scholarly journals Unification of Independence in Quantum Probability

1998 ◽  
Vol 01 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Romuald Lenczewski
Keyword(s):  
Free Product ◽  
Tensor Products ◽  
Quantum Probability ◽  
Product Formula ◽  
Free Algebras ◽  
Noncommutative Probability ◽  
Free Variables ◽  
Probability Spaces

Let [Formula: see text], be the conditionally free product of unital free *-algebras [Formula: see text], where ϕl, ψl are states on [Formula: see text], l∈I. We construct a sequence of noncommutative probability spaces [Formula: see text], m∈N, where [Formula: see text] and [Formula: see text], m∈N, [Formula: see text], and the states [Formula: see text], ϕl are Boolean extensions of ϕl, ψl, l∈I, respectively. We define unital *-homomorphisms [Formula: see text] such that [Formula: see text] converges pointwise to *l∈I(ϕl,ψl). Thus, the variables j(m)(w), where w is a word in [Formula: see text], converge in law to the conditionally free variables. The sequence of noncommutative probability spaces [Formula: see text], where [Formula: see text] and Φ(m) is the restriction of [Formula: see text] to [Formula: see text], is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product [Formula: see text] into a cocommutative *-bialgebra associated with m-freeness.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

Quantum Probability

2017 ◽  
Author(s):  
Guido Bacciagaluppi
Keyword(s):  
Quantum Mechanics ◽  
Hidden Variables ◽  
Quantum Probability ◽  
Classical Probability ◽  
Discrete Probability ◽  
Joint Distributions ◽  
Finite Dimensional ◽  
Main Emphasis ◽  
Probability Spaces ◽  
Incompatible Observables

The topic of probability in quantum mechanics is rather vast. In this chapter it is discussed from the perspective of whether and in what sense quantum mechanics requires a generalization of the usual (Kolmogorovian) concept of probability. The focus is on the case of finite-dimensional quantum mechanics (which is analogous to that of discrete probability spaces), partly for simplicity and partly for ease of generalization. While the main emphasis is on formal aspects of quantum probability (in particular the non-existence of joint distributions for incompatible observables), the discussion relates also to notorious issues in the interpretation of quantum mechanics. Indeed, whether quantum probability can or cannot be ultimately reduced to classical probability connects rather nicely to the question of 'hidden variables' in quantum mechanics.

THE FIVE INDEPENDENCES AS QUASI-UNIVERSAL PRODUCTS

2002 ◽  
Vol 05 (01) ◽  
pp. 113-134 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Boolean Product ◽  
Probability Spaces

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

(𝜃,ω)-Twisted Radford’s Hom-biproduct and ϖ-Yetter–Drinfeld modules for Hom-Hopf algebras

2020 ◽  
Vol 19 (03) ◽  
pp. 2050046
Author(s):  
Xiao-Li Fang ◽  
Tae-Hwa Kim
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Product Formula ◽  
Necessary And Sufficient Conditions ◽  
Drinfeld Modules ◽  
Associative Algebras ◽  
Necessary And Sufficient

To unify different definitions of smash Hom-products in a Hom-bialgebra [Formula: see text], we firstly introduce the notion of [Formula: see text]-twisted smash Hom-product [Formula: see text]. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product [Formula: see text] and the twisted smash Hom-coproduct [Formula: see text] to afford a Hom-bialgebra, which generalize the well-known Radford’s biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65]. Furthermore, we introduce the notion of the category of [Formula: see text]-Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the [Formula: see text]-twisted Radford’s Hom-biproduct [Formula: see text] is a Hom-bialgebra if and only if [Formula: see text] is a Hom-bialgebra in the category of [Formula: see text]-Yetter–Drinfeld modules [Formula: see text], generalizing the well-known Majid’s conclusion.

ON PERIODIC PRODUCTS OF GROUPS

1995 ◽  
Vol 05 (01) ◽  
pp. 7-17 ◽  
Author(s):  
SERGEI V. IVANOV
Keyword(s):  
Free Product ◽  
Product Formula ◽  
Remarkable Property ◽  
Defining Relations ◽  
Products Of Groups ◽  
Inductive Construction ◽  
Burnside Groups

Adian introduced periodic n-products of groups which are given by imposing of defining relations of the form An=1 on the free product [Formula: see text] of groups Gα, α∈I, without involutions. The defining relations An=1 are constructed by a complicated induction which is quite similar to the inductive construction of free Burnside groups due to Novikov and Adian. This periodic n-product [Formula: see text] of groups Gα, α∈I, has the remarkable property that for every [Formula: see text] either xn=1 or x is conjugate to an element of Gα for some α. The main result of the article is that this property of periodic n-product can be taken as its definition. This gives a new non-inductive characterization of periodic n-products. An analogous characterization of periodic [Formula: see text]-products due to Ol’shanskii is also given.

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