This is a question related to a quantum causal model suggested by Allen et. al (PRX 7, 031021, 2017) and extended by Barrett et. al (arXiv:1906.10726), which is a quantization of Reichenbach's common cause principle and causal models.

For the classical case of Reichenbach's principle, if X is a complete common cause of Y and Z, then the conditional independence is denoted as \(P(YZ|X) = P(Y|X)P(Z|X)\). The quantum version of the conditional independence is given by \(\rho_{YZ|X} =\rho_{Y|X}\rho_{Z|X}\), where \(\rho_{YZ|X} \) is the Choi-Jamiolkowski state of the quantum channel \(\mathcal{E}_{YZ|X}\), etc.

According to Theorem 3 of PRX (2017), the quantum conditional independence \(\rho_{YZ|X} =\rho_{Y|X}\rho_{Z|X}\) holds iff \(\mathcal{H}_{X}\) can be decomposed as \(\mathcal{H}_{X} = \oplus_{i}\mathcal{H}_{X^i_L}\otimes \mathcal{H}_{X^i_R}\) and \(\rho_{YZ|X} = \sum_{i}(\rho_{Y| X_i^L}\otimes \rho_{Y| X_i^R})\).

The most fundamental and simple case seems unitary transformations, e.g. a unitary trasformation from A & B to C & D (A,B,C, & D are quantum systems). I may use the result of Theorem 3 to show that \(\rho_{CD|AB} = \rho_{C|AB}\rho_{D|AB}\) holds for the unitary transformation. However, even if the authors said that it can be directly verified, I cannot find a specific way of decomposing \(\mathcal{H}_{CD|AB}\) to as \(\oplus_{i}\mathcal{H}_{AB^i_L}\otimes \mathcal{H}_{AB^i_R}\) to show this.

Can anybody teach me how to decompose such Hilbert spaces in general to prove the quantum conditional independence of unitary transformations? Thanks in advance.