement of mutual consistency of the algebraic and the semigroup structures completely determines both the composition classes and the two-product algebras they consist of. The solutions are labelled by a single parameter which in the physical case is proportional to the square of the quantum of action.     

Variational processes from the weak forward equation

In this paper the author constructs Markov diffusion processes from a given system of Borel probability measures on ad-dimensional Euclidean space. He constructs a, so-called, variational process which does not always coincide with a Nelson process. He also discusses Schrödinger's problem in quantum mechanics.     

Group-theoretical structure of quantum continuous measurements

The group-theoretical structure of continuous measurements is investigated in the framework of the path-integral phenomenological theory of quantum continuous measurements. The “transversal” group transforming alternative measurement results (outputs) into each other and the “longitudinal” semigroup describing the evolution of a quantum system subject to continuous measurement are introduced as well as their unification in a single semigroup. The resulting group-theoretical scheme generalizes the scheme describing the evolution of a nonrelativistic particle in an external field.