Cellular automata in operational probabilistic theories

Paolo Perinotti

QUIT Group, Dipartimento di Fisica, Università degli studi di Pavia, and INFN sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

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The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion of locality for update rules. Cellular automata are then defined as homogeneous and local update rules. Finally, we prove a general version of the wrapping lemma, that connects CA on different Cayley graphs sharing some small-scale structure of neighbourhoods.

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Cited by

[1] Robin Lorenz and Jonathan Barrett, "Causal and compositional structure of unitary transformations", Quantum 5, 511 (2021).

[2] Paolo Perinotti, "Causal influence in operational probabilistic theories", Quantum 5, 515 (2021).

[3] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classicality without local discriminability: Decoupling entanglement and complementarity", Physical Review A 102 5, 052216 (2020).

[4] Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469.

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