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Dissipative dynamics for infinite lattice systems Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2024-03-11 Shreya Mehta, Boguslaw Zegarlinski
We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated
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Combinatorial aspects of weighted free Poisson random variables Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2024-02-17 Nobuhiro Asai, Hiroaki Yoshida
This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted
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Quantum properties of classical Pearson random variables Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-12-27 Luigi Accardi, Abdon Ebang Ella, Un Cig Ji, Yun Gang Lu
This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable X with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied
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On near-martingales and a class of anticipating linear stochastic differential equations Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-12-16 Hui-Hsiung Kuo, Pujan Shrestha, Sudip Sinha, Padmanabhan Sundar
The goals of this paper are to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. For a class of anticipating linear stochastic differential equations, we prove the existence and uniqueness of solutions using two approaches: (1) Ayed–Kuo differential formula using an ansatz, and (2) a
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Hamiltonian of free field on infinite-dimensional hypercube Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-11-20 Lixia Zhang, Caishi Wang
The infinite-dimensional hypercube (IDH) is an infinite connected graph with infinite degree at each its vertex, and can be viewed as an infinite-dimensional analog of finite-dimensional hypercubes. In this paper, we investigate a self-adjoint operator determined by the topology of the IDH and a function on the nonnegative integers, which can be interpreted as the Hamiltonian of a free fermion field
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Uniqueness and superposition of the space-distribution-dependent Zakai equations Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-11-15 Meiqi Liu, Huijie Qiao
The work concerns nonlinear filtering problems of McKean–Vlasov stochastic differential equations with correlated noises. First of all, we establish the space-distribution-dependent Kushner–Stratonovich equations and the space-distribution-dependent Zakai equations. Then, the pathwise uniquenesses of their strong solutions are shown. Finally, we prove a superposition principle between the space-di
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Stochastic averaging principle for McKean–Vlasov SDEs driven by Lévy noise Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-11-10 Tingting Zhang, Guangjun Shen, Xiuwei Yin
In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a
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An infinite-dimensional nonlinear equation related to Gibbs measures of a SOS model Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-11-04 U. A. Rozikov
For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the
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Relative entropy via distribution of observables Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-09-06 George Androulakis, Tiju Cherian John
We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the
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Operator-valued Gaussian processes and their covariance kernels Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-08-30 Palle E. T. Jorgensen, James Tian
In this paper, we discuss a new framework for operator-valued Gaussian processes and their covariance kernels. Our emphasis is four-fold: (i) starting with a positive operator-valued measure (POVM) Q, we present algorithms for constructing an associated centered, operator-valued, Gaussian process X with Q as its covariance kernel; (ii) we present different classes of POVMs, and we examine the corresponding
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Exponential convergence in the trace norm and time-inhomogeneous Markovian evolutions on ℬ(ℋ) Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-08-29 Zheng Li
We consider a sequence of operators τn that describes the time evolution of normal states of an open quantum system. Our objective is to analyze the asymptotic behavior of τn∘⋯∘τ1(ω0), where ω0 is an arbitrary initial normal state. Our focus is on specific situations in which the operators τn are produced by various quantum Markov semigroups of weak coupling limit-type. To tackle this problem, we introduce
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Analogues of Poisson-type limit theorems in discrete bm-Fock spaces Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-08-29 Lahcen Oussi, Janusz Wysoczański
In this paper, we present analogues of the Poisson limit distribution for the noncommutative bm-independence, which is associated with several positive symmetric cones. We construct related discrete Fock spaces with creation, annihilation and conservation operators, and prove Poisson type limit theorems for them. Properties of the positive cones, in particular the volume characteristic property they
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Refinements of asymptotics at zero of Brownian self-intersection local times Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-08-29 A. A. Dorogovtsev, Naoufel Salhi
In this paper, we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension d≥4 the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide
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Factoriality of mixed q-deformed Araki-Woods von Neumann algebras Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-08-29 Panchugopal Bikram, R. Rahul Kumar, Kunal Mukherjee
In this paper, we prove that the mixed q-deformed Araki-Woods von Neumann algebras are factors for dim(ℋℝ)≥3.
