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I see a tautology here, |ud>-|du> is en entangled state because it cannot be put in the form ##| a>|b>## where a and b are vectors in their respective spaces. I you redefine |ud>-|du> as a new vector in the product space you have simply mathematically erased entanglement by redefining the Hilbert space that now, in a sense, is no longer a product space because now |ud>-|du> is by definition a single vector.Consider the 4 maximally entangled states, eg |ud>-|du>. If a Martian chose them for basis states then he would see them as separable

This way of looking at entanglement have no physical implications.

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