Entanglement and products

We consider a sequence of quantum states, $ρ_N$, associated with a submanifold $Λ$ of product of two integral compact Kähler manifolds. We show that, when $Λ$ is a product submanifold, then, in the semiclassical limit, these states are not entangled. We discuss whether geometry of $Λ$ (specifically $Λ$ being a product submanifold) has any relationship with entanglement properties of $ρ_N$.

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