Infinite Dimensional Analysis, Quantum Probability and Applications
QP41 Conference, Al Ain, UAE, March 28–April 1, 2021
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Article
The Expected Markov Property for Quantum Markov Fields
After a short introduction to the Algebraic formulation of classical stochastic processes and fields, we extend to the quantum case the notion of expected classical Markov field and we prove the equivalence of...
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Article
Infinite Volume Limits of Entangled States
We study the structure of entangled states on infinite tensor products. We prove that the infinite volume limit of entangled states is, in many cases, a separable state (or even a product state). This result w...
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Book and Conference Proceedings
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Chapter and Conference Paper
The Non–linear and Quadratic Quantization Programs
After a short outline of the notion of canonical quantum decomposition of a classical random field and of its connection with the program of non–linear quantization, we concentrate our attention on quadratic q...
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Article
Characterization of Product States on Polynomial Algebras in Terms of Scalar Products of the Associated n-Particle Vectors
We prove that, in the framework of generalized quantizations, orthogonality of \(n\) -particle vectors corresponding to different multi-indexes, together with an additional condition on the norms of \(n\) -par...
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Article
Orthogonal Polynomial Decomposition for Random Fields with All Moments
We discuss the difference between orthogonal polynomials on finite and infinite dimensional vectors spaces. In particular we prove an infinite dimensional extension of Favard lemma.
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Chapter
Complementarity and Stochastic Independence
A mathematical approach to the notion of complementarity in quantum physics is described and its historical development is shortly reviewed. After that, the notion of n-complementarity is introduced as a natur...
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Article
\(C^*\) -Quadratic Quantization
In the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual Heisenberg group introduced in Accardi et al. (Infin Dimens Anal Quantum Probab Rel...
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Article
C*-Non-Linear Second Quantization
We construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previ...
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Chapter and Conference Paper
\(*\) –Lie Algebras Canonically Associated to Probability Measures on \({\pmb {\varvec{\mathbb {R}}}}\) with All Moments
In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on
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Article
On Quantum Markov Chains on Cayley Tree III: Ising Model
In this paper, we consider the classical Ising model on the Cayley tree of order \(k\) ...
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Chapter
Models of Time
The first part of the chapter describes, in a qualitative way, a scheme of axiomatic approach to the notion of time. It is shown that, even restricting the physical requirements to a minimum, a multiplicity of...
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Article
On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three
In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of su...
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Article
Quantum Probability: New Perspectives for the Laws of Chance
The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery mar...
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Chapter and Conference Paper
Itô Calculus and Quantum White Noise Calculus
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Article
Entangled Markov chains
Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated and unique...
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Article
A Stochastic Limit Approach to the SAT Problem
There exists an important problem whether there exists an algorithm to solve an NP-complete problem in polynomial time. In this paper, a new concept of quantum adaptive stochastic systems is proposed, and it i...
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Chapter and Conference Paper
Square of White Noise Unitary Evolutions on Boson Fock Space
With the help of Mathematica we deduce an explicit formula for bringing to normal order the product of two normally ordered monomials in the generators of the Lie algebra of SL(2, ℝ). We use this formula to prove...
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Article
Phase Measurement in Interacting Fock Space
In this paper we discuss probability operator measure and phase measurement in one mode interacting Fock space.
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Article
Control of Quantum Langevin Equations
The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply th...
