Fabio Costa’s research while affiliated with KTH Royal Institute of Technology and other places

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Figure 1: Process repeatability. Repetitions of an experiment are modelled as a one-shot scenario, comprising multiple operations that are only performed once. A de Finetti theorem for processes seeks to group these operations such that they represent independent trials under equivalent conditions, with each trial involving multiple operations in arbitrary, possibly unknown or indefinite, causal relations.
Figure 2: Operations and processes (a) Quantum operations, represented as CJ operators M A a , M B b , transform an input to an output quantum system-here depicted as wires. The labels A, B function as generalised coordinates, identifying the operations without necessarily referring to a background causal structure, while a, b denote measurement outcomes. (b) A process matrix W represents the most general way to connect operations. (c) Inserting operations into a process, with no open wires left, returns the probability for observing outcomes a, b through the Born rule for processes, Eq. (4).
Figure 3: Exchangeable and i.i.d. processes (a) A priori, n repetitions of an experiment constitute a oneshot process W (n) . (b) The process can be interpreted as n i.i.d. trials if it is a product of identical processes, W (n) = W ⊗n . Our result guarantees that exchangeable processes are always mixtures of i.i.d. processes.
A de Finetti theorem for quantum causal structures
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February 2025

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Quantum

Fabio Costa

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Jonathan Barrett

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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 with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical probabilities, quantum states, and quantum channels are beautifully answered by so-called "de Finetti theorems", which connect a simple and easy-to-justify condition – symmetry under exchange – with a very particular multipartite structure: a mixture of identical states/channels. Here we extend the result to processes with arbitrary causal structure, including indefinite causal order and multi-time, non-Markovian processes applicable to noisy quantum devices. The result also implies a new class of de Finetti theorems for quantum states subject to a large class of linear constraints, which can be of independent interest.

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