Conjugate index vector ($|{\varphi }_B^{\ast }⟩={\sum }_j{c}_j^{\ast }|j{⟩}_B$)

Fix paired orthonormal bases and let $|\varphi {⟩}_A={\sum }_j{c}_j|j{⟩}_A$. Its conjugate index vector in the other Hilbert space uses the conjugated coefficients. For the unnormalized maximally entangled state $|\tilde{\Psi }⟩={\sum }_j|j{⟩}_A|j{⟩}_B$, the partial inner product is $(I\otimes ⟨{\varphi }_B^{\ast }|)|\tilde{\Psi }⟩=|\varphi {⟩}_A$. Both the conjugation and the lack of normalization of $\tilde{\Psi }$ matter. Applying a linear operator on the first subsystem gives the corresponding relative state of the operated vector.