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. The bility to introduce thermodynamic limits to the latter presents with some difficulties, stemming extension of the theory of C∗ and von Neumann from the fact that, as we mentioned above, we algebras. are defining update rules prior to any mechanical We then start with a review of the framework notion. This implies that even space-time is not of OPTs, and build the necessary tools to define available at this fundamental stage, and without infinite composite systems. The starting point geometry we cannot define homogeneity as trans- is the construction of mathematical objects that lational invariance under the group correspond- describe measurements on finitely many systems ing to a given space-time. On the contrary, we within an infinite array— the OPT counterpart will define homogeneity by formalising the idea of the space of local effects of quantum theory. that every single cell has to be treated equally Effects for the infinite system are then defined as by the update rule, and this notion will be de- limits of Cauchy sequences of local effects. Since fined operationally, requiring that no experiment the introduction of a suitable topology for the def- made of local operations will allow establishing inition of limits is needed, we open the paper with any difference between cells. As a consequence section3, where a review of OPTs is provided, of homogeneity, one can prove that every homo- along with a few new results that will be useful, geneous update rule is underpinned by the Cay- and a rather consistent part of the section will be ley graph of some group, generalising a result of dedicated to the introduction and discussion of Refs. [25, 46]. The mathematical theory known as norms that will provide the necessary topological geometric group theory then tells us that the Cay- framework for a consistent definition of limits. ley graph representing the causal connections of The subject of the subsequent Section4 is then cells in the memory array, being a metric space, the construction of the Banach space of effects for uniquely identifies an equivalence class of met- the infinite composite system, and consequently ric spaces, that captures both the algebraic and the construction of the space of states as suitable geometric essential features of the group. Aston- linear functionals on effects. An important sub- ishingly, for Cayley graphs of finitely presented section will be dedicated to the construction of groups—which is the case for cellular automata

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 3 as we define them here—the equivalence class al- the framework. ways contains a smooth manifold of dimension at Quantum theory is about system types A 1 most four (see Ref. [47], pag. 90). This result —namely complex Hilbert spaces HA that are means that we can always think of a cellular au- classified by their dimension dA = dim(HA)—and tomaton as if it was embedded in a Riemannian transformations that can occur on systems as a manifold, in such a way that the Riemannian dis- consequence of undergoing a test. Tests are rep- tance between nodes is almost the same as the resented by quantum instruments EX = {Ei}i∈X, distance between nodes given in terms of steps i.e. a collection of Completely Positive (CP) maps along the graph edges. Ei that sum to a trace-preserving one (a channel): P Also the notion of locality, presented in sec- E = i∈X Ei. The quantum operation Ei—a CP tion7, comes with its own difficulties, that can be trace non-increasing map—represents the change overcome by proving a generalisation of the result in the system occurring upon the event of a spe- known as “unitarity plus causality implies local- cific outcome i ∈ X in the test. In the diagram- izability" [32]. This is where the theory, which is matic language of OPTs systems are represented inspired by that of QCAs, deeply differs from the as labelled wires, and instruments or quantum op- theory of classical cellular automata. In particu- erations as boxes with an input and output wire, lar, considering the collection of transformations e.g. VAV −1 for all A acting on a given system g, we A B A B will define the neighbourhood of g as the set of EX , Ei , systems on which transformations VAV −1 act non trivially. respecitvely. States can be considered as special Once the theory of cellular automata is fully transformations where the input system is I, hav- developed, some results are proved in Section8. ing HI = C, i.e. dI = 1. Notice that the a prepa- In particular, a very useful theorem is the wrap- ration test, i.e. the most general test from I to A, ping lemma, which under very wide hypotheses— represents a probabilistic preparation procedure though not universal—allows for the classification where some state in the collection PX = {ρi}i∈X of automata on a given infinite graph by classi- is prepared, with the sub-normalised density ma- fying automata on any suitably “wrapped" finite trix ρi occurring upon reading the outcome i ∈ X. version of the same graph. The probability of occurrence of ρi is Tr[ρi], and P In Section9, a few examples are reviewed. In ρ = i∈X ρi has unit trace. Preparation tests particular, using the general notion of locality, or states for system A are represented by special we apply the definition of causal influence and diagrams the neighbourhood scheme to the case of classical A A cellular automata. With the above definitions at PX , ρi , hand, we show that allegedly local automata are actually non-local. This solves a long standing respectively. The space HA of Hermitian opera- 2 puzzle about the connection between classical and tors on HA, whose dimension is DA = dA, is the quantum automata—the so-called quantisation of real span of states of system A. classical automata [31, 48]. A POVM QY = {Qj}j∈Y is a collection of ef- The paper is concluded by Section 10 with a fects 0 ≤ Qj ≤ IA that sum to the identity opera- P summary of the results and some closing remarks. tor: j∈Y Qj = IA. This kind of test can be seen as a quantum instrument from A to I, namely a collection of linear functionals on the real space 3 Operational Probabilistic Theories spanned by density matrices, where the functionals aj(ρ) are defined as aj(ρ) := Tr[ρQj]. The In this section we will provide a thorough introduction to OPTs, which is partly a review of the 1We remark that for the purpose of information pro- literature, and partly presentation of new results cessing the type of a system is captured by the dimension of the corresponding Hilbert space, e.g. the electron spin that will be used in the remainder. We provide is of the same type as the photon polarisation. This is here a sketchy introduction before going through slightly different than the usual notion of system type in formal definitions, and use Quantum Theory as physics, where the two types in the above example are an illustrative example for the main notions in different.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 4 diagrammatic representation of POVMs and ef- We warn the reader that this presentation has fects is the following some elements that are slightly diﬀerent from other reviews in the literature, and is tailored A A Q QY , j , to ease the presentation of subsequent material. respectively. Most of the results in the present sections are Composite systems, such as AB, correspond to new, and their proof will be given. Those theorems that are not original are not proved, and to the Hilbert space HA ⊗HB of dimension dAB = due reference is provided. dAdB. Tests {Ei}i∈X from A to B and {Fj}j∈Y from B to C can be run in sequence, obtaining a An operational theory Θ consists in i) a collection Test(Θ) of tests TA→B, each labelled by new test: {Dij}(i,j)∈X×Y where Dij := FjEi. On X input and output letters from a collection Sys(Θ) the other hand, tests {Ei}i∈X from A to B and 0 denoting system types, e.g. A → B—that will {Ej}j∈Y from B to B on subsystems of a composite system AA0 can be run in parallel, obtaining be systematically omitted—and by a ﬁnite set of outcomes X; for every pair of types A, B ∈ Sys(Θ) the test {Dij}(i,j)∈X×Y with Dij := Ei ⊗ Fj. In diagrams, we draw for quantum operations the set of tests of type A → B is denoted hhA → Bii; ii) a very basic associative rule for sequential A C A B C FjEi = Ei Fj , composition: the test TX ∈ hhA → Bii can be fol- 0 0 0 lowed by the test SY ∈ hhA → B ii if A ≡ B, thus

A B obtaining the sequential composition STX×Y ∈ Ei 0 AA0 BB0 hhA → B ii; iii) a rule ⊗ : (A, B) 7→ AB for com- Ei ⊗ Fj = , A0 B0 posing labels in parallel, and a corresponding rule Fj for tests ⊗ :(SX, TY) 7→ (S ⊗ T)X×Y, with the fol- and analogously for instruments. lowing properties Very relevant structures in the theory are the 1. Associativity: (AB)C = A(BC). real space HA of Hermitian operators on HA, spanned by quantum states, the cone of positive 2. For every SX ∈ hhA → Bii and TY ∈ hhC → operators in PA ⊆ HA, the convex set of states Dii, one has SX ⊗ TY ∈ hhAC → BDii. Asso- [[A]], obtained by intersecting the positive cone ciativity of ⊗ holds: with the half-space Tr[X] ≤ 1, and the convex (S ⊗ T ) ⊗ W = S ⊗ (T ⊗ W ). set of deterministic states [[A]]1, obtained by in- X Y Z X Y Z tersecting the positive cone with the aﬃne hy- perplane Tr[X] = 1. Similar structures can be 3. Unit: there is a label I such that IA = AI = generalised to the space spanned by quantum op- A for every A ∈ Sys(Θ). erations. In the general framework of OPTs, we 4. Identity: for every A ∈ Sys(Θ), a test I ∈ will systematically refer to the generalisation of A hhA → Aii such that I S = S I = S , for the above concepts. B X X A X every SX ∈ hhA → Bii.

3.1 Formal framework 5. For every AX ∈ hhA → Bii, BY ∈ hhB → Cii, D ∈ hhD → Eii, E ∈ hhE → Fii, one has The framework of OPTs is meant to capture the Z W main traits of Quantum Theory (shared e.g. by (BY ⊗ EW)(AX ⊗ DZ) = (BYAX) ⊗ (EWDZ). Classical Theory, or Fermionic Theory, etc.) sum- (1) marised above, and use them as deﬁning properties of a family of abstract theories that might 6. Braiding: for every pair of system types A, B, be candidates for an alternative representation of ∗ there exist tests SAB, SAB ∈ hhAB → BAii elementary physical systems and their transfor- ∗ ∗ such that SBASAB = SBASAB = IAB, and mations. In the remainder of this section we pro- SAB(AX ⊗ BY) = (BY ⊗ AX)SAB. Moreover, vide a brief review of the framework of OPTs (for reference see e.g. [15,8,9]). Some of the most S(AB)C = (IA ⊗ SBC)(SAC ⊗ IB), relevant diﬀerences with respect to the quantum SA(BC) = (SAB ⊗ IC)(IB ⊗ SAC). case stem from the fact that systems might not ∗ compose with the tensor product rule. When SAB ≡ SAB, the theory is symmetric.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 5 All the theories developed so far are symmetric. are independent. While it is immediate that 1 ∈ All tests of an operational theory are (finite) [[I → I]]—since {1}∗ = {II} is the only singleton collections of events: hhA → Bii 3 RX = {Ri}i∈X. test—we will assume that 0 ∈ [[I → I]]. This If hhA → Bii 3 RX = {Ri}i∈X and hhB → Cii 3 means that we can consider e.g. tests of the form TY = {Tj}j∈Y, then {1, 0, 0}. Events in [[A → B]] are called transformations. hhA → Cii 3 (TR)X×Y := {TjRi}(i,j)∈X×Y. As a consequence of the above definitions, every set [[A]] := [[I → A]] can be viewed as a set of Similarly, for R ∈ hhA → Bii and T ∈ hhC → X Y functionals on [[A]]¯ . As such, it can be viewed as Dii, a spanning subset of the real vector space [[A]]R of linear funcitonals on [[A]]¯ . On the other hand (R ⊗ T)X×Y := {Ri ⊗ Tj}(i,j)∈X×Y. [[A]]¯ := [[A → I]] is a separating set of positive The set of events of tests in hhA → Bii is de- linear functionals on [[A]], which then spans the ∗ ¯ noted by [[A → B]]. By the properties of se- dual space [[A]] =: [[A]]R. The dimension DA of R ¯ quential and parallel composition of tests, one [[A]]R (which is the same as that of [[A]]R) is called can easily derive associativity of sequential and size of system A. One can easily prove that in parallel composition of events, as well as the ana- any OPT Θ, I is the unique system with unit size logue of Eq. (1). For every test TX ∈ hhA → Bii DI = 1. Using the properties of parallel compo- with TX = {Ti}i∈X, and every disjoint partition sition, one can also prove that DAB ≥ DADB. S Events in [[A]] are called states, and denoted by {Xj}j∈Y of X = j∈Y Xj, one has a coarse grain- 0 lower-case greek letters, e.g. ρ, while events in [[A]]¯ ing operation that maps TX to TY ∈ hhA → Bii, 0 0 0 are called effects, and denoted by lower-case latin with TY = {Tj }j∈Y. We define TXj := Tj . The parallel and sequential compositions distribute letters, e.g. a. When it is appropriate, we will over coarse graining: use the symbol |ρ) to denote a state, and (a| to denote an effect. We will also use the circuit no-

TXj ⊗ Rk = (T ⊗ R)Xj ×{k}, tation, where we denote states, transformations and eﬀects by the symbols AlTXj Bk = (ATB){l}×Xj ×{k}. ρ A A B A Notice that for every test TX ∈ hhA → Bii there , A , a , 0 exists the singleton test T∗ := {TX}. One can respectively. Sequential composition of ∈ easily prove that the identity test IA is a sin- A [[A → B]] and ∈ [[B → C]] is denoted by the gleton: IA = {IA}, and IBT = TIA for B every event T ∈ [[A → B]]. Similarly, for diagram ∗ ∗ SAB = {SAB} and SAB = {SAB} we have A C A B C ∗ ∗ BA = A B . SBASAB = SBASAB = IAB. The collection of events of an operational theory Θ will be denoted For composite systems we use diagrams with mul- by Ev(Θ). The above requirements make the col- tiple wires, e.g. lections Test(Θ) and Ev(Θ) the families of mor- phisms of two braided monoidal categories with A B . the same objects—system types Sys(Θ). C A D An operational theory is an OPT if the tests hhI → Iii are probability distributions: 1 ≥ Ti = A A P The identity will be omitted: I = pi ≥ 0, so that i∈X pi = 1, and given two tests A . The swap SAB and its inverse will be de- S , T ∈ hhI → Iii with Si = pi and Ti = qi, the X Y noted as follows following identities hold A B A B Si ⊗ Tj = SiTj := piqj, = B S A X B A TXj := pi, i∈Xj A B A B meaning that events in the same test are mutually ∗ = B S A exclusive and events in diﬀerent tests of system I B A

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 6 In the present paper we will always assume not true that [[AB]]R = [[A]]R ⊗ [[B]]R, but only that the theory under consideration is symmet- [[A]]R ⊗[[B]]R ⊆ [[AB]]R. As a consequence, the lin- ric, however all the results will be straightfor- ear map representing T on [[A]]R is not sufficient wardly generalisable. We will consequently draw to determine the linear map representing T ⊗IC the swap SAB as on [[AC]]R. One can easily prove that [[A → B]] spans a real A B A B A B vector space [[A → B]] . Being [[A → B]] spanning = ∗ = R B S A B S A B A for [[A → B]]R, the criteria of definition1 and lemma1 hold for A , B ∈ [[A → B]]R. Elements An OPT Θ is specified by the collections of of [[A → B]] are called generalized events. Every systems and tests, along with the parallel com- R space [[A → B]] has a zero event 0A→B := 0I→I ⊗ position rule ⊗ R T ∈ [[A → B]], where T is an arbitrary event in [[A → B]]. As a consequence of the coarse graining Θ ≡ (Test(Θ), Sys(Θ), ⊗). rule for tests on [[I → I]], one can easily show that Definition 1 (Equal transformations). Let coarse graining of two or more transformations is A , B ∈ [[A → B]]. Then we define A = B represented by their sum. Precisely, given a test if for every system C and every ρ ∈ [[AC]] and {Ti}i∈X ⊆ [[A → B]], for X0 = {0, 1} ⊆ X one has 0 a ∈ [[B¯C]]¯ , one has T0 = T0 + T1, namely for every system C it is 0 T0 ⊗ IC = T0 ⊗ IC + T1 ⊗ IC. Finally, every A B A B A B set [[A → B]] has a subset consisting in singleton ρ a = ρ a C C events, that we denote by [[A → B]]1, and call deterministic. For A, B 6= I, a deterministic event Notice that, since states separate effects and is called channel. A channel U ∈ [[A → B]]1 is viceversa effects separate states, the above defi- reversible if there exists a channel V ∈ [[B → A]]1 nition is equivalent to the two following equality such that VU = IA, UV = IB. criteria. Let us now define the cones

Lemma 1. Let A , B ∈ [[A → B]]. Then the [[A → B]]+ := {λT | λ ≥ 0, T ∈ [[A → B]]}. following conditions are equivalent. We will often write A . 0 as a shorthand for 1. A = B. A ∈ [[A → B]]+. The cone [[A → B]]+ introduces 2. For every C and every ρ ∈ [[AC]], one has a partial ordering in [[A → B]]R, deﬁned by

A B A B A B A . B ⇔ (A − B) . 0. ρ = ρ . C C OPTs are assumed to have all sets [[A → B]] (and (2) thus also cones [[A → B]]+) closed in the operational norm. 3. For every C and every a ∈ [[B¯C]]¯ , one has Two systems may be operationally equivalent if they can be mapped one onto the other via a A B A B A B reversible transformation. Clearly, in this case a = a . C C every processing of the first system is perfectly (3) simulated by a processing of the other, and viceversa.We define operationally equivalent systems as follows. One can show [8] that events T ∈ [[A → B]] can be identified with a family of linear maps, Definition 2. Let A, B be two systems. We say one for every C, that characterize the action of that A and B are operationally equivalent, in for- T ⊗ IC on [[AC]]R as a linear map to [[BC]]R. mula A =∼ B, if there exists a reversible transfor- As anticipated in the introductory paragraph, we mation U ∈ [[A → B]]1. remind the reader that some of the difficulties that we will be faced with in the remainder orig- Definition 3. If A1 and A2 are operationally inate from the fact that in a general theory it is equivalent through U and B1 and B2 through V ,

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 7 then, for every system C, A1 ∈ [[A1C → B1C]]R where signals can propagate only in the direc- and A2 ∈ [[A2C → B2C]]R are operationally tion defined by input and output of processes, −1 equivalent if A2 = (V ⊗ IC)A1(U ⊗ IC). In within a cone of causal influence determined by particular, ρ1 ∈ [[A1C]]R and ρ2 ∈ [[A2C]]R are interactions between systems. These are the only operationally equivalent if ρ2 = (U ⊗IC)ρ1, and theories where one can consistently use informa- ¯ ¯ ¯ ¯ a1 ∈ [[A1C]]R and a2 ∈ [[A2C]]R are operationally tion acquired in a set of tests to condition the −1 equivalent if a2 = a1(U ⊗ IC). choice of subsequent tests, where the partial ordering we are referring to is that determined by Lemma 2. Let AB be operationally equivalent the input/output direction. to A. Then B must be operationally equivalent to the trivial system I. Definition 4 (Causal theories). A theory (T , A) is causal if for every test {Ti}i∈X and every col- i Proof. First of all, since DAB ≥ DADB ≥ DA, lection of tests {Sj }j∈Y labelled by i ∈ Y, the if AB =∼ A the chain of inequalities is satu- generalized test {Ci,j}(i,j)∈X×Y with rated, and thus DB = 1. This implies that ev- A C A B i C ery state of system AB is of the form τ ⊗ 1B, Ci,j := Ti Sj , where τ ∈ [[A]] and 1B is the unique deterministic state of system B. The same result holds is a test of the theory. for states of ABC for arbitrary C. Now, let Notice that this notion of causality is strictly U ∈ [[AB → A]]1 be the reversible channel imple- stronger than the one usually adopted in the lit- menting operational equivalence. Let η ∈ [[AC]], erature about OPTs, that can be summarised and (U ⊗ I )(1 ⊗ η) =η ˜ ∈ [[AC]]. One can C B as uniqueness of the deterministic effect (see choose U as to identify η˜ = η. Indeed, there e.g. Refs. [8,9, 15]). Indeed, a first result of is a reversible transformation V ∈ [[A → A]] crucial importance derives uniqueness of the de- such that (V ⊗ I )˜η = η. In particular, one C terministic effect for every system type in causal can choose V := 1 U −1, where now 1 repre- B theories. sents the unique deterministic effect of B. Right- reversibility of V is trivial. On the other hand, Theorem 1 ([8, 15]). In a causal theory, for ev- −1 let σ ∈ [[AC]], and let 1B ⊗ τ := (U ⊗ IC)σ ∈ ery system type A the set of deterministic effects [[ABC]]. Now, after applying the deterministic ef- [[A]]¯ 1 is the singleton {eA}. fect 1B one has (V ⊗ IC)σ = τ. One can reverse V by composing τ with 1B, obtaining 1B ⊗ τ, A proof of the above theorem, that proceeds and then applying U ⊗ IC, finally obtaining by contradiction, can be found in the mentioned (U ⊗ IC)(1B ⊗ τ) = σ. With a suitable choice of references. We remark that, being the notion of U , one then has (U ⊗IC)(1B ⊗η) = η. Let now causality in these references different form the one 1B ⊗ η = τBC ⊗ νA, with ν ∈ [[A]]1. Discarding defined here, the theorem has a slightly different system A on both sides via some deterministic statement. In the same references, one can find effect eA ∈ [[A]]¯ 1, one obtains τBC = 1B ⊗ θ, with the equivalence of uniqueness of the deterministic θ = (eA ⊗ IC)η. Finally, this implies that every effect with non-signalling from output to input. state of system BC is of the form τ = 1 ⊗ θ, BC B Theorem 2 ([8, 15]). In an OPT, every system with θ ∈ [[C]]. Once we fix η ∈ [[A]] , we can then 1 type A has a unique deterministic effect if and define the reversible channel ∈ [[BC → C]] W only if the marginal probabilities of preparation given by W := eA(U ⊗ IC)(η ⊗ IBC). Clearly, −1 tests cannot depend on the choice of subsequent W (1B⊗θ) = θ, and the inverse map W ∈ [[C → −1 −1 observation tests. BC]] is obtained as W := eA(U η ⊗ IC). We then have BC ' C for every C, i.e. B ' I. As a consequence of our notion of causality, in a non-deterministic theory (i.e. a theory where 3.2 Causality and the no-restriction hypothesis [[I]] 6= {0, 1}), every convex combination of tests is a test (see e.g. [15]). This implies that the sets In the remainder of the paper we will focus on [[A]]∗, [[A]]¯ ∗, [[A → B]]∗ are convex, where ∗ can be causal theories. The causality property, that replaced by nothing, 1 or +. Moreover, the sets we define right away, characterizes those theories with ∗ = + are convex cones.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 8 0 0 Another important consequence of causality is [[A → B ]]+ ≡ [[A → B]]+. Thus the ordering . is that a transformation C ∈ [[A → B]] is determin- the same for the two theories. As a consequence, istic if and only if it maps the deterministic effect the preservation of cones under the two composi- 0 eB to the deterministic effect eA. tions is inherited by Θ from the same property in 0 0 0 0 Θ. Now, if A ∈ [[A → B ]]1 and B ∈ [[B → C ]]1, Theorem 3 ([8, 15]). In a causal theory, a trans- then formation C ∈ [[A → B]] is a channel iff

A B A (eC|BA = (eB|A = (eA|, C e = e . 0 0 In causal theories, one can always assume that thus BA ∈ [[A → C ]]1. Moreover, thanks to every state is proportional to a deterministic one. causality of the theory Θ one has eAB = eA ⊗ eB, 0 0 0 0 Indeed, including in the theory every state ρ ∈ thus for A ∈ [[A → B ]]1 and B ∈ [[C → D ]]1, [[A]]+ such that (eA|ρ) = 1 does not introduce any inconsistency with the set of transformations, as (eBD|A ⊗ B = (eB|A ⊗ (eD|B we now prove. Let Θ be a causal OPT, and define = (eA| ⊗ (eC| = (eAC|, the theory Θ0 through the bijection κ : Sys(Θ) → 0 0 0 0 Sys(Θ0) :: A 7→ A0, with and consequently A ⊗ B ∈ [[A C → B D ]]1. Now, given events A , B in the theory Θ0, by defi- 0 0 0 [[A → B ]]R ≡ [[A → B]]R, nition we have C , D deterministic in Θ such that 0 0 [[A → B ]]1 := {T . 0 | (eB|T = (eA|}, C . A and D . B, thus 0 0 0 0 [[A → B ]] := {T . 0 | ∃C ∈ [[A → B ]]1, C . T }, F :=(D − B)(C − A ) + (D − B)A hhA0 → B0ii := + B(C − A ) . 0, {T ⊆ [[A0 → B0]] | X ∈ [[A0 → B0]] }. T 1 G :=(C − A ) ⊗ (D − B) T ∈T + (C − A ) ⊗ B + A ⊗ (D − B) . 0. where . denotes the ordering induced by the cones of the theory Θ, and the cardinality of sets T Finally, F + BA . BA and G + A ⊗ B . A ⊗ in the last definition is implicitly assumed to be B are channels, and thus BA and A ⊗ B are finite. events. Theorem 4. Let Θ be a causal OPT, and con- One can now show that the new theory Θ0 is sider the theory Θ0 defined above. Then Θ0 is an causal. OPT with a unique deterministic effect eA0 for 0 every A . Corollary 1. Let Θ be a causal OPT, and Θ0 as 0 Proof. All the compositional structures of Θ are in theorem4. Then Θ is a causal OPT. 0 inherited by Θ . The defining conditions in the 0 0 0 0 0 Proof. Let {Ti}i∈X be a test in [[A → B ]], and special case of [[A ]] = [[I → A ]] give i 0 0 {Sj }j∈Y be tests in [[B → C ]] for every i ∈ X. 0 Then [[A ]] = {ρ ∈ [[A]]+ | (eA|ρ) ≤ 1},

0 0 0 X i X while for [[A¯ ]] = [[A → I ]] they give ∀i ∈ X (eC0 |Sj = (eB0 |, (eB0 |Ti = (eA0 |. j∈Y i∈X 0 [[A¯ ]]1 = {eA}. This implies that What remains to be proved is that the set of X X transformations is closed under sequential and (eC0 |Wi,j = (eA0 |, parallel composition. First of all, we observe that i∈X j∈Y 0 0 i 0 0 by theorem3 [[A → B]]1 ⊆ [[A → B ]]1, thus Wi,j := Sj Ti ∈ [[A → C ]] ∀i ∈ X, j ∈ Y. 0 0 [[A → B]] ⊆ [[A → B ]]. This makes [[A → B]]+ ⊆ 0 0 [[A → B ]]+. On the other hand, since by def- As a consequence, every conditional test Wi,j := 0 0 i 0 0 inition it is also [[A → B ]] ⊆ [[A → B]]+, we Sj Ti ∈ [[A → C ]] is admitted in the theory 0 0 0 have [[A → B ]]+ ⊆ [[A → B]]+, which makes Θ .

