Cellular Automata in Operational Probabilistic Theories

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 The theory of cellular automata in oper- 5.3 Causal influence . 37 ational probabilistic theories is developed. 5.3.1 Relation with signalling . 38 We start introducing the composition of 5.4 Block decomposition . 39 infinitely many elementary systems, and then use this notion to define update rules 6 Homogeneity 41 for such infinite composite systems. The notion of causal influence is introduced, 7 Locality 47 and its relation with the usual property of signalling is discussed. We then introduce 8 Cellular Automata 51 homogeneity, namely the property of an 8.1 Results . 51 update rule to evolve every system in the same way, and prove that systems evolving 9 Examples 55 by a homogeneous rule always correspond 9.1 Classical case . 55 to vertices of a Cayley graph. Next, we de- 9.2 Quantum case . 55 fine the notion of locality for update rules. 9.3 Fermionic case . 56 Cellular automata are then defined as homogeneous and local update rules. Finally, 10 Conclusion 56 we prove a general version of the wrapping lemma, that connects CA on different Cay- Acknowledgments 57 ley graphs sharing some small-scale struc- A Identification of the sup- and operational ture of neighbourhoods. norm for effects 57 Contents B Quasi-local states 57 1 Introduction1 C Proof of identity 94 64 2 Detailed outlook3 D Homogeneity and causal influence 64 3 Operational Probabilistic Theories4 E Proof of right and left invertibility of a 3.1 Formal framework . 5 locally defined GUR 64 3.2 Causality and the no-restriction hypothesis . 8 F Proof of theorem 10 65 3.3 Norms . 10 G Proof of lemmas 66 and 67 66 arXiv:1911.11216v4 [quant-ph] 8 Jul 2021 4 The quasi-local algebra in OPTs 13 4.1 Quasi-local effects . 14 References 67 4.2 Extended states . 19 4.3 Quasi-local transformations . 22 1 Introduction 5 Global update rules 31 5.1 Update rule . 32 In the last two decades a new approach to quan- 5.2 Admissibility and local action . 35 tum foundations arose, grounded on the quantum information experience [1,2,3,4]. This line Paolo Perinotti: [email protected], of research on the fundamental aspects of quan- http://www.qubit.it tum theory benefits from new ideas, concepts and Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 1 methods [5,6,7] that lead to a wealth of remark- ence. The notion of a cellular automaton for able results [8,9, 10, 11]. In particular, quan- quantum systems was first devised as the quantum theory can be now understood as a theory tum version of its classical counterpart, e.g. in of information processing, that is selected among Refs. [27, 28, 29, 30], but turned out to give rise to a universe of alternate theories [12, 13,1, 14] by a rather independent theory, developed starting operational principles about the possibility or im- from Ref. [31]: the theory of Quantum Cellular possibility to perform specific information pro- Automata (QCAs). The latter counts presently cessing tasks [9, 15]. various important results—see e.g. Refs. [32, 33], The scenario of alternate theories among which just to mention a few. We stress that, most the principles select quantum theory is called the commonly, QCAs are defined to be reversible al- framework of Operational Probabilistic Theories gorithms, and most results in the literature are (OPTs) [8, 15], and has connections with the less proved with this hypothesis. It is known, how- structured concept of Generalized Probabilistic ever, that many desirable preoperties fail to hold Theory (GPT) [5, 16, 17], as well as the dia- in the irreversible case (see e.g. Ref. [34]). In the grammatic category theoretical approach often present work we will not consider irreversible cel- referred to as quantum picturialism [18, 19]. In- lular automata, and leave this subject for further spiration for the framework came also from quan- studies. tum logic [20, 21]. Quantum theory, namely the theory of Hilbert spaces, density matrices, completely positive In order to use cellular automata as candi- maps and POVMs (in particular we refer to the date physical laws in the foundational perspec- elegant exposition of Ref. [22]), can thus be re- tive based on OPTs, one has two choices at formulated as a special theory of information pro- hand. The first one is to start treating automata cessing. Besides the many advantages of this re- within a definite theory, and this is the approach sult, one has to face a main issue: the theory adopted so far, in particular within Fermionic as such is devoid of its physical content. El- theory [35, 36, 37]. In the linear case, Fermionic ementary systems are thought