 Partial inner product

A partial inner product contracts one factor of a bipartite state against a fixed bra, leaving a vector in the other factor. For $|\Psi ⟩={\sum }_{i,j}{c}_{ij}|i⟩\otimes |j⟩$ and $|a⟩={\sum }_i{a}_i|i⟩$, the result is ${\sum }_{i,j}\overline{a_i}{c}_{ij}|j⟩$. It is linear in $|\Psi ⟩$ and conjugate-linear in $|a⟩$. For a normalized rank-one projective outcome on the first factor, its squared norm is the outcome probability; divide by that norm to obtain the conditional pure state when it is nonzero.