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 9 Remark 1. In the remainder, we will focus on [[A → B]]+, since the latter implies that C = λC0 theories satisfying causality and the further re- for some C0 ∈ [[A → B]] and λ ≥ 0. Then, quirements C 0. Moreover, since C sends eB to eA, one has (C ⊗ IC)[[AC]]1 ⊆ [[BC]]1, and thus [[A → B]]1 = {T . 0 | (eB|T = (eA|}, (C ⊗ IC)[[AC]] ⊆ [[BC]]. By the no-restriction [[A → B]] = {T . 0 | ∃C ∈ [[A → B]]1, C . T }, hypothesis, this is tantamount to C ∈ [[A → B]]. X We can now prove a theorem that is a very hhA → Bii = {T ⊆ [[A → B]] | T ∈ [[A → B]]1}. T ∈T important consequence of the no-restriction hypothesis, along with causality. In particular, this implies that [[A]] = {ρ ∈ [[A]]+ | (eA|ρ) ≤ 1}. Thanks to corollary1, this is not a Theorem 5. In a theory satisfying the no- signiﬁcant restriction. restriction hypothesis, let T ∈ [[A → B]]R. Then T , C − T 0 for some channel C ∈ [[A → B]]1 We now narrow down focus on theories that iﬀ T ∈ [[A → B]]. satisfy the no-restriction hypothesis. Let us consider the set of preparation-tests for every system Proof. The hypothesis T , C − T 0 is equiva- of the theory Θ. Then we will complete the set lent to T , C − T ∈ [[A → B]]+. Equivalently, of tests allowing for all those tests that transform for every C and every P ∈ [[AC]]+, one has preparation tests to preparation tests, even when (X ⊗ IC)P ∈ [[BC]]+ for X = T , C − T . Let applied locally. This requirement is the general- now P ∈ [[AC]]. Then one has isation of the assumption made in quantum theory that a map is a transformation if and only (eBC|[(C − T ) ⊗ IC]|P ) ≥ 0, if it is completely positive and trace non increasing, and a collection of transformations is a test and thus if and only if the sum of its elements is a channel, 0 ≤ (e |( ⊗ )|P ) ≤ (e |P ) ≤ 1. i.e. completely positive and trace-preserving. BC T IC AC

This implies that (T ⊗ IC)P ∈ [[AC]]. Finally, Assumption 1. Let A ∈ [[A → B]]R. The no- restriction hypothesis consists in the requirement by the no-restriction hypothesis, T ∈ [[A → B]]. that T ∈ hhA → Bii if and only if for every C and The converse statement is trivial. every P ∈ hhI → ACii, one has (T ⊗ I )P ∈ hhI → C In the literature [10, 11] one can often find a dif- BCii. ferent requirement under the name “no-restriction We will write A 0 if for every C and every hypothesis”, namely that for every system A the ¯ P ∈ [[AC]]+, one has (A ⊗ IC)P ∈ [[BC]]+. The set [[A]] coincides with the set of functionals a set of transformations A 0 is a cone, that we on [[A]]R that satisfy 0 ≤ (a|ρ) ≤ 1 for ev- denote by ery ρ ∈ [[A]]. The last condition may be nei- ther necessary nor sufficient for the no-restriction K(A → B) := {A ∈ [[A → B]]R | A 0}. hypothesis as we state it, despite counterexam- ples are still unknown for both cases. Indeed, The above defined cone introduces a (partial) or- the no-restriction hypothesis for effects imposes dering in [[A → B]] , defined by R the requirement that for every C and every state A B ⇔ A − B 0. (4) P ∈ [[AC]], one has Notice that, under the no-restriction hypothe- A a ∈ [[C]] . (6) sis, the following identity holds P C

K(A → B) = [[A → B]] , + 3.3 Norms and thus for every , ∈ [[A → B]] , A B R As the ﬁrst step in the present work is to con- A B ⇔ A . B. (5) struct inﬁnite composite systems, we will need a thorough notion of sequences and limits. From a We remark that, under assumption1, the hy- topological point of view the vector spaces that pothesis of theorem3 can be relaxed to C ∈ we constructed so far have no special structure,

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 10 however the operational procedures for discrim- Proof. By definition it is ination of processes provide a natural definition of distance between events. Such a distance is re- kρ ⊗ σkop = sup (a0 − a1|ρ ⊗ σ) (a ,a ) lated to the success probability in discriminating 0 1 events. Making the space of events into a metric = sup (b0 − b1|ρ), (b ,b )∈M space, the operational distance immediately pro- 0 1 vides a topological structure through the induced where M := {(b0, b1) | bi = ai(IA ⊗ σ)}. Thus operational norm. However, in order to make the space of events of infinite composite systems into kρ ⊗ σkop ≤ kρkop. a Banach algebra, a stronger norm is needed, that On the other hand, for every binary observation- is introduced here: the sup-norm. In the quan- test (a , a ) in [[A]]¯ one has tum and classical case the two norms coincide. 0 1 The operational norm k·kop for states is defined ai = (ai ⊗ eB)(IA ⊗ σ), as follows [15]. and thus M actually contains every possible bi- Definition 5. The operational norm on [[A]]R is nary observation-test on A. Finally, this implies that kρkop := sup (2a − e|ρ) = sup (a0 − a1|ρ). a∈[[A]]¯ a0,a1∈[[A]]¯ a0+a1=eA kρ ⊗ σkop = kρkop. (7) For more details on k·kop see Refs. [8, 15]. The operational norm k·kop for transformations We now proceed to define the sup-norm. A ∈ [[A → B]]R on finite systems is then defined Definition 6. The sup-norm kA ksup of A ∈ as [[A → B]]R is defined as

kA kop := sup k(A ⊗ IC)Ψkop kA ksup := inf J(A ), C,Ψ∈[[AC]]1 ˜ J(B) := {λ | ∃C ∈ [[A → B]]1, λC B −λC }. = sup sup (2a − e|A ⊗ IC|Ψ). C,Ψ∈[[AC]] a∈[[B¯C]]¯ We now show that k·ksup actually deﬁnes a ¯ norm. In the special case of eﬀects a ∈ [[A]]R we have

Proposition 1. The sup-norm on [[A → B]]R is kakop := sup k(a ⊗ IC)Ψkop C,Ψ∈[[AC]] well deﬁned:

= sup sup (a ⊗ [2b − eC]|Ψ). 1. kA ksup is non-negative, and it is null iﬀ C,Ψ∈[[AC]] b∈[[C]]¯ A = 0, As a consequence, 2. for µ ∈ R one has kµA ksup = |µ|kA ksup, kakop = sup (a|ρ0 − ρ1) 3. kA + Bksup ≤ kA ksup + kBksup. ρ0,ρ1∈[[A]] ρ0+ρ1∈[[A]] Proof. 1. Let A ∈ [[A → B]]R. Suppose that = sup {p sup (a|ρ) − (1 − p) inf (a|σ)} j := inf J(A ) < 0. Then there exists 0 < ε < |j| p∈[0,1] ρ∈[[A]] σ∈[[A]] such that j + ε ∈ J(A ), namely = max{ sup (a|ρ), inf (a|σ)} σ∈[[A]] ρ∈[[A]] − (|j| − ε)C A (|j| − ε)C = sup |(a|ρ)|. ⇒ −2(|j| − ε)C 0, ρ∈[[A]] for some ∈ [[A → B]] , which is absurd. Then Here we only prove one new result about the op- C 1 inf J( ) ≥ 0. Suppose now that J( ) = 0. erational norm, that we will use in the following. A A This implies that for every n ∈ N there exists ∈ [[A → B]] such that 1 − 1 . Lemma 3. Let ρ ∈ [[A]]R and σ ∈ [[B]]1. Then Cn 1 n Cn A n Cn Now, by the closure of [[A]]+ in the operational 1 kρ ⊗ σkop = kρkop. (8) norm, and considering that the sequences ( n Cn ±

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 11 A ) converge to ±A , it must be ±A ∈ [[A]]+. Corollary 3. For A ∈ [[A → B]]R and B ∈ This is possible if and only if A = 0. 2. The [[C → D]]R, kA ⊗ Bksup ≤ kA ksupkBksup. proof is trivial for µ = 0. Let then µ 6= 0. If Proof. The result follows straightforwardly from x ∈ J(A ), then xC ± A 0 for some C ∈ proposition2 and corollary2. [[A → B]]1, and thus |µ|(xC ± A ) 0, namely |µ|x ∈ J(µA ). Thus kµA ksup ≤ |µ|kA ksup. For An important result regarding the sup-norm is the same reason, kA ksup ≤ (1/|µ|)kµA ksup, and provided by the following proposition. finally kµA ksup = |µ|kA ksup. 3. Let now x ∈ J(A ) and y ∈ J(B). Then xC ± A 0, and Proposition 3. For A ∈ [[B → C]]R and B ∈ yD ± B 0, for C , D ∈ [[A → B]]1. Thus (x + [[A → B]]R, kABkop ≤ kA ksupkBkop. y) ± ( + ) 0, where := x/(x + y) + F A B F C Proof. By definition we have y/(x+y)D ∈ [[A → B]]1. Thus, x+y ∈ J(A +B), and then k + k ≤ k k + k k . A B sup A sup B sup kABkop The following property makes ([[A → = sup sup (2a − eCD|AB ⊗ ID|Ψ). D,Ψ∈[[AD]]1 a∈[[C¯D]]¯ A]]R, k·ksup) a Banach algebra. Now, for every a ∈ [[C¯D]]¯ , and λ ∈ J( ), upon Proposition 2. For A ∈ [[B → C]]R and B ∈ A defining a := a, a := e − a, we have [[A → B]]R, kABksup ≤ kA ksupkBksup. 0 1 CD

Proof. Let x ∈ J(A ), and y ∈ J(B). Then there (a0 − a1|A ⊗ IC are C , D such that xC ± A 0, yD ± B 0. = (a0|[A ⊗ ID] − (a1|[A ⊗ ID] Now, we have = λ[(ã0| − (ã1|], 1 [(xC + A )(yD − B) + (xC − A )(yD + B) −1 1 2 where (ãi| := λ {(ai|[A ⊗ID]+ 2 (e|[(λC −A )⊗ = xyCD − AB 0 ID]}, with C : [[B → C]]1 such that λC ± A 0. 1 Clearly, [(xC + A )(yD + B) + (xC − A )(yD − B) 2 a˜0, a˜1 ∈ [[B¯D]]¯ , a˜0 +a ˜1 = eBD, =xyCD + AB 0, thus and then xy ∈ J(AB). Thus, kABksup ≤ k ksupk ksup. A B (a0 − a1|AB ⊗ ID|Ψ) = λ[(ã | − (ã |] ⊗ |Ψ) Lemma 4. Let A ∈ [[A → B]]R, and for an arbi- 0 1 B ID trary C, let D ∈ [[C → D]]1. Then kA ⊗ Dksup = ≤ λ sup (2b − eBD|B ⊗ ID|Ψ) ¯ ¯ kA ksup. b∈[[BD]]

Proof. Let x ∈ J(A ). Then by deﬁnition there This implies that kABkop ≤ λ supb∈[[B¯D]]¯ (2b − exists C ∈ [[A → B]]1 such that xC ± A 0. eBD|B ⊗ ID|Ψ), and then for every λ ∈ J(A ) This implies that kABkop xC ⊗ D ± A ⊗ D 0, ≤ λ sup sup (2b − eBD|B ⊗ ID|Ψ) D,Ψ∈[[AD]]1 b∈[[B¯D]]¯ namely x ∈ J(A ⊗ D), and then J(A ) ⊆ J(A ⊗ D). Now, let y ∈ J(A ⊗ D). Then there exists = λkBkop. 0 0 C ∈ [[AC → BD]]1 such that yC ± A ⊗ D 0. Finally, taking the inﬁmum over λ ∈ J( ) we Composing the l.h.s. of the latter relation with A 00 get ψ ∈ [[C]]1 and eD, we obtain yC ± A 0, where 00 := (e | 0|ψ) is a channel, and then C D C kABkop ≤ kA ksupkBkop. y ∈ J(A ). Thus, J(A ⊗ D) ⊆ J(A ). Finally, since J(A ⊗D) = J(A ), we have kA ⊗ Dksup = Corollary 4. The sup-norm is stronger than the kA ksup. operational norm.

Corollary 2. For A ∈ [[A → B]]R, and for arbi- Proof. It is suﬃcient to observe that kA kop = trary C, it is kA ⊗ ICksup = kA ksup. kAI kop ≤ kA ksupkI kop = kA ksup.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 12 Lemma 5. Let A ∈ [[A → B]]1 be a channel. Proof. For k·kop, the equality trivially follows Then kA ksup = 1. from the definition. For k·ksup, it is a special case of lemma4. Proof. Clearly, A A −A , thus 1 ∈ J(A ), and inf J(A ) ≤ 1. On the other hand, suppose From lemma5, the following consequence fol- that there exists 1 > λ ∈ J(A ). This implies lows. that there exists a channel C ∈ [[A → B]]1 such Corollary 7. For the deterministic effect eA one that D := λC − A 0. However, this implies has keAksup = keAkop = 1. that (e|BD = −(1 − λ)(e|A 0. This is absurd, and then λ ∈ J(A ) must be λ ≥ 1. Thus, We now prove a result that will be very useful inf J(A ) ≥ 1. Finally, this implies kA ksup = later. 1. Lemma 7. Let A ∈ [[A → B]]+, and (a| := ¯ Corollary 5. The sup-norm of the identity chan- (eB|A ∈ [[A]]+. Then kA ksup = kaksup. nel ∈ [[A → A]] is 1. I 1 Proof. First of all, by proposition2, one has The sup-norm for effects is just the special case kaksup ≤ keAksupkA ksup = kA ksup. On the of the sup-norm of transformations with the out- other hand, let λ ∈ J(a). This implies that put system equal to I. b := λeA − a 0. Definition 7. Let a ∈ [[A]]¯ , and let us define R Let us then define B := |ρ)(b| ∈ [[A → B]]+, for the half-line some ρ ∈ [[B]]1. By theorem3, it is easy to check that A + B = λC for C ∈ [[A → B]]1. Then, J(a) := {λ ∈ R+ | λeA a −λeA}. λC − A = B 0, (9) The sup-norm kaksup is defined as which means that λ ∈ J(A ). Then, J(a) ⊆ kaksup := inf J(a). J(A ), and finally kA ksup ≤ kaksup. ¯ In the case of Classical and Quantum Theory, Proposition 4. The sup-norm on [[A]]R is well defined: phrasing the definitions of sup- and operational norm in terms of semidefinite programming prob- 1. kaksup is non-negative, and it is null iff a = lems, one can conclude that they are strongly 0, dual. As a consequence, in these special cases they define the same norm. 2. for µ ∈ R one has kµaksup = |µ|kaksup,

3. ka + bksup ≤ kaksup + kbksup. 4 The quasi-local algebra in OPTs

Proof. The case of effects is just a special case of Following the definition of Ref. [31], we de- the result of Proposition1. fine a cellular automaton in a general OPT as a triple (G, A, V ), where G is a denumerable As an immediate consequence of proposition3, set of labels for the systems that compose the we have the following result. automaton—addresses of the memory cells—, AG is a (possibly infinite) composite system corre- Corollary 6. The sup-norm is stronger than the sponding to the collection of systems A labelled operational norm on [[A]]¯ . g R by elements g ∈ G—i.e. the memory array—, and V is a reversible transformation on [[A¯ G]], such In the special case of [[I → I]]R = R, one has that · −1 is an automorphism of [[A → A ]]. [[I → I]]1 = {1}, and [[I → I]]+ = R+, thus J(x) = V V G G However, this is definition is incomplete, for many {λ ∈ R | λ±x ≥ 0}. Clearly, kxksup = inf J(x) = |x|. We now need the following lemma. reasons. In the first place, most of the objects mentioned above are not thoroughly defined. The ¯ Lemma 6. Let a ∈ [[A]]R. Then ka ⊗ eBkop = purpose of the present section is to set the ground kakop and ka ⊗ eBksup = kaksup. for the rigorous definition of a cellular automaton.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 13 We start fixing some notation. For every R ⊆ is particularly simple, thanks to causality, that G let provides us with a preferential local effect, the O unique deterministic one, without the need of in- AR := Ag, troducing any arbitrary choice of a reference local g∈R effect. Moreover, while the construction of the with the convention that A∅ := I. The full sys- space of quasi-local states is not logically nec- N tem is then AG = g∈G Ag. This purely formal essary, as we will find them as a subspace of notion will now be thoroughly substantiated. bounded functionals on quasi-local effects, the With a slight abuse of notation, we will often same is not true of effects. use R = g instead of R = {g}, dropping the If we defined effects as the dual space of quasi- braces. The set of finite regions of G will be de- local states, we would end up with far more effects noted as than needed, while lacking sectors of the state space, that are usually reached by the evolution of (G) R := {R ⊆ G | |R| < ∞}, a quasi-local state through a cellular automaton. These are the main reasons why our construction while the set of arbitrary regions of G will be begins with effects. denoted as The first notion we introduce is that of a lo- (G) R := {R ⊆ G}. cal effect. Let AG denote the formal composition of countably many systems from a general OPT: (G) (G) N Clearly R ⊆ R . In the remainder, when we AG := g∈G Ag. Intuitively speaking, a local ef- (G) write e.g. eR or IR for R ∈ R , we mean the fect of AG is an event within a test that discards deterministic effect or the identity transformation all the systems in G but the finite region R, where for the system AR, respectively. a non-trivial measurement is performed. We then N The following construction of g∈G Ag is in- define the local effect (a, R) as a pair made of an ¯ spired by that of Refs. [49, 50], however with sig- effect a ∈ [[AR]] and the region R. To lighten the nificant differences. notation, in the remainder of the paper we will use the symbol aR instead of (a, R). The set of 4.1 Quasi-local effects local effects is denoted by In this subsection we start the mathematical con- Pre[[A¯ G]]L struction of the system A , the parallel compo- G : G ¯ (G) ¯ sition of infinitely many finite systems. The first = [[AR]] = {aR | R ∈ R , a ∈ [[AR]]}. (G) object that we will define is the space of gener- R∈R alised effects, along with its convex cone of pos- The above definition can be widened encompass- itive effects, and the convex set of effects. The ing generalised effects: construction is based on the notion of a local effect, that is an effect which acts non trivially only ¯ G ¯ Pre[[AG]]LR := [[AR]]R. on finitely many systems labelled by g ∈ R, with R∈R(G) R ∈ R(G). We will say that such an effect acts on the finite region R. We will then introduce a Let us now consider a partition of the region R real vector space structure over the set of local into two disjoint regions S0 ∪ S1 = R. Since in effects, and a norm that is induced by the sup the definition of a local effect aR we did not set norm for local effects. The last step is then the ¯ constraints on the effect a ∈ [[AR]]R, it might be topological closure of the space of local effects in 0 0 ¯ that a = eS0 ⊗ a , with a ∈ [[AS1 ]]R. It is clear the sup-norm. The Banach space thus obtained from our intuitive notion of a local effect that aR is the space of generalised quasi-local effects, and and a0 should represent the same effect. This S1 the positive cone along with the convex set of ef- observation leads us to the equivalence relation fects contain the limits of sequences of elements of defined as follows. local cones or convex sets of effects, respectively. As we will see in the next section, the definition Definition 8 (Equivalent local effects). We say 0 ¯ of the space of quasi-local states is much more in- that the effects aR and aS in Pre[[AG]]LR are 0 volved than that of quasi-local effects. The latter equivalent, and denote this as aR ∼ aS, if there

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 14 ¯ exists a0 ∈ [[AR∩S]]R such that the following iden- Proof. As to existence, we remark that, by tities hold Eq. (11), Ra is the set of all g ∈ G such that for all regions S ∈ R one has g ∈ S. Thus, ( (a,R) a = a0 ⊗ e , S\R if h 6∈ Ra, there must exist S ∈ R(a,R) such that 0 (10) a = a0 ⊗ eR\S. h 6∈ S. This implies that i) by definition there exists fS ∼ aR, and ii) by lemma8 cS∪h ∼ aR, It is clear that the real notion of a local ef- with fect is captured by the equivalence classes modulo the above equivalence relation. We thus quotient c = f ⊗ eh. (12) ¯ Pre[[AG]]R and define the obtained set as the set of local effects of AG. As a consequence, if cT ∼ aR and h ∈ T , but h 6∈ Ra, by definition (10) cT must be of the ¯ Definition 9. A generalised local effect is an form of Eq. (12), for some f ∈ [[AT \h]]R. Clearly, (G) equivalence class [aR]∼. The set of generalised Ra ∈ R , since for any S ∈ R(a,R) one has Ra ⊆ local effects of AG is S. Now, let S ∈ R(a,R), and cS ∼ aR. One has 0 0 (G) S = Ra ∪ S , where S := (S \ Ra) ∈ R , thus [[A¯ G]]L := Pre[[A¯ G]]L / ∼ . 0 R R Ra ∩ S = ∅. By Eq. (12), we then have

With a slight abuse of notation, in the following 0 ¯ c = b ⊗ eS , b ∈ [[ARa ]]R. (13) we will write aR instead of [aR]∼, unless the context requires explicit distinction of the two sym- We now define a˜Ra := bRa , and finally, since bols. One can easily prove the following result Eq. 13 holds for any cS ∼ aR, one can easily verify that a˜ ∼ a . ¯ Ra R Lemma 8. Let aR ∈ Pre[[AG]]LR. Then for every As to uniqueness, we remark that the re- (G) finite region H ∈ R such that H ∩ R = ∅, gion R is uniquely defined. Now, suppose that ¯ a one has (a ⊗ eH )R∪H ∈ Pre[[AG]]LR, and (a ⊗ ¯ there were two different b, c ∈ [[ARa ]]R such that eH )R∪H ∼ aR. bRa , cRa ∼ aR. Then by Eq. (10) it must be b = c. The proof of the above lemma is straightforward, and we do not report it here. We now make local effects into a vector space. Let us now come back to our initial goal, that ¯ is to define a local effect as an event that acts Definition 11. Let aR, bS ∈ [[AG]]LR, and h ∈ R. non-trivially only on a finite region R. Intuition Then we define leads again to figure out what is the preferred ( representative of the class of a local effect: within (ha)R h 6= 0, haR := the equivalence class, it is the element defined on 0∅ h = 0, the smallest region. We now provide a formal aR + bS := cR∪S definition of such a minimal representative, and c := a ⊗ e + b ⊗ e . show that it is well posed. S\R R\S

Definition 10. The minimal representative of Notice that it is not always true that Rc = Ra∪ Rb. As an example, consider a = fg ⊗fg and b = the equivalence class aR, denoted as a˜R , is de- 1 2 a f ⊗ (e − f ), with f 6= e , 0 6= f 6= e , and fined through g1 g2 g1 g1 g2 g2 Ra = Rb = {g1, g2}. Then c = a + b = fg1 ⊗ eg2 , \ and clearly Rc = {g1}, which is strictly included Ra := S, a˜Ra ∼ aR, (11) S∈R in Ra = Rb = Ra ∪ Rb. (a,R) ¯ It is easy to check that [[AG]]LR is a real vector where R(a,R) is the set of all those finite regions space with null element given by the equivalence (G) ¯ class of 0 . S ∈ R for which there exists b ∈ [[AS]]R such ∅ that bS ∼ aR. We now equip the real vector space of local effects with a norm, and we then close it to ob- Lemma 9. The minimal representative exists tain the Banach space of quasi-local effects. The and is unique. definition is based on a norm for effects of finite

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 15 ¯ systems, as every local effect aR ∈ [[AG]]LR re- constant, and viceversa. In all the presently ¯ duces to the effect a˜ ∈ [[ARa ]]R. A natural norm known theories the latter condition is satisfied. one might think of is then the operational norm, The two above conditions in Eqs. (15) and (16) that induces the following definition. are discussed in detail in AppendixA, where we prove that condition (16) implies condition (15). Definition 12. The operational norm ka k of R op Our new norm is an order-unit norm. As we a ∈ [[A¯ ]] is defined by the following expres- R G LR will discuss in subsection 4.2, its dual coincides sion with the operational norm on quasi-local states.

kaRkop := ka˜kop, (14) Let us now see the definition of the sup-norm in detail. ¯ where a˜ ∈ [[ARa ]]R. Definition 13. The sup-norm ka k of a ∈ Unfortunately, if one completes the space of R sup R [[A¯ ]] is defined by the following expression local effects in the operational norm, in general G LR the dual norm on the space of bounded linear ka k := ka˜k , (17) functionals—which will be our state space—does R sup sup not coincide with the operational norm on the where a˜ ∈ [[A¯ ]] . state space, for those states that can be inter- Ra R preted as quasi-local preparations. We will then We now want to prove that the operational choose a different norm on our space of effects. norm and the sup-norm are well-defined norms The new norm will be referred to as sup-norm, as ¯ on [[AG]]LR. it is the extension of the sup-norm to the infinite case. Proposition 5. Let aR ∼ a˜Ra . Then kaRkop = As far as finite-dimensional systems are con- kakop, and kaRksup = kaksup. cerned, this choice does not represent a problem, 0 as all norms are equivalent in finite-dimensional Proof. Let aR ∼ a˜Ra , and R = Ra ∪ R , with 0 vector spaces. However, the sup-norm is stronger Ra ∩ R = ∅. Then by Eq. (10) we have than the operational one—see corollary4—, and thus the space that we construct, completing our a =a ˜ ⊗ eR0 , normed vector space of local effects with sup- norm Cauchy sequences, might contain distinct and thus by definitions 12 and 13 and lemma6, limits that are operationally equivalent. This it is point is a delicate one, and to avoid an unreason- ka k = ka˜k = ka˜ ⊗ e 0 k = kak , able construction where there exist different ef- R op op R op op fects that are operationally equivalent, we impose kaRksup = ka˜ksup = ka˜ ⊗ eR0 ksup = kaksup. a sufficient constraint for the operational norm and the sup-norm to be equivalent also for infi- We can then prove the desired result. nite systems: we restrict attention to those the- Proposition 6. The functionals k·k and k·k ories where the following property holds: there op sup are norms on [[A¯ ]] . exists a finite constant k such that for every sys- G LR ¯ tem A and every a ∈ [[A]]R ¯ Proof. 1. For every aR ∈ [[AG]]LR it is clear that ka k ≥ 0 and ka k ≥ 0. Now, ka˜k = kaksup ≤ kkakop. (15) R op R sup op ka˜ksup = 0 if and only if a˜ = 0, namely, re- This implies that not only the bound (15) holds minding definition 11, aR = 0∅. 2. For every for a fixed system A, but it holds with a fixed ¯ aR ∈ [[AG]]LR, by definitions7, 11, and 12, one constant independent of the system A. In turn, straightforwardly has that, for every µ ∈ R, a sufficient condition for (15) is that for every system A kµaRkop = kµa˜kop = |µ|ka˜kop = |µ|kaRkop, ∗ ¯ kµaRksup = kµa˜ksup = |µ|ka˜ksup = |µ|kaRksup. [[A]]+ ≡ [[A]]+, (16) namely every positive functional on the cone of 3. Let now cR∪S = a ⊗ eS\R + eR\S ⊗ b for ¯ ¯ states is proportional to an effect by a positive a ∈ [[AR]]R and b ∈ [[AS]]R, as from definition 11.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 16 ¯ Then, by lemma6, proposition5 and by the tri- the sup-norm on [[AG]]QR thus defined is indepen- angle inequality, we have dent of the specific sequence within an equiva- ¯ lence class. The space [[AG]]QR is by construction ka + b k = kc k = kck ¯ R S op R∪S op op a real Banach space, and [[AG]]LR can be identi- ≤ kakop + kbkop = kaRkop + kbSkop, fied with the dense submanifold containing constant sequences aR = [aR]. Clearly, in this case kaR + bSksup = kcR∪Sksup = kcksup kaksup = kaRksup. ≤ kaksup + kbksup = kaRksup + kbSksup. ¯ Definition 14. Let a ∈ [[AG]]QR. If there is a We remark that, in a theory that satisfies as- Cauchy sequence anR in the class defining a, sumption (16), one has k·k ≡ k·k on [[A]]¯ , n op sup R such that, for some n0 ∈ N, for every n ≥ n0 and thus k·kop = k·ksup on [[A¯ G]] (the proof can ¯ R one has añ ∈ [[ARn ]], then we call a a quasi-local be found in appendixA). In general, however, the effect. We denote the set of quasi-local effects by sup-norm is stronger than the operational norm, [[A¯ G]]Q. as we now prove. The cone [[A¯ G]]Q+ contains all elements that are proportional to an element in [[A¯ ]] by a pos- Lemma 10. Let a ∈ [[A¯ ]] . Then ka k ≤ G Q R G LR R op itive constant. kaRksup. The deterministic quasi-local effect eG is the Proof. Let us consider µ ∈ J(a). Then for every class eG := 1∅. ρ ∈ [[AR]]1 we have The cone [[A¯ G]]Q+ introduces a partial ordering in [[A¯ G]]Q , that we denote by (µeR ± a|ρ) ≥ 0, R a b ⇔ a − b ∈ [[A¯ ]] . i.e. |(a|ρ)| ≤ µ. Then, taking the supremum over G Q+ states on l.h.s. we obtain kaRkop ≤ µ, and finally, Quasi-local effects can be interpreted as effects taking the infimum over J(a) we obtain the the- that can be arbitrarily well approximated by local sis. observation procedures. The effect eG plays an important role, as it is the unique deterministic The space of local effects that we constructed effect in [[A¯ ]] . This will be proved shortly. Let so far is a normed real vector space. We now G Q us start providing the first important property of make it into a Banach space, the space of quasi- e . local effects, by the usual completion procedure G for normed spaces. We first introduce the space Lemma 11. Let a ∈ [[A¯ G]]Q. Then also eG − a ∈ [[A¯ ]] of Cauchy sequences G CR [[A¯ G]]Q. a : → [[A¯ ]] :: n 7→ a . Proof. Let a = [a ]. Let a0 := (e − a ) ∈ N G LR nRn nRn nRn n Rn ¯ 0 0 [[ARn ]]. Then one has (an − am)Rn∪Rm = (am − We then define the equivalence relation between 0 an)R ∪R , and thus a is a Cauchy sequence ∼ m n nRn Cauchy sequences in [[A¯ G]]C defined by a = b 0 R whose class we call a ∈ [[A¯ G]]Q. Finally, since iff lim ka − b k = 0. Finally, we de- n→∞ nRn nSn sup a+a0 is the class [a ]+[a0 ] = [(a +a0 ) ] = nRn nRn n n Rn fine the space [[A¯ G]]Q of generalised quasi-local 0 R [1∅] = eG, we have that a + a = eG, namely effects, by taking the quotient 0 eG − a = a ∈ [[A¯ G]]Q. ¯ ¯ ∼ [[AG]]QR := [[AG]]CR/ = . (18) By the above lemma we know that not only eG a for every quasi-local effect a, but also that The elements of this space will be denoted by every quasi-local effect a can be complemented a = [anR ]. In this context, the class of a con- 0 0 n by a quasi-local effect a such that a + a = eG. stant Cauchy sequence (Rn = R and anR = aR) n In other words (a, eG − a) is a binary quasi-local will be denoted by a := [a ]. Clearly, since R nRn observation test. |ka k − ka k | ≤ ka − a k , nRn sup mRm sup nRn mRm sup We can also prove the converse result. for a Cauchy sequence a = [anRn ] also kanRn ksup is a Cauchy sequence, and we define kaksup := Lemma 12. Let a ∈ [[A¯ G]]Q+, and eG − a ∈ ¯ ¯ limn→∞ kanRn ksup. One can easily verify that [[AG]]Q+. Then a, eG − a ∈ [[AG]]Q.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 17 Proof. Indeed, the statement is true for finite Proof. In the case of local effects a = aR the systems thanks to the no-restriction hypothesis statement is a trivial recasting of the definition (see theorem5), and thus it holds for a given of sup-norm. Let us then consider the class a ∈ ¯ Cauchy sequence in the class of a, namely if [[AG]]QR with a = [anRn ]. One has, by definition, a0 := (e − a ) ∈ [[A¯ ]] for all n, then kak = lim ka k . Let us consider the nRn n Rn Rn + sup n→∞ n sup 0 ¯ ¯ sequence µ := ka k + ε ∈ J(a ). Clearly, an, an ∈ [[ARn ]]. Thus, a = limn→∞ an ∈ [[AG]]Q, n n sup n 0 0 ¯ and a = limn→∞ an = eG − a ∈ [[AG]]Q. limn→∞ µn = kaksup + ε. Moreover, since

k{(µn − µm)e ± (an − am)}R ∪R ksup We now prove that eG lies in the interior of n m [[A¯ G]]Q+, namely there is a ball of radius r around ≤ |kamksup − kanksup| + kam − anksup ≤ 2ε, eG in sup-norm that is contained in [[A¯ G]]Q+. the sequence {(µn + ε)e ± an}Rn converges to (kak +ε)e ±a ∈ [[A¯ ]] , thus for every ε ≥ 0 Lemma 13. There exists an open ball Br(eG) sup G G Q+ ¯ one has kaksup + ε ∈ J(a). This implies that of radius r in [[AG]]QR that is fully contained in [[A¯ G]]Q+. inf J(a) ≤ kaksup. Proof. Let ka − e k < r < 1/2. Then, by G sup On the other hand, let µ ∈ J(a), then deﬁnition, there exist a sequence anRn such that lim a = a, and n ∈ such that for n ≥ n n→∞ n 0 N 0 µeG ± a 0. kan − eGksup ≤ kan − aksup + ka − eGksup < 2r. By corollary8, we have that, for ε > 0,

This implies that (ε + µ)eG ± a ∈ Bεr(εeG + µeG ± a) ⊆ [[A¯ G]]Q+.

