Unification of independence in quantum probability

Yang-Baxter映射、概率分布与独立性保持性质的超离散化

Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property

本文从超离散化视角研究Yang-Baxter映射与独立性保持性质的关系，引入概率分布的超离散化概念，证明这两类性质在超离散化下保持不变，并生成了具有IP性质的分段线性映射新实例。

What is Stochastic Independence?

The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections. In this context, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Schürmann become classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. The notion of a reduction of one independence to an…

The Theory of Quantum Levy Processes

Various recent results on quantum Lévy processes are presented. The first part provides an introduction to the theory of Lévy processes on involutive bialgebras. The notion of independence used for these processes is tensor independence, which generalizes the notion of independence used in classical probability and corresponds to independent observables in quantum physics. In quantum probability there exist other notions of independence and Lévy processes can also be defined for the five so-called universal independences. This is the topic of the second part. In particular, it is shown that bo…

Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. Th…

Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces

We investigate questions related to the notion of traffics introduced by the author C. Male as a noncommutative probability space with numerous additional operations and equipped with the notion of traffic independence. We prove that any sequence of unitarily invariant random matrices that converges in noncommutative distribution converges in distribution of traffics whenever it fulfills some factorization property. We provide an explicit description of the limit which allows to recover and extend some applications (on the freeness from the transposed ensembles by Mingo and Popa and the freene…

Categorial Independence and Lévy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its …