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The matrix desired is at most rank two and can be easily expressed as the product of a matrix and its transpose. I will number the elements for convenience and assume that state > 1 (otherwise we have imaginary elements, which would require further thought). n = 3; v1 = Prepend[Array[a, n], 1]; v2 = Prepend[Array[0 &, n], Sqrt[state - 1]]; M1 = ...

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