Now, this implies that there are two open balls, [2re + (an − e)]Rn = [an − (1 − 2r)e]Rn 0. centred at (µ + ε)eG ± a, that are fully contained ¯ Now, this implies that for n ≥ n0 it is an in [[AG]]Q+. There exists then n0 ∈ N such that ¯ for n ≥ n0 one has (1 − 2r)eRn 0, and thus an ∈ [[AG]]L+. Finally, by definition, one obtains a ∈ [[A¯ G]]Q+. Thus, the (µ + ε)eG ± an 0, ball Br(eG) is contained in [[A¯ G]]Q+. namely (µ + ε) ∈ J(a ). Thus, J(a) ⊆ J(a ) − ε, ¯ n n Corollary 8. Let a ∈ [[AG]]Q+. Then for ε > 0, and Bεr(εeG + a) ⊆ [[A¯ G]]Q+ for some r > 0. inf J(a) ≥ kanksup − ε. (19) Proof. Let b ∈ Bεr(εeG + a). Then Finally, this implies that inf J(a) ≥ kaksup. kb − a − εeGksup < εr As the cone [[A¯ ]] is closed, we can easily ⇔ k(b − a)/ε − e k < r. G Q+ G sup prove that the infimum in the expression of the sup norm is actually a minimum. Then (b − a)/ε = c ∈ [[A¯ G]]Q+, ad thus ¯ Lemma 15. Let a ∈ [[AG]]QR. Then b = εc + a ∈ [[A¯ G]]Q+. kaksupeG ± a 0. (20) We now prove that the deterministic effect eG allows for an extension of the defining property Proof. By lemma 14, for every ε > 0 one has of the sup-norm. (kaksup + ε)eG ± a 0.

Lemma 14. The sup-norm of a ∈ [[A¯ G]]Q can ± R The sequences b := (kaksup+1/n)eG±a are both be expressed as n Cauchy, and their limits are both in [[A¯ G]]Q+, then kaksup = inf J(a),

J(a) := {µ ≥ 0 | µeG a −µeG}. kaksupeG ± a 0.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 18 Before concluding, we remark that the opera- aR = [aR] = [(aR ⊗ eH )] = aR ⊗ eH , which is ¯ tional norm can be extended to [[AG]]QR, by sim- the meaning of Eq. (22). ply defining for a = [anRn ] As a final remark, we observe that for every denumerable set G, and every subset R ⊆ G, in- (G) kakop := lim kanRn kop. ¯ n→∞ cluding infinite ones, one can define [[AR ]]QR as the closed subspace spanned by those quasi-local Indeed, since effects a = anRn such that, for every n ∈ N, Rn ⊆ N R. If one constructs the system AR := Ag, |kanR kop − kamR kop| ≤ kanR − amR kop g∈R n m n m it is straightforward to construct an ordered Ba- ≤ ka − a k , nRn mRm sup nach space isomorphism the sequence ka k is Cauchy, and the limit nRn op † ¯ (G) ¯ ¯ JR : [[AR ]]QR → [[AR]]QR :: [anRn ]G 7→ [anRn ]R, is well defined. Moreover, also on [[AG]]QR the sup-norm is stronger than the operational (25) norm, as can be straightforwardly concluded from with the norm coinciding with the sup-norm in lemma 10. However, in theories where there ex- † ¯ both spaces. The left-inverse of JR is ists k > 0 such that k·ksup ≤ kk·kop in [[A]]R for every finite system A, the same inequality holds −1† (G) J : [[A¯ R]] → [[A¯ ]] . (26) in the limit, and the operational and sup-norm R QR R QR are equivalent. In particular, this is the case un- † and −1† can be diagrammatically denoted der assumption (16). JR JR as We now introduce a diagrammatic notation for ¯ R R effects in [[A]]QR that will make part of the subse- a † = R a , quent proofs and arguments more intuitive. First JR G\R ¯ e of all, we denote a quasi-local effect a ∈ [[AG]]QR by the symbol (27) R a G G −1† R a . (21) J a = G\R . (28) R e ¯ In the case of a local effect aR ∈ [[AG]]LR, we will draw 4.2 Extended states R Now that we defined the space of quasi-local ef- G a aR = G\R . (22) fects, we can define the space of states as the e ¯ space of bounded linear functionals on [[AG]]QR. The set of states is then defined considering those Notice that, since for (a ⊗ eR )R with a ∈ [[A¯ R ]] 1 0 linear functionals that, acting on local effects of and R1 := R \ R0, it is (a ⊗ eR )R ∼ aR , we have 1 0 an arbitrary finite region R, behave as a state in R0 a [[AR]]. R0 a R1 e = . (23) (G\R)∪R1 Definition 15 (Generalised extended states). G\R e e The space [[AG]]R of generalised extended states of A is the topological dual of [[A¯ ]] , i.e. the From the above identity, we can intuitively con- G G QR Banach space [[A¯ ]]∗ of bounded linear function- clude that G QR ¯ als on [[AG]]QR, equipped with the norm R1 (G\R)∪R1 e e = G\R . (24) e kρk∗ := sup |(a|ρ)|. (29) kaksup=1 Indeed, let G = R ∪ H, with |R| < ∞. Then The operational norm on [[AG]]R, defined as [eR] = [1∅] = eG, which is the equation represented by the diagram in Eq. 24. Similarly, let kρkop := sup (2a − eG|ρ), (30) aR ⊗ bH := limn→∞(a ⊗ bn), where [bnSn ] = b ∈ a∈[[A¯ ]] ¯ G Q [[AH ]]QR. By corollary3, one has [(a⊗bn)R∪Sn ] = (aR ⊗ bH ). Thus, in G = R ∪ H, one has coincides with the norm k·k∗, as we now prove.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 19 ˆ−1 Lemma 16. Let ρ ∈ [[AG]]R. Then Deﬁning JS by duality as

−1† kρk∗ = sup (2a − eG|ρ). (31) (a|Jˆ−1ρ) := (J a|ρ), (35) ¯ S S a∈[[AG]]Q we can then equivalently express Eq. (34) as Proof. Let 0 ≤ ε ≤ kρkop. We then have ˆ−1 ρ|S = J ρ. (36) 0 ≤ kρkop − ε ≤ (2a − eG|ρ), S Representing generalised extended states through for some a ∈ [[A¯ ]] . Invoking lemma 11, one has G Q diagrams, and reminding eq. (28), the above

eG + (2a − eG) = 2a 0, equations can be recast as

eG − (2a − eG) = 2(eG − a) 0, S R S ρ|S ρ −1 a = JS a and, by lemma 14, k2a − eGksup ≤ 1. Then it is

S (2a − eG|ρ) a kρkop − ε ≤ ≤ kρk∗. = ρ G\S , k2a − eGksup e ¯ On the other hand, let us now pick a ∈ [[AG]]R which can be taken as the definition of the equa- with kaksup = 1. Then, by lemma 15, eG ±a 0. tion 1 If we define a± := 2 (eG ± a), we have a± 0, S and ρ S |S := ρ G\S . (37) e a+ + a− = eG, (a+ − a−) = a, (32) We can now introduce the set of states [[AG]] ¯ which by lemma 12 implies that a± ∈ [[AG]]Q. as the special set of generalised extended states Then whose restrictions are all states. In other words, an element ρ of the space [[A ]] is a state if, ±(a|ρ) = (2a − e |ρ). G R ± G restricted to any finite region R ∈ R(G), it defines a state for that region. Thus, |(a|ρ)| ≤ kρkop, for every ε > 0, and taking the supremum on the l.h.s. we obtain kρk∗ ≤ Definition 18 (Extended states). The set [[AG]] kρkop. of states in the space of generalised extended Technically speaking, the two norms k·k and states [[AG]]R is the set of those elements ρ ∈ sup (G) [[AG]] such that for every R ∈ R the local re- k·k∗ are a base and order-unit norm pair [51]. R What is missing now is the notion of a convex striction of ρ to R is a state ρ|R ∈ [[AR]]. Deter- set of proper states, the extended preparations of ministic states, whose set is denoted by [[AG]]1, are those states ρ ∈ [[AG]] with (eG|ρ) = 1. our infinite system AG. Let us then give the following definition. One can easily verify that the set [[AG]] is con- Definition 16 (Local restriction). Given a state vex, as a consequence of convexity of [[A]] for ev- (G) ery finite system A. As we did for finite systems, ρ ∈ [[AG]]R, the local restriction of ρ to S ∈ R we also define a positive cone generated by [[AG]]. is the functional ρ|S ∈ [[AS]]R defined through Moreover, it is easy to check that the restriciton ¯ (G) (a|ρ|S) := (aS|ρ), ∀a ∈ [[AS]]. (33) of a state to an infinite region S ∈ R is s state in [[AS]]. The notion of a restriction can be brought further, considering infinite regions, by invoking the Definition 19 (Extended positive cone). The set isomorphism JR defined in Eq. (25), as follows. [[AG]]+ in the space of generalised extended states [[A ]] is the cone of those elements ρ ∈ [[A ]] Definition 17 (Restriction). Given a state ρ ∈ G R G R (G) such that there exist ρ¯ ∈ [[AG]] and µ ≥ 0 such [[AG]]R, the restriction of ρ to S ∈ R is the that ρ = µρ¯. functional ρ|S ∈ [[AS]]R defined through We can now show an important result about −1† ¯ (a|ρ|S) := (JS a|ρ), ∀a ∈ [[AS]]. (34) the restriction of states.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 20 ˆ−1 Lemma 17. The restriction JR [[AG]] of the set On the other hand, being eG a legitimate el- of states of AG coincides with [[AR]]. ement of [[A¯ G]]Q with keGksup = 1, one has kρk∗ = sup |(a|ρ)| ≥ (eG|ρ). Proof. It is a straightforward exercise to verify kaksup=1 ˆ−1 that JR [[AG]] ⊆ [[AR]]. For the converse, let Thanks to lemma 12, we have the following ρ ∈ [[AR]], and let us now construct a state σ ∈ corollary. [[A ]] such that σ = ρ. We define R¯ := G \ R. G |R Corollary 9. Let ρ ∈ [[A ]]. Then 0 ≤ (a|ρ) ≤ There are two possible situations: R¯ ∈ R(G), or G (e |ρ) for every a ∈ [[A¯ ]] . ¯ (G) ¯ (G) ¯ G G Q R ∈ R . If R ∈ R , let aT ∈ [[AG]]LR, with T ∈ R(G). We define We can now show that also in the infinite case every state is proportional to a deterministic one. (a|σ) := (a|ρ|T ∩R ⊗ ν|T ∩R¯), (38) Let us first prove a preliminary lemma. Lemma 20. Let ρ ∈ [[A ]] . If (a|ρ) = 0 for for an arbitrary ν ∈ [[AR¯]], where we introduced G R 0 (G) ¯ the notation σ|S0 for S ⊆ S ∈ R , and σ a state every a ∈ [[AG]]Q, then ρ = 0. of the composite system σ ∈ [[A ]] = [[A 0 A 0 ]], S S S\S Proof. If (a|ρ) = 0 for every a ∈ [[A¯ ]] then, with G Q using lemma 11, we have S0 0 σ 0 S ¯ |S := σ S\S0 . (2a − eG|ρ) = (a|ρ) − (eG − a|ρ) = 0, ∀a ∈ [[AG]]Q. e This implies that kρk∗ = 0, and thus ρ = 0. Clearly, the functional σ is bounded, and thus it can be extended to a functional on the full Proposition 7. Every state ρ ∈ [[AG]] is propor- ¯ space [[AG]]QR, i.e. σ ∈ [[AG]]R. Moreover one can tional to a deterministic state ρ¯ ∈ [[AG]]1. (G) easily verify that σ|T ∈ [[AT ]] for every T ∈ R , Proof. Let ρ ∈ [[AG]]. If ρ = 0, then for every thus σ is the desired state in [[AG]]. A similar construction can be carried out for the case R¯ ∈ ρ¯ ∈ [[AG]]1 it is ρ = 0¯ρ. Let then ρ 6= 0. By (G) lemmas 18 and 20, there must exist a ∈ [[A¯ G]] R , taking ν ∈ [[A ¯]]. By construction, also in R such that (a|ρ) > 0. By corollary9, one then this case σ ∈ [[AG]], as can be straightforwardly checked. has (eG|ρ) ≥ (a|ρ) > 0. By definition, for every S ∈ R(G), we have The next result that we prove is that quasi- (eA |ρ|S) = (eS|ρ) = (eG|ρ) > 0. local effects a ∈ [[A¯ G]]Q are positive on the set S [[AG]]. (G) Thus, if we set ρ¯ := ρ/(eG|ρ), for every S ∈ R Lemma 18. Let a ∈ [[A¯ G]]Q, and ρ ∈ [[AG]]. we obtain Then (a|ρ) ≥ 0 (eAS |ρ¯|S) = (eG|ρ¯) = 1, Proof. Let a ∈ [[A¯ G]]Q be a local effect aR = (G) implying that ρ¯ ∈ [[AS]]1 for every S ∈ R . [aR]. Then, by definition, (aR|ρ) = (a|ρ|R) ≥ |S Then, ρ¯ ∈ [[AG]]1, and ρ¯ is such that ρ = (eG|ρ)¯ρ. 0. Now, let a = [anRn ]. We have (a|ρ) = limn→∞(anRn |ρ) ≥ 0. The following result draws a first analogy be- We now show that the operational norm is ¯ tween the properties of the sup-norm for finite equivalently defined on [[AG]]QR by systems and that for infinite parallel composi- kakop := sup |(a|ρ)|. tions. ρ∈[[AG]]

Lemma 19. Let ρ ∈ [[AG]]+. Then kρk∗ = Indeed, let a = [anRn ]. Then (eG|ρ). |(a|ρ)| = lim |(anR |ρ)| = lim |(an|ρ )|. n→∞ n n→∞ |Rn Proof. Let ρ ∈ [[AG]]+. Then, by virtue of lemma 18, (a|ρ) ≥ 0 for all a ∈ [[A¯ G]]. Let now This implies that kaksup = 1. By lemma 15, one has eG ± a 0, |(a|ρ)| ≤ |(an|ρ )| + ε ≤ kanR kop + ε, and thus (eG|ρ) ≥ |(a|ρ)|. Then (eG|ρ) ≥ kρk∗. |Rn n

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 21 thus sup |(a|ρ)| ≤ kak . On the other Proposition 8. Let a ∈ [[A¯ ]] . Then (a|ρ) = 1 ρ∈[[AG]] op G Q hand, for every anRn and every ε, there exists for all ρ ∈ [[AG]]1 if and only if a = eG.

ρε ∈ [[ARn ]] such that kanRn kop ≤ |(an|ρε)| + ε. Now, by lemma 17, this implies that for every ε Proof. Notice that, since eG a for every a ∈ [[A¯ ]] , and (b|ρ) ≥ 0 for every b ∈ [[A¯ ]] and there exists ρ˜ε ∈ [[AG]] such that G Q G Q ρ ∈ [[AG]]1, we have

≤ |(anRn |ρ˜ε)| − ε (eG − a|ρ) = 0, ∀ρ ∈ [[AG]]1, ≤ |(a|ρ˜ε)|. (39) which by theorem6 implies a = eG. Thus, in conclusion, kakop ≤ supρ∈[[A ]] |(a|ρ)|. G A class of states of particular interest is that A crucial property of [[AG]] is that extended of quasi-local states, which are intuitively under- states are separating for [[A¯ G]]Q, namely if for every state two generalised effects give the same stood as states whose preparation can be arbi- value, then they coincide. trarily approximated by local procedures. These live in small subspaces of the space [[AG]]R. Their Theorem 6 (States separate effects). Let a ∈ construction is similar to that of quasi-local ef- ¯ [[AG]]QR. If (a|ρ) = 0 for all ρ ∈ [[AG]]1, then fects, but differs form it in a relevant respect, a = 0. that is the necessity of defining arbitrary reference states. The construction is inspired by pio- Proof. Let a = [an ] ∈ [[A¯ G]]Q , and (a|ρ) = 0 Rn R neering works on infinite tensor products by von for every ρ ∈ [[AG]]1. Then for every ρ ∈ [[AG]]1 one has Neumann and Murray [49], and can be found in appendixB. |(an|ρ)| = |(an − a|ρ)| ≤ kan − aksup, 4.3 Quasi-local transformations and thus for every ε > 0 there exists n0 such that, for n ≥ n0, kankop ≤ ε. Thus, we have Now we are going to define the quasi-local alge- kakop = 0. As a consequence, for every ε > 0 bra [[AG → AG]]QR of transformations. In usual there exists n0 such that for n ≥ n0 one has approaches to quantum infinite systems, one usually introduces the effect algebra, that in the OPT kanksup ≤ kkankop ≤ ε, case would correspond to the space of quasi-local effects. However, one can easily understand that, and thus kaksup = 0. Now, this implies that a = 0. unlike effects in a general OPT, the quantum effect algebra contains far more information than Remark 2. The above result is made possible the mere structure of the space of effects. Indeed, by requirement (15) that the sup-norm and the in the quantum effect algebra one can find every operational norm for effects of finite systems A operator on the Hilbert space, and in turn, the are bounded as operational role of a linear operator is to provide a Kraus representation of a transformation. In this ¯ ∀a ∈ [[A]]R kaksup ≤ kkakop, perspective, one can also understand why multiplication of effects has a meaning at all: there with k independent of the system A. In par- is no point in multiplying effects, but multiplica- ticular, this is true under the assumption (16), tion of Kraus operators is the mathematical rep- i.e. that [[A]]¯ ∗ ≡ [[A]] . + + resentation of sequential composition. Thus, the We can now prove the main consequence of algebraic structure of effects conveniently sum- uniqueness of the deterministic effect eG for AG, marises information about every kind of event in i.e. that eG is the unique effect that amounts to the theory: effects, with their coarse-graining rep- 1 on [[AG]]1. resented by the sum, and transformations, with

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 22 sequential composition represented by multiplica- We can now define an equivalence relation be- tion. tween local transformations as follows Cellular automata are defined in quantum the- ( ory by specifying their action on the effect alge- A = C ⊗ IS\T , AS ∼ BT ⇔ (41) bra. Such an action thus provides information B = C ⊗ IT \S, about how both effects and transformations are transformed by the cellular automaton. For a for some C ∈ [[AS∩T ]]R. The set of local transfor- general OPT, however, such a compact algebraic mations is then defined as structure embodying all the relevant information [[A → A ]] := Pre[[A → A ]] / ∼ is absent, and the definition of a CA on the space G G LR G G LR of effects is not sufficient: we need to specify how Lemma 22. Let AR ∈ Pre[[AG → AG]] . Then the CA transforms transformations. LR for every finite region H ∈ R(G) such that H ∩ For these reasons we construct now the Banach R = ∅, one has (A ⊗ I )R∪H ∈ Pre[[AG → algebra of quasi-local transformations, in a way AH AG]] , and (A ⊗ I )R∪H ∼ AR. that is very closely reminiscent of the construc- LR AH tion of quasi-local effects. Here, in order to define Proof. It is straightforward to verify that (A ⊗ a local transformation, we recur to the notion of a IAH )R∪H ∼ AR by direct inspection of the defin- transformation that acts non-trivially on finitely ing equation (41). many systems, while it acts as the identity on the remaining ones. In other words, the role that is We can now provide a way to identify a canon- played by the deterministic effect in the case of ical representative of the equivalence class of AR, local effects is played here by the identity trans- which is defined in analogy to the case of effects formation. Also in this case, the topological clo- as follows. sure will be taken in the sup-norm, and this choice is of great relevance to ensure a Banach algebra Definition 21. The minimal representative of the equivalence class , denoted as ˜ , is de- structure for the closure. AR ARA Let us then introduce the Banach algebra of fined through quasi-local transformations. : \ ˜ RA = S, ARA ∼ AR, (42) Definition 20. Let R ⊆ G be an arbitrary finite S∈R(A ,R) region of G. We define local transformation AR where R is the set of all those finite regions of AG any pair (A ,R), where A ∈ [[AR → AR]]R. (A ,R) † ¯ S ⊆ G for which there exists B ∈ [[AS → AS]] The action AR of AR on Pre[[AG]]LR is defined as R follows such that BS ∼ AR. R\S R\S R\S e Lemma 23. The minimal representative exists AR R∩S † := R∩S R∩S . (40) and is unique. ARaS S\R S\R aS The proof follows step by step that of lemma 73. We also define Pre[[A → A ]] as the set of G G LR The first result that we need to prove is the local transformations AR. following. † † Lemma 21. Let aS ∼ bT . Then ARaS ∼ ARbT . Lemma 24. Let AR ∼ BS. Then for every ef- Proof. By hypothesis, there exists c ∈ [[A¯ S∩T ]] R fect aT ∈ Pre[[A¯ G]]L , one has such that R † † A aT ∼ B aT . (43) a = c ⊗ eS\T , b = c ⊗ eT \S. R S Then we have ˜ Proof. If (C )RC is the minimal representative of † AR ∼ BS, by Eq. (41), one has (ARaS| = (c ⊗ eS\T ⊗ eR\S|(A ⊗ IS\R), † = ˜⊗ , (ARbT | = (c ⊗ eT \S ⊗ eR\T |(A ⊗ IT \R). A C IR\RC = ˜⊗ . By Eq. (10) the thesis follows. B C IS\RC

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 23 Now, by the defining equation 40, we have that In order to close the algebra of local operations we introduce the topology given by the sup-norm, ( †a | = (ã ⊗ e |( ˜⊗ ) AR T (T ∪R)\Ra C I(T ∪R)\RC given in the following, in analogy to the case of ˜ effects. We also discuss the interplay of the sup- =(ã ⊗ e[T ∪(R∩S)]\R |(C ⊗ I[T ∪(R∩S)]\R ) a C norm topology with that given by the operational ⊗ (e |, R\S norm, that we define right away. ( † a | = (ã ⊗ e |( ˜⊗ ) BS T (T ∪S)\Ra C I(T ∪S)\RC Definition 25. The operational norm kARkop =(ã ⊗ e |( ˜⊗ ) [T ∪(R∩S)]\Ra C I[T ∪(R∩S)]\RC ˜ of ARA = AR ∈ [[A → A]]LR is defined by the ⊗ (eS\R|, following expression ˜ and finally by Eq. (123) this implies the thesis. kARkop := kA kop. (44)

Proposition 9. Let AR ∈ [[A → A]]LR. Then k k = k k . By virtue of lemmas 21 and 24, we can define AR op A op the action of AR ∈ [[AG → AG]]L on local effects ˜ 0 R Proof. Let AR ∼ ARA , and let R = RA ∪ R , [[A¯ ]] as 0 G R with RA ∩ R = ∅. Then by Eq. (41) we have † † = ˜⊗ , AR[aT ] := [ARaT ], A A IAR0 where we leave the square braces to denote equiv- Now, by definition of k·kop (see [8]) it straightfor- alence classes for ∼ in Pre[[A¯ ]] , for the sake of wardly follows that kB ⊗ ICkop = kBkop. Thus, G LR ˜ clarity. We can now make local transformations kARkop = kA kop = kA kop. into a vector space as we did for local effects, as Unfortunately, the operational norm does not follows. enjoy the basic property that would make the algebra of transformations into a Banach algebra, Definition 22. Let AR, BS ∈ [[AG → AG]] , LR i.e. it is not true that k k ≤ k k k k . and h ∈ . Then we define AB op A op B op R For this reason, in analogy with the case of quasi- ( local effects, we introduce a second norm, the sup- (hA )R h 6= 0, hAR := norm, whose interplay with the operational norm 0 h = 0, ∅ will make it possible to define the closed Banach AR + BS := CR∪S algebra of quasi-local transformations.

C := A ⊗ IS\R + B ⊗ IR\S. This is obtained extending the definition of sup-norm to the algebra of local transformations Moreover, the following operation makes the of AG. set of local transformations into an algebra. Definition 26. The sup-norm kARksup of AR ∈ [[AG → AG]]LR is defined as Definition 23. Let AR, BS ∈ [[AG → AG]]LR. Then we define ˜ ˜ kARksup := inf J(A ) = kA ksup. (45)

ARBS := ({A ⊗ IS\R}{B ⊗ IR\S})R∪S. Proposition 10. Let AR ∈ [[AG → AG]]LR. Then kARksup = kA ksup. ˜ 0 Proof. Let AR ∼ AR , and let R = R ∪ R , We omit the straightforward proof that the A A with R ∩ R0 = ∅. Then by Eq. (41) we have above definition is well defined, i.e. independent A of the choice of representatives in the classes of = ˜⊗ , A A IAR0 AR and BS. ˜ and by corollary2, kA ksup = kA ksup = Definition 24. The algebra of generalized lo- kARksup. cal transformations is the unital algebra [[AG → AG]]LR of finite real combinations of local trans- Corollary 10. Let A , B ∈ [[AG → AG]]LR. formations, with unit 1∅ and null element 0∅. Then kABksup ≤ kA ksupkBksup.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 24 Proof. The result is a straightforward conse- 1. AnRn BnSn is a Cauchy sequence. quence of propositions 10 and2. 0 0 2. For every A 0 , B 0 in the equivalence nRn nSn classes deﬁning and , respectively, one Corollary 11. On [[AG → AG]]LR, one has A B ∼ 0 0 has An Bn = A 0 B 0 . kARkop ≤ kARksup. Rn Sn nRn nSn Proof. The result is a straightforward conse- Item1 can be easily proved since we have quence of propositions9 and 10, and corol- kAnBn − AmBmksup lary4. =kAnBn − AmBn + AmBn − AmBmksup

The above results allow us to extend the lo- ≤kAn − AmksupkBnksup cal algebra of events into the quasi-local alge- + kAmksupkBn − Bmksup. bra, which is a Banach algebra. The construction is analogous to that of quasi-local states, and Similarly, for item2 we have proceeds as follows. First we define the algebra 0 0 kAnBn − A B ksup [[AG → AG]]C of sup-Cauchy sequences of local n n R 0 0 0 0 transformations, i.e. its elements are sequences =kAnBn − AnBn + AnBn − AnBnksup 0 A : N → [[AG → AG]]LR :: n 7→ AnRn such that ≤kAn − AnksupkBnksup for every ε > 0 there exists n0 ∈ N such that 0 0 + kAnksupkBn − Bnksup. for all m, n ≥ n0, one has kAn − Amksup < ε. Now we define the equivalence relation between The quasi-local algebra is actually a real Ba- ∼ elements of [[AG → AG]]CR: A = B if for every nach algebra if equipped with the norm k·ksup, as ε > 0 there exists n0 ∈ N such that for every we now prove. n ≥ n0 one has kAn − Bnksup < ε. Finally, we Proposition 11. The algebra [[A → A ]] define the quasi-local algebra as follows. G G QR equipped with the norm k·ksup is a Banach al- Definition 27. The space of quasi-local trans- gebra. formations of A is defined as Proof. We only need to prove that kABksup ≤

∼ kA ksupkBksup. Let A = [AnRn ] and B = [[AG → AG]]QR := [[AG → AG]]CR/ = . [BnSn ], respectively. Now, for every AnRn and BnS one has Definition 28. An element A in [[AG → AG]]QR n is an event if A = [AnRn ] for a sequence AnRn k(An ⊗ ISn\Rn )(Bn ⊗ IRn\Sn )ksup such that An ∈ [[ARn → ARn ]] for all n ∈ N. The ≤ k ⊗ k k ⊗ k , set of events will be denoted by [[AG → AG]]Q. An ISn\Rn sup Bn IRn\Sn sup An element A ∈ [[AG → AG]]Q is a channel and by proposition 10 kABksup ≤ if A = [AnR ] for a sequence AnR such that n n kA ksupkBksup. An ∈ [[ARn → ARn ]]1 for all n ∈ N. The set of channels will be denoted by [[AG → AG]]Q1. The subsets [[AG → AG]]Q and [[AG → AG]]Q1 The subset of [[AG → AG]]QR containing ele- are convex, and [[AG → AG]]Q+ is a convex cone. ments A = [AnRn ] for a sequence AnRn such Moreover, they are closed in the sup-norm. that An ∈ [[ARn → ARn ]]+ for all n ∈ N will be Proposition 12. The sets [[AG → AG]]Q, denoted by [[AG → AG]]Q+. Equivalently, we will [[AG → AG]]Q1, and [[AG → AG]]Q+ are closed write A 0 for A ∈ [[AG → AG]]Q+ in the sup-norm. In the following, in analogy with the finite case, Proof. The argument is the same in both cases. we will write A B if A − B 0. The algebra Let {Am}m∈N be a Cauchy sequence with of quasi-local transformations [[AG → AG]]QR is Am ∈ [[AG → AG]]Q∗ for all m, with ∗ = defined as follows. “nothing”, 1, +. By definition, Am = [AmnRmn ],

with Amn ∈ [[ARmn → ARmn ]]∗. Then one eas- Definition 29. Let A = [AnRn ], B = [BnSn ]. ily proves that the sequence {AmmRmm }m∈N is We define AB := [AnRn BnSn ]. Cauchy, and its limit is A := limm→∞ AmmRmm , The product AB is well defined. Indeed: which by definition is in [[AG → AG]]Q∗.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 25 Proposition 13. Let A ∈ [[AG → AG]]QR. Then Proof. By proposition2 and lemma 25, the thesis ¯ is true for A ∈ [[AG → AG]]LR and a ∈ [[AG]]LR. kA kop ≤ kA ksup. (46) Let now a = [apRp ] and A ∈ [[AG → AG]]LR. Then Proof. By deﬁnition kA kop = limn→∞ kAnkop † and kA ksup = limn→∞ kAnksup, where kA apksup ≤ kapksupkA ksup, {AnRn }n∈N ⊆ [[AG → AG]]LR is a Cauchy sequence converging to A , thus by proposition3 and taking the limit for p → ∞ we obtain the one has thesis. Finally, let A = [AnRn ]. In this case, † ¯ by lemma 25 An a ∈ [[AG]]QR for every n ∈ N. k k ≤ k k , ∀n ∈ , † ¯ An op An sup N Moreover, {An a}n∈N ∈ [[AG]]CR. Indeed, by the result we just proved which implies the inequality in the limit. † † k(A − A )aksup ≤ kaksupkAn − Amksup. Now, we can prove the following lemma. n m ¯ Finally, by lemma 25, we have Lemma 25. Let a ∈ [[AG]]CR be a Cauchy sequence, and let A ∈ [[AG → AG]]L . We then † R kAn aksup ≤ kaksupkAnksup, have and taking the limit for n → ∞ we obtain the † ¯ A a ∈ [[AG]]CR. (47) thesis.

Proof. We just need to evaluate A remarkable result that will be important in † † kA am − A anksup, and using proposition2 one the following is given by the following lemma. has Lemma 27. Let A ∈ [[AG → AG]]Q. Then for † ¯ † ¯ kA (am − an)ksup ≤ kam − anksupkA ksup. every a ∈ [[AG]]Q one has A a ∈ [[AG]]Q.

