Entanglement of Observables: Quantum Conditional Probability Approach

Foundations of Physics 53 (5):1-22 (2023)
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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, _AB_-entanglement implies standard entanglement, but not vise verse. _AB_-entanglement for dichotomous observables is equivalent to their correlation, i.e., ABψAψBψ.\langle AB\rangle _{\psi} \not = \langle A\rangle _{\psi} \langle B\rangle _{\psi}. We describe the class of quantum states that are AuBuA_{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.

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