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Let us consider the product Fock space $\Gamma(\mathbb{C}^m) \otimes \Gamma(\mathbb{C}^n)$ and consider Gaussian states in that space. While reading some literature on Gaussian state entanglement (namely Simon, and Werner and Wolf) , I realised that by separability they mean separability when each components are Gaussian states themselves. So, my question is the following.

Does there exist any Gaussian state which is separable such that in its separable representation there always exist term(s) which are tensor products of non Gaussian state? In other words, does separability means Gaussian separability for Gaussian states?

Recall that, in general sum of two Gaussian states need not be Gaussian. Gaussian states, by themselves do not form a convex body. I could not construct such an example. However, it may be possible that such example may not exist, by some clever application of weak law. Advanced thanks for any help/suggestion.

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This is an open problem, and a special case of a more general problem, namely the question whether the entanglement of formation (i.e., the minimum average pure state entanglement needed to form a mixed state) for Gaussian states can always be achieved by a decomposition into Gaussian states. AFAIK the problem has only been settled (in the positive) for the case of symmetric 1+1 mode systems.

See Problem 29 on the list of open problems in quantum information of the Hannover QI group, https://qig.itp.uni-hannover.de/qiproblems/Entanglement_of_formation_for_Gaussian_states.

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