† Proof. By hypothesis, = [ ] and a = Since the sequence a is Cauchy, also A a is A AnRn [a ], with ∈ [[A → A ]] and a ∈ Cauchy. mSm An Rn Rn m ¯ † ¯ [[ASm ]]. Thus An an ∈ [[ARn∪Sn ]]. Now, one can ¯ † † Lemma 26. Let a, b ∈ [[AG]]CR be equivalent easily verify that limn→∞ An an = A a, which Cauchy sequences, and let A ∈ [[AG → AG]]LR. proves the statement. We then have In particular, one has the following condition. † † [A anRn ] = [A bnSn ]. (48) Lemma 28. Let A ∈ [[AG → AG]]Q+. Then † † † A ∈ [[AG → AG]]Q iﬀ A eG ∈ [[A¯ G]]Q. Proof. We can bound kA an − A bnksup using proposition2, obtaining Proof. By deﬁnition, one has A ∈ [[AG → AG]]Q

† if and only if A = [AnRn ] with An ∈ [[ARn → kA (an − bn)ksup ≤ kan − bnksupkA ksup. ARn ]], and by our assumptions the latter is equivalent to a := †e ∈ [[A¯ ]]. Now, since n An ARn Rn Since [anRn ] = [bnSn ], the thesis follows. kanRn − amRm ksup ≤ kAnRn − AmRm ksup, one As a consequence of the above results, all local has that A ∈ [[AG → AG]]Q if and only if ¯ † ¯ transformations leave the space [[AG]]QR invari- A eG = [anRn ] ∈ [[AG]]Q. ant. We now prove a further important result: Lemma 29. Let A ∈ [[A → B]]Q+, and a := the above statement can be extended to all the † quasi-local algebra. A eG. Then kA ksup = kaksup.

Proof. By deﬁnition, A = [AnRn ], with An ∈ Theorem 7. The quasi-local algebra [[AG → ¯ [[ARn → ARn ]]+. Now, kAnRn ksup = kAnksup, AG]]QR leaves the space [[AG]]QR invariant, and † and by lemma7, kAnksup = kAn eRn ksup = for every A ∈ [[AG → AG]]QR and every a ∈ † ¯ kAnR eGksup. Then, [[AG]]QR, it is n † † † kA ksup = lim kA eGksup = kA eGksup. kA aksup ≤ kaksupkA ksup. (49) n→∞ n Rn

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 26 On the same line, we can provide the following One can take a further step, and prove the fol- condition for A ∈ [[AG → AG]]Q+ to be a channel. lowing result that will be of crucial importance in the remainder. Lemma 30. Let A ∈ [[AG → AG]]Q+. Then † A ∈ [[AG → AG]]Q1 iff A eG = eG. ¯ Corollary 12. Let a ∈ [[AG]]QR. Then there exists a sequence of quasi-local transformations Proof. First, let A be a channel. By definition † AnR such that limn→∞ A eG = a. If a ∈ it must be A = limn→∞ AnR , where AnR are n nRn n n ¯ local channels. Thus, by theorem3, † e = [[AG]]Q the sequence can be found in [[AG → AnRn G † A ]] . e . It follows that †e = lim e = G Q G A G n→∞ AnRn G eG. Now, for the converse, by definition A ∈ Notice that the sequence AnR is not necessar- [[A → A ]] iff = [ ] with ∈ n G G Q+ A AnRn AnRn ily Cauchy, thus one cannot conclude that for ev- [[ARn → ARn ]]+ for all n ∈ N. Let us take n ≥ n0 ¯ ery a ∈ [[AG]]QR there is A ∈ [[AG → AG]]QR such so that kAn − A ksup ≤ ε. Then we have † Rn that a = A eG. However, the following lemma completes the picture. |kAnRn ksup − kA ksup| ≤ ε.

† Proposition 14. Let a ∈ [[A¯ G]]Q . Then there Now, by lemma 29, since A eG = eG, we R have k k = 1. Let us deﬁne 0 := exists A ∈ [[AG → AG]]QR such that A sup AnRn

AnRn /kAnRn ksup. Clearly, † a = A eG. (51) k 0 − k = k k |1 − 1/k k | ≤ ε. AnRn AnRn sup AnRn sup AnRn sup Moreover, for a ∈ [[A¯ G]]Q∗ with ∗ ∈ This implies that 0 is a sequence converg- AnRn {“nothing”,1,+} one can find the above A in the ing to , with k 0 k = 1. As a con- A AnRn sup set [[AG → AG]]Q∗. sequence, there exist channels CnRn such that 0 Proof. Let us define the sets (Cn − An)Rn 0. Again by lemma7, k( − 0) k = k( − 0)† e k ε : † Cn An Rn sup Cn An Rn Rn sup Ta = {A ∈ [[AG → AG]]QR | kA eG − aksup ≤ ε}. = ke − 0† e k G AnRn G sup ε The sets Ta are not empty for every a and every 0 † = k(A − AnRn ) eGksup ε > 0, as a consequence of corollary 12. More- 0 ≤ k( − )k ≤ ε. over, they are closed. Indeed, let {An}n∈ be a A AnRn sup N Cauchy sequence in Tε . This implies that by defi- 0 a This implies that A = [A ] = [Cn ], and † ¯ ¯ nRn Rn nition an := An eG ∈ Bε(a). Since the ball Bε(a) then ∈ [[A → A ]] . † A G G 1 is closed, also limn→∞ an = (limn→∞ An) eG ∈ ¯ ε Bε(a), namely limn→∞ An ∈ Ta. Finally, one The following weak version of the converse of ε δ clearly has that for ε < δ it is Ta ⊆ Ta, and the lemma 28 can be easily proved. ε diameter dε := sup{kA − Bksup | A , B ∈ Ta} is ¯ Lemma 31. Let a ∈ [[AG]]LR. Then there exists a non-decreasing function of ε. Let then A ∈ [[AG]]LR such that T0 := \ Tε . † a a a = A eG. (50) ε>0

¯ 0 Moreover, for a ∈ [[AG]]L, Eq. (50) is satisfied Either limε→0+ dε > 0, in which case Ta is non † with A ∈ [[AG → AG]]L. empty, and each of its elements satisfies A eg = a, or the limit is 0, in which case, by Cantor’s Proof. Let a = aR ∈ [[A¯ G]] . Let us define LR intersection theorem for complete metric spaces, AR := |σ)(a|, where σ ∈ [[AR]]1 is an arbitrary T0 = { } is a singleton, with †e = a. The deterministic state. Then clearly a A A G same argument can be applied in each of the cases † ¯ AReG = aR. where a ∈ [[AG]]Q∗, considering the closed sets ε∗ ε Ta := Ta ∩ [[AG → AG]]Q∗. It is also clear that, for a = aR ∈ [[A¯ G]]L, the above defined transformation AR is in We now prove a result that is analogous to [[AG]]L. lemma 12.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 27 Lemma 32. Let A ∈ [[AG → AG]]Q+, and Proof. As to linearity, it is sufficient to consider C − A ∈ [[A¯ G → AG]]Q+ for some C ∈ [[AG → that AG]]Q1. Then A ∈ [[AG → AG]]Q. ˆ † (a|A [xρ1 + yρ2]) = (A a|xρ1 + yρ2) † † Proof. By definition, there are three sequences = x(A a|ρ1) + y(A a|ρ2) , 0 in [[A → A ]] and in [[A → AnRn AnSn G G L CnTn G = (a|x ˆρ ) + (a|y ˆρ ), 0 A 1 A 2 AG]]L1 such that limn→∞ An = A , limn→∞ An = which by definition implies C − A , and limn→∞ Cn = C . This implies that ˆ ˆ ˆ A (xρ1 + yρ2) = xA ρ1 + yA ρ2. 0 kCn − (An + A )ksup ≤ ε, n In order to prove boundedness, it is sufficient to and consequently there is a sequence of channels consider Eq. (49), to conclude that ˆ † † Dn in [[AG → AG]]L1 such that |(a|A ρ)| = |(A a|ρ)| ≤ kA aksupkρk∗

0 ≤ kaksupkA ksupkρk∗, εDn + Cn − (An + An) 0. which implies that By the above equation, it is easy to check that ˆ kA ρk∗ ≤ kA ksupkρk∗.

1 1 0 We now prove that [[AG]]Q is invariant under Fn − An 0, Fn − A 0, R 1 + ε 1 + ε n the quasi-local algebra, and more precisely, every (B) space [[AG]]ρ Q is invariant. where Fn is the channel ε/(1+ε)Dn+1/(1+ε)Cn. 0 R 0 Theorem 8. Let ∈ [[A → A ]] . Then Thus, by theorem5, both An/(1+ε) and An/(1+ A G G QR ε) belong to [[AG → AG]]L. Let εm be a real ˆ A [[AG]]Q ⊆ [[AG]]Q , Cauchy sequence converging to 0, and consider R R and for every (B, ρ0), the corresponding sequences Bm := Anm /(1 + ε ), 0 := 0 /(1 + ε ). Since (B) (B) m Bm Anm m ˆ[[A ]] ⊆ [[A ]] . A G ρ0QR G ρ0QR

εm Proof. We start observing that, by Eq. (40), kBm − Anm ksup = kAnm ksup, † 1 + εm when we apply ARaS to a local state τT in ε (B) 0 0 m 0 [[AG]]ρ Q , we obtain kB − A ksup = kA ksup, 0 R m nm 1 + ε nm m AS\(R∪T )

A(R∩S)\T one has A = [AnRn ] = [BmRm ] and C − A = ρ0(R∪S)\T [ 0 ] = [ 0 ], with and 0 beloning to AR\(S∪T ) aS AnSn BmSm Bn Bn A [[AG → AG]]L. Thus, A , C − A ∈ [[AG → AG]]Q. R∩S∩T AR A (R∩T )\S e , τ A T (T ∩S)\R e We will now deﬁne the action of the quasi-local AT \(R∪S) e algebra of transformations on quasi-local states. The action is deﬁned by duality as follows.

Definition 30 (Dual action of the quasi-local ¯ for every S and every aS ∈ [[AS]]R, which implies algebra). Let A ∈ [[AG → AG]]QR. We define the AB(R\T ) ˆ map A : [[AG]]R → [[AG]]R as follows ˆ AR∩T ARτT ˆ † ¯ AS\R (a|A ρ) := (A a|ρ), ∀a ∈ [[AG]]QR. (52) (53) The first thing we need to prove is that the A ˆ B(R\T )\(R\T ) map A : [[AG]]R → [[AG]]R is linear and bounded. ρ 0B(R\T ) AR\T AR\T = . Proposition 15. For every A ∈ [[AG → AG]]QR AR∩T AR AR∩T the map ˆ : [[A ]] → [[A ]] is linear and A G R G R τT AT \R bounded.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 28 This means that local states in [[A ]](B) are Remark 4. Notice that every quasi-local trans- G ρ0QR mapped to local states in the same sector un- formation in [[AR → AR]]QR does not only rep- ˆ ¯ der AR. Now, by proposition 15, Cauchy se- resent a linear map on [[AR]]QR, but a family of quences are mapped to Cauchy sequences, so that maps that represent the same transformation act- (B) (B) ¯ ˆ [[A ]] ⊆ [[A ]] . Moreover, again by ing on [[AG]] for every R ⊆ G. This is due to the AR G ρ0QR G ρ0QR proposition 15, also Cauchy sequences of local isomorphism of [[AR → AR]]QR with the subalge- (G) (G) transformations satisfy the same condition, thus bra [[AR → AR ]]QR given in Eq. (55). ˆ[[A ]](B) ⊆ [[A ]](B) . Similarly, one can eas- A G ρ0QR G ρ0QR (G) ˆ The following result shows that [[AR → ily prove that A [[AG]] ⊆ [[AG]] . QR QR (G) ¯ (G) AR ]]QR preserves the subspace [[AR ]]QR. As a consequence of the above theorem, the ac- (G) (G) tion of [[A → A]]Q on the space [[AG]]QR of quasi- Lemma 33. Let A ∈ [[AR → AR ]]QR. Then local states is decomposed into many irreducible (B) † ¯ (G) ¯ (G) representations, one for every space [[A ]] . A [[AR ]]Q ⊆ [[AR ]]Q . G ρ0Q R R

Remark 3. Notice that, given A , B ∈ [[AG → Proof. Let AnRn be a sequence in the class of

AG]]QR, one has A , with Rn ⊆ R for every n. Let amSm be a ¯ (G) sequence in the class of a ∈ [[AR ]]QR, with Sm ⊆ ( )† = † †, (\) = ˆ ˆ. (54) † † AB B A AB A B R for all m. Then A a = [(An an)Rn∪Sn ], and for every n, Rn ∪ Sn ⊆ R. This implies that (G) Let R ∈ R , and let A ∈ [[AG → AG]]Q. † ¯ (G) A a ∈ [[AR ]]QR. Suppose that A = [AmRm ] and for every m ∈ N, R∩Rm = ∅. This implies that, for Sm = Rm ∪R, The following intuitive result states that an ef- = [ 0 ], with 0 = ( ⊗ ) , for fect that is localised in the region R ⊆ G is ob- A AmSm AmSm Am IR Sm 0 Am ∈ [[ASm\R]]R. We then write A = B ⊗ IR. tained by a quasi-local transformation that is lo- Notice that in this case, given ρ ∈ [[AG]]R, one has calised in the same region. This has a straightforward proof that will be omitted. ˆ ˆ ˆ kρ|R − (A ρ)|Rkop = k(Amρ)|R − (A ρ)|Rkop ¯ (G) ˆ ˆ Lemma 34. Let a ∈ [[A ]]Q∗, with ∗ ∈ ≤ k(Am − A )ρk∗ ≤ εkρk∗, R {“nothing”, 1, +, R}. Then ˆ and then (A ρ)|R = ρ|R. † a = A eG, For R ∈ R(G), there is a straightforward ordered Banach algebra isomorphism between (G) (G) (G) (G) with A ∈ [[AR → AR ]]Q∗. [[AR → AR]]QR and [[AR → AR ]]QR, where the latter is deﬁned as We conclude the section with a result that con- ﬁrms, under restrictive hypotheses, an obvious (G) (G) [[AR → AR ]]QR := {AR | A ∈ [[AR → AR]]R}. expectation.

A similar isomorphism can be found also in the Lemma 35. Let A ∈ [[AG → AG]]QR. Then (G) case of infinite regions R ∈ R , where by defi- kA ksup = inf J(A ), where (G) (G) nition [[AR → AR ]]Q is the closed subalgebra R J(A ) of Cauchy classes [AnRn ] with Rn ⊆ R for every n ∈ N. The isomorphism in this case is given by := {λ ∈ R | ∃C ∈ [[AG → AG]]Q1, λC ± A 0}. † † (G) (G) † −1† Proof. Let λ ∈ J(A ). Then there exists a quasi- [[AR → AR]]Q = JR[[AR → AR ]]Q JR , R R local channel C such that λC ± A 0. By defi- (55) ± nition, this implies that λC ± A = [B ± ] with nRn † (G) ± + − where J : [[A¯ ]]Q → [[A¯ R]]Q is defined in B 0. Defining An := 1/2(Bn − Bn ) and R R R R + − Eq. (25). In the following, when we want to spec- Cn := 1/(2λ)(Bn + Bn ), by construction one ify that a given quasi-local transformation A is has in the subalgebra [[AR → AR]]QR, we will write [C + ] = C , [A − ] = A , AR. nSn nSn

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 29 ± where the domains Sn are suitably defined. On Lemma 36. Given A ∈ [[AG → AG]]QR, let λ the other hand, C = [DnTn ] for DnTn ∈ [[ATn → SA ⊆ [[AG → AG]]Q1 be the set of channels C λ ATn ]]1. Thus, one has kCn − Dnksup ≤ ε, taking such that λC ± A 0. Then SA is closed in the + suitable care in defining the domain Un := Sn ∪ sup-norm. Tn of Cn − Dn. In other terms, one can find a Proof. The empty set is closed, so we will con- channel Fn such that λ sider the case where SA is not empty. Let λ {Cn}n∈N be a Cauchy sequence in SA . Then εFn + Dn − Cn 0, ± D := λCn ± A 0 are both Cauchy sequences, ε − + 0. n Fn Dn Cn that by definition converge to λC ± A 0. By proposition 12, C ∈ [[AG → AG]]Q1. In turn, this implies that

Proposition 16. Let A ∈ [[AG → AG]]QR. Then (1 + ε)eUn − cn 0, there exists a quasi local channel C such that

(ε − 1)eUn + cn 0, kA ksupC ± A 0. (58) : † where cn = CneUn . Considering the ﬁrst of the Proof. By lemma 35, for every ε > 0 there exists above relations, we can deﬁne Cε such that (kA ksup+ε)Cε±A 0. Let us then : kA ksup+ε 1 consider the non empty sets Sε = SA , that Gn := Cn + |ρn)(dn|, 1 + ε are closed by lemma 36. It is straightforward to verify that for δ < ε it is Sδ ⊆ Sε. Let dε be the where dn := eUn − 1/(1 + ε)cn 0 and ρn is an diameter of Sε, namely dε := sup{kC1 − C2ksup | arbitrary state in [[AUn ]]1. By our assumptions, C1, C2 ∈ Sε}. Notice that 0 ≤ dε ≤ 2. Being † since Gn 0 and GneUn = eUn , Gn is a channel. dε a non-decreasing function of ε, d0 = limε→0+ Moreover, by construction we have that exists. The limit d0 is the diameter of \ ± S0 := Sε. λ(1 + ε)Gn ± An = Bn + λ(1 + ε)|ρn)(dn| 0, ε>0 thus λ(1 + ε) ∈ J(An), i.e. λ(1 + ε) ≥ kAnksup. If d0 > 0, then S0 is clearly not empty, and each of In the limit, we then have λ ≥ kA ksup. On the its elements is a channel C such that kA ksupC ± other hand, if λ 6∈ J(A ), for every C ∈ [[AG → A 0. If d0 = 0, then by Cantor’s intersection AG]]Q1 one has theorem for complete metric spaces the set S0 is a singleton S0 = {C∞}, with kA ksupC∞ ± A λC + A 6 0 ∨ λC − A 6 0. (56) 0.

Since the sequence λC ±An converges to λC ±A We ﬁnally prove a crucial result, that is the for every sequence An converging to A , and the converse of lemma 32. This shows that every positive cone is closed, for every C and every An quasi-local event is an element of a quasi-local as above, there must be n0 such that for n ≥ n0 test.

Proposition 17. Let A ∈ [[AG → AG]]Q+. λC + An 6 0 ∨ λC − An 6 0. (57) Then A ∈ [[AG → AG]]Q if and only if there exists C ∈ [[AG → AG]]Q1 such that C − A ∈ This holds, in particular, for every local Cn with [[AG → AG]]Q+. the same domain as An, which implies λ 6∈ J(An). As a consequence, λ < kAnksup, and Proof. Suﬃciency is proved in lemma 32. Let us ﬁnally λ ≤ kA ksup. It is now easy to conclude then prove that if A ∈ [[AG → AG]]Q then there that inf J(A ) = kA ksup. exists C ∈ [[AG → AG]]Q1 such that C − A 0.

First of all, notice that A = [AnRn ] with An ∈ We now prove that the infimum defining the [[ARn → ARn ]], and then by sup norm is actually a minimum, analogously to kA ksup = lim kAnksup the case of effects. The proof in this case is less n→∞ straightforward, and we first need the following † = lim kA eGksup ≤ 1. lemma. n→∞ nRn

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 30 By proposition 16, there exists C ∈ [[AG → system AGC. For this reason we need to define ¯ AG]]Q1 such that an automorphic transformation of [[AG]]QR with care. One further delicate issue is the follow- − k k − 0. C A A supC A ing. When one conjugates a quasi-local transformation A with an automorphism V †, obtianing 5 Global update rules A 0 = V −1†AV †, the latter is a linear bounded ¯ map on [[AG]]QR, but there is no guarantee that We have now set the theoretical background it is still an element of the quasi-local algebra needed for the definition of a cellular automaton. [[AG → AG]]QR, which is surely a desideratum for In the present section we give the definition of candidate cellular automata. In the remainder, an update rule (UR), that along with the system when we write AGC we mean to deal with a sys- 0 AG—the array of cells, whose definition has been 0 ∼ tem AG with G = G ∪ h0 and Ah0 = C. In analyzed in detail in the previous sections—will view of the above observations, we introduce the provide the backbone of the notion of a cellu- following notion lar automaton. We then introduce the notion of a global update rule (GUR), as a family of up- Definition 31 (Automorphic family). An auto- ¯ date rules satisfying suitable admissibility condi- morphic family of maps on [[AG]]QR is a collec- † ¯ ¯ tions in order to represent a local action when tion of automorphisms VC of [[AGC]]QR, one for every C ∈ Sys(Θ), with the properties i) for every extended to composite systems AGC. A GUR ∈ [[A → A ]] one has −1† † † = † ; thus describes the evolution occurring when we A C C R VC Ah0 VC Ah0 apply the global update rule on a joint state of ii) for fixed C0 and every choice of system C1, AG and C, leaving C unaffected. We then analyse setting C = C0C1 and AG0 := AGC0, and for ev- 0 00 the main features of GURs, and in particular we ery A ∈ [[AG0 → AG0 ]]QR, there exist A , A ∈ focus on the causal influence relation given by a [[AG0 → AG0 ]]QR such that global update rule, and the block decomposition −1† † † 0† V A 0 V = A 0 , (60) that allows one to calculate the action of a GUR C G C G † † 00† −1† on local effects using only local transformations. AG0 = VC AG0 VC , (61) Remark 5. In the remainder we will assume where 0† = ( −1† † † ), and 00† = A VC0 A VC0 A that for every g ∈ G the system A is not trivial, −1† † g ( † ) VC0 A VC0 namely Ag 6' I. We remark that, as a special case of item ii) First of all let V † be a bounded automorphism ¯ in definition 31, taking C0 ≡ I and C = C1, for of the space [[AG]]QR of quasi-local effects. Then, every choice of system C, and every A ∈ [[AG]] , with a notation that is reminiscent of the one QR one has adopted for quasi-local transformations, we will −1† † † −1† † † denote its action as VC AGVC = (VI AGVI )G, † ¯ ¯ † † † −1† = ( † † −1†) . V : [[AG]]QR → [[AG]]QR :: a 7→ V a. VC AGVC VI AGVI G The action on effects allows us to define the dual When referring to an automorphic family we map that acts on the space of extended states, as will use the shorthand V †. One has to remind, follows however, that this symbol refers to a family of automorphisms. In particular, the definition is (a| ˆρ) := ( †a|ρ), ∀a ∈ [[A¯ ]] . (59) V V G QR meant to allow for formulas such as VAV −1, −1 One can easily verify that Vˆ is linear and V [[AG → AG]]$∗V , for $ = Q, L and ∗ = bounded, thanks to boundedness of the map V †. R, +, 1 or nothing. The meaning of the two ex- Unlike the usual notion of a map in quantum the- pressions above is given by the following defini- ory or theories with local discriminability, how- tions: ˆ † ever, the action of V or V is not sufficient to −1 := { ˆ ( ˆ⊗ ˆ ) ˆ−1 | C ∈ Sys(Θ)}, VAV VC0C A IC VC0C identify uniquely the corresponding transforma- (62) tion. In particular, those actions are not suffi- −1 V [[AG0 → AG0 ]] V := cient to determine that of V ⊗ IC, when sys- $∗ −1 tem AG is considered as a part of the composite {VAV | A ∈ [[AG0 → AG0 ]]$∗}. (63)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 31 −1 Notice that by definition of an automorphic fam- Proof. In the first place, the family VC is an au- ily and by Eq. (62), one has tomorphic family: items i) and ii) in definition 31 † are trivially proved. Now, if kVC aksup = kaksup −1 ˆ ˆ ˆ−1 ˆ −1† VAV = {(VC0 A VC ) ⊗ IC | C ∈ Sys(Θ)}, ¯ ¯ 0 for every a ∈ [[AGC]]QR, then kVC aksup = (64) † −1† kVC VC aksup = kaksup, which proves isometric- −1† ity of VC . From the definition in Eq. (59) one namely an automorphic family maps [[AG0 → ˆ−1 ˆ −1 can easily verify that (V ) = (VC) . Now, it AG0 ]]QR onto itself surjectively. In particular, this C is true of [[AG → AG]]Q . This implies that for is a straightforward observation that by Eq. (65), R ˆ−1 −1 V [[AGC]] = [[AGC]]. Finally, multiplying both A ∈ [[AGC → AGC]]QR, VAV represents a C sides of Eq. (66) by V −1 to the left and by V to quasi-local transformation in [[AGC → AGC]]QR. An automorphic family is defined as to satisfy the right, for every A ∈ [[AGC → AGC]]L+ one some necessary conditions that we require for the has representatives of a transformation of the form A = V −1A 0V , V ⊗ IC. However, the definition of an automorphic family is not sufficient to ensure consistency. which expresses the same condition as in Eq. (67) This fact must be taken into account in the next for V −1, and similarly, multiplying both sides of subsection. Sufficient conditions will be only pro- Eq. (67) by V −1 to the left and by V to the right, vided through the notion of admissibility. one obtains the same condition as in Eq. (66) for V −1.

5.1 Update rule † In the following we will often use V to de- We can now use the notion of automorphic family note a UR, when G and AG are clear from the ˆ † to introduce update rules. An update rule is de- context. The symbols V and V will be often ˆ † fined as a family of automorphisms, one for each used instead of VC and VC , respectively. Accord- ingly, the symbols a, b, . . . and , ,... will de- extended system AGC, with the constraint that A B the rule must act reversibly on the set of states note a ⊗ eC, b ⊗ eC,... and A ⊗ IC, B ⊗ IC,..., 0 and, by conjugation, on the cone of positive local respectively. Finally, we will sometimes write G ∼ 0 transformations. for G ∪ h0 with Ah0 = C, so that AG = AGC. The next result states that an update rule (G∪h0) (G∪h0) Definition 32 (Update rule). An update rule leaves the cone [[AG → AG ]]L+ invariant. † † (UR) is a triple (G, A, V ), where V is an au- † ¯ Lemma 38. Let (G, A, V ) be a UR, and define tomorphic family of isometric maps on [[AG]]QR AG0 := AGC for arbitrary C ∈ Sys(Θ). Then the such that 0 0 −1 (G ) (G ) map V · V leaves the cone [[AG → AG ]]L+ 1. the maps ˆ leave [[A C]] invariant, i.e. (G0) (G0) VC G invariant, namely for A ∈ [[AG → AG ]]L+ 0 0 00 (G ) ˆ Eqs. (66) and (67) hold with A , A ∈ [[AG → VC[[AGC]] = [[AGC]]; (65) (G0) AG ]]L+. 0 0 −1 G G 2. the map V · V leaves the cones [[AGC → Proof. Let A ∈ [[AG → AG ]]L+ ⊆ [[AG0 → AGC]]L+ invariant, i.e. for every A ∈ AG0 ]]L+. By definition of UR 0 00 [[AGC → AGC]]L+ there are A , A ∈ −1 0 VAGV = AG ∈ [[AG0 → AG0 ]]L+. [[AGC → AGC]]L+ such that Moreover, by definition of an automorphic family −1 0 VAV = A , (66) of maps and Eq. (64), we have = 00 −1. (67) A VA V 0 (G0) (G0) A ∈ [[AG → AG ]]LR. The first result is that the inverse of a UR is a The same holds exchanging V −1 and V . UR. ¯ ¯ As a consequence of isometricity on [[AGC]]QR, Lemma 37. Let (G, A, V †) be a UR. Then a UR V † has a dual Vˆ that acts isometrically on −1† (G, A, V ) is a UR. [[AGC]]R.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 32 † ˆ ¯ Lemma 39. Let V be a UR. The maps VC on Proof. Let a ∈ [[AG0 ]]L+. By proposition 14, [[AG0 ]]R are isometric. there exists A ∈ [[AG0 → AG0 ]]L+ such that † a = A eG0 . Then Proof. Let ρ ∈ [[AGC]]R, and remind that † † † V a = V A eG0 kρk∗ = sup |(a|ρ)|. † † −1† † kaksup=1 = V A V V eG0 † † −1† Then we have = V A V eG0 0† 0 ˆ ˆ † = A eG0 = a . kVCρk∗ = sup |(a|VCρ)| = sup |(VC a|ρ)| kaksup=1 kaksup=1 0 We observe that, by definition of UR and by = sup |(a |ρ)| = kρk∗, −1 lemma 38, · preserves the cones [[A 0 → ka0k =1 V V G sup 0 ¯ † ¯ AG0 ]]L+. Thus, a ∈ [[AG0 ]]L+, and V [[AG0 ]]L+ ⊆ † −1† [[A¯ 0 ]] . Since also is a UR, we have thanks to isometricity and invertibility of VC . G L+ V † ¯ ¯ V [[AG0 ]]L+ = [[AG0 ]]L+. Now, isometricity grants Corollary 13. Let † be a UR. Then ¯ V that Cauchy sequences of local effects in [[AG0 ]]L+ ˆ [[A C]] = [[A C]] ¯ VC G 1 G 1 are mapped to Cauchy sequences in [[AG0 ]]L+, † † Proof. By eqs. (65) and (29) and isometricity of and that limn→∞ V an = V limn→∞ an. Fi- † ¯ ¯ ˆ ˆ nally, by proposition 12, V [[AG0 ]]Q+ ⊆ [[AG0 ]]Q+. VC, one has VC[[AGC]]1 ⊆ [[AGC]]1. The same is ˆ−1 ˆ Since also V −1† is a UR, we conclude that true for V , thus V [[AGC]]1 = [[AGC]]1. C C † ¯ ¯ V [[AG0 ]]Q+ = [[AG0 ]]Q+. The next results will allow us to conclude that † a UR preserves [[A¯ ]] . We start proving that an Proposition 18. Let V be a UR. Then G Q † ¯ ¯ update rule † preserves the unique deterministic V [[AG0 ]]Q = [[AG0 ]]Q. In particular, for C = I, V † V [[A¯ G]]Q = [[A¯ G]]Q effects eG0 .

