A de Finetti theorem for quantum causal structures

What does it mean for a causal structure to be “unknown”? Can we even talk about “repetitions” of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes are independent and identically distributed (i.i.d.)? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by “de Finetti theorems”, which connect a simple and easy-to-justify condition—symmetry under exchange—to a very particular multipartite structure: a mixture of identical states/channels. Practically, they provide the foundations for principle-based Bayesian methods, e.g., in tomography. Apart from the foundational relevance, de Finetti representations for general causal structures would be useful in the analysis of multi-time, non-Markovian processes, with applications to state-of-the-art quantum devices.

At face value, it appears that each causal structure or assumption on causal structure requires its own de Finetti theorems. Fortunately, I will show that each scenario can be mapped to a linear constraint on quantum states. By proving a de Finetti representation for states subject to a sufficiently large class of constraints, we can derive all the desired results for a broad class of processes.