This paper shows that every quantum process whose causal order is well-defined (even if it is chosen during the experiment) can be realised as a quantum circuit in which the order of operations is controlled by ordinary classical information. In simple terms, if the order in which separate parties act on quantum systems is not genuinely “indefinite” but is compatible with a definite causal order, then that process can be implemented by a circuit that decides who goes next using classical signals as it runs.

The work is set in the process matrix framework. In that framework, one models several parties that each receive a quantum system, perform a local operation, and send a system away. A process matrix is a mathematical object that connects those local operations and encodes the possible causal relations between the parties. A process is called causally separable when, in every run, one can view some party as acting first (possibly only with some probability), and then the remaining parties follow in an order that again has the same property. Earlier work gave a decomposition that is sufficient to guarantee causal separability, but it was not known whether that decomposition was also necessary in the general multipartite case.

The authors solve that open problem. They show that the sufficient condition identified before is also necessary, completing the characterisation of multipartite causal separability. The main technical tool is what they call a “coherent teleportation technique.” Teleportation in quantum theory is a way to move quantum information from one place to another. Here they use a version that transfers the initial party’s systems coherently to all remaining parties rather than to a single chosen party. This coherent choice makes the recursive argument work: if a process looks causally separable when you pick operations that teleport coherently, then it must admit the decomposition previously known to be sufficient.