† † Proof. Let a ∈ [[A¯ 0 ]] . Then e 0 − a ∈ [[A¯ 0 ]] , Lemma 40. If V is a UR, then V eG0 = eG0 , G Q G G Q+ by lemma 12, and thus, by virtue of lemmas 40 for AG0 := AGC and arbitrary C ∈ Sys(Θ). † † ¯ and 41, V a, eG0 − V a ∈ [[AG0 ]]Q+. Again by Proof. By deﬁnition of a UR, for every ρ ∈ [[A 0 ]] † ¯ G lemma 12, this implies that V a ∈ [[AG0 ]]Q, and one has ˆρ ∈ [[A 0 ]], and lemma 19, gives for † ¯ ¯ V G thus V [[AG0 ]]Q ⊆ [[AG0 ]]Q. Finally, since also every ρ ∈ [[AG0 ]] V −1† is a UR, the thesis follows. † ˆ ˆ (V eG0 |ρ) = (eG0 |V ρ) = kV ρk∗ = kρk∗ = (eG0 |ρ). Let now V † be an update rule. By our def- −1 † inition, the map V · V preserves the cones Thanks to theorem6, we then have V eG0 = eG0 . −1 [[AG0 → AG0 ]]L+. We will now prove that V · V also preserves the cones [[AG0 → AG0 ]]Q+, as well † † as the sets of transformations [[AG0 → AG0 ]]L and Corollary 14. Let V be a UR. Then V ah0 = 0 0 (G ) (G ) [[AG0 → AG0 ]]Q. ah for all ah ∈ [[A → A ]]Q . 0 0 h0 h0 R We start with the following lemma † Proof. By proposition 14, ah = A eG0 . Thus, 0 h0 Lemma 42. Let V † be a UR. Then V · V −1 by deﬁnition 31 and lemmas 40 and 56, we have leaves the sets [[AG0 → AG0 ]]L1 invariant, namely † † † −1† 0 0 V ah = V A V eG0 for A ∈ [[AG → AG ]]L1 Eqs. (66) and (67) hold, 0 h0 0 00 † with A , A ∈ [[AG0 → AG0 ]]L1. In particular, = A eG0 = ah . 0 h0 0 for C = I the thesis holds with G = G.

¯ 0 We now prove that the cones of eﬀects [[AG ]]L+, Proof. Let C ∈ [[AG0 → AG0 ]]L1. By deﬁni- ¯ ¯ ¯ [[AG]]L+, [[AG0 ]]Q+ and [[AG]]Q+ are invariant un- tion and by lemma 38, VCV −1 = C 0, and 00 −1 0 00 der the action of any UR. C = VC V , for C , C ∈ [[AG0 → AG0 ]]L+. By lemmas 37 and 40, Lemma 41. Let V † be a UR. Then † ¯ ¯ † ¯ V [[AG0 ]]L+ = [[AG0 ]]L+, and V [[AG0 ]]Q+ = 0† † † −1† C eG0 = V C V eG0 = eG0 , [[A¯ 0 ]] . In particular, for C = I it is G Q+ 00† −1† † † † C e 0 = V C V e 0 = e 0 , V [[A¯ G]]Q+ = [[A¯ G]]Q+ G G G

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 33 0 00 and due to lemma 30, C , C ∈ [[AG0 → AG0 ]]L1. The last results finally allow us to prove that for a UR V †, the map V ·V −1 preserves the cones [[AG0 → AG0 ]]Q+. Given a UR V †, the above result allows us to −1 † prove that V ·V maps local transformations to Lemma 46. Let V be a UR. Then V [[AG0 → −1 local transformations, and is isometric on [[AG0 → AG0 ]]Q+V = [[AG0 → AG0 ]]Q+. AG0 ]]LR. Proof. By lemmas 38 and 44, along with the defi- † Lemma 43. Let V be a UR, and A ∈ [[AG0 → nition of a UR, · −1 maps Cauchy sequences in −1 V V AG0 ]]L . Then V [[AG0 → AG0 ]]L V = [[AG0 → R R [[AG0 → AG0 ]]L+ to Cauchy sequences in [[AG0 → AG0 ]]L , and −1 R AG0 ]]L+, thus V [[AG0 → AG0 ]]Q+V ⊆ [[AG0 → −1† −1 A 0 ]] . Since is a UR, the thesis fol- kVAV ksup = kA ksup. (68) G Q+ V lows. Proof. By proposition 16, there exists C ∈ † † ¯ [[AG0 → AG0 ]]L1 such that Next, we show that for a UR V , V [[AG0 ]]LR = ¯ [[AG0 ]]LR. kA ksupC ± A 0. Lemma 47. Let V † be a UR. Then By definition of a UR, we then have † ¯ ¯ V [[AG0 ]]LR = [[AG0 ]]LR. −1 −1 kA ksupVCV ± VAV 0, (69) ¯ Proof. Let aR ∈ [[AG0 ]]LR. Then by proposi- −1 and being VCV ∈ [[AG0 → AG0 ]]L1 ⊆ tion 14 −1 [[A 0 → A 0 ]] , it follows that ∈ G G LR VAV † aR = A eG0 , [[AG0 → AG0 ]]LR. Consequently, we have R −1 V [[AG0 → AG0 ]]LRV ⊆ [[AG0 → AG0 ]]LR. for some ∈ [[A 0 → A 0 ]] . Now, we have Finally, since V −1† is a UR, we have that AR G G LR −1 V [[AG0 → AG0 ]]LRV = [[AG0 → AG0 ]]LR. † † † † † −1† a = e 0 = e 0 , Moreover, by lemma 42, Eq. (69) implies that V R V AR G V ARV G −1 −1 kA ksup ∈ J(VAV ). Thus kVAV ksup ≤ † † −1† 0† 0 −1† and by lemma 43 V ARV = AR0 with AR0 ∈ kA ksup. Now, being V a UR, the equality † 0 0 −1 [[AG0 → AG0 ]]LR. Then V aR = aR0 , with R ∈ kVAV ksup = kA ksup follows. 0 R(G ). As to surjectivity, it can be straightfor- † Lemma 44. Let V be a UR. Then V [[AG0 → wardly proved exploiting lemma 37. −1 AG0 ]]QRV = [[AG0 → AG0 ]]QR, and Finally, for a UR V †, the map V ·V −1 preserves −1 −1 lim VAnV = V ( lim An)V . (70) the set of quasi-local transformations. n→∞ n→∞ Proof. Since V · V −1 is isometric and surjective Lemma 48. Let V † be a UR. Then one has on the dense submanifolds [[AG0 → AG0 ]]L , by R −1 lemma 43 it maps Cauchy sequences to Cauchy V [[AG0 → AG0 ]]QV = [[AG0 → AG0 ]]Q, (71) −1 sequences, and their equivalence classes to equiv- V [[AG0 → AG0 ]]LV = [[AG0 → AG0 ]]L. (72) alence classes, thus it maps [[AG0 → AG0 ]]QR to itself. Surjectivity follows from surjectivity Proof. Let A ∈ [[AG0 → AG0 ]]Q. Then kA ksup ≤ on [[AG0 → AG0 ]]LR. Eq. (70) can be indepen- 1, and by proposition 17, there is C ∈ [[AG0 → dently checked applying both sides to arbitrary AG0 ]]Q1 such that ¯ a ∈ [[AG0 ]]QR. C − A 0. † Lemma 45. Let V be a UR. Then V [[AG0 → −1 −1 AG0 ]]Q1V = [[AG0 → AG0 ]]Q1. By lemma 46, VAV ∈ [[AG0 → AG0 ]]Q+. −1 Moreover, by lemma 42,VCV ∈ [[AG0 → Proof. Let {Cn}n∈N be a Cauchy sequence in AG0 ]]Q1. By the above observations, we have [[AG0 → AG0 ]]L1. Then by definition of UR and lemma 44 we have that { −1} is a VCnV n∈N VAV −1 0, Cauchy sequence in [[A 0 → A 0 ]] , and its limit G G L1 −1 −1 −1 VCV − VAV 0. VCV is in [[AG0 → AG0 ]]Q1.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 34 Finally, by lemma 32, one has This implies that, for every finite region R ∈ 0 R(G ), the UR V † induces a linear map V (R)† : −1 −1 −1 , − ∈ [[A 0 → A 0 ]] . ¯ ¯ VAV VCV VAV G G Q [[AR]]R → [[AR0 ]]R. The following definition of admissible UR is now given in terms of the proper- The same argument holds for A ∈ [[AG0 → ties of the induced maps V (R)†. −1† A 0 ]] . Since is a UR, surjectivity on both G L V † sets is proved. Definition 33. Let (G, A, V ) be a UR, and let V (R)† be the induced maps on finite regions † 0 ∼ Clearly, given a UR V , the image of the alge- R ⊆ G , for arbitrary choice of Ah0 = C. We call −1 (G) bra of local transformations of Ag under V ·V , the UR admissible if for every finite S ∈ R the † i.e. family of maps V (S ∪ h0) represents a transformation VS ∈ [[AS0 → AS]]R, i.e. (G0) (G0) −1 (G0) (G0) V [[Ag → Ag ]]QRV ⊆ [[AG → AG ]]QR, † † † V (S ∪ h0) = VS ⊗ IC. (73) (G0) (G0) is a subalgebra of [[AG → AG ]]QR isomorphic Consider now a partition G = R ∪ R¯, where 0 0 (G ) (G ) ¯ : to [[Ag → Ag ]]QR. R = G \ R. We will write AG = AR ⊗ AR¯. ¯ By definition, the space [[AG]]QR of quasi-local effects contains the spaces [[A¯ (G)]] and [[A¯ (G)]] 5.2 Admissibility and local action R QR R¯ QR as closed subspaces. However, if the theory does As we have seen, independently of the nature of not satisfy local discriminability, it is not true −1 system C, the map V · V does more than map- in general that [[A¯ ]] = [[A¯ (G)]] ⊗ [[A¯ (G)]] . G QR R QR R¯ QR ping linear maps to linear maps: it sends transfor- This makes it difficult to define a UR of the form mations to transformations. This is an important V ⊗ W . result, as the definitions of automorphic family In the remainder of the section we close this and UR are meant to embody necessary condi- gap. For this purpose, we start defining what it tions for a family of maps to represent V ⊗ IC. † means, for a UR V on AG, to act locally on R— However, URs are not yet consistent with such in symbols † = 0† ⊗ † . For our purpose, V V IR¯ a factorisation. The reason for this is that lo- we then need to provide a suitable definition of cal discriminability in our theory is not assumed. † the identity V † = V 0† ⊗ I , which satisfies the Thus, while knowledge of the linear maps † is R¯ VC necessary (but not sufficient) condition sufficient to determine the action of the UR on factorised effects a ⊗ b and, by conjugation, on ( 0† ⊗ † )(a ⊗ b ) = ( 0†a ) ⊗ (b ), G C V IR¯ S T V S T generalised transformations AG ⊗ I , and such C (G) ¯ actions are consistent with a factorised action of for every S, T ∈ R such that S ⊆ R and T ⊆ R. † the form V ⊗ IC, still we do not have enough in- Definition 34. Let V be a UR for AG. We † † formation to determine the action of VC on those say that V acts locally on the region R ⊆ G ¯ ¯ (G0) elements of [[AGC]]QR that lie outside the sub- if for every C and every finite region S ∈ R space [[A¯ ]] ⊗ [[C]]¯ (and analogously for the 0 ∼ G QR QR with G = G ∪ h0 and Ah0 = C, the induced map conjugate action on transformations). For this V (S)† has the form purpose we introduce in this section the notion † † † of admissibility. V (S) = WS∩R ⊗ IS\R. (74) 0 We remind that G stands for G ∪ h0, with As a consequence of the above definition we ∼ (G0) Ah0 = C for some system C. Let R ∈ R be now prove the following result, for which we do ¯ (G0) a finite region. Since [[AR ]]QR is isomorphic to not provide the proof, that is straightforward. ¯ : [[AR]]R, whose size is DR = DAR , we can find a † 0 Lemma 49. Let (G, A, V ) be a UR acting lo- DR ¯ (G ) basis {bjR}j=1 ⊆ [[AR ]]QR. Thus, by lemma 47, 0 0 cally on R. Then † ¯ (G ) ¯ (G ) V [[AR ]]Q ⊆ [[A 0 ]]Q for some finite region 0 R R R † † R0 ∈ R(G ), that can be obtained as V (aS ⊗ bT ) = (WS aS) ⊗ bT , (75) for S ⊆ R and T ⊆ R¯. In particular, DR 0 0 0 [ 0 † ¯ (G0) †[[A¯ (G )]] = [[A¯ (G )]] , and †b = b for every R := Rj, V bjR ∈ [[A 0 ]]Q . V R QR R QR V Rj R 0 j=1 b ∈ [[A¯ (G )]] . R¯ QR

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 35 † † ¯ Lemma 50. Let (G, A, V ) be a UR act- Lemma 51. Let V be a UR for [[AG]]R that acts ing locally on R. Then there exists a UR locally on R, and W † a UR that acts locally on 0† † −1† −1† 0† R¯. Then 0† ⊗ 0† := † † = † †. (R, A, V ) such that V JR0 a = JR0 V a V W V W W V and −1† −1†a = −1† 0−1†a for every a ∈ V JR0 JR0 V We can now deﬁne a global update rule as fol- [[A¯ 0 ]] , with A 0 := A C. R QR R R lows.

Proof. Linearity and isometricity of the restric- Definition 35 (Global update rule). Let † (G0) † ¯ † † tion VR0 of V to [[AR0 ]]QR are straightforward (G, A, V ) be a UR. We call (G, A, V ) a Global 0† † † −1† by lemma 49. Thus, also V := JR0 VR0 JR0 Update Rule (GUR), or global rule for short, if † −1† is linear and isometric. Let now A ∈ [[AR0 → (G, A, V ) and (G, A, V ) are admissible. AR0 ]]Q. Then one has Remark 6. The set of GURs (G, A, V †) is a 0† † 0−1† † † −1† † † −1† −1† group. The proof of closure under composi- V A V = JR0 VR0 JR0 A JR0 VR0 JR0 tion and associativity are tedious but straightfor- = † † † −1† −1† JR0 VR0 AR0 VR0 JR0 ward, and we omit them here. Moreover, given † † † −1† −1† † † = JR0 (V AR0 V )R0 JR0 , (G, A, V ) and (F, A, W ) one can define their parallel composition as (G ∪ F, A, V † ⊗ W †). † −1† † † where AR0 := JR0 A JR0 . In the above chain The analysis of properties of parallel composition of equalities we used the fact that, since V †, is beyond the scope of the present work, and will 0 −1† † ¯ (G ) V and AR0 preserve the space [[AR0 ]]QR, then be the subject of further studies. also † † −1† does, and one can thus define V AR0 V We now provide an important example of † † −1† its restriction (V AR0 V )R0 . Finally, since GUR. Let G = H ×{0, 1}, and let H := (H, i) for † i † † −1† 0† (G0) V is a UR, one has V AR0 V = A with ¯ ∼ 0 0 i = 0, 1. Let aR ∈ [[AR ]]QR with Ah0 = C. Let 0 (G ) (G ) 0−1 0 A ∈ [[AR0 → AR0 ]]Q. Thus, V AV ∈ R ∩ G = (R0, 0) ∪ (R1, 1). Then we define R˜ := [[AR0 → AR0 ]]Q. A similar argument leads to R0 ∪ R1 ⊆ H, and R¯ := R˜ × {0, 1} ∪ R. Clearly, 00 the existence of A ∈ [[AR0 → AR0 ]]Q such that R ⊆ R¯. If (h, i) ∈ R¯, then also (h, i⊕1) ∈ R¯. Let = 0−1 00 0. Now, let ρ ∈ [[A ]]. Then ρ A V A V G then a¯R¯ := aR ⊗ eR¯\R. defines a state ρ 0 on [[A¯ 0 ]] , by lemma 17, (H) |R R QR Now, for every T ∈ R let us define the GUR with † (G, A, ST ) by setting −1† (a|ρ|R0 ) = (aR0 |ρ) = (JR0 a|ρ), † h † i S [aR] := S a¯ , (76) T R˜∩T R¯ ¯ (G0) for every aR0 ∈ [[A 0 ]]Q . Now, we have R R † N † (H) where SX := h∈X Sh for X ∈ R , and Sh −1† −1† ˆ ˆ † is the map that swaps A(h,0) and A(h,1). It is (a|[V ρ]|R0 ) = (JR0 a|V ρ) = (V JR0 a|ρ) (H) † −1† −1† † † −1† straightforward to verify that, for every T ∈ R , = (VR0 JR0 a|ρ) = (JR0 JR0 VR0 JR0 a|ρ) ST is a UR. Being defined in terms of induced −1† 0† ˆ0 † = (J 0 V a|ρ) = (a|V ρ|R0 ). R maps, it is immediate to verify that ST admissi- −1 ble. Moreover, ST is involutive, i.e. S = ST . ˆ ˆ0 T We then showed that [V ρ]|R0 = V (ρ|R0 ). More- Of particular interest for the following is the GUR † over, since preserves the set of states, [ ˆρ] 0 V V |R (G, A, SH ). ˆ0 is a state. Thus, V [[AR0 ]] ⊆ [[AR0 ]]. The same Notice that, for a finite disjoint partition H = argument for 0−1 brings to the conclusion that ∪k S , we have † = Nk † . Thus, by V i=1 k SH i=1 SSk ˆ0 0† † V [[AR0 ]] = [[AR0 ]]. Thus, V is a UR for AR. lemma 51, for every GUR (G, A, V ), and for ev- Moreover ery i, j, we have

† −1† † −1† −1† 0† −1 0 a = 0 0 a = 0 a, V JR VR JR JR V V (SSi∪Sj ⊗ IH\(Si∪Sj ))V −1 −1 −1 = V (SS ⊗ I )V V (SS ⊗ I )V and the same holds for V . i H\Si j H\Sj −1 −1 = V (SSj ⊗ IH\S )V V (SSi ⊗ IH\S )V , In the following, under the hypothesis of j i lemma 50, we will write † = 0† ⊗ † . V V IR¯ where IS denotes I(S,0) ⊗ I(S,1).

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 36 0 For A ∈ [[AR → AR]], it is easily veriﬁed that from g to g if there exists F ∈ [[AgC → AgC]] such that A = SH (A )SH = SR(A ⊗ IR)SR, (R,1) (R,0) C C C C A(R,0) = SH (A(R,1))SH = SR(IR ⊗ A )SR. g F g 0 0 0 6= g¯ F g¯ , (77) −1 g¯ V V g¯ g0 g0

Finally, let us consider a GUR (G, A, U † ⊗ V †). (78) One can prove that SH (U ⊗ V )SH = V ⊗ U , and even better in the form by the following argument. Let us consider a ∈ (G0) C C C C [[A¯ ]]Q . Then R R 0 g F g g¯0 g¯0 6= F . † † † † † † ( ⊗ ) a = ( ⊗ ) ( a) g¯ g¯ g0 g0 SH U V SH R SH U V SR˜ R¯ V V † ˜ † ˜ † † = SH [(U (R, 0) ⊗ V (R, 1) )(S ã)]R¯ R Definition 37. We define forward and backward † † † † = [S (U (R,˜ 0) ⊗ V (R,˜ 1) )(S a)] ¯ + 0 0 R˜ R˜ R g-neighbourhoods the sets Ng := {g |g g } ˜ † ˜ † and N − := {g0|g0 g}, respectively. = [V (R, 0) ⊗ U (R, 1) a]R¯ g † = (V ⊗ U ) aR. In terms of diagrams, the forward g- neighbourhood A + of g can be understood as Ng N 5.3 Causal influence the smallest region K = A 0 such that for k∈K gk (G) In the present subsection we define the notion of every k ∈ K there exists C and F ∈ [[Ag C → (G) causal influence that establishes, given a GUR, Ag C]]QR such that whether the evolution allows interventions on one C C system to affect other systems after one step, or g F g more generally how many steps are needed for such an influence to occur. We will often use the g¯ V g¯ (G) (G) (G0) notation [[AR C → AR C]]∗∗ to denote [[AR → 0 (79) (G ) 0 ∼ AR ]]∗∗, with G = G ∪ h0 and Ah0 = C. In the C C following we also adopt the convention that for + 0 + Ng F Ng (G) ¯ = . R ∈ R , R denotes the complementary region ¯ + ¯ + Ng V Ng R¯ := G \ R. Notice that, consistently with the notation R = g, we will write g¯ to denote G \ g. 0 0 0 with F acting non trivially on gk, namely F 6∈ (G) (G) Definition 36 (Causal influence). We say that, [[A 0 C → A 0 C]]Q . Notice that g may or may g¯k g¯k R + according to the global rule V , system g does not not belong to Ng . causally influence g0 if for every external system ± Analogously to the definition of Ng , one can C ± give a definition of NR for every finite region R ⊆ G. (G) (G) −1 (G) (G) V [[Ag C → Ag C]]QRV ⊆ [[Ag¯0 C → Ag¯0 C]]QR. Definition 38. We say that the system g0 belongs On the other hand, we will say that according to the forward neighbourhood of R, and write g0 ∈ + to the global rule V system g causally influences NR , if 0 0 system g , and write g g , in the opposite case, (G) (G) i.e. when ∃C, F ∈ [[AR C → AR C]]QR : −1 (G) (G) (G) (G) (V ⊗ IC)F (V ⊗ IC) 6∈ [[Ag¯0 C → Ag¯0 C]]QR. ∃C, F ∈ [[Ag C → Ag C]]QR : −1 (G) (G) − (V ⊗ IC)F (V ⊗ IC) 6∈ [[Ag¯0 C → Ag¯0 C]]QR. The backward neighbourhood NR of R is defined as The above definition gets a clear meaning as a − 0 + diagram: according to V there is causal influence NR := {g ∈ G | Ng0 ∩ R 6= ∅}. (80)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 37 Clearly, one has the former definition of causal influence, which provides also this piece of information. N + ⊇ [ N +. (81) R g Given a GUR (G, A, V †), we can now construct g∈R a graph Γ(G, E) with vertices g ∈ G and edges 0 0 We will prove that the latter is actually an equal- E := {(g, g ) ∈ G × G|g g }. We call such a ity in the next section. graph the graph of causal influence of the GUR One might define causal influence without in- V †. In the next section we will prove that, if we ±∗ voking any external system C, as follows. denote by Ng the neighbourhoods for the GUR V −1, the following relations hold Definition 39 (Individual influence). We say that the system g does not individually influence N ± = N ∓∗. (82) g0 if f f

(G) (G) −1 We also define the local perspective present P + V [[Ag → Ag ]]QRV R of R ∈ R(G) as the set P + := N − ⊇ R, (G) (G) R N + ⊆ [[A 0 → A 0 ]]Q . R g¯ g¯ R namely the set of systems that can causally in- + On the other hand, we will say that according to fluence the one-step future NR of R. Similarly the global rule V system g individually influences one can define the local retrospective present of R 0 0 − + system g , and write g g g , in the opposite as PR := N − ⊇ R, namely the set of systems NR case, i.e. when − that can be influenced by the one-step past NR + − (G) (G) of R. In general, PR 6= PR . We will also need to ∃F ∈ [Ag → Ag ]QR : +∗ +∗ refer to the sets PR := N + . −1 (G) (G) NR VFV 6∈ [[Ag¯0 → Ag¯0 ]]QR.

The individual forward neighbourhood of g is 5.3.1 Relation with signalling + 0 0 Ng (g) := {g ∈ G | g g g }. The term influence that we used above is precisely defined in Eq. (78). In the present sub- A second definition that can be given without section we will make a clear point that this def- referring to external systems C is the following. inition is stronger than the notion of signalling Definition 40 (Cooperative influence). We say that is customary in the literature on quantum that the region S does not cooperatively influence information theory [52, 53, 54] or in general op- g0 if erational probabilistic theories [15]. In that context, a channel C ∈ [[AB → A0B0]] is called non- (G) (G) −1 (G) (G) 0 V [[AS → AS ]]QRV ⊆ [[Ag¯0 → Ag¯0 ]]QR. signalling (or semicausal) from B to A if there exists ∈ [[A → A0]] such that 0 F In the opposite case we will write S S g . The cooperative forward neighbourhood of a region S 0 A A0 A A F is defined as = . (83) B C B0 e B e + 0 0 NS (S) := {g ∈ G | S S g }. We will now show that, under the hypothesis that Notice now that in a theory without local dis- N +∗ is finite for every g ∈ G, the condition of criminability it may happen that a system g does g Eq. 79, in a suitably defined sense, implies that not individually influence g0 but it does cooper- 0 + of Eq. (83). Let us consider the state ρ ∈ [[AG]]R. atively, namely one can have g ∈ NS (S), but ˆ 0 † Then (aS|AT ρ) := (a |ρ) = (A aS|ρ). In the g0 6∈ N +(g) for any g ∈ S. In other words, S∪T T g case where a = e , ( †e |ρ) = (b |ρ), where N +(S) ⊃ S N +(h). We call this phenomenon S G AT G T S h∈S h † ˆ non-local activation of causal influence. If such b = A eT . Let us now consider (V ρ)|R, which by definition is the state in [[A ]] such that, for a phenomenon occurs, however, it can be hard R R every a ∈ [[A¯ ]] to establish which of the “cooperating” systems R R h ∈ S is responsible for causal influence on ˆ † ˆ ˆ 0 + (a|[V ρ]|R) = (JR a|V ρ) = (aR|V ρ). g ∈ NS (S). This is the reason why we prefer

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 38 Now, by lemma 31, one has Then, by Eq. (76), given R ⊆ AG and a ∈ [[A¯ (G)C]]¯ , with R ∩ G = (R , 0) ∪ (R , 1), ˆ ˆ ˆ R QR 0 1 (aR|V ρ) = (eG|ARV ρ) † † ˆ ˆ−1 ˆ ˆ W aRC = (W a)R0C = (eG|V V ARV ρ) † −1 † † † 0 =[ ( ⊗ ) ( ⊗ ) ]a ˆ ˆ SH VH0 IH1 SH VH0 IH1 RC = (eG|V AN +∗ ρ) R † −1 † † 0 =[ ( ⊗ ) ]a 00 ˆ0 SH VH0 IH1 SH R C = (eG|A +∗ ρ) NR † −1 † † 0 =[S (V ⊗ IH1 ) S +∗ ]aR00C H H0 N ∪R1 = (b +∗ |ρ) = (b|ρ +∗ ). R0 NR |NR † −1 † † † =[S (V ⊗ IH1 ) S +∗ (VH0 ⊗ IH1 ) ]aRC, H H0 N ∪R1 ˆ R0 In order to determine [V ρ]|R it is sufficient to know ρ +∗ . Indeed, the linear map VR : where a0 := ( ⊗ )†a, R0 := |NR VH0 IH1 +∗ + 00 +∗ [[A +∗ ]] → [[AR]] gives (N , 0) ∪ (N , 1), and R := (N , 0)∪(R1, 1). NR R R R0 R1 R0 (H) Now, defining for S ∈ R ˆ (V ρ)|R = VR(ρ|N +∗ ). (84) R 0 Y 0 SS := Sh, (86) Diagrammatically, the equality can be recast, h∈S 0 := ( ⊗ ) ( −1 ⊗ ), (87) with a slight abuse of notation, as in Eq. 83, as Sh VH0 IH1 Sh VH0 IH1 follows thanks to eq. (77) we have +∗ N +∗ NR R R R VR −1 † † † +∗ N = +∗ . (V ⊗ IH1 ) S +∗ (VH0 ⊗ IH1 ) R V R¯ N H0 N ∪R1 e R e R0 0† = S +∗ . N ∪R1 Notice that, by the no-restriction hypothesis, for R0 an admissible V the map VR is linear and admis- We remark that, since ( ⊗ )·( −1 ⊗ ) (G) VH0 IH1 VH0 IH1 sible, and is thus a transformation in [[A +∗ → is an automorphism of [[A → A ]] , one has NR G G QR (G) AR ]]. Moreover, 0 0 0 0 Sf Sg = SgSf , ∀f, g ∈ H. (88) † † V aR = (VRa)N +∗ . (85) Q 0† R Thus, the product +∗ Sh in Eq. (86) is well h∈NR defined. Notice also that, by definition † Finally, notice that by definition VReR = eN +∗ , R 0 0 (G) (G) ShSh = IG. (89) namely VR ∈ [[A +∗ → AR ]]1. The argument NR +∗ 0 (G) (G) can be repeated for the case where Ng is not Since ∈ [[A → A ]] Sh (N +,0)∪(h,1) (N +,0)∪(h,1) Q1 finite, as long as N +∗ ⊂ G. h h g 0 (G) (see Fig.1), one has that S +∗ ∈ [[A → We remark that in the case of Quantum or N ∪R1 S R0 Fermionic cellular automata, the condition for (G) A ]]Q1, with no causal influence is equivalent to no-signalling, S + + +∗ while in the case of Classical cellular automata S := (N +∗ ∪ N , 0) ∪ (N ∪ R1, 1). N R1 R0 no causal influence is strictly stronger than no- R0 signalling. In a different context, a difference be- Then we can write tween the two notions was pointed out in Ref. [55] † † 0† W aRC = S S +∗ aRC H N ∪R1 R0 5.4 Block decomposition † 0† = S + + +∗ S +∗ aRC. N ∪N ∪N ∪R1 N ∪R1 † N+∗ R1 R0 R0 Let us now consider the GUR (G, A, W ) where R0 we set G = H × {0, 1}, and = ⊗ −1, W VH0 VH1 † † However, due to the structure of W , one actually for an arbitrary GUR (H, A, V ), Hi denoting a has shorthand for (H, i). We observe that 0 † 00 (S +∗ ⊗ IC) aRC = a + +∗ , −1 −1 N ∪R1 (N ,0)∪(N ,1)C = ⊗ = ( ⊗ ) ( ⊗ ) . R0 R1 R0 W VH0 VH1 VH0 IH1 SH VH0 IH1 SH

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 39 (H) + R . We have the following (see AppendixC) N ⇤

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 g −1 † [(VAV ) ⊗ IH1 ] aS˜C −1 † =[W (A ⊗ IH1 )W ] aS˜C 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 g + 0† † † † 0† † N ⇤ † 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 g =(SR SR ⊗ IC)(A ⊗ IH0 ) (SRSR ⊗ IC)aS˜C.

