= Conditional-vector product-state criterion
A nonzero bipartite <pure state> is a <product state> if and only if all its <partial inner products> on the first subsystem lie in a common one-dimensional <vector subspace> of the second. For a product $|a\rangle\otimes|b\rangle$, every contraction equals $\langle u|a\rangle|b\rangle$. Conversely, contract against each <vector> of an <orthonormal basis> to write $|\Psi\rangle=\sum_i|i\rangle\otimes|\chi_i\rangle$. If $|\chi_i\rangle=c_i|b\rangle$ for a common nonzero $|b\rangle$, then $|\Psi\rangle=(\sum_ic_i|i\rangle)\otimes|b\rangle$. Zero conditional <vectors> are allowed. A nontrivial <linear dependence> relation requires multipliers not both zero; allowing both multipliers to vanish would give no test at all.
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