The qq-bit (I): Central limits with left q-Jordan–Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q

Abstract

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.