Quantum state exclusion

A task in which a quantum measurement outputs a label of a preparation that can be ruled out. If preparations are ${\rho }_j$ and the outcome labelled $j$ has effect $E_j$, error-free exclusion requires $\mathrm{Tr}(E_j{\rho }_j)=0$ for every $j$, together with $E_j\ge 0$ and ${\sum }_j{E}_j=I$. A family admitting a complete error-free exclusion measurement consists of antidistinguishable quantum states. Exclusion need not identify the actual preparation.

For $d$ labelled pure states in dimension $d$, an orthonormal basis $\{|b_j⟩{\}}_{j=0}^{d-1}$ supplies error-free projective state exclusion when $⟨b_j|{\psi }_j⟩=0$ for every $j$. The Born rule gives zero probability to reporting $j$ on input ${\psi }_j$; orthonormal-basis completeness ensures that an outcome is always obtained. Conversely, zero error for this labelled basic measurement forces precisely those zero overlaps. The matrix of transition probabilities is therefore a matrix of column probability distributions with zero diagonal.