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  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.

asked May 24 in Theoretical Physics by beomseungjeen (5 points) [ no revision ]

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