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Perfect identification on every trial is impossible, because the family contains a nonorthogonal pair. Any physical identification device induces a POVM whose effect $E_i$ corresponds to its declaration of label $i$. Certainty of correct identification would require $⟨{\psi }_i|E_i|{\psi }_i⟩=1$ and $⟨{\psi }_j|E_i|{\psi }_j⟩=0$ for every $j\ne i$.

Since $0\le E_i\le I$, the first identity gives $(I-E_i)|{\psi }_i⟩=0$, while positivity and the second identity give $E_i|{\psi }_j⟩=0$. For example these conclusions follow by writing the zero expectations as squared norms of $(I-E_i{)}^{1/2}{\psi }_i$ and $E_i^{1/2}{\psi }_j$. ThusPerfect identification would force the entire family to be orthogonal, contrary to the hypothesis. This argument permits ancillas, general measurements and disturbance of the input; it does not assume that the device must preserve the state.

Unambiguous identification with a nonzero success probability for every input is possible. Normalize the given kets and let $G_{ij}=⟨{\psi }_i|{\psi }_j⟩$ be their Gram matrix. Their linear independence makes $G$ positive definite. Define reciprocal vectors within their span byThen $⟨{\psi }_k|{\chi }_i⟩={\sum }_j{G}_{kj}(G^{-1}{)}_{ji}={\delta }_{ki}$, and hence also $⟨{\chi }_i|{\psi }_k⟩={\delta }_{ik}$. Put $R={\sum }_i|{\chi }_i⟩⟨{\chi }_i|$. It is positive and has finite operator norm. Choosing $ϵ=1/(2\Vert R\Vert )>0$, setAll these effects are positive, and $E_{?}\ge I/2$, so they sum to the identity and define a POVM. The Born rule givesThus every label is correct whenever it occurs and every input has the strictly positive success probability $p_k=ϵ$. This is the reciprocal-state construction of unambiguous discrimination.

For an explicit success constant, collect the input kets into a matrix $\Psi$, so ${\Psi }^{†}\Psi =G$. Its polar factorization is $\Psi =U{G}^{1/2}$ with $U^{†}U=I_N$. The reciprocal columns are $\Psi G^{-1}=U{G}^{-1/2}$, and therefore $R=U{G}^{-1}{U}^{†}$. Its largest eigenvalue is $1/{\lambda }_{\min }(G)$, so one can take $ϵ={\lambda }_{\min }(G)/2$. Since the normalized Gram matrix has trace $N$, this is at most $1/2$.

To realize the device, use measurement operators $M_i=\sqrt{E_i}$ and $M_{?}=\sqrt{E_{?}}$. The map $|v⟩\mapsto {\sum }_i{M}_i|v⟩|i⟩+M_{?}|v⟩|?⟩$ preserves inner products because the effects sum to $I$. Extend this linear isometry to a unitary on the input and an initially blank ancilla, then projectively measure the ancilla's orthogonal outcome labels. Its outcome probabilities are precisely those above. This gives a physical unambiguous quantum state discrimination device rather than only a formal set of effects.