From Classical to Quantum: Polymorphisms in Non-Commutative Probability
Palle E. T. Jorgensen, James Tian
TL;DR
The paper addresses extending classical polymorphisms and Rokhlin disintegration to a noncommutative probability setting suitable for quantum measurements and entanglement. It develops a quantum polymorphism framework built from POVMs and operator-valued kernels, and proves a noncommutative Rokhlin-type disintegration (via Theorems 4-2 and 4-5) together with explicit dilations to PVMs, linking quantum conditioning to entanglement. It then explores two extension avenues—polymorphisms defined by quantum states (density operators) and $\mathscr{L}(\mathscr{H})$-valued kernels—addressing how scalar marginals arise as traces with density operators and posing open questions about representability, extremal structure, and the role of entanglement. The framework provides mathematical tools for conditional reasoning in quantum information and lays groundwork for operator-valued approaches to quantum statistics and quantum graphs.
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.