# Examples for the unitary transformation of quantum causal models

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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.

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