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Stochastic quantization of laser propagation models Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-07-27 Sivaguru S. Sritharan, Saba Mudaliar
This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear
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Recurrence and transience of quantum Markov semigroups constructed form quantum Bernoulli noises Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-07-27 Jinshu Chen, Shexiang Hai
Quantum exclusion semigroup constructed form quantum Bernoulli noises can be written on diagonal and off-diagonal operators space, respectively. The diagonal parts describe a classical Markov process. In this paper, we first produce a decomposition into the sum of irreducible components determined by the class states of the associated classical Markov process and prove that each irreducible component
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Quantum Wasserstein distance of order 1 between channels Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-30 Rocco Duvenhage, Mathumo Mapaya
We set up a general theory leading to a quantum Wasserstein distance of order 1 between channels in an operator algebraic framework. This gives a metric on the set of channels from one composite system to another, which is deeply connected to reductions of the channels. The additivity and stability properties of this metric are studied.
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Strong uniqueness of finite-dimensional Dirichlet operators with singular drifts Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-30 Haesung Lee
We show Lr(ℝd,μ)-uniqueness for any r∈(1,2] and the essential self-adjointness of a Dirichlet operator Lf=Δf+〈1ρ∇ρ,∇f〉, f∈C0∞(ℝd) with d≥3 and μ=ρdx. In particular, ∇ρ is allowed to be in Llocd(ℝd,ℝd) or in Lloc2+𝜀(ℝd,ℝd) for some 𝜀>0, while ρ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence
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A Feynman–Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-21 Hyun-Jung Kim, Ramiro Scorolli
We consider the (unique) mild solution u(t,x) of a one-dimensional stochastic heat equation on [0,T]×ℝ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of u(t,x), denoted by ∂xu(t,x), and its representation as a Feynman–Kac type closed form. The chaos expansion of ∂xu(t,x) makes it possible to find its (optimal)
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On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-21 Matthijs Vernooij
Cipriani and Sauvageot have shown that for any L2-generator L(2) of a tracially symmetric quantum Markov semigroup on a C*-algebra 𝒜 there exists a densely defined derivation δ from 𝒜 to a Hilbert bimodule H such that L(2)=δ∗∘δ¯. Here, we show that this construction of a derivation can in general not be generalized to quantum Markov semigroups that are symmetric with respect to a non-tracial state
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Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-21 Mihai Popa
A recent work by Mandarino, Linowski and Życzkowski left open the following question. If μN is a certain permutation of entries of an N2×N2 matrix (“mixing map”) and UN is an N2×N2 Haar unitary random matrix, then is the family UN,UNμN,(UN2)μN,…,(UNm)μN asymptotically free? (Here by Aμ we understand the matrix resulted by permuting the entries of A according to the permutation μ.) This paper presents
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Metrics induced by certain Hilbert-Schmidt fidelities on positive semi-definite matrices Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-21 Ho Minh Toan, Vu The Khoi
Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces 𝒫 of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the
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Effects of permutation functions on woven and non-woven frames Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-21 Ahmad Ahmadi, Abbas Askarizadeh
When a signal measured by woven frames is transmitted, the order of some of these coefficients may change, resulting in undesirable signal reconstruction. In this paper, we investigate the permutations that if two frames are woven, one of the frames will be woven with a permutation of the other frame. We also examine the existence of permutations that a Riesz basis and reordered Riesz basis are woven
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The Stockwell transform on locally compact abelian groups Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-06-06 Fatemeh Esmaeelzadeh
Let G be a locally compact abelian (LCA) group and α be a topological automorphism on G. For φ∈L2(G), the Stockwell transform on the locally compact group G with respect to the automorphism α is defined and denoted by Sφ,α. The inversion formula for the Stockwell transform Sφ,α is established and as a result it is concluded that Sφ,α is an isometry. Moreover, some properties of the Stockwell transform
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On stopping rules for tree-indexed quantum Markov chains Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-05-31 Abdessatar Souissi
In the present paper, we introduce stopping rules and related notions for quantum Markov chains on trees (QMCT). We prove criteria for recurrence, accessibility and irreducibility for QMCT. This work extends to trees the notion of stopping times for quantum Markov chains (QMC) introduced by Accardi and Koroliuk, which plays a key role in the study of many properties of QMC. Moreover, we illustrate
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Least squares type estimators for the drift parameters in the sub-bifractional Vasicek processes Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-04-04 Nenghui Kuang, Huantian Xie
In this paper, we consider least squares type estimators for the drift parameters in the sub-bifractional Vasicek processes defined by dXt=𝜃(μ+Xt)dt+dStH,K,t≥0,X0=0, with unknown parameters 𝜃>0 and μ∈R, where SH,K is a sub-bifractional Brownian motion with indices H∈(0,1) and K∈(0,1]. The strong consistency results as well as the asymptotic distributions of these estimators are obtained.
