Quantum Probability and Related Topics

The following sections are included:

• FOREWORD

• CONTENTS

The following sections are included:

• Introduction and statement of the problem

• Random variables in CeHeis_ℂ(1)

• Infinite divisibility

• Kernels and matrices

• Positive definite kernels

• Functions of positive definite matrices and kernels

• Conditionally positive definite kernels

• Infinitely divisible kernels

• The Kolmogorov decomposition theorem for ℂ–valued PD kernels

• Boson Fock spaces

• Infinitely divisible kernels and Boson Fock spaces

• References

We review some recent results concerning Landau levels and Tomita-Takesaki modular theory. We also extend the general framework behind this to quasi *-algebras, to take into account the possible appearance of unbounded observables.

By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.

We introduce a one-mode type interacting Fock space naturally associated to the negative binomial distribution μ_r,α. The Fourier transform in generalized joint eigenvectors of a family {J_ϕ ; ϕ ∈ ε} of Pascal Jacobi fields provides a way to explicit a unitary isomorphism between and the so-called Pascal white noise space L²(ε′, Λ_r,α). Then, we derive a chaotic decomposition property of the quadratic integrable functionals of the Pascal white noise process in terms of an appropriate wick tensor product.

We review the basic concepts of quantum stochastic analysis using the universal Itô B*-algebra approach. The thermal quantum white noise as a noncommutative analog of the Lévy-Itô process is characterized in terms of the modular B*-Itô algebra. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The Belavkin master equation for a quasi-Markov quantum stochastic dynamics on the predual space of a W*-algebra is presented as a noncommutative version of the Zakai equation driven by the modular quantum white noise. This is done by the noncommutative analog of the Girsanov transformation which we introduce in full QS generality.

In this note we consider a twisted quantum wave guide i.e. a domain of the form Ω_θ := r_θω × ℝ where ω is a connected open and bounded subset of ℝ² and r_θ = r_θ(x₃) is a rotation by an angle θ(x₃) depending on the longitudinal variable x₃. We are interested in the spectral analysis of the Dirichlet Laplacian acting in Ω_θ. The results discussed here come from a joint paper between : P.Briet, H. Kovarik, G. Raikov, E.Soccorsi, Comm. P.D.E., (2009).

Let be a quantum Markov semigroup on with a faithful normal invariant state ρ whose generator is represented in a generalised GKSL form , with possibly unbounded H, L_ℓ. We show that the biggest von Neumann-subalgebra of where acts as a semigroup of automorphisms coincides with the generalised commutator of under some natural regularity conditions.

The proof we present here does not involve dilations .

We give a sufficient condition for all elements in the set of invariant states of a Quantum Markov Semigroup (QMS) to be diagonal. We apply our result to complete the characterization of the set of invariant states for the asymmetric exclusion Quantum Markov Semigroup introduced in Ref.¹⁰

This paper presents mathematically well grounded statistical methods for state estimation in the indirect measurement setting, when the measurement is performed on an ancilla system that is put into interaction with the unknown one. Both the unknown and the ancilla quantum systems are assumed to be quantum bits. The measurements applied on the ancilla qubit are the classical von Neumann measurements using the Pauli matrices as observables. The repeated measurements performed on the ancilla enables us to construct estimators of the initial state of the unknown system. Based on the statistical properties of the considered indirect measurement scheme,¹⁶ three related but different approaches are proposed and investigated: (i) a direct estimation procedure that is based on the estimated relative frequencies of the characterizing conditional probability densities, (ii) Bayesian recursive approach for state estimation, and (iii) a martingale approach that bases the estimator on the stopping times of the state evolution as a martingale driven by the repeated measurements.

The statistical properties, i.e. the unbiasedness and the efficiency of the proposed procedures are investigated both analytically and experimentally using simulation.

We study the invariant state of generic quantum Markov semigroups using the Fagnola-Rebolledo criterion. This class arising from the stochastic limit of a discrete system with generic Hamiltonian H_S, acting on h, interacting with a Gaussian, gauge invariant, reservoir.

We define conditionally monotone independence in two states which interpolates monotone and Boolean ones. This independence is associative, and therefore leads to a natural probability theory in a non-commutative algebra.

White Noise analysis may be thought of a well-established theory. This is true in a sense, however we are surprised to find that there are so many profound properties still remain undiscovered. In this report, we shall have a quick review of white noise theory, then we shall propose some of future directions to be investigated, from our viewpoint. Further, we shall discuss a new noise which is of Poisson type.

In a note we discuss several problems and difficulties arising during the study of stochastic Volterra equations of convolution type. The resolvent approach to such equations enables us to obtain results in an analogous way like in the semigroup approach used to stochastic differential equations. However, in the resolvent case new difficulties arise because the solution operator corresponding to the Volterra equation in general does not create a semigroup. In this paper we present some consequences and complications coming from deterministic and stochastic convolutions connected with the stochastic Volterra equations under consideration.

Entanglement is a fundamental concept in Quantum Mechanics. In this paper, we study various aspects of coherence and entanglement, illustrated by several examples. We relate the concepts of loss of coherence and disentanglement, via a model of two two-level atoms in different types of reservoir, including both cases of independent and common bath. We also relate decoherence and disentanglement, by focussing on the sudden death of the entanglement and the dependence of the death time with the "distance" of our initial condition, from the decoherence free subspace. In particular, we study the "sudden death of the entanglement", in a two-atom system with a common reservoir. Finally we study the creation of entanglement under various initial conditions.

The subject of this paper is a mathematical transition from the Fisher information of classical statistics to the matrix formalism of quantum theory. If the monotonicity is the main requirement, then there are several quantum versions parametrized by a function. In physical applications the minimal is the most popular. There is a one-to-one correspondence between Fisher informations (called also monotone metrics) and abstract covariances. The skew information and the χ²-divergence are treated here as particular cases.

While the notion of complementarity in quantum systems goes back to the beginning of quantum theory, the concept of complementary subalgebras is quite new. Here we overview the recent developments of the field, characterizations of complementarity and complementary decompositions are reviewed, and we point to some still open questions.

We consider semigroups {α_t : t ≥ 0} of normal, unital, completely positive maps α_t on a von Neumann algebra . The (predual) semigroup ν_t(ρ) := ρoα_t on normal states ρ of leaves invariant the face supported by the projection , if and only if α_t(p) ≥ p (i.e., p is sub-harmonic). We complete the arguments showing that the sub-harmonic projections form a complete lattice. We then consider r_o, the smallest projection which is larger than each support of a minimal invariant face; then r_o is sub-harmonic. In finite dimensional cases sup α_t(r_o) = 1 and r_o is also the smallest projection p for which α_t(p) → 1. If {ν_t : t ≥ 0} admits a faithful family of normal stationary states then r_o = 1 is useless; if not, it helps to reduce the problem of the asymptotic behaviour of the semigroup for large times.

We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *–algebras are Hilbert modules. As a result of our considerations we discuss requirements a notion of positivity on a *–algebra should fulfill and derive some basic consequences.

We extend the non-commutative constructions outlined in Ref.^1–3 to the general n parameter case, these being integrals over an n - dimensional parameter space, employing martingales as integrand. We extend previous results in type one,^1,4 type two^1,5 stochastic integrals and introduce new integrals for the Clifford sheet. These stochastic integrals are orthogonal, centred L² - martingales, obeying isometry properties. We develop the construction to consider general martingale representations for the Clifford sheet.