Finally, since this holds for every a, we have that 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 g for A ∈ [[A(R,0)C → A(R,0)C]] + + R N ⇤ N ⇤

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 gg 0 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 gg0 −1

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 g0 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 g0 (VAV ) ⊗ IH1

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 g 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 g 0 0 =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 =(SR ⊗ IC)(IH0 ⊗ A )(SR ⊗ IC). (94) + + N ⇤ N ⇤

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 gg 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 gg0 0

AAAITXicdZVbb9s2FMfV7uIs67Z2e9wLO7VABwSB5Q3bHoKgseNcml58TZyUQUFSlCPoGpIuYqj6BHttP1ef+0H2VhSlRFKsm06AYZ3f+Z8Lj0QK53HIRbv97sbNr77+5tvW2nfr39/64cefbt/5+ZhnC0bolGRxxmYYcRqHKZ2KUMR0ljOKEhzTExz1Kv/JS8p4mKUTsczpeYLmaRiEBAmJhvMXt932Zru+wPUbT9+4jr4GL+60/oJ+RhYJTQWJEefPvXYuzgvEREhiWq7DBac5IhGa06LurwT3JfJBkDH5SwWo6YoOJTxB4mJD/vNlgje4SBBbMn8DJzK6cvEVfUUYD3j5pSxfini+EME/50WY5gtBU6JaChYxEBmoxgL8kFEi4qW8QYSFci2AXCCGiJDDW6kyZyi/CMlVuX5/paWlT+uG6hrgagng1TLLqykXKI6NQ+qGJGRkEQqJZAjkOXG9ooB1zwFwvdI4DorKCQ60PWBZFkgd9XkU5jBHDKZZmPryQRQQB6D2bwITfkn94h7cgDiWHfLLBWL0XhVUAuXvJXl8pcpiDHo6aruADKXz+kFW9lYBY2XXZhQVcBveBXBb2RgXcAvehVslqG3BUh6wRC5IrYewakG1K04/c7Kk8qk4tiMnIMFOqcSsq+yuWQ3rKdAzgl1l75ZmPayvSL80OfcU2LOSfUX2G8mhAodNGaIAMWWGyh7aHCNFRk2OsQJjK5kpMmskpwqcWsmZImcNETuFmZ6cgkJdi7q6QdGzrKdluxbtNunGFo61bt+ifZPusWWPteypRU+N7MiyI/OoDy07NLqJZROdbmrRtOnu2MJjrTux6MSke2LZEy2bWTQzslPLTrXszKIzI9uzTL4Uij2z7JnR9S3rm8U++iT2kRGOLBzpfAOLBqD8/2exMiizI1PBqWiOgqIfBGXzfoxXfWNhXROZ6FPfRO5gXoXq8kNba6jZnOX1oYOz2K9O3CwuqoNHr3aco9RWkoZJFQkZ9cr15PYHSopZtZ23XO+VlviRlkSR0fi40mBsRUdK87tRdKX9wLrHPddzOzVxO41owtSembDmhCSoFtb13Y7t6rq0L1BFigO9seXA6gGsnLgjimJeNGfiyHD8kqozGgdqTBrLIVUukBoQaRAZcKnBZbkuv7Te59/V6zfHnU3vj83O8E/3YVd/c9ecX53fnAeO5/ztPHQOnIEzdYhDnX+d186b1tvWf633rQ9KevOGjvnFWbnWWh8BrCrr0Q== g that Figure 1: In the top picture we provide a graphical representation of the transformation 0, with input wires + [ + Sg NR ⊆ Ng . on the top left and output on the bottom right. In the g∈R bottom picture we provide a graphical illustration of the 0 0 0 0 identity SgSg0 = Sg0 Sg. Along with Eq. (81) the latter proves that actually and ﬁnally this implies that + [ + NR = Ng . (95) † † 0† † W aRC = (S +∗ + S +∗ ⊗ I )aRC. g∈R N ∪N N ∪R1 C R0 R1 R0 (90) − 0 + Analogously, since NR = {g | ∃g ∈ R, g ∈ Ng0 }, We now observe that it is −1 −1 [ W = V ⊗ VH = SH WSH , − 0 0 − − H0 1 NR = {g | ∃g ∈ R, g ∈ Ng } = Ng . (96) g∈R and thus, by Eq. (90), it is −1† 0 † 0 Let us ﬁnally consider the maps (V −1BV ) for W aRC = (S +∗ ⊗ IC) aR0C NR ∪R0 (H) 1 B ∈ [[A(R,1)C → A(R,1)C]]R, with R ∈ R . 0† † † = (S +∗ S ⊗ I )aRC, (91) In this case, using Eq. (89) and manipulating N ∪R0 R0∪R1 C R1 Eq. (94) we obtain 0 † where aR0C = [S a](R1,0)∪(R0,1)C. Let us then ⊗ ( −1 ) consider R1 = ∅. In this case for [a(R,0)C] ∈ IH0 V BV ¯ (G) ¯ 0 † 0 [[A C]], being R = (R, 0) ∪ (∅, 1), it holds that = (S +∗ ⊗ IC)(B ⊗ IH1 )(S +∗ ⊗ IC). (R,0) NR NR (97) † † 0† † [(V a)(N +∗,0)C] = S +∗ S +∗ ⊗ IC[a(R,0)C], R NR NR 0 (92) Since the Sh’s that do not commute with B ⊗ − −1† 0† † † IH1 in Eq. (97) correspond to h ∈ Ng for some [(V a)(N +,0)C] = SR SR ⊗ IC[a(R,0)C]. (93) − R g ∈ R, namely h ∈ NR , we can conclude that

In view of the above results, we now consider the +∗ − −1 NR ⊆ NR . (98) maps VAV for A ∈ [[A(R,0)C → A(R,0)C]]R, (H) with R ∈ R . First of all, we observe that ±∗ This observation is suﬃcient to prove that NR = −1 −1 N ∓, as anticipated in the previous section. (VAV ) ⊗ IH1 = W (A ⊗ IH1 )W R

(G) † is a local transformation in ∈ [[A + C → Lemma 52. Let (G, A, V ) be a GUR. Then (NR ,0) (G) (G) ¯ ¯ ±∗ ∓ A + C]]QR. Let then a ∈ [[A ˜ C]]R for S ∈ N = N . (99) (NR ,0) S R R

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 40 ± ∓ Proof. We remind that g ∈ Nf iff f ∈ Ng . Now, two following ingredients that are needed for the considering Eq. (98) for R = g we have that definition of homogeneity: i) a notion of “running the same test on different regions”, and ii) a cri- g0 ∈ N −∗ ⇔ g ∈ N +∗ ⇒ g ∈ N − ⇔ g0 ∈ N +. g g0 g0 g terion for equivalence of two homologous regions −∗ + provided by the requirement that the events of This implies that Ng ⊆ Ng . Exchanging V −1 +∗ − the same test in the two regions have equal prob- and V , we can conclude that Ng ⊆ Ng ⊆ +∗ +∗ − abilities, if the regions are prepared in the same Ng , i.e. Ng = Ng . The thesis follows invoking Eqs. (95) and (96), which then give us state. However, we are in a situation where the struc- − +∗ NR = NR . ture of G must emerge from the evolution given by the global rule V . The detailed idea is thus to In the remainder we will then use N ± and P ±, R R turn the above argument on its head, and define and abandon the symbols N ±∗ and P ±∗. R R two regions to be homologous if they are “treated We observe that the block decomposition of in the same way" by V . The precise construction ⊗ −1 that we achieved through the block V V is the following. transformations S 0 is a generalisation of a result g First of all, in order for the dynamics to treat that was proved for quantum cellular automata two regions in the same way, the two regions need in Ref. [32]. Local rules of other kinds are used to have the same structure, in such a way that we in the literature, with nice properties that the can make sense of “performing the same range of present block decomposition lacks, such as com- tests in the two different regions”. This concept is posability (see e.g. Ref [56]). However, for later captured by the extension of the notion of oper- purposes the block form of Eq. (86) is particularly ational equivalence (see definition2) to regions. suitable in our case, to cope with update rules in Intuitively, two operationally equivalent regions theories without local discriminability. must correspond to operationally equivalent systems. However, they also need to“have the same 6 Homogeneity structure”, where the term structure refers to the fact that a system AR associated with a region The principle of homogeneity, as it is usually for- R is decomposed in a preferred way into subsys- mulated in physics, regards equivalence of points tems Agi corresponding to the cells that compose in space-time. Here, however, space-time is not the region R = {g1, g2, . . . , gk}. The notion of the background for occurrence of physical events, operational equivalence of regions must then also on the contrary it emerges as a convenient de- capture the idea that two equivalent regions ad- scription of relations between information pro- mit equivalent decompositions. cessing events. We then provide a definition The definition is thus recursive, starting from of homogeneity for a global update rule, that elementary regions R = {g}, and then defining regards the way in which different systems are equivalence of R1 and R2 as the equivalence of treated by the evolution. In particular, roughly any strict subregion S1 ⊂ R1 with a strict subre- speaking, every system must be treated in the gion S2 ⊂ R2, and viceversa. same way by the GUR. We recall Remark5, namely that for the re- It would be easy to state the homogeneity prin- mainder every elementary region g ∈ G corre- ciple if we knew in advance that the set G has sponds to some non-trivial system, Ag 6' I. some geometric structure that is invariant under the action of a group—technically speaking, if G Definition 41. Two elementary regions g and h ∼ were a symmetric space—so that we could de- are operationally equivalent if Ag = Ah. Two fi- fine pairs of homologous regions as those mapped nite regions R1,R2 ⊆ G are operationally equiv- into one another by some group element. In this alent if AR1 ' AR2 , and for every S1 ⊂ R1 there case, we could provide a criterion for the evolu- exists S2 ⊂ R2, and viceversa for every S2 ⊂ R2 tion to treat every system in the same way, con- there exists S1 ⊂ R1, such that AS1 and AS2 are sisting in the existence of some representation of operationally equivalent. the group of symmetries of the set G that connects the evolution of every two homologous re- Two operationally equivalent regions have the gions. The group in this case would provide the same structure, namely they can be decomposed

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 41 into pairwise equivalent subsystems, as we now Deﬁnition 42. Given two operationally equiv- prove. alent regions R1 = {g1, g2, . . . , gk},R2 = (G) {h1, h2, . . . , hk} ∈ R , let us choose a special re- Lemma 53. An elementary region R1 = {g} versible transformation U ∈ [[AR1 → AR2 ]] such cannot be operationally equivalent to a non- Nk that U = i=1 Ui, where Ui ∈ [[Agi → Ahi ]] elementary one R2 = {h1, . . . , hk}. is reversible for every i. We then say that R1 Proof. The statement is trivial, since R1 = {g1} is operationally equivalent to R2 through U , and does not have proper subregions, while any non- (2) n that {Bl }l=1 ⊆ [[AR2 C → AR2 C]] represents the elementary region R2 does, so operational equiv- (1) n same test as {B } ⊆ [[AR C → AR C]] if for alence cannot hold. l l=1 1 1 every 1 ≤ l ≤ n one has

Lemma 54. The regions R1 and R2 are op- (2) (1) −1 B = (U ⊗ IC)B (U ⊗ IC). (100) erationally equivalent if and only if R1 = l l ∼ {g1, g2, . . . , gk}, R2 = {h1, h2, . . . hk}, and Agi =

Ahi . In order to discriminate the way in which two Proof. The condition is clearly sufficient, as one systems Ag1 and Ag2 evolve, one needs to per- can immediately realise by considering the re- form a testing-scheme, consisting of a stimulus- versible transformation TS ∈ [[AS1 → AS2 ]]1 de- (x) m N test {Ak }k=1 ⊆ [[ARx C → ARx C]]Q on the re- fined as TS := Tj, with Tj ∈ [[Ag → gj ∈S1 j gion R 3 g , possibly involving an ancillary sys- A ]] reversible for every j = 1, . . . , k. We now x x hj 1 tem C, and, after the evolution step V , a control- prove that the condition is necessary. Let R1 (x) n test {Bl }l=1 ⊆ [[AN + C → AN + C]]Q on the re- and R2 be operationally equivalent. By defini- Rx Rx gion N + (where x ∈ {1, 2}) and the ancilla C, in tion, and by lemma 53, for every gl ∈ R1 there Rx ∼ order to detect a “response”. is hm ∈ R2 such that Agl = Ahm , and viceversa. We now introduce the equivalence relation Two operationally equivalent regions are homologous if there is a GUR T that interchanges g ∼ g ⇔ A =∼ A . i j gi gj them while preserving equivalence of local tests

We can then partition R1 and R2 into equivalence of finite regions. In rigorous words, we have the classes [gi] and [hj], respectively, obtaining R1 = following definition. S S g [gil ] and R2 = h [hil ]. Let us start consid- il il Definition 43 (Homologous regions). Let ering the case where R1 contains a unique equiv- † (G) (G, A, V ) be a GUR, and let R1,R2 ∈ R alence class: R1 = [g1]. Then R2 must contain at be operationally equivalent regions through ∈ ∼ Ti least one element h1 with Ah = Ag . Moreover, + + 1 1 [[AR → AR ]], such that N and N are op- R cannot contain h such that h 6∼ h , because 1 2 R1 R2 2 0 0 1 erationally equivalent through ∈ [[A 0 → To R1 h0 must be equivalent to an elementary subregion AR0 ]]. We say that R2 is homologous to R1 if of R , by lemma 53. Then, R = [h ]. Finally, 2 1 2 1 there exists an admissible GUR (G, A, T †), such since A⊗h ∼ A ∼ A ∼ A⊗k, by lemma2 g1 = R2 = R1 = g1 (i) m (i) n it must be h = k. In the general case where that for all schemes ({Ak }k=1, {Bl }l=1) ⊆ [[ARi C → ARi C]]Q × [[AN + C → AN + C]]Q rep- R1 contains more than one equivalence class, one Ri Ri resenting the same pair of tests for i = 1, 2, one can simply consider the subregions S1l := [gil ]. (1) (1) −1 (2) (2) Clearly, for each S1l there must be S2l = [hjl ], has Bl VAk = T Bl VAk T . We de- with |S2l | = |S1l |. note this relation as R2 ./ R1. Remark 7. Notice that, as a consequence of the We remark that, while the relation ./ defined above result, two operationally equivalent regions above clearly depends on V , we will generally not (G) R1,R2 ∈ R have the same cardinality |R1| = make this fact explicit in the notation, as we will |R2|. always consider contexts where a given V is considered, and there will not be room for ambiguity. Given two operationally equivalent regions On the other hand, when we want to specify the R1,R2 ⊆ G, we now have a way of defining the GUR T that makes R1 homologous to R2, we notion of “running the same test”, that resorts to write R ./ R . the identification of a canonical isomorphism of 2 T 1 the two regions. Lemma 55. If R2 ./T R1 then R1 ./T −1 R2.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 42 (i) m (i) n Proof. For every scheme ({Ak }k=1, {Bl }l=1) We remark that, straightforwardly, with (1) = ( −1 ⊗ ) (2)( ⊗ ) and Ak Ti IC Ak Ti IC ∼ (1) −1 (2) (R2 ./ R1) ⇒ (AR1 = AR2 ). (101) Bl = (To ⊗ IC)Bl (To ⊗ IC), one has (1) (1) = −1 (2) (2) . Now, invert- Bl VAk T Bl VAk T Now, the discrimination of systems g1 and g2 ing on the right and −1 on the left, one T T is successful if there exists a region R1 3 g1 such (2) (2) (1) (1) −1 obtains Bl VAk = TBl VAk T . that for any operationally equivalent region R2 3 g2, the region R2 is not homologous to R1. In Lemma 56. Let V be a global rule, and let the precise terms, we have the following definition. region R2 be homologous to R1 through T . Then T −1VT = V . Definition 44 (Absolute discrimination). The evolution given by the global rule V discriminates Proof. It is sufficient to consider the special case two systems g1, g2 ∈ G if for any GUR T there where the test scheme is ({ (1)}, { (1)}), with A0 B0 exists a region R1, containing g1, for which there (1) (1) A0 = IR1C and B0 = IN + C. Then, we is no region R2, containing g2, that is homologous R1 (2) (2) to R1 through T , with g1 ./T g2. have A0 = IR2C, B0 = IN + C, and V = R2 T −1VT . In simple words, the evolution given by the global rule V allows for discrimination of the two Lemma 57. Let V be a global rule, and let R2 ./ systems g1, g2 ∈ G if there exists a test that gives R1 for some suitable T . Then different probabilities depending on whether it is applied at g1 or at g2, or if the tests that one can [[A(G)C → A(G)C]] −1 = [[A(G)C → A(G)C]], T R1 R1 T R2 R2 run on R1 cannot be performed on any region R2 (G) (G) (G) (G) containing g , or viceversa. [[A C → A C]] −1 = [[A C → A C]]. 2 T N + N + T N + N + R1 R1 R2 R2 Now, the notion of discrimination of systems g1 and g2 with respect to a third system e is given Proof. Let us choose the test scheme as as follows. (1) m (1) (1) ({ } , { }), with = 0 . Then, Ak k=1 B0 B0 IR1C (2) we have = 0 , and Definition 45 (Relative discrimination). The B0 IR2C evolution V discriminates two systems g1, g2 ∈ G (1) −1 (2) relatively to a reference system e, with e ∈ G, if VAk = T VAk T for every T there exists a region R1, containing −1 (1) −1 =T V (Ti ⊗ IC)Ak (Ti ⊗ IC)T {g1, e}, for which there is no region R2, contain- −1 −1 (1) −1 ing {g2, e}, homologous to R1 through T with =T VTT (Ti ⊗ IC)Ak (Ti ⊗ IC)T e ./T e and g1 ./T g2. −1 (1) −1 =VT (Ti ⊗ IC)Ak (Ti ⊗ IC)T , We can now require the evolution described where in the last equality we used the result of by a global rule to be homogeneous by a precise lemma 56. If we now invert V to the left on both statement. sides, we obtain Principle 1 (Homogeneity). The global rule V (1) −1 (1) −1 discriminates any two systems g 6= g ∈ G, rela- Ak = T (Ti ⊗ IC)Ak (Ti ⊗ IC)T , 1 2 tively to an arbitrary reference system e ∈ G, but and finally not absolutely.

(1) −1 Before providing a convenient restatement of TAk T (1) −1 (2) the principle, we analyse some of its aspects and =(Ti ⊗ IC)Ak (Ti ⊗ IC) = Ak . consequences. In the ﬁrst place, the principle says that if we do not choose a reference element A similar argument leads us to conclude that e ∈ G, every two systems g1 and g2 cannot be (1) −1 discriminated. If we take the negation of deﬁni- TBk T tion 44, we obtain a straightforward consequence (1) −1 (2) =(To ⊗ IC)Bk (To ⊗ IC) = Bk . of homogeneity.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 43 Lemma 58. Let V satisfy the homogeneity prin- Lemma 61. Let V be non trivial and satisfy the ciple1. For every pair g1, g2 ∈ G, there exists homogeneity requirement, and R2 ./ R1. Then T such that for every choice of region R1 3 |R1| = |R2|. g1, there exists an operationally equivalent re- Proof. This follows from lemmas 54 and 60, since gion R2 3 g2 through Ti ∈ [[AR1 → AR2 ]] such that also N + and N + are operationally equiv- R1 and R2 are operationally equivalent by deﬁ- R1 R2 nition. alent through ∈ [[A 0 → A 0 ]], and for all To R1 R2 (i) m (i) n schemes ({Ak }k=1, {Bl }l=1) representing the Let now g1 and g2 be an arbitrary pair of sys- (1) (1) same tests for i = 1, 2, one has Bl VAk = tems in G, and V a homogeneous GUR. By the (2) (2) −1 homogeneity principle1, there exists T such that TBl VAk T . for every region R1 3 g1 there is a region R2 3 g2 Proof. The statement follows negating deﬁni- homologous through T to R1. The set of these tion 44. transformations is denoted as TV , i.e.

We can now draw a ﬁrst non trivial conse- TV := {T | ∃g1, g2 ∈ G quence from the statement of principle1. : ∀R1 3 g1 ∃R2 3 g2,R2 ./T R1}.

Lemma 59. For a homogeneous GUR Lemma 62. Let satisfy homogeneity, and let † V (G, A, V ), all local systems Ag for g ∈ G T ∈ TV . Then there exists a permutation π : are operationally equivalent. G → G such that if R2 ./T R1 it is

Proof. Let us consider two elements g1, g2 ∈ G. R2 = π(R1) (103) As a consequence of the homogeneity principle1, for the special choice R1 = {g1}, there exists Proof. By definition, if T ∈ TV there exist R 3 g such that R ./ {g } for a suitable . 2 2 2 1 T g1, g2 ∈ G such that for every S1 3 g1 there Now, by lemma 54, it must be |R2| = |{g1}| = 1, is S2 3 g2 such that S2 ./T S1. Let now i.e. R2 = {g2}. Thus, we have R2 ./T R1, and f1 ∈ R1. Consider the finite region V := {f , g }. Then there is a region ∼ 1 1 1 Ag2 = Ag1 . V2 3 g2 such that V2 ./T V1, and by lemma 61, one has |V | = |V | = 2. Consequently, it must Corollary 15. If the region R containing g is 2 1 2 2 be V = {f , g }. Moreover, since we know that homologous to {g } through , then R = {g }, 2 2 2 1 T 2 2 g ./ g , by definitions 41, 42, and 43, it is and 2 T 1 f2 ./T f1. Thus, for every f1 ∈ R1 there is a unique f ∈ R such that f ./ f . We then T −1[[A(G)C → A(G)C]]T = [[A(G)C → A(G)C]]. 2 2 2 T 1 g1 g1 g2 g2 define π : G → G by setting π(f ) := f . By (102) 1 2 lemma 55 the map π is both injective and surjec- Proof. By the proof of lemma 59 one has that tive, and thus it is a permutation of G. R2 = g2. The thesis then follows from lemma 57. In the following, given T ∈ TV , we will denote T = Tπ where π is the permutation of G in lemma 62. The second result that we prove is a technical lemma that will play a crucial role in the following Lemma 63. Let (G, A, V †) be a homogeneous results. GUR, and π ∈ ΠV . Then the GUR Tπ is unique. Lemma 60. Let (G, A, †) satisfy the homo- V Proof. Let Tπ, Sπ ∈ TV correspond to the same geneity requirement. Then for every region R one ¯ ¯ permutation π. Let a ∈ [[AGC]]QR. Then by N ∼ ⊗|R| ∼ † has AR = x∈R Agx = A0 , where A0 = Ag for lemma 28 one has a = A eG0 , with A ∈ [[AGC → every g ∈ G. AGC]]QR. Thus

Proof. By lemma 59 we know that for every g ∈ −1 † −1 † † † (SπTπ ) a = [(SπTπ ) ⊗ I ]A eG0 G one has A ∼ A . The remaining part of the C g = 0 −1† † † † −1† † † statement thus follows straightforwardly. = (Tπ Sπ ⊗ IC)A (Sπ Tπ ⊗ IC)eG0 ,

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 44 0 ∼ where G = G ∪ h0 and Ah0 = C. Now, by deﬁ- We can now summarise all the results that nition we drew from homogeneity as follows: for every g , g ∈ G there exists a permutation π of G such † † † −1† † 1 2 (Sπ ⊗ IC)A (Sπ ⊗ IC) ¯ that π(g1) = g2, and a GUR Tπ of [[AG]]QR such † † † −1† † (G0) = (Tπ ⊗ IC)A (Tπ ⊗ IC) that for every R ∈ R and C † † † −1† † = (Ti ⊗ IC)A (Ti ⊗ IC), (G) (G) [[AR C → AR C]]Q∗ which implies −1 (G) (G) = Tπ [[Aπ(R)C → Aπ(R)C]]Q∗Tπ, (104) −1 † (SπTπ ) a = a. and Tπ leaves the evolution invariant, i.e.

Since the latter holds for every C and every a, we −1 Tπ VTπ = T . conclude that Sπ = Tπ. The transformations π form a group Π that acts Corollary 16. The set TV = {Tπ} is a repre- V regularly (i.e. transitively and freely) on G. sentation of a group ΠV = {π | Tπ ∈ TV } of permutations of G. Now, it is clear that, chosen e ∈ G, the elements of G can be identified with the elements We can summarise the above results as follows. of ΠV , as the action of ΠV on G is regular. For every pair g1, g2 there exists a reversible map In other therms, G is a principal homogeneous π : G → G such that π(g ) = g and a reversible 1 2 space for ΠV , and there is a bijection ΠV ↔ G. −1 transformation Tπ of [[AG]]Q such that Tπ ·Tπ is However, the homogeneity principle implies an an automorphism of [[AG → AG]]Q and for every even stronger result: the set G can be seen as R ⊂ G and every system C the set of vertices of a special Cayley graph of Π . The following arguments, largely inspired to [[A(G)C →A(G)C]] V R R Q∗ Refs. [25, 46], make the mentioned result rigor- −1 (G) (G) = Tπ [[Aπ(R)C → Aπ(R)C]]Q∗Tπ, ous. As a consequence of homogeneity, one has and Tπ leaves the GUR V invariant, i.e. 0 0 −1 π(g) π(g ) ⇔ g g . (105) Tπ VTπ = V . ± ± In other words, Nπ(g) = π(Ng ) for π ∈ ΠV , as The transformations π clearly form a group ΠV can be checked starting from the definition of N ±. whose action ΠV × G → G is transitive. g We now proceed considering the second part of The proof is provided in AppendixD. the homogeneity principle, i.e. the statement that Let now g ∈ G and π ∈ ΠV such that π(e) = g. One can then label the elements g0 ∈ N ± by discrimination of every pair g1 6= g2 is possible g −1 0 ± ±1 k relative to some third element e ∈ G. The main π (g ) ∈ Ne = {hi }i=1. Notice that for the consequence is that the action ΠV × G → G is moment we have no reason to claim that k < ∞. free. We will associate all the pairs (g, g0) ∈ E with 0 the color hi if g = π(e) and g = π(hi) for Lemma 64. Let V satisfy the homogeneity prin- 2 some π ∈ ΠV . This construction enriches the ciple1. If e 6= π ∈ ΠV , there is no element g of graph Γ(G, E), which becomes a vertex-transitive G such that π(g) = g. coloured directed graph, with colours corresponding to the labels h ∈ N + Proof. Let Tπ ∈ T . Suppose that there is i e V 0 g ∈ G such that π(g) = g. Then for every re- Given two elements g, g ∈ G, a path p from g to g0 is an array of elements p = (g , g , . . . , g ) ∈ gion R1 3 g, the region R2 := π(R1) contains g 1 2 k and one has R2 ./ R1, with g ./ g. This is T T 2In principle, the colour of an edge (g, g0) might in contradiction with definition 45, i.e. with the be ambiguously defined if there existed two permuta- homogeneity principle1. −1 −1 tions π, σ ∈ ΠV such that π (g) = σ (g) = e and −1 0 −1 0 ± −1 0 −1 0 π (g ), σ (g ) ∈ Ne , but π (g ) 6= σ (g ). How- Corollary 17. Let ΠV be the group of permu- ever, due to lemma 64, this is never the case, since it −1 0 0 −1 tations of G such that TV is a representation of would imply that σπ (g ) 6= g , and thus σπ 6= e, but [σπ−1](g) = σ(e) = g. ΠV . The action of ΠV on G is regular.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 45 ×k 0 G such that, setting g0 := g and gk+1 := g , This notation is particularly handy, since one has g(hp1 hp2 . . . hpm ) = (ghp1 )hp2 . . . hpm . i1 i2 im i1 i2 im ∀0 ≤ i ≤ k (gi gi+1) ∨ (gi+1 gi). For every w ∈ F , and for every g ∈ G and The lenght of a path p, denoted `(p) is k +1 if p ∈ for every permutation π ∈ ΠV , we now show that G×k. In the following we will restrict attention π(gw) = π(g)w. The first step consists in proving ±1 0 to the case where Γ(G, E) is connected, i.e. for the result for w = hi . Let g = ghi, namely by 0 −1 −1 0 every two elements g, g ∈ G there exists a path definition σ (g) = e and σ (g ) = hi for some 0 0 0 from g to g . The set of paths from g to g will (unique) σ ∈ πV . By Eq. (105) it is π(g) π(g ). + 0 0 be denoted by pg,g0 . We can now introduce the Moreover, if we define f := π(g) and f := π(g ) graph metric of Γ(G, E) as follows. Let g, g0 ∈ G. we have Then we define −1 0 −1 −1 0 hi = σ (g ) = σ π (f ), 0 dΓ(g, g ) := min `(p) −1 −1 −1 p∈p+ e = σ (g) = σ π (f), g,g0 0 0 0 namely f = fhi. This implies We remark that dΓ(g, g ) = dΓ[π(g), π(g )] for all π ∈ Π , since g , . . . , g is a path from g to g0 if 0 V 1 k π(ghi) = π(g ) = π(g)hi. (106) and only if π(g1), . . . , π(gk) is a path from π(g) 0 0 −1 −1 to π(g ) due to condition (105). Similarly, if g = ghi by definition σ (g) = hi + − −1 0 We now use the alphabet Ne ∪ Ne to form and σ (g ) = e for some (unique) σ ∈ πV . Thus, arbitrary words, obtaining a free group F: com- setting again f := π(g) and f 0 := π(g0), we have position corresponds to word juxtaposition, with −1 −1 −1 the empty word λ representing the identity, and hi = σ (g) = σ π (f), −1 −1 −1 0 −1 −1 0 the formal rule hihi = hi hi = λ. An element e = σ (g ) = σ π (f ), p1 p2 pn w = h h . . . h of F —with pj ∈ {−1, 1}— i1 i2 in 0 −1 thus corresponds to a path on the graph, where namely f = fhi , and the symbol h−1 ∈ N − denotes a backwards step i e π(gh−1) = π(g0) = π(g)h−1. (107) along the arrow h . For every hp1 hp2 . . . hpm = i i i i1 i2 im w ∈ F , one has w−1 = h−pm . . . h−p2 h−p1 . im i2 i1 Notice that the last result implies that for every The action M : F × G → G of words w = pair f, g ∈ G, given π ∈ ΠV such that π(f) = g, hp1 . . . hpm−1 hpm ∈ F on elements g ∈ G can be ± ± i1 im−1 im there is a bijection Nf ↔ Ng , given by defined as follows ± ± Nπ(g) = π(Ng ). (108) M(w, g) = M(hp1 hp2 . . . hpm , g) i1 i2 im := M[hp2 . . . hpm ,M(hp1 , g)], One can now prove that π(fw) = π(f)w by in- i2 im i1 duction on the length l(w) of the word w. Indeed, where the action of hi ∈ S on the elements g ∈ G we know that it is true for l(w) = 1. Suppose now 0 is defined as M(hi, g) = g if there exists π ∈ ΠV that for l(w) = n − 1 one has π(fw) = π(f)w, −1 −1 0 0 0 0 p such that π (g) = e and π (g ) = hi, while and consider w with l(w ) = n. Then w = whi −1 0 M(hi , g) = g if there exists π ∈ ΠV such that with l(w) = n − 1 and p = ±1. We have π−1(g) = h and π−1(g0) = e 3. In the following i 0 p p we will use the shortcut π(fw ) = π(fwhi ) = π[(fw)hi ] p p 0 = π(fw)hi = π(f)whi = π(f)w , gw := M(w, g). where the induction hypothesis is used in the 3The action is well defined. Suppose indeed that there fourth equality. are two elements f, f 0 both satisfying the definition of Let us now suppose that for some f ∈ G and M(hi, g). This means that there exist π, σ such that −1 −1 −1 −1 0 some word w ∈ F one has fw = f. Then for π (g) = σ (g) = e, while π (f) = σ (f ) = hi. 0 0 However, according to lemma 64, the first condition im- every f ∈ G one can take π such that π(f) = f , plies that σ = π. By the same lemma, the second condi- thus obtaining tion then implies that f = f 0. A similar argument shows −1 0 0 that M(hi , g) is well defined. f w = π(f)w = π(fw) = π(f) = f .