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Matrix-valued Schrödinger operators over finite adeles Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-03-30 R. Urban
Let K be an algebraic number field. With K we associate the ring of finite adeles 𝔸K. In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group 𝔸Kn. Specifically, we consider matrix-valued Hamiltonian operators H𝔸Kn=Δ𝔸Kn⊗Id+V, where Δ𝔸Kn is the Vladimirov operator and V is a non-negative definite potential. The free part of the Hamiltonian
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Central limit theorems for heat equation with time-independent noise: The regular and rough cases Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-03-10 Raluca M. Balan, Wangjun Yuan
In this paper, we investigate the asymptotic behavior of the spatial average of the solution to the parabolic Anderson model with time-independent noise in dimension d≥1, as the domain of the integral becomes large. We consider three cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the
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A note on the rate of convergence in the Boolean central limit theorem Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2023-03-10 Mauricio Salazar
Several results giving upper bounds for the speed of convergence in non-commutative central limit theorems are known. A natural question is whether there also exist lower bounds, or whether there exist classes of probability measures where the speed of convergence is faster. In this paper, we answer this for the Boolean central limit theorem in the bounded support case and using the Lévy distance.
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Algebraic central limit theorems: A personal view on one of Wilhelm’s legacies Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-31 Michael Skeide
Bringing forward the concept of convergence in moments from classical random variables to quantum random variables leads to what can be called algebraic central limit theorem for (classical and) quantum random variables. I reflect in a very personal way how such an idea is typical for the spirit of doing research in mathematics as I learned it in Wilhelm von Waldenfels’s research group in Heidelberg
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Sufficient statistic and Rao–Blackwell theorem in quantum probability Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-31 Kalyan B. Sinha
In the decision-theoretic foundation of Classical Statistics, the idea and concept of a sufficient statistic play a central role. Here a parallel concept of a sufficient statistic in quantum probability is proposed and as a consequence, various levels of “quantum Rao–Blackwell theorem” are proven.
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Two-dimensional quantum Bernoulli process and the related central limit theorem Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-31 Yungang Lu
In this paper, we introduce a quantum decomposition of a two-dimensional Bernoulli random variable (ξ1,ξ2), where E(ξ1)=E(ξ2)=0, E(ξ12)=E(ξ22)=1 and E(ξ1ξ2)=c∈(−1,1). Based on this quantum decomposition, we defined the two-dimensional quantum Bernoulli process and set a corresponding central limit theorem, both in the sense of quantum moments and in the sense of characteristic function.
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Some thoughts on Wilhelm von Waldenfels and on universal second order constructions Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-27 Roland Speicher
This paper contains some thoughts on Wilhelm von Waldenfels and on universal second order constructions. It is my contribution to the Special Commemorative Issue of IDAQP in Honour of Professor Robin Lyth Hudson and Professor Wilhelm von Waldenfels.
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On the non-commutative multifractional Brownian motion Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-27 Marco Dozzi, René Schott
We define a non-commutative analogue of a real gaussian Volterra-type multifractional Brownian motion (NC-mfBm for short) and show that its trajectories behave locally like non-commutative fractional Brownian motion. We determine the pointwise Hölder exponent as well as a random matrix approximation of this process.
