Suppose a quantum mechanical particle enters a beam-splitter, which sends its wave packets into two mutually orthogonal channels, $C_a$ and $C_b$. Suppose that $C_a$ also contains System A, with observable $|\Psi_A\rangle$, with which the particle's (in general, different) observable $|\Psi_P\rangle$ interacts, via Hamiltonian $H_{AP}$. $C_b$ does not contain system A, nor any other system (just free space).
Without System A, once within the beam splitter, we could use the simple description:
\begin{equation} |\Psi\rangle = (|C_a\rangle + |C_b\rangle) / \sqrt 2 \end{equation}
With System A located in $C_a$, after the particle enters the beam splitter, the wave function of the composite system evolves to:
\begin{equation}|\Psi\rangle_{t = t_{A+\tau}} = \exp \left( -i\int^{t_{A+\tau}}_{t_A} H_{AP} \mathrm{d}t \right)|\Psi_{A_i}\rangle|\Psi_{P_i}\rangle |C_a\rangle + |\Psi_{A_i}\rangle|\Psi_{P_i}\rangle |C_b\rangle \end{equation}
where the notation "$|\alpha\rangle |\beta\rangle$" represents the tensor product of $|\alpha\rangle$ and $ |\beta\rangle$. $|\Psi_{A_i}\rangle$ is the initial state of System A's observable before interacting with the particle, $|\Psi_{P_i}\rangle$ is the initial state of the particle's observable before interacting with System A, $t_A$ is the time when the particle begins interacting with System A, and $t_{A+\tau}$ is when the particle stops interacting with System A. Normalization is suppressed for simplicity.
In the second equation, the $C_a$ term contains a Hamiltonian that associates specific states of the particle with specific states of System A (entanglement of the particle with system A). The $C_b$ term contains the "null" Hamiltonian, which associates each state of the particle with all states of System A (for all intents and purposes, no entanglement of the particle with System A). Further, the second equation implies that if both channels were later recombined, there would be no interference between their components of the wave function of the particle.
Does the above description sound correct?