= Conjugate index vector
{title2=$|\phi_B^*\rangle=\sum_jc_j^*|j\rangle_B$}
Fix paired orthonormal bases and let $|\phi\rangle_A=\sum_jc_j|j\rangle_A$. Its conjugate index vector in the other <Hilbert space> uses the conjugated coefficients. For the unnormalized <maximally entangled state> $|\widetilde\Psi\rangle=\sum_j|j\rangle_A|j\rangle_B$, the partial inner product is $(I\otimes\langle\phi_B^*|)|\widetilde\Psi\rangle=|\phi\rangle_A$. Both the conjugation and the lack of normalization of $\widetilde\Psi$ matter. Applying a <linear operator> on the first subsystem gives the corresponding <relative state> of the operated vector.
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