Entanglement of observables: Quantum conditional probability approach
Abstract
This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it is meaningless to speak about entanglement without pointing to the fixed observables A and B, so this is AB-entanglement. Dependence of quantum observables is formalized as non-coincidence of conditional probabilities. Starting with this probabilistic definition, we achieve the Hilbert space characterization of the AB-entangled states as amplitude non-factorisable states. In the tensor product case, -entanglement implies standard entanglement, but not vice verse. AB-entanglement for dichotomous observables is equivalent to their correlation. We describe the class of quantum states that are A_u B_u-entangled for a family of unitary operators (u). Finally, observables entanglement is compared with dependence of random variables in classical probability theory.
Cite
@article{arxiv.2303.12393,
title = {Entanglement of observables: Quantum conditional probability approach},
author = {Andrei Khrennikov and Irina Basieva},
journal= {arXiv preprint arXiv:2303.12393},
year = {2023}
}