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Infinite-horizon risk-sensitive performance criteria for translation invariant networks of linear quantum stochastic systems Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-21 Igor G. Vladimirov, Ian R. Petersen
This paper is concerned with networks of identical linear quantum stochastic systems which interact with each other and external bosonic fields in a translation invariant fashion. The systems are associated with sites of a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). The block Toeplitz coefficients of these QSDEs are specified by the
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Twisted convolution quantum information channels, one-parameter semigroups and their generators Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-19 K. R. Parthasarathy
Using the tool of quantum characteristic functions of n-mode states in the boson Fock space Γ(ℂn) we construct a semigroup of quantum information channels. This leads to a special class of one-parameter semigroups of such channels. These semigroups are concrete but their generators have unbounded operator coefficients. These one-parameter semigroups are also quantum dynamical semigroups and the form
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De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-19 Vitonofrio Crismale, Stefano Rossi, Paola Zurlo
Local actions of ℙℕ, the group of finite permutations on ℕ, on quasi-local algebras are defined and proved to be ℙℕ-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann
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Motzkin path decompositions of functionals in noncommutative probability Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-12-14 Romuald Lenczewski
We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional Φ are naturally described in terms of Motzkin paths identified with reduced Motzkin words M. Using this fact, we demonstrate that the mixed moments of order n of free random variables
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On irreducibility of Gaussian quantum Markov semigroups Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Franco Fagnola, Damiano Poletti
The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a d-mode Fock space is represented in a generalized GKLS form with an operator G quadratic in creation and annihilation operators and Kraus operators L1,…,Lm linear in creation and annihilation operators. Kraus operators, commutators [G,Lℓ] and iterated commutators [G,[G,Lℓ]],… up to the order 2d−m, as linear
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Non-commutative stochastic processes with independent increments Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Michael Schürmann
This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed
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On the relationships between covariance and the decoherence-free subalgebra of a quantum Markov semigroup Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Emanuela Sasso, Veronica Umanità
In this paper, we investigate if the presence of symmetries in the evolution of an open quantum system gives information about the invariant subspaces. Unfortunately, the answer is, in general, negative, but under suitable conditions we can observe that the representation through which we describe the symmetry determines a privileged family of orthogonal projections strongly correlated with the structure
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The average values of a kind of functionals in LP and concentration without measure Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Cheng-Shi Liu
This paper focuses on the average values of functionals like Y=∫01g(x(t))dt on the set M={x|∥x∥p≤R,x∈C[0,1]} in Lp[0,1]. The densities of coordinates of points in M are derived out. The formula of average value EY of functional Y is obtained. The variance DY of Y is proven to be zero, which shows the phenomenon of concentration without measure, and then the nonlinear commutation identity Eh(Y) = h(EY)
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A white noise approach to stochastic currents of Brownian motion Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Martin Grothaus, Herry Pribawanto Suryawan, José Luís Da Silva
In this paper, we study stochastic currents of Brownian motion ξ(x), x∈ℝd, by using white noise analysis. For x∈ℝd∖{0} and for x=0∈ℝ we prove that the stochastic current ξ(x) is a Hida distribution. Moreover for x=0∈ℝd with d>1 we show that the stochastic current is not a Hida distribution.
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Positivity of Gibbs states on distance-regular graphs Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-28 Michael Voit
We study criteria which ensure that Gibbs states (often also called generalized vacuum states) on distance-regular graphs are positive. Our main criterion assumes that the graph can be embedded into a growing family of distance-regular graphs. For the proof of the positivity we then use polynomial hypergroup theory and translate this positivity into the problem whether for x∈[−1,1] the function n↦xn
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Dilation, functional model and a complete unitary invariant for C.0Γn-contractions Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-22 Sourav Pal
A commuting tuple of operators (S1,…,Sn−1,P), defined on a Hilbert space ℋ, for which the closed symmetrized polydisc Γn=∑1≤i≤nzi,∑1≤i
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Wilhelm von Waldenfels (2-3-1932–12-3-2021), a pioneer of quantum probability Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-11 Luigi Accardi
This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels with particular attention to the contributions he gave to quantum probability, a field in which he was one of the pioneers.
