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It is found that the problem of existence of bound entangled states withnon-positive partial transpose (NPPT) has the intriguing relation to theHilbert's 17th problem. More precisely, we compute the expectation value of thepartially transposed Werner states by Schmidt rank-2 vectors for NPPT and1-copy undistillable region. It is the positive polynomial but shown not to beexpressed as the sum of squares of polynomials. A remedy for such pathologicalbehavior as well as a similar but different approach to the problem is alsomentioned.

A problem of existence of bound entangled states with non-positive partial transpose and the Hilbert's 17th problem