= Reciprocal-state construction of unambiguous discrimination
{title2=$|\chi_i\rangle=\sum_j(G^{-1})_{ji}|\psi_j\rangle,\quad E_i=\epsilon|\chi_i\rangle\langle\chi_i|$}
For normalized linearly independent <pure states>, their <Gram matrix> $G_{ij}=\langle\psi_i|\psi_j\rangle$ is positive definite. The displayed reciprocal vectors satisfy $\langle\chi_i|\psi_j\rangle=\delta_{ij}$. The operator $R=\sum_i|\chi_i\rangle\langle\chi_i|$ has nonzero <eigenvalues> equal to those of $G^{-1}$: use the polar factorization of the matrix whose columns are $\psi_i$. Choosing $0<\epsilon\leq\lambda_{\min}(G)$ gives the <POVM> effects $E_i$ and $E_?=I-\epsilon R$. Each conclusive outcome has <probability> $\epsilon$ on its own state and zero on every other input. The inconclusive effect preserves positivity, so the construction is a physical <quantum measurement>.
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