I Quantum Causal Modelling: Mathematics

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I have been trying to read through the following paper: https://iopscience.iop.org/article/10.1088/1367-2630/18/6/063032/pdf, but am stuck at the parts from page 6 onwards.

What is the "approach/formalism" used in the following sections of the paper, and are there any gentle introductions (available online, preferably) to this topic I can read through that may help me understand what's in the paper? I have seen similar mathematics appear in quite a few QM papers now and can't help but wonder if it's important.

Thanks in advance.
 
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Perhaps I should have been more specific in my question, apologies:

Could anyone recommend some introductory materials to
1. "Process Matrix Formalism" - as mentioned in the abstract
2. Completely Positive Trace Preserving maps, Choi–Jamiołkowski isomorphism, and how they are used in QM

Thanks in advance.
 

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