An arbitrary twoqubit computation In 23 elementary gates or less
Proceedings of the 40th annual Design Automation Conference
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As separable states are a convex combination of product states, the geometry of the manifold of product states, Σ+, is studied. Prior results by Sanpera, Vidal and Tarrach are extended. Furthermore, it is proven that states in the set tangent to Σ+ at the maximally mixed state are separable; the set normal contains, among others, all extended GHZ states. A canonical decomposition is given. A surprising result is that for the case of two particles, the closest product state to the maximally entangled state is the maximally mixed state. An algorithm is provided to find the closest product state.