Preprint
Non-commutative probability, joint distributions, conditioning, and the associated polymorphisms
ArXiv.org
Cornell University
07/16/2024
DOI: 10.48550/arxiv.2407.11846
Abstract
We present a parallel between commutative and non-commutative polymorphisms.
Our emphasis is the applications to conditional distributions from stochastic
processes. In the classical case, both the measures and the positive definite
kernels are scalar valued. But the non-commutative framework (as motivated by
quantum theory) dictates a setting where instead now both the measures (in the
form of quantum states), and the positive definite kernels, are operator
valued. The non-commutative theory entails a systematic study of positive
operator valued measures, abbreviated POVMs. And quantum states (normal states)
are indexed by normalized positive trace-class operators. In the
non-commutative theory, the parallel to the commutative/scalar valued theory
helps us understand entanglement in quantum information. A further implication
of our study of the non-commutative framework will entail an interplay between
the two cases, scalar valued, vs operator valued.
Details
- Title: Subtitle
- Non-commutative probability, joint distributions, conditioning, and the associated polymorphisms
- Creators
- Palle E. T JorgensenJames Tian
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- DOI
- 10.48550/arxiv.2407.11846
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 07/16/2024
- Academic Unit
- Mathematics
- Record Identifier
- 9984657357102771
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