From Classical to Quantum: Polymorphisms in Non-Commutative Probability

TL;DR

This paper explores the parallels between classical and quantum polymorphisms, focusing on their applications to conditional distributions and the role of operator-valued measures in quantum information theory.

Contribution

It introduces a systematic study of non-commutative polymorphisms, extending classical concepts to operator-valued measures and quantum states, with implications for understanding entanglement.

Findings

Establishes a parallel between scalar and operator-valued measures.

Highlights the role of positive operator valued measures (POVMs) in quantum theory.

Provides insights into the interplay between classical and quantum probabilistic frameworks.

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.