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 46 Thus, if a path w ∈ F is closed starting from we will draw an undirected edge to represent hi. f ∈ G, then it is closed also starting from any The presentation can be chosen by arbitrarily di- −1 other g ∈ G. viding S into S+ ⊆ S and S− := S+ in such a We can now easily see that the subset R of F way that S+ ∪ S− = S. The above arbitrariness corresponding to words r such that gr = g for is inherent the very notion of group presentation all g ∈ G is a normal subgroup. Indeed, R is a and corresponding Cayley graph, and is exploited subgroup because the juxtaposition of two words in the literature, in particular in the definition of s, s0 ∈ R is again a word ss0 ∈ R, and for every isotropy [25, 57]. word s ∈ R also s−1 ∈ R. To prove that R is nor- For convenience of the reader we remind the mal in F we just show that it coincides with its definition of Cayley graph. Given a group G and normal closure, i.e. for every w ∈ F and every r ∈ a set S+ of generators of the group, the Cayley R, we have wrw−1 ∈ R. Indeed, defining for arbi- graph Γ(G, S+) is defined as the colored directed trary g the element g0 := gw, we have g0w−1 = g, graph having vertex set G, edge set {(g, gh) | g ∈ and thus gwrw−1 = g0rw−1 = g0w−1 = g, namely G, h ∈ S+}, and a color assigned to each gener- wrw−1 ∈ R. ator h ∈ S+. Notice that a Cayley graph is reg- We thus identified a normal subgroup R con- ular—i.e. each vertex has the same degree—and taining all the words r corresponding to closed vertex-transitive—i.e. all vertices are equivalent, paths. If one takes the quotient F/R, one obtains in the sense that the graph automorphism group a group whose elements are equivalence classes of acts transitively upon its vertices. The Cayley words in F . If we label an arbitrary element of graphs of a group G are in one to one correspon- G by e, it is clear that the elements of G are in dence with its presentations, with Γ(G, S+) cor- one-to-one correspondence with the vertices of G, responding to the presentation hS+|Ri. since for every g ∈ G there is one and only one class in F/R whose elements lead from e to g. 7 Locality We can then write g = w for every w ∈ F such that w represents a path leading from e to g. No- If a GUR were to represent a physical law, we tice that the elements of F/R, i.e. equivalence would like that, in order to determine it, it is classes of words [w]R, correspond to elements of sufficient to test it in a finite region and for a fi- G, i.e. for every g ∈ G there is a unique class nite amount of time. This possibility is granted [w]R such that g = [w]R. The criterion for class by homogeneity in conjunction with a second membership of words is very simple: w ∈ g iff w principle—locality—to which the present section connects the vertex e ∈ G to the vertex g ∈ G. is devoted. The locality requirement can be As a consequence of the above arguments, we phrased in simple words by stipulating that in can now show that for every π ∈ ΠV and for every order to calculate the effects of a physical law, i) g = [w]R ∈ G, one has π(g) = [sw]R for a fixed it is possible to adopt a reductionist procedure, word s ∈ F . Indeed, consider g0 = [w0]R. Let decomposing an arbitrary system into elementary 0 0 0 −1 [w ]R = π(g0). Then w = w w0 w0 = sw0, with parts and then calculating its evolution by evolv- 0 −1 s := w w0 , and π(g0) = [sw0]R. Let now f ∈ G. ing every part separately, considering only pair- We can always find t ∈ F so that f = [w0t]R, wise interactions between parts, and ii) if the re- −1 e.g. by setting t := w0 z, f = [z]R. Then we have duction is operated, only finite systems will be π(f) = π([w0t]R) = π(g0t) = π(g0)t = [sw0t]R = involved in the calculation for every elementary [sz]R, where we use the definition π(k)r := [xr]R part. with π(k) = [x]R. This definition clearly makes In the present section we introduce the precise sense, since when [a]R = [b]R and [r]R = [u]R, statement of the locality principle, and analyse its also [ar]R = [bu]R. main consequences. In order to make this notion In technical terms, the graph Γ(G, E) = independent of homogeneity, we formulate it for Γ(G, S) is the Cayley graph of the group G = general admissible GURs. In particular, this im- F/R. Homogeneity thus implies that the set G is plies that the mathematical statement of locality a group G that can be presented as G = hS|Ri, alone will not allow for determination of a local where S is the set of generators of G and R is the GUR by observation of a finite region. −1 group of relators. In the following, if hi = hi Before giving a formal definition, we provide a

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 47 (G) ± S ± few heuristic steps leading to the final formula- For X ∈ R we then define NX := g∈X Ng , ± tion. Let us start considering an admissible GUR and the sets Pg are then defined exactly as for a (G, A, V †), with uniformly bounded neighbour- GUR. Finally, we suppose that a family of maps 0 hood, namely such that there exists k < ∞ so Sg is defined, with the properties (88) and (89), ± that for every g ∈ G it is |Ng | ≤ k. We remark and an analogue of (94) and (97). ± ± 2 that, since |Ng | ≤ k, it is |Pg | ≤ k . This also 0 Definition 46 (Local rule). A k-local rule implies that the transformations Sg are transfor- + 0 + (G) mations of finitely many systems. (N , A, S ) on G is i) a map N : G → R + In the remainder, we will often omit the labels that associates g ∈ G with a finite region Ng ∈ (G) + − 0, 1 on the wires in diagrams, and adopt the con- R such that |Ng |, |Ng | ≤ k for every g ∈ G; 0 0 0 vention that the upper half is labelled 0 and the ii) a map S : g 7→ Sg, where Sg are maps in lower one 1. [[A + A → A + A ]]1 with the follow- (Ng ,0) (g,1) (Ng ,0) (g,1) Now, reminding Eq. (92), given a ∈ ing properties for every f, g ∈ G and R,S ∈ R(G): ¯ (G) [[AR C]]QR, one has 0 0 1. SgSg = IA + Ag ; † † † Ng [(V a) − ] = [{(V ⊗ I )a} − ] (NR ,1)C R C (NR ,1)C 0 0 0 0 0† † 2. SgSf = Sf Sg; = [{(S − ⊗ IC)a}(N −,1)C]. NR R 3. for every C ∈ [[A C → A C]] Thus, we have (R,0)∪(S,1) (R,0)∪(S,1) − PR R + ψ − + N − \R − + P \N N ∪S P \N − R S R R S N P −\R R R 0 R + + − VR = S e . NS NS NR R R N − N − R R e 0 C C 0 SN −∪S C SN −∪S − R R − (109) NR S S NR S\N − N −\S S\N − Notice that, by admissibility of the GUR V , and R R R thanks to the no-restriction hypothesis, − + PR \NS VR ∈ [[A − → AR]]1. NR + + NS NS Moreover, the GUR V can be equivalently de- ¯ = C 0 C , (110) fined starting from its action on [[AG]]LR, by set- − C − † † NR NR ting V [aR] := [(VRa)N − ]. Indeed, following R S\N − this definition, one can prove that V is right- R invertible, by the argument discussed in Ap- pendixE. We remark that the construction of the 0 Q 0 where SX := g∈X Sg for every X ⊆ G. inverse provides a new GUR W . As a consequence, we have that W †V † = In the following, we will often omit the sub- † † † 0 V W = IG for some GUR W , and thus W = script X in SX in the diagrams, since X corre- V −1. We want to stress here that in proving sponds to the label of the lower half wires. More- this result we only use properties (88), (89), (94) over, in order to make the equations lighter, from 0 and (97) of Sg, derived in section 5.4, along with now on we will always write or draw transfor- uniform boundedness of the neighbourhoods for mations and effects that do not involve explicitly the GUR V (see AppendixE for the details). the external system C, however it will always be Let us now abstract our focus from the situa- assumed that the conditions hold also locally on tion where a GUR (G, A, V †) is given. Instead, extended systems, unless explicitly stated other- we will suppose that, given the pair (G, A), a fi- wise. + nite neighbourhood Ng ⊆ G is defined for every g ∈ G, and for f ∈ G we set Remark 8. Notice that for a transformation C of the form C0 ⊗I(R\R0,0)∪(S\S0,1), due to the def- − + 0 0 Nf := {h ∈ G | f ∈ Nh }. inition of S − and the property2 of Sg, one NR ∪S

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 48 has that Proof. The equalities follow taking S = ∅ and − + PR \NS R = ∅ in Eq. (110), respectively. P −\N + N +\N + R S S S0 Remark 9. Given a local rule (N +, A, S 0) we + + N + N + NS NS S0 S0 define the forward and backward evolution of local transformations F ∈ [[ARC → ARC]] for C 0 C = C 0 C C C0 a finite region R ∈ R(G) through the expres- N − N − N − N − R R R0 R0 sions (111) and (112), as follows − − − S\N NR \N 0 R R ± ± − [VL (F )]N ±C := [F ]N ±C. (113) S\NR R R Now, once we established the notion of a local [P −∪N +]\[P − ∪N + ] R S R0 S0 rule and its application to local transformations, we can define the property of a global rule of be- N + \R0 ing reducible to a local rule. P − \N + N− ∪S0 P − \N + R0 S0 R0 R0 S0 + + Definition 47 (Reduction to a local rule). We N 0 0 0 N 0 S R R S † = . say that the GUR (G, A, V ) is reducible to a 0 C C 0 + 0 S C0 S local rule if there exists a k-local rule (N , A, S ) N − N − (G) R0 S0 S0 R0 for some k < ∞ such that for every R ∈ R and S0\N − N − \S0 S0\N − every F ∈ [[ARC → ARC]] R0 R0 R0 [N −∪S]\[N − ∪S0] −1 − R R0 (V ⊗ IC)F (V ⊗ IC) = V (F ) ⊗ I − , L G\NR −1 + The first property that we prove is the follow- (V ⊗ IC)F (V ⊗ IC) = V (F ) ⊗ I + . L G\NR ing. (114) + 0 Lemma 65. Given a local rule (N , A, S ) on We now prove the two main results of this sec- G, the following identities hold: tion, which relate local rules and GURs in a one- (G) to-one correspondence. 1. For every S ∈ R , and every A ∈ [[ASC → ASC]], Theorem 9. Every local rule (N +, A, S 0) on G + + + 0 † 0 (NS ,0) (NS ,0) (NS ,0) identifies a GUR (G , A, W ), with G = G × −1 {0, 1}, and W = SGWSG. 0 C C 0 S − S − NS NS (S,1) (S,1) A (S,1) (S,1) Proof. First of all, one can define the linear maps † ¯ ¯ ¯ ¯ WC on [[AGC]]LR as follows. For aZC ∈ [[AGC]]LR,

+ + with Z = (R, 0) ∪ (S, 1), we set (NS ,0) (NS ,0) + † − C A C W [aZC] := [a − + ], = ; (111) (NR ,0)∪(NS ,1)C (S,1) where

(G) 2. for every R ∈ R and every B ∈ [[ARC → N˜ + N˜ + N−∪S N−∪S N\−∪S ARC]] R e R R e − − − N + \R (P ,0) (P \R,0) (P ,0) S\N − S\N − P −\N + N−∪S R R R R e R R S R e − − + (R,0) (R,0) NR NR NS R 0 0 S − S − N C B C N C C R R a− := S S 0 a . − − − + + − (NR ,1) (NR ,1) (NR ,1) NS NS NR S

P −\N + P −\N + S\N − N −\S R S e R S R R e − N˜ + (P ,0) N\−∪S N\−∪S N−∪S R R e R R e C C = . (112) We omitted the subscript for S and S 0, as the (N −,1) − (N −,1) R B R systems on which the transformations act are

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 49 ˜ + ¯ ¯ clear from the context. Finally, NX is a short- a(R,0)∪(S,1)C ∈ [[AG0 C]]LR. Then + + hand for NX \X, and Xb for X \NX . Writing a = † ˆ † A e, for A ∈ [[A(R,0)∪(S,1)C → A(R,0)∪(S,1)C]], (a(R,0)∪(S,1)C|W ρ) = (W a(R,0)∪(S,1)C|ρ) and using Eq. (110), one can easily check that † 0† =(S − + S − a |ρ) − ¯ ¯ N ∪S∪N N ∪S (S,0)∪(T,1)C the effect a is actually in [[A − + C]]. We R N−∪S R (NR ,0)∪(NS ,1) R remark that the map W † is well defined: for ˆ0 ˆ =(a(R,0)∪(S,1)C|SN −∪SSN −∪S∪N + ρ) R R N−∪S a ⊗ eW ∈ [aZC], with W = (W0, 0) ∪ (W1, 1), R and W0 ∩ R = W1 ∩ S = ∅, one has ˆ0 ˆ =(a|[SN −∪SSN −∪S∪N + ρ]|(N −,0)∪(N +,1)C), R R N−∪S R S R † † W [(a ⊗ eW )(R∪W0,0)∪(S∪W1,1)C] = W [aZC]. ˆ and clearly this implies that W ρ|(R,0)∪(S,1) is a (G) ˆ state for every R,S ∈ R . Then W [[AGC]] ⊆ The above identity is verified exploiting ˆ−1 [[AGC]]. The same argument holds for W , Eq. (110), along with the identity in Re- ˆ thus W [[AGC]] = [[AGC]]. Let now AT C ∈ mark8. As to reversibility, for [a ] with ZC [[AGC → AGC]]L, with T = (R, 0) ∪ (S, 1). One Z = (R, 0) ∪ (S, 1), one can define ˆ−1 ˆ can evaluate W AT CW by dually evaluating † † −1† W AT CW . The calculation is a straightfor- †[a ] := [a+ ], ward application of Eq. (110), and shows that Z ZC (N +,0)∪(N −,1)C R S ˆ−1 ˆ W AT CW ∈ [[AGC → AGC]]L. The same argu- ˆ ˆ−1 ment holds for WAT CW , and thus where ˆ−1 ˆ W [[AGC → AGC]]LW = [[AGC → AGC]]L.

N˜ + All the above observations prove that W is a UR. \− N−∪R \+ NR ∪S S NS ∪R e e + Finally, being constructed from local transforma- − + N − \S − − + P \N N ∪R N \R P \N S R S S tions, one can easily check that the family { | S R e e WC + N + C ∈ Sys} is an admissible UR. This concludes the NR R S R proof that (G0, A, W †) is a GUR. Now, comparing C + := 0 C a , ± a S S the equations defining a ± ∓ , one can − (N ,0)∪(N ,1)C N N − R S S S R S −1 + easily conclude that Z = W = SGWSG. − − − N − \S R\NS R\N N \R N ∪R e S S S + e N˜ ˜ + N−∪R N + 0 S N−∪R N\−∪S Theorem 10. Let (N , A, S ) be a local rule on e S R e G. Then the GUR (G0, A, W †), with G0 = G × {0, 1} in theorem9, is of the form W = V ⊗V −1, and, similarly to what was done for W , check where (G, A, V †) is reducible to (N +, A, S 0). that Z † is well defined. One can now verify that † † † † W Z = Z W = I ¯ . The above observa- The detailed proof can be found in AppendixF. [[AG]]LR tions show that W † and Z † = W −1† act isomet- The properties used in the proof are (88), (89), ¯ (94), and (97). The first two are required as rically on [[AG]]LR, since items1 and2 in the definition 46 of a local rule, while the remaining two are a consequence of ka k = k † †a k RC sup Z W RC sup item3 as shown by lemma 65. † ≤ kW aRCksup ≤ kaRCksup, In the case of a homogeneous update rule, does † † kaRCksup = kW Z aRCksup the existence of a local rule allow one to determine the full (G, A, V †) by local tests? One would in- ≤ k †a k ≤ ka k . Z RC sup RC sup tuitively expect that the answer is positive, since 0 0 Sg = Se are the same for every g ∈ G, by virtue † −1† 0 Being isometric, both W and W preserve of homogeneity, and thus, knowing Se, one could Cauchy sequences and their equivalence classes. calculate the action of V through Eq. (112). On As a consequence, W † and W −1† can be uniquely the other hand, after giving the question another ¯ extended to invertible isometries of [[AG0 ]]QR, thought, one can realise that calculating the ac- † †−1 ± with Z = W . Let now ρ ∈ [[AG0 C]], and tion of V through the maps VL requires another

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 50 0 piece of information besides knowing Se. Actu- relators. This fact has the very important con- ally, determining the cellular automaton with lo- sequence that every cellular automaton defines cal tests also requires knowledge of the regions a Cayley graph that is quasi-isometric to some + (G) NR for every R ∈ R —or, in other words, smooth Riemannian manifold of dimension d ≤ 4 knowledge of the structure of the graph of in- (see [47], sec. IV pag. 90). fluence relations. This information is indeed nec- An important remark is in order. Thanks to essary in order to know how to correctly calcu- homogeneity, one can easily show that the local ± 0 late the transformation VL (AR) for a given re- rule Sg for a cellular automaton on the Cayley gion R. Indeed, it might happen that knowledge graph Γ(G, S ) has the property that 0 = 0 + Sg SeG of finitely many closed paths from the elements for every g ∈ G. Thus, cellular automata can be g ∈ R is not sufficient to reconstruct all the new identified by their Cayley graph Γ(G, S+) and by 0 closed paths that appear as the size of the region a reversible transformation ∈ [[A + Ae → SeG Ne G R increases. This means that one needs infinitely G AN + AeG ]]1. ± ± eG many rules to identify systems in Ng1 ∩Ng2 . How- ever, this is not the case if closed paths in the 8.1 Results graph of influence relations can be decomposed into elementary closed paths having a uniformly We now prove two important theorems regarding bounded size, let us say by a constant length l. In cellular automata. The first one shows that, in this case, knowledge of local rules and of the local order to obey the properties of homogeneity and structure of the graph up to some finite distance locality, a cellular automaton cannot be in the l is sufficient to calculate the action of V in an quasi-local algebra. arbitrary finite region. This is the reason for the Theorem 11. Let |G| = ∞. The homogeneous following definition. and local cellular automaton V is an element of Definition 48 (Decomposability into bounded [[AG → AG]]Q1 if and only if it is trivial: V = regions). We say that a global update rule IG. (G, A, V †) is decomposable into bounded regions Proof. Let V be in the quasi-local algebra, and if the closed paths of the influence graph of V be a sequence of local transformations in can be decomposed into elementary closed paths VnRn the class of . Then for ε > 0 there exists n of uniformly bounded length. V 0 such that for every n ≥ n0 one has We are finally in position to formulate the locality principle, which can be thought of as the re- kVnRn − V ksup ≤ ε. quirement that finite, local information is needed −1 Moreover, for π ∈ ΠV , one has TπVTπ = V . in order to reconstruct the update rule. Then Principle 2 (Locality). The GUR (G, A, V †) is k − −1k ≤ ε. reducible to a local rule and decomposable into VnRn TπVTπ sup bounded regions. Since Tπ is a GUR, it is isometric and thus −1 kTπ VnRn Tπ − V ksup 8 Cellular Automata −1 = kVnRn − TπVTπ ksup ≤ ε. (115) The principles of homogeneity and locality single Now, this implies that out the special class of global rules that we will −1 call cellular automata. kVnRn − Tπ VnRn Tπksup ≤ 2ε. Definition 49. Let (G, A, V †) be a global up- It is easy to prove that, if |G| = ∞, for every (G) date rule obeying the principles of homogeneity R ∈ R there must exist π ∈ ΠV so that R ∩ and locality. We say that (G, A, V †) is a cellular π(R) = ∅. Suitably choosing π, condition (115) automaton. then takes the form Notice that the influence graph of a cellular kW ⊗ I − I ⊗ W ksup automaton is the Cayley graph of a finitely pre- = kW ⊗ I − W ⊗ W + W ⊗ W − I ⊗ W ksup sented group, i.e. a group that is finitely generated and can be presented through finitely many ≤ 2kW − I ksup ≤ ε.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 51 Being ε arbitrary, we must conclude that W = group G satisfy the mentioned requirements. We

I , namely VnRn = 1∅. now explain the hypotheses in detail, and ﬁnally prove the main theorem. The second result pertains the following types We start remarking that if the group G has a of theory. ﬁnite presentation

1. Theories with local discriminability [8,9, G = ha1, . . . , an|r1, . . . , rki, 15, 58], or more generally every theory where the algebra of transformations A ∈ it is straightforward to verify that any quotient H of G has a finite presentation of the form [[A1 ... An → A1 ... An]] is generated by local transformations, i.e. transformations of H = hϕ(a1), . . . , ϕ(an)|ϕ(r1), . . . , ϕ(rk), b1, . . . , bji, the form where the new relations bk do not belong to the ∈ ⊗ [[A → A ]], Ai IA¯ i i i group R, conjugate closure of {ϕ(r1), . . . , ϕ(rk)}. When H is a quotient of G, referring to the ¯ where Ai is the composite system made of presentation above, we will denote by R0 the all A1 ... An except from Ai. This class in- conjugate closure of the group generated by cludes, among others, classical information k j {φ(rl)}l=1 ∪ {bm}m=1. Moreover, the neighbour- theory and quantum theory. 0± hoods of f in Γ(H, φ(S+)) will be denoted by Nf for f ∈ H. We now state two lemmas that we use 2. Theories where the algebra of transforma- in the following. The proofs are provided in Ap- tions ∈ [[A ... A → A ... A ]] is gener- A 1 n 1 n pendixG ated by bipartite transformations, i.e. transformations of the form Lemma 66. Let N + be a neighbourhood system corresponding to the Cayley graph Γ(G, S+) of a ∈ ⊗ [[A A → A A ]], Ai,j IA¯ iA¯ j i j i j a finitely generated group G, and let H = φ(G) be a finite quotient of G such that for every ¯ ¯ −1 0 where AiAj is the composite system made of ha, hb, hc, hd ∈ S+ one has φ(hahb ) ∈ R , if and −1 −1 −1 0 all A1 ... An except from Ai and Aj. This only if hahb ∈ R and φ(hahb hchd ) ∈ R if −1 −1 case includes the Fermionic theory as well as and only if hahb hchd ∈ R. real quantum theory. 0± ± 0± ± 1. Nφ(g) = φ(Ng ), and |Nφ(g)| = |Ng |. In the above cases, for groups G with suitable 2. P 0± = φ(P ±), and |P 0± | = |P ±|. properties that we will immediately specify, the φ(g) g φ(g) g admissible local rules for a cellular automaton can 3. Given f1, f2 ∈ H, one can choose gi ∈ be sought considering finite-dimensional systems. φ−1(f ) such that N 0± ∩ N 0± = φ(N ± ∩ N ±) i f1 f2 g1 g2 In the remainder we will then restrict to the fam- and |N 0± ∩ N 0±| = |N ± ∩ N ±|. ilies of theories introduced above. f1 f2 g1 g2 Before proving this result, we need some pre- 4. Given f1, f2 ∈ H, one can choose gi ∈ φ−1(f ) such that P 0∓ ∩ N 0± = φ(P ∓ ∩ N ±) liminary definition and lemma. i f2 f1 g2 g1 and |P 0∓ ∩ N 0±| = |φ(P ∓ ∩ N ±)|. Definition 50 (Quotient group). Let G, H be f2 f1 g2 g1 two groups, and suppose that there exists a group 5. If g 6∈ N ± and φ(g ) ∈ N 0± , it is P ± ∩ 2 g1 2 φ(g1) g1 homomorphism ϕ : G → H. Then H is a quo- ∓ Ng = ∅. tient of G. If H is finite, it is called a finite 2 quotient of G. Lemma 67. In addition to the hypothe- −1 −1 −1 ses of lemma 66, let φ(hahb hchd hehf ) ∈ The reason why we are interested in finite quo- 0 −1 −1 −1 R iff hahb hchd hehf ∈ R for every tients is that, under suitable assumptions that ha, hb, hc, hd, he, hf ∈ S+. Then will be made rigorous shortly, the same local rule 0± 1. Given f1, f2 ∈ H with f2 ∈ P , one has defines an automaton on G and an automaton on f1 P 0± ∩ P 0± = φ(P ± ∩ P ±), for some g ∈ any of its finite quotients H. This result makes f1 f2 g1 g2 i φ−1(f ), with g ∈ P ±, and |P 0± ∩ P 0±| = it much easier to provide unitarity conditions in i 2 g1 f1 f2 ± ± all those cases where the finite quotients of the |Pg1 ∩ Pg2 |.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 52 2. Let g 6∈ P ± and φ(g ) ∈ P 0± . Then N ± ∩ local rule (N +, A, 0) with neighbourhood sys- 2 g1 2 φ(g1) g1 S + + ± ± ± tem N such that |N | = |S+|, the local Ng2 = Pg1 ∩ Pg2 = ∅. eG rule (N 0+, A, ˜0), with ˜0 := 0 is well- S SeH SeG Corollary 18. Under the hypotheses of defined on the Cayley graph Γ(H, φ(S+)), with lemma 66, the homomorphism φ is invertible on 0+ + Nφ(g) = φ(Ng ). Viceversa, given a local N ± and P ± for every g ∈ G. g g rule (N 0+, A, S˜0) whose neighbourhood system 0+ In order to make the hypotheses of lemmas 66 N corresponds to a Cayley graph Γ(H, φ(S+)), + 0 0 : ˜0 and 67 clearer, in Fig.2 we show two examples the local rule (N , A, S ), with Se = Se is 2 well-defined on the Cayley graph Γ(G, S ), with of finite quotients of the group Z , the first one + φ(N +) = N 0+ satisfying the hypotheses of lemma 66 but not g φ(g) those of lemma 67, and the second one satisfying Proof. We remark that, by definition of the maps both. 0 (see Definition 46) a given choice of 0 Sg SeH is in principle compatible with every Cayley graph with |N 0± | = |N ± |, i.e. the same num- eH eG ber of generators (we remind that 0 = 0 ∈ Sg SeH [[AN + AeG → AN + AeG ]]1). By lemma 66, the eG eG cardinality of N 0+ is the same as that of N + . eH eG Then ˜0 = 0 is well defined on Γ(H, φ(S )) SeH SeG + iff it is well defined on Γ(G, S+). What remains to be proved is that the map ˜0 = 0 identi- SeH SeG (a) fies a local rule on the system of neighbourhoods N 0+—namely that Eq. (110) as well as items1 and2 in definition 46 hold—iff it does on the system of neighbourhoods N +. Item1 is satisfied by S˜0 = S 0 by definition in both cases. As to item2, it is trivial to verify that the only important feature in order to decide whether it holds in any neighbourhood system N 0+ is the eH cardinality of N 0+ ∩N 0+ (or N + ∩N +) for every eH f eG g f ∈ P 0+ (or g ∈ P − ), since this identifies the eH eG (c) systems where both ˜0 = 0 and ˜0 = 0 (b) SeH SeG Sf Sg act non-trivially, namely those where the prod- 2 Figure 2: (a) The Cayley graph of Z presented as ucts S˜0 S˜0 or S 0 S 0 might not be commuta- ha, b|aba−1b−1i; (b) the Cayley graph of the finite eH f eG g tive. Lemma 66 thus ensures that item2 is sat- quotient of 2 hφ(a), φ(b)|φ(aba−1b−1), φ(a)6, φ(b)5i, Z ˜0 satisfying the hypotheses of lemma 66 but not those isfied by S iff it is by S . Let us then focus on of 67; (c) the Cayley graph of the finite quotient Eq. (110). For theories of type1, it is sufficient of Z2 hφ(a), φ(b)|φ(aba−1b−1), φ(a)8, φ(b)7i, satisfying to verify that Eq. (110) holds for |R| = 1 and the hypotheses of both lemmas 66 and 67. Notice that |S| = 0 or |R| = 0 and |S| = 1. Both conditions −1 −1 we implicitly assume that S+ contains also a and b , easily follow if we consider lemma 66. but we do not express the presentation accordingly, to keep the formulas readable. In principle there should Lemma 69. For theories of type2, under the be two extra generators c, d (and φ(c), φ(d) in the fi- hypotheses of lemma 67, given a homogeneous nite quotients), along with the relations ac and bd (and + 0 φ(ac), φ(bd), accordingly). local rule (N , A, S ) with neighbourhood system N + such that |N + | = |S |, the local eG + rule (N 0+, A, S˜0), with S˜0 := S 0 is well- We can now prove the following results, which eH eG defined on the Cayley graph Γ(H, φ(S )), with are the core of the final theorem at the end of the + N 0+ = φ(N +). Viceversa, given a local section. φ(g) g rule (N 0+, A, S˜0) whose neighbourhood system 0+ Lemma 68. For theories of type1, under the N corresponds to a Cayley graph Γ(H, φ(S+)), + 0 0 ˜0 hypotheses of lemma 66, given a homogeneous the local rule (N , A, S ), with Se := Se is

