 Solution

The linear operator $M_A$ acts on the first factor, while the partial inner product acts on the second. They therefore commute in the contraction: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 $\Vert M_A{\varphi }_A\Vert$ when that norm is nonzero. If $M_A{\varphi }_A=0$, 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.