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 ρYZ|X=ρY|XρZ|X, where ρYZ|X is the Choi-Jamiolkowski state of the quantum channel EYZ|X, etc.
According to Theorem 3 of PRX (2017), the quantum conditional independence ρYZ|X=ρY|XρZ|X holds iff HX can be decomposed as HX=⊕iHXiL⊗HXiR and ρYZ|X=∑i(ρY|XLi⊗ρY|XRi).
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 ρCD|AB=ρC|ABρD|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 HCD|AB to as ⊕iHABiL⊗HABiR 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.