of as elementary cellular automata reduce to Quantum Walks, and information carriers—brutally speaking, memory this brings in the picture all the tools from such cells—rather than elementary particles or fields in widely studied topic [38, 39, 40, 41, 42, 43]. The space-time. While this framework is satisfactory non-linear case is far less studied [44, 45], and for an effective, empirical description of physical does not offer as many results for the analy- experiments, when it comes to provide a theo- sis. Needless to say, this approach faces difficul- retical foundation for the physics of elementary ties that are specific of the theory at hand, and systems, the informational approach at this stage prevents a comparison of different theories on a calls for a way to re-embrace mechanical notions ground that is genuinely physical. such as mass, energy, position, space-time, and complete the picture encompassing the dynamics of quantum systems. The second approach is initiated in the present A recent proposal for this endeavour is based paper, and consists in defining cellular automata on the idea that physical laws have to be ulti- in the general context of OPTs. This perspective mately understood as algorithms, that make sys- offers the possibility of extracting the essential tems evolve, changing their state, exactly as the features of the theory of quantum or Fermionic memory cells of a computer are updated by the cellular automata, those that are not specific of run of an algorithm [23, 24, 25]. Such a program the theory but are well suited in any theory of already achieved successful results in the recon- information processing. As a consequence, one struction of Weyl’s, Dirac’s and Maxwell’s equa- can generalise some results, and figure out why tions (for a comprehensive review see Ref. [26]). and how others fail to extend to the broader sce- The most natural candidate algorithm for de- nario. Moving a much less structured mathemat- scribing a physical law in this context is a cellular ical context, this approach can use only few tools, automaton. The theory of cellular automata is but provides results that have the widest appli- a wide and established branch of computer sci- cability range. Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 2 2 Detailed outlook the algebra of quasi-local transformations, that allows for the description of transformations on In this section we provide a short, non technical an infinite system that can be arbitrarily well ap- discussion of the main results. The purpose of proximated by local operations on finitely many this work is to define cellular automata in OPTs, subsystems. Indeed, an important part of the and prove some general results that will help ap- theory of CAs in OPTs is built by ruling the way plying the theory to special cases of interest. in which quasi-local transformations are trans- A cellular automaton is an algorithm that up- formed by the CA. In particular, the way in which dates the information stored in an array of mem- the CA propagates the effects of a local transfor- ory cells in discrete steps, in such a way that one mation on surrounding subsystems will be the key needs to read the content of a few neighbouring to the definition of the neighbourhood of the sub- cells to determine the state of a given cell at the system, a concept that is central to the theory of next step. Cellular automata in the literature CAs. are often, but not always, defined to be homogeneous: in this case the local rule for the update Once this is done, the next step consists in of the cell is the same for every cell. Here we defining update rules, and their admissibility con- will adopt homogeneity, but the subject is pre- ditions, which are the subject of Section5, along sented so that the generalisation of definitions to with causal influence and a block-decomposition inhomogeneous CA is straightforward. theorem. Update rules are defined in the first place as automorphisms of the space of quasi- Typically, the interesting case of a cellular au- V local effects, but an important request they must tomaton is the one involving an infinite memory abide is that when they act on a quasi-local trans- array. The first challenge we have to face is then formation by conjugation as −1, the ob- to extend the theory of OPTs from finite, arbi- A VAV tained transformation is again quasi-local. trarily large composite systems to actually infinite ones. This piece of theory has an interest The next step is taken in section6, and con- per se, for many reasons ranging from the possi- sists in defining the property of homogeneity.

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