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Noncommutative quantum decomposition of Gegenbauer white noise process Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-11-11 Anis Riahi
The main purpose of this paper is to derive a general structure of Gegenbauer white noise analysis as a counterpart class of non-Lévy white noise. First, we start with a new detailed construction of the Gegenbauer Fock space Γβ(ℋ) which serves to obtain the quantum decomposition associated with the Gegenbauer white noise processes. More precisely, based on the notion of quantum decomposition and the
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Ergodic theorems for higher order Cesàro means Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-09-30 Luigi Accardi, Byoung Jin Choi, Un Cig Ji
We investigate the convergence of higher order Cesàro means in Banach spaces. The main results of this paper are: (1) The proof of mean and Birkhoff-type ergodic theorems for higher order Cesàro means. (2) The existence of a one-to-one correspondence between convergent Cesàro means of different orders. (3) The proof of strong laws of large numbers for higher order sums of independent and identically
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Universality theorems for asymmetric spaces Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-09-30 A. R. Alimov
Spaces with asymmetric metric and asymmetric norm are considered. It is shown that any metrizable separable asymmetrically normed linear space (X,∥⋅|) can be isometrically isomorphic imbedded, as an affine linear manifold, into the classical space C[0,1] with uniform norm ∥⋅∥C. A similar result is obtained for spaces of density 𝔞. For spaces with asymmetric metric, it is shown that each such space
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Rates of convergence for laws of the spectral maximum of free random variables Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-09-30 Yuki Ueda
Let {Xn}n be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence {Wn}n of the renormalized spectral maximum of random variables X1,…,Xn. It is known that the renormalized spectral maximum Wn converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence
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A new construction of shearlets Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-09-30 Pooran Ghaderihasab, Ahmad Ahmadi
In order to achieve optimally sparse approximations of signals exhibiting anisotropic singularities, the shearlet systems that are systems of functions generated by one generator with dilation, shear transformation and translation operators applied to it were introduced. In this paper, we will construct the shearlet systems that are not only Parseval frames for L2(ℝ2) but they are also obtained from
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Analysis of space-dependent noise functionals with an application to linearly correlated processes Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-09-07 Yun-Ching Chang, Hsin-Hung Shih
Recently, Hida et al.5, 11 introduced a noise of new type, called the Poisson space noise. It depends only on the space parameter, quite different from the time derivative of a Poisson process. In this paper, without employing the Minlos theorem, we shall construct Poisson space noise from the view of its path behavior, which will be realized by the space derivative of an additive process. Then the
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Characterization of Gaussian quantum Markov semigroups Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-07-30 Damiano Poletti
We give a characterization of QMSs on the Bosonic Fock Space Γ(ℂd) whose predual preserves the set of gaussian states. We show they can be obtained via certain generalized GKLS generators and they satisfy an explicit formula for their action on Weyl operators.
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Free CIR processes Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-07-26 Holger Graf, Henry Port, Georg Schlüchtermann
For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive
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Quantum Ising model with generalized competing XY-interactions on a Cayley tree Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-05-21 Farrukh Mukhamedov, Soueidy El Gheteb
In this paper, the phase transition phenomena for the Ising model (with nearest-neighbor interaction J0) but with quantum generalized competing XY-interactions (J1 and J2 coupling constants) are treated by means of a quantum Markov chain (QMC) approach. We point out that the case J1=J2 has been carried out in Ref. 32. Note that if J2=0, then it turns out that phase transition exists, for any value
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Computation of sandwiched relative α-entropy of two n-mode Gaussian states Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-04-30 K. R. Parthasarathy
A formula for the sandwiched relative α-entropy D̃α(ρ||σ) = 1 α − 1ln Tr σ1−α 2αρσ1−α 2αα for 0 < α < 1, of two n-mode Gaussian states ρ, σ in the boson Fock space Γ(ℂn) is presented. This computation extensively employs the ℰ2-parametrization of Gaussian states in Γ(ℂn) introduced in Ref. 10.
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Combinatorics of NC-probability spaces with independent constants Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-04-23 Carlos Diaz-Aguilera, Tulio Gaxiola, Jorge Santos, Carlos Vargas
The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as
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A note on the infinite-dimensional quantum Strassen’s theorem Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-04-18 Luigi Accardi, Abdallah Dhahri, Yun Gang Lu
In Ref. 3, the quantum Strassen’s theorem has been extended to the infinite-dimensional case. This theorem consists in the solution of the coupling problem for two states on the algebra of bounded operators on two Hilbert spaces ℋ1, ℋ2 with the additional constraint that the coupling state has support in a pre-assigned sub-space of ℋ1 ⊗ℋ2. In this paper, we give an alternative proof of the main theorem
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Full error analysis for the training of deep neural networks Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-04-05 Christian Beck, Arnulf Jentzen, Benno Kuckuck
Deep learning algorithms have been applied very successfully in recent years to a range of problems out of reach for classical solution paradigms. Nevertheless, there is no completely rigorous mathematical error and convergence analysis which explains the success of deep learning algorithms. The error of a deep learning algorithm can in many situations be decomposed into three parts, the approximation
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Stochastic integrals and Gelfand integration in Fréchet spaces Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IF 0.9) Pub Date : 2022-03-30 Fred Espen Benth, Luca Galimberti
We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results