Entanglement of Observables: Quantum Conditional Probability Approach

Abstract

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., $$\langle AB\rangle _{\psi} \not = \langle A\rangle _{\psi} \langle B\rangle _{\psi} .$$ ⟨ A B ⟩ ψ ≠ ⟨ A ⟩ ψ ⟨ B ⟩ ψ . We describe the class of quantum states that are $$A_{u} B_{u}$$ 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.