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Infinite Dimensional Analysis Quantum Probability and Related Topics

A CENTRAL LIMIT THEOREM FOR BOSE ${\mathcal Z}$-INDEPENDENT QUANTUM RANDOM VARIABLES

Michael Skeide

Published Jun 1, 1999• 3 citations
Publisher

Abstract

We present a central limit theorem for Bose [Formula: see text]-independent operator-valued random variables. Furthermore, we show that the central limit distribution may be represented by an algebra of creators and annihilators on a symmetric Fock module. As an example we recover the distribution of creators and annihilators on the truncated Fock space, i.e. the central limit distribution of Boolean independence.

DOI
10.1142/s0219025799000163

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We present a central limit theorem for Bose [Formula: see text]-independent operator-valued random variables. Furthermore, we show that the central limit distribution may be represented by an algebra of creators and annihilators on a symmetric Fock module. As an example we recover the distribution of creators and annihilators on the truncated Fock space, i.e. the central limit distribution of Boolean independence.
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We consider random linear continuous operators Ω→L(H,H)Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})Ω→L(H,H) on a Hilbert space H\mathcal{H}H. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents eAi/ne^{A_i/n}eAi​/n. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.
The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem [Formula: see text] of the present paper provides such a criterium. Theorem [Formula: see text] deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem [Formula: see text]. Theorem [Formula: see text] naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.