M_{A}

acts on the first factor, while the partial inner product acts on the second. They therefore commute in the contraction:

(I_{A} \otimes ⟨ ϕ_{B}^{*} ∣) (M_{A} \otimes I_{B}) ∣ Ψ_{A B} ⟩ = M_{A} (I_{A} \otimes ⟨ ϕ_{B}^{*} ∣) ∣ Ψ_{A B} ⟩ = M_{A} ∣ ϕ_{A} ⟩ .

(1)

Thus the operated vector is exactly the relative state obtained from the operated bipartite vector using the same conjugate index vector. For a physical normalized pure state, divide by

∥ M_{A} ϕ_{A} ∥

when that norm is nonzero. If

M_{A} ϕ_{A} = 0

, the vector identity still holds, but there is no normalized output state for that zero-probability branch. An arbitrary operator is not assumed unitary or norm preserving.