 Relative state of a bipartite vector

For a bipartite vector $|v⟩\in {\mathcal{H}}_R\otimes {\mathcal{H}}_A$ and a chosen orthonormal basis of $R$, the generally unnormalized relative state on $A$ is obtained by the partial inner product with one basis vector. It gives the expansion $|v⟩={\sum }_j|j⟩\otimes |{\eta }_j⟩$. The squared norm of a relative state is the probability of its corresponding basis outcome when $|v⟩$ is normalized. Relative states of eigenvectors of a Choi matrix construct the columns of its Kraus operators.