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Finding the closest separable state to a given target state is a notoriouslydifficult task, even more difficult than deciding whether a state is entangledor separable. To tackle this task, we parametrize separable states with aneural network and train it to minimize the distance to a given target state,with respect to a differentiable distance, such as the trace distance orHilbert--Schmidt distance. By examining the output of the algorithm, we obtainan upper bound on the entanglement of the target state, and construct anapproximation for its closest separable state. We benchmark the method on avariety of well-known classes of bipartite states and find excellent agreement,even up to local dimension of $d=10$, while providing conjectures and analyticinsight for isotropic and Werner states. Moreover, we show our method to beefficient in the multipartite case, considering different notions ofseparability. Examining three and four-party GHZ and W states we recover knownbounds and obtain additional ones, for instance for triseparability.

Building separable approximations for quantum states via neural networks