13 - Probabilistic Entanglement of Quantum Observables

Summary

In this chapter contextual probabilistic entanglement is represented withinthe Hilbert space formalism. The notion of entanglement is clarified anddemystified through decoupling it from the tensor product structure andtreating it as a constraint posed by probabilistic dependence of quantum observablesA and B. In this framework, it is meaningless to speak aboutentanglement without pointing to the fixed observables A and B, so thisis AB-entanglement. Dependence of quantum observables is formalized asnon-coincidence of conditional probabilities. Starting with this probabilisticdefinition, we achieve the Hilbert space characterization of the AB-entangledstates as amplitude non-factorisable states. In the tensor productcase, AB-entanglement implies standard entanglement, but not vice versa.AB-entanglement for dichotomous observables is equivalent to their correlation. Finally, observables entanglement is compared with dependence of random variables in classical probability theory.