Pure state

A pure quantum state is represented by a unit vector $|\psi ⟩$ up to global phase, or equivalently by the rank-one density operator $|\psi ⟩⟨\psi |$.

The Euclidean norm distance between normalized pure state vector representatives in a common Hilbert space. Its square is $2-2\mathrm{Re}⟨\psi |\varphi ⟩$. If vectors differing by a global phase are identified, minimize over that phase to obtain $\sqrt{2-2|⟨\psi |\varphi ⟩|}$. This is a different distance convention from trace distance or one minus squared quantum fidelity; stating the convention is necessary when defining $\epsilon$-closeness.

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.

An index state is a chosen orthonormal basis vector in the indexing subsystem of a relative state decomposition. It labels the corresponding partial inner product; it need not be an eigenvector of the subsystem's reduced density matrix.

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.