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 53 well-defined on the Cayley graph Γ(G, S ), with Then, we apply ˜0 , obtaining + SN 0− φ(N +) = N 0+ f2 g φ(g) (N 0−,1) (N 0−,1) f1 f1 (N 0−,1) 0 (N 0−,1) f (f2,0) C (f2,0) f Proof. The first part of the proof proceeds ex- 2 2 actly as that of lemma 68. The non trivial part ˜0 0− ˜0 S 0− (N ,1) S 0− of the proof regards the equivalence of Eq. (110) N f2 N f2 f2 0 ˜0 0− 0− 0− for both S on G and S on H. Suppose that (P ,0) (P \f2,0) (P ,0) f2 f2 f2 S 0 defines a local rule for the neighbourhood sys- (P 0−,0) + f1 tem Ng . In theories of type2, it is sufficient 0− 0− (P \[P ∪f2],0) to verify that Eq. (110) holds for regions with f2 f1 |R| = 2, |S| = 0, or |R| = 1, |S| = 1, or |R| = 0, (P 0−∪P 0−,0) |S| = 2. Let then R,S ⊆ Γ(H, φ(S+)), with f1 f2 ˜ ˜ R = φ(R) and S = φ(S). In the first case we have (N 0−,1) (N 0−,1) f1 f1 R = {f1, f2}, with R˜ = {g1, g2} and fi = φ(gi), = , (116) (N 0−,1) 00 (N 0−,1) while S = ∅. The relevant informations in or- f2 C f2 der to verify Eq. (110) are the cardinalities i) 0− 0− 0− 0− 0− where on the l.h.s. we reversed the ordering of |N ∩ N | and ii) |P ∩ P |. If f2 ∈ P , f1 f2 f1 f2 f1 the wires for the sake of simplicity of the di- by lemma 67 we know that we can choose g1 0 0− 0− agram. Overall, considering that ˜ = and g so that |N ∩ N | = |N − ∩ N −| and SN 0−∪N 0− 2 f1 f2 g1 g2 f1 f2 |P 0− ∩ P 0−| = |P − ∩ P −|. Thus, since Eq. (110) ˜0 ˜0 = ˜0 ˜0 , we then proved f1 f2 g1 g2 SN 0− SN 0− SN 0− SN 0− (G) (G) f1 f2 f2 f1 holds for C ∈ [[A → A ]], and ˜0 R˜ R˜ Eq. (110) for the local rule Sf on the neighbourhood scheme given by the graph Γ(H, φ(S+)). (G) ∼ (H) ˜0 0 In the second case, let R = f1 = φ(g1), S = A ˜ = AR , S = S , R f2 = φ(g2). The informations that matter are whether one can choose g , g so that f ∈ N 0− 1 2 2 f1 we conclude that Eq. (110) holds also for C ∈ iff g ∈ N −, and |P 0− ∩N 0+| = |P − ∩N +|. Again 2 g1 f1 f2 g1 g2 [[A(H) → A(H)]]. On the other hand, if f 6∈ P 0−, R R 2 f1 by lemma 66, we know that the answer is posi- we can calculate ˜0 ˜0 in two steps. tive. Then, also for |R| = |S| = 1, since Eq. (110) SN 0− C SN 0− f1,f2 f1,f2 (G) (G) ˜0 0 holds for C ∈ [[A ˜ ˜ → A ˜ ˜ ]], and First we apply S 0− = S − , treating the system (R,0)∪(S,1) (R,0)∪(S,1) N Ng f1 1 f as an external system. Reminding that f 6∈ (G) (H) 0 0 2 2 A =∼ A , S˜ = S , P 0−, it is N 0− ∩ N 0− = ∅. By Eq. (110), that (R,˜ 0)∪(S,˜ 1) (R,0)∪(S,1) f1 f1 f2 0 holds for S − we then have we conclude that Eq. (110) holds also for C ∈ Ng1 [[A(H) → A(H) ]]. The argument for 0− 0− (R,0)∪(S,1) (R,0)∪(S,1) (P \[P ∪f2],0) f2 f1 |R| = 0, |S| = 2 is the same as for |R| = 2,

(f2,0) (f2,0) |S| = 0. For the converse, one can follow the (P 0−,0) (P 0−,0) same argument as for the direct statement. The f (f1,0) C (f1,0) f 1 1 only differences are that in the case |R| = 2 − and |S| = 0 one might have g2 6∈ Pg1 while ˜0 0− ˜0 S 0− (P \f1,0) S − − 0− N f1 N f = φ(g ) ∈ φ(P ) = P , and in the case f1 f1 2 2 g1 f2 (N 0−,1) (N − ,1) (N 0−,1) − f1 f1 f1 |R| = |S| = 1 it might happen that g2 6∈ Ng1 0− 0− while f ∈ N . By lemma 67, the first case (Nf ,1) 2 f1 2 + + − − can occur only if Ng1 ∩ Ng2 = Pg1 ∩ Pg2 = ∅, 0− 0− while by lemma 66 the second case occurs only (P ∪P \f2,0) f1 f2 − + if Pg1 ∩ Ng2 = ∅. In both cases, the different (f2,0) (f2,0) structure of the neighbourhoods between G and (N 0−,1) 0 (N 0−,1) = f1 C f1 . H does not preclude the derivation of Eq. (110) in the case of G from the same identity for the case of H, as one can immediately realise by di- (N 0−,1) f2 rect inspection of Eq. (110).

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 54 Theorem 12 (Wrapping lemma). For a theory though we cannot even try to provide an exhaus- of type1, under the hypotheses of lemma 66, tive account of the relevant bibliography—it is the local rule (N +, A, S 0) defines a cellular au- customary to express V through its local rule, tomaton (G, A, V ) on the Cayley graph Γ(G, S+) which is specified by its action on local states. if and only if the same local rule defines a cel- The only point we want to discuss here is the lular automaton (H, A, W ) on the Cayley graph apparent clash of our general wrapping lemma Γ(H, φ(S+)). The same is true of a theory of with results in the literature. In particular, in type2, under the hypotheses of lemma 67. Ref. [31] a counterexample to the classical version of the wrapping lemma is provided, referring Proof. The thesis now easily follows from lem- to [48]. The local rule for the counterexample is mas 68 and 69, and theorem 10. the following. Let G = Zr with r 6∈ 3Z. The rule defined by The above result allows us, at least in theories 0 of the two types considered so far, and for CAs bj = bj r1 ⊕ bj ⊕ bj⊕r1 (117) on Cayley graphs of groups with finite quotients What we claim here is that, according to our satisfying the hypotheses of lemmas 66 or 67, to notion of neighbourhood based on definition 36, reduce the classification of CAs on graphs of infi- the above automaton has a neighbourhood sys- nite groups to that of CAs on finite groups, which tem where N + is actually much larger than is in principle a much easier task. Moreover, once i {j 1, j, j ⊕ 1}. Indeed, if one considers the a suitable finite quotient is found, one can simu- r r action of on the algebra of transformations in- late the evolution of a finite region and for a finite V stead of its action on states, one can easily realise number of steps considering the same evolution that N + = G for every i. However, by checking on the finite wrapped graph instead of having to i the conditions for no-signalling, it is easy to re- consider the actual infinite graph. alise that the rule in Eq. (117) only signals to sites i 1, i, and i ⊕ 1. Thus, while no-signalling 9 Examples is necessary for no causal influence, the converse is not true. 9.1 Classical case We remark that the definition of causal influence that we give, based on transformations, is Classical systems d can be classified in terms of inspired by the quantum definition, which indeed the number d of pure states ρy, y = 1, . . . , d which involves the generators of the local algebra, and coincides with the dimension of the state space thus the Kraus representatives of transformations d = dim[[d]]R. All the pure states of a classical rather than the mere set of local states. In this system are jointly discriminable by atomic effects respect, the notion of neighbourhood used in the ax, x = 1, . . . , d, i.e. (ax|ρy) = δxy. For the sake literature on CA is different from the quantum of simplicity, we will restrict attention to classi- one, and we point to this relevant difference as cal computation, i.e. the sub-theory of classical the responsible for the apparent incompatibility theory where there is only one kind of non-trivial discussed in Ref. [31]. The subject was exten- elementary system, the bit having d = 2, and all sively studied in Ref. [60] and later in Ref.[61], other systems are composite systems of n bits, where the authors maintain the notion of neigh- n ¯ having dimension d = 2 . Every effect in [[AG]]Q bourhood for classical CA based on the signalling for an infinite system |G| = ∞ can be decom- condition, and prove a bound on the mismatch posed as a series of effects corresponding to finite, between the classical neighbourhood of a given arbitrarily large bit strings bR, with CA and the neighbourhood of the quantum version of the same CA. bR := [aR], a = ab ⊗ ab ⊗ ... ⊗ ab , r1 r2 r|R| 9.2 Quantum case with R = {r1, r2, . . . , r|R|}. In this case, let bR denote a local effect. A global rule V can be spec- Quantum cellular automata represent now a † 0 ified by the input/output relation V bR = bR0 . widely studied field of research. After the very However, in the extensive literature about classi- general analysis of Refs. [31, 32, 33, 61], there are cal cellular automata—see e.g. [59] for reference, various results in the literature about quantum

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 55 cellular automata. Here we cite the very general theory being of type2, they satisfy the wrap- classification of Ref. [62, 63], while for extensive ping lemma in the form of lemma 68, instead of reviews we refer to Refs. [64, 65]. lemma 69. This is due to the fact that Fermionic transofrmations, like quantum transformations, 9.3 Fermionic case admit a Kraus decomposition, and can thus be represented through Kraus operators instead of For the presentation of fermionic theory as an completely positive maps. It turns out that local OPT we refer to [36, 37]. The Fermionic systems † Fermionic field operators ψj,g and ψj,g provide a are fully specified by the algebra A generated by basis for the full algebra of Fermionic operators. local field operators ψi. In fact, A is the alge- Thus, in the Fermionic case, if Eq. (110) is sat- bra of field operator polynomials. This algebra is isfied for local transformations, it is always sat- Z2-graded, with two modules AE and AO, given isfied. Real quantum theory, which shares many by the span of even and odd polynomials, respec- features with Fermionic theory, is different in this tively. No combination of even and odd poly- respect, and we conjecture that a counterexam- nomials is allowed. As in all quantum theories, ple to the thesis of lemma 68 can be found in the A provides a simplified representation in terms latter case. of Kraus operators of the algebra of transformations, which is not graded, but rather reducible to a direct sum. Linear combinations of transfor- 10 Conclusion mations indeed do not correspond to linear combinations of the corresponding Kraus operators. Summarising the content of the manuscript, we The local systems for a site g of a cellular au- defined the composition of denumerably many tomaton in this case are registers made of one systems from an OPT, starting from the space of or more local Fermionic modes. Local field op- quasi-local effects, and defining quasi-local trans- erators are then labelled ψi,g, where g ∈ G and formations. We then define states of such system i ∈ Jg = {1, . . . , ng}, |Jg| representing the size in terms of suitable functionals on the space of of the register at site g. Since the local algebra effects. Among these, there are some that can Ag is finitely generated, the locality requirement be interpreted as the result of local preparations, implies that V is given by a local rule, which in and that we deem quasi-local states. turn is completely specified by its action on the We defined global update rules for finite or in- local generators ψi,g for every g ∈ G. Thus, one finite systems, and proved a generalisation of a has block-decomposition theorem for GURs. We then analysed those GURs that are homogeneous. The V (ψi,f ) definition of homogeneity is a big chapter on its X (g,j,s,t) sj †tj †ti = T ψ 1 ψ 1 . . . ψsk ψ k , own, and deserves a careful treatment. As a con- i,f j1,g1 j1,g1 jk,gk jk,gk + ×k sequence of the definition, we proved that for ho- g∈(Nf ) j∈Jg mogenous GURs the memory array is organised s,t∈{0,1}∗ as the Cayley graph of a group. We then defined locality for GURs, and studied its consequences where Jg := ×g∈N + Jg. Most of the automata f in detail. studied so far in the literature are linear, namely In the last section we used homogeneity and lo- 0 cality to define cellular automata, and proved the (ψ ) = X T j,g ψ . V i,f i,f j,g wrapping lemma under very general assumptions. g∈N + f The theory developed here is of fundamental j∈Jg importance for an approach to physical laws as Remarkable exceptions are represented by information-processing algorithms, and will be Refs. [44, 66, 45]. In particular, in Ref. [45] the used for the construction of statistical mechanics full classification of Fermionic cellular automata in post-quantum theories, as well as the recon- with |Jg| = 1 on Cayley graphs of Z2 × Z2 and Z struction of dynamical laws in space-time beyond was carried out. the now established approach based on quantum As regards Fermionic cellular automata, we walks, thus encompassing interacting quantum would like to remark that, despite Fermionic and post-quantum field theories.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 56 Acknowledgments that λ ± (a|ρ) ≥ 0 for every ρ ∈ [[A]]. Thus, |(a|ρ)| ≤ λ for every ρ ∈ [[A]]. Finally, we can con- This publication was made possible through the clude that kakop ≤ kaksup. On the other hand, if support of a grant from the John Templeton kakop < kaksup, then there exists ε > 0 such that Foundation under the project ID#60609 Causal for every ρ ∈ [[A]] one has |(a|ρ)| ≤ kaksup − ε. Quantum Structures. The opinions expressed In particular, this is true for ρ1 ∈ [[A]]1. Conse- in this publication are those of the authors and quently do not necessarily reflect the views of the John Templeton Foundation. The author wishes to |(a|ρ)| = κ|(a|ρ1)| ≤ κ(kaksup − ε) ∀ρ ∈ [[A]], thank G. M. D’Ariano, A. Bisio, A. Tosini, and F. Buscemi for many inspiring discussions. Sug- where 0 ≤ κ ≤ 1 and ρ = κρ1. Then (e|ρ) = κ, gestions about the terminology by G. Chiribella and are also acknowledged that were useful to sharpen the notion that is now defined as causal influence. ({kaksup − ε}e ± a|ρ) ≥ 0, ∀ρ ∈ [[A]]. A particular thank to J. Barrett for fruitful con- versations about various facets of causal influence Finally, this implies that kaksup −ε ∈ J(a), which and signalling, and to M. Erba for his help in tun- is absurd. Then, kaksup = kakop. ing the definitions and dissipating misconceptions on the same subject. B Quasi-local states

We start considering arbitrary finite disjoint par- A Identification of the sup- and oper- ∞ titions of the system AG. Let B := {Bi}i=1 de- ational norm for effects note a disjoint partition of the set G into finite blocks, i.e. 0 < |B | < ∞ for every i ∈ . The We show that, under the hypothesis of assump- i N set of such partitions (quotiented by the irrele- tion (16), one has kakop = kaksup for every ¯ ¯ ∗ vant ordering of regions) is denoted by B. a ∈ [[A]]R. Indeed, if [[A]]+ = [[A]]+, then, by theorem5, one has Definition 51. For a given partition B ∈ B, we ¯ ¯ define the set of finite B-regions of G as [[A]] = {a ∈ [[A]]R | 0 ≤ (a|ρ), (e − a|ρ), ∀ρ ∈ [[A]]}. (118)    k  R(G,B) := R ⊆ G | R = [ B ; 0 ≤ k < ∞ , First of all, one can rephrase the above condition ij   as j=1 (120) ¯ ¯ [[A]] = {a ∈ [[A]]R | 0 ≤ (a|ρ), (e − a|ρ), ∀ρ ∈ [[A]]1}. (119) and the set of B-regions of G as   Indeed, since [[A]] ⊆ [[A]], the set on the l.h.s. in k 1 (G,B)  [  Eq. (118) is clearly a subset of that on the l.h.s. in R := R ⊆ G | R = Bij ; 0 ≤ k ≤ ∞ ,   Eq. (119). Moreover, since every state ρ ∈ [[A]] j=1 is proportional by a constant κ ∈ [0, 1] to a (121) deterministic one ρ ∈ [[A]] (see remark1), if 0 1 where for k = 0 we define 0 ≤ (a|σ), (e − a|σ) for all σ ∈ [[A]]1, then 0 ≤ (a|ρ), (e − a|ρ) for all ρ ∈ [[A]]. Then, by 0 [ the identity in Eq. (119), one can identify the set Bij := ∅. ¯ ¯ j=1 [[A]] as the subset of [[A]]R containing only those functionals a on [[A]]R such that 0 ≤ (a|ρ) and Sk (G,B) (B) 0 ≤ 1 − (a|ρ) for ρ ∈ [[A]]1, and, in turn, the Given R = j=1 Bij in R , let NR := k above condition is equivalent to {ij}j=1.

[[A]]¯ = {a ∈ [[A]]¯ | 0 ≤ (a|ρ) ≤ 1, ∀ρ ∈ [[A]]}. (G,B) R Remark 10. Clearly, R(G,B) ⊆ R . It is also (G,B) Now, we remind that λ ∈ J(a) if and only if trivial to prove that R and R(G,B) are closed λe ± a 0. Under our hypotheses, this means under ∪, ∩, \.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 57 Given a finite region R ∈ R(G), for any parti- A reference local state allows one to attribute a tion B ∈ B it is possible to cover R with regions state to any finite region R ∈ R(G) by the follow- of B. ing procedure. First, let us consider the B-cover Definition 52. Given B ∈ B, the cover in B of B(R) of R. Then, the state of B(R) is given by a region R ⊆ G, denoted B(R), is the minimal ρ [B(R)] := O ρ . B-region C = Sk B such that R ⊆ C. 0 0ij j=1 ij (B) ij ∈N The above notion of cover is well defined, as B(R) proved by the following lemma. Finally, we discard all the extra systems in the Lemma 70. Let R ⊆ G. The cover B(R) in B region B(R) \ R. of R exists and is unique. ρ0|R := ρ0[B(R)]|R. Proof. First of all, the set of B-regions S ∈ (G,B) R such that R ⊆ S is not empty, since The above construction is then formalised in (G,B) R ⊆ G ∈ R . We just need to prove that, the following definition. (G,B) having defined C(R,B) := {S ∈ R | R ⊆ S}, Definition 54. The state of a finite region R ∈ there is always a unique minimal element B(R) ∈ R(G,B) in the reference local state ρ is defined as C(R,B), i.e. such that no B-region strictly con- 0 tained in B(R) covers R. As to existence, let R R ρ0|R := , h ∈ R. Then h ∈ Blh for some lh. One can eas- ρ0[B(R)] B(R)\R S e ily realise that R ⊆ C := h∈R Blh . Moreover, ρ := 1. if we remove a given B-region Blh from C, and 0|∅ h ∈ Blh , then h 6∈ C \ Blh , thus C is a minimal cover of R in B. As to uniqueness, suppose that Given a partition B and a reference local state there are two different minimal covers of R in B, ρ0, let us now define the set of primitive local states as follows. say C1 and C2. In this case, it is not restrictive to suppose that there is h ∈ G such that h ∈ C 1 Definition 55. The set of primitive local states and h 6∈ C . Let B ∈ B be the set such that 2 i is h ∈ Bi. One has Bi ⊆ C1 and Bi ∩ C2 = ∅. Since R ⊆ C2, we can conclude that R ∩ Bi = ∅, and Pre[[A ]](B) := {(σ, R) | σ ∈ [[A ]] ; R ∈ R(G,B)}. G R R R thus R ⊆ (C1 \ Bi) ∈ C(R,B). Thus, C1 cannot be minimal in C(R,B). This set collects all the generalised local prepa- The cover B(R) can also be thought of as the rations of finite regions of B. As in the case intersection of all covers of R in B. of effects, we introduce the simplified notation σS := (σ, S). A choice of reference local state Lemma 71. The cover B(R) of a finite region ρ0, allows us to interpret a given element σS of R is finite. Pre[[A ]](B) as a preparation of the region S in G R Proof. Every element gk ∈ R belongs to some the state σ. Thus, the element σS allows one to

Bik ⊆ B, where k 7→ ik is generally not injec- attribute a generalised state τ ∈ [[AT ]]R to any S|R| region T ∈ R(G) as follows (see fig.3) tive. It is then clear that R ⊆ k=1 Bik , and S|R| thus B(R) ⊆ k=1 Bik . τ := σ|S∩T ⊗ ρ0|(T \S). (122) Given one partition B ∈ B, we can now define the set of states that differ on finitely many blocks Definition 56. Given a reference local state ρ from a given assignment of states of blocks in B. 0 over B we define the following equivalence rela- First, we define the reference state as follows. (B) tion in Pre[[AG]] Definition 53 (Reference local state). Given a R ( partition B ∈ B, a reference local state over B is σ = ν ⊗ ρ0|(S\T ), a map σS ∼ρ0 τT ⇔ (123) τ = ν ⊗ ρ0|(T \S), G ρ0 : N → [[ABj ]]1 :: j 7→ ρ0j ∈ [[ABj ]]1. j∈N for some ν ∈ [[AS∩T ]]R.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 58 The symbol σSρ0 will denote the equivalence

⇢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 01 class of σ ∈ Pre[[A ]] under the relation ∼ . S G R ρ0

⇢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 05 Notice that in Eq. (123), thanks to closure of

⇢AAAIVXicdZVpb9s2GMfVY42bHT32sm/YqQU6IDAsJz0GBEFjxzmaHj4TO2UQUBTlCDpD0kUMVZ9ib7fPNezDDBglkmK9tQIM6/k9/+fgI5FysyhgvNX6+8bNW7e/u7PWuLv+/Q8//nTv/oOHJyxdUEwmOI1SOnURI1GQkAkPeESmGSUodiNy6obd0n/6iVAWpMmYLzNyHqN5EvgBRlygGaSX6UXeahcX9+1Wc7P9/JWzCeTNiy158/K3LeA0W9VlW+rqXzxYewG9FC9iknAcIcY+Oq2Mn+eI8gBHpFiHC0YyhEM0J3nVZwGeCuQBP6Xil3BQ0RUdilmM+OWG+GfL2N1gPEZ0Sb0NNxbRpYut6EtCmc+Kr2X5WsTHBfdfnedBki04SbBsyV9EgKegHA/wAkowj5biBmEaiLUAfIkowlwMcaXKnKLsMsDXxfrTlZaWHqkaqmqA6yWA18s0K6edoyjSDqEb4IDiRcAFEiGQZdh28hxWPfvAdgrtOMxLJzhUdp+mqS90xGNhkMEMUZikQeKJB5FD1weVvwl0+BXx8idwA7qR6JBdLRAlT8qgAkh/N86ia1nWdUFXRe3kkKJkXj3I0t7OYSTtygzDHO7AxwDuSNt1c7gNH8PtAlQ2pwnzaSwWJNeDabmgyhUl/3HSuPTJOLorJiDAbiHFtCPtjl4N7UrQ1YI9ae8Vej20J0mv0Dn3Jdg3kgNJDmrJkQRHdRksAdZlBtIemBxDSYZ1jpEEIyOZSjKtJTMJZkZyJslZTfhurqcnpiBRx6COapB3Desq2Z5Be3W6kYEjpTsw6ECne2vYWyV7b9B7LTs27Fg/6iPDjrRubNhYpZsYNKm7OzHwROlODTrV6d4Z9k7JpgZNtWxm2EzJzgw607J9w8RLIdkHwz5oXc+wnl7smy9i32jh0MChytc3qA+Kbz+LlUHpHZlwRnh9FOQ93y/q92O06htx4xqLRF/6xmIHszJUlR+YWgPF5jSrDh03jbzyxE2jvDx41GpHGUpMJWHoVCEXUZ9tR2x/IKUuLbfztu18VhIvVJIw1BrPLTWua0THUvOrVnSE/cy4R13bsdsVsdu1aEzlnhnT+oTEqBJW9e226er/0h5HJckP1cYWA6sGsHLiDgmKWF6fiUPN3U9EntGuL8eksBhS6QKJBqECoQZXClwV6+JLqz+n4Ns3J+2ms9lsD7bs1x31zW1Yj6xfrGeWY720XluHVt+aWNiKrd+tP6w/1/5a+6dxu3FHSm/eUDE/WytX496/uWTtag== 02 (G,B) BAAAIT3icdZVbb9s2FMfV7uI0u7TdHvfCTi3QAYFhOellQBA0dpxL04tt2YmTKghIinIEXUPSRQxVH2Gv2+fa4z7J3oZRIinWWysgCM/v/M+FRyaF8jhkvNP569btL7786uvW2p31b7797vu79+7/cMKyBcVkirM4ozMEGYnDlEx5yGMyyymBCYrJKYr6lf/0PaEszNIJX+bkIoHzNAxCDLlAbu/yyeU9u9Pe7D557mwCuXi6JRfPft0CTrtTP7alnuHl/dZTz8/wIiEpxzFk7J3TyflFASkPcUzKdW/BSA5xBOekqDsswSOBfBBkVPylHNR0RQcTlkB+tSH+s2WCNhhPIF1SfwMlIrpysRV9RSgLWPmpLJ+KeLfgwfOLIkzzBScpli0FixjwDFSDAX5ICebxUiwgpqHYC8BXkELMxfhWqswpzK9CfFOuP1ppaemTuqG6BrhZAu9mmeXVnAsYx9ohdCMcUrwIuUAixGM5tp2i8OqeA2A7pXYcFpUTHCp7SLMsEDrisyjMvRxSL83C1BcvovBQAGp/G+jwa+IXD70ND8WiQ3a9gJQ8rIJKIP39JI9vZFmEQF9F7RQehem8fpGVvV14sbRrM4oKb8d7ALwdaSNUeNveA2+7BLXNacoCmogNyf1gWm2odsXpf5w0qXwyju6KCQiwW0ox7Um7p3dD+xL0tWBP2nul3g8dSDIodc59CfaN5ECSg0ZyJMFRUwZLgHWZkbRHJsdYknGTw5XANZKZJLNGcibBmZGcS3LeEL5b6OmJKUjUM6inGuR9w/pKtmfQXpPONdBVugODDnS6V4a9UrI3Br3RsmPDjvWrPjLsSOsmhk1UuqlB06a7EwNPlO7UoFOd7rVhr5VsZtBMy84MO1Oyc4POtWzfMPGjkOytYW+1bmDYQG/25UexL7VwbOBY5RsaNATl59/FyqD0iUw5I7y5CopBEJTN78Nd9bncuCYi0ce+iTjBrApV5Uem1kixOc3rSwdlsV/duFlcVBeP2q2bw9RUEoZOFXER9cF2xPEHUopodZy3beeDkviRkkSR1vio0iBkRMdS84tW9IT92Ljdvu3Y3ZrY3UY0ofLMTGhzQ2JYC+v6dtd09X/pgMOKFIfqYIuB1QNYuXHHBMasaO7EseboPZF3NArkmBQWQ6pcINUgUiDS4FqB63JdfGn15xR8fnHSbTub7e5oy37RU9/cNesn62frseVYz6wX1qE1tKYWtubWb9bv1h+tP1t/t/5ZU9Lbt9TiR2vlWbvzL8Qc66U= 5 R under set diﬀerence, one has (S \ T ), (T \

BAAAIT3icdZVbb9s2FMfV7uI0u7Xr417YqQU6IDAsJ+sFCILGjnNperEtO3FSBQFJUY6ga0i6iKHqI+x1+1x73CfZ2zBKJMV6awUE4fmd/7nwyKRQHoeMdzp/3br9xZdffd1au7P+zbffff/D3Xs/nrBsQTGZ4izO6AxBRuIwJVMe8pjMckpggmJyiqJ+5T99TygLs3TClzm5SOA8DYMQQy6Q27vsXt61O+3N7q/PnE0gF0+25OLp8y3gtDv1Y1vqGV7eaz3x/AwvEpJyHEPG3jmdnF8UkPIQx6Rc9xaM5BBHcE6KusMSPBLIB0FGxV/KQU1XdDBhCeRXG+I/WyZog/EE0iX1N1AioisXW9FXhLKAlZ/K8qmIdwsePLsowjRfcJJi2VKwiAHPQDUY4IeUYB4vxQJiGoq9AHwFKcRcjG+lypzC/CrEN+X6o5WWlj6pG6prgJsl8G6WWV7NuYBxrB1CN8IhxYuQCyRCPJZj2ykKr+45ALZTasdhUTnBobKHNMsCoSM+i8LcyyH10ixMffEiCg8FoPa3gQ6/Jn7x0NvwUCw6ZNcLSMnDKqgE0t9P8vhGlkUI9FXUTuFRmM7rF1nZ24UXS7s2o6jwdrwHwNuRNkKFt+098LZLUNucpiygidiQ3A+m1YZqV5z+x0mTyifj6K6YgAC7pRTTnrR7eje0L0FfC/akvVfq/dCBJINS59yXYN9IDiQ5aCRHEhw1ZbAEWJcZSXtkcowlGTc5XAlcI5lJMmskZxKcGcm5JOcN4buFnp6YgkQ9g3qqQd43rK9kewbtNelcA12lOzDoQKd7ZdgrJXtj0BstOzbsWL/qI8OOtG5i2ESlmxo0bbo7MfBE6U4NOtXpXhv2WslmBs207MywMyU7N+hcy/YNEz8Kyd4a9lbrBoYN9GZffhT7UgvHBo5VvqFBQ1B+/l2sDEqfyJQzwpuroBgEQdn8PtxVn8uNayISfeybiBPMqlBVfmRqjRSb07y+dFAW+9WNm8VFdfGo3bo5TE0lYehUERdRH2xHHH8gpYhWx3nbdj4oiR8pSRRpjY8qDUJGdCw1v2hFT9iPjdvt247drYndbUQTKs/MhDY3JIa1sK5vd01X/5cOOKxIcagOthhYPYCVG3dMYMyK5k4ca47eE3lHo0COSWExpMoFUg0iBSINrhW4LtfFl1Z/TsHnFyfdtrPZ7o627Bc99c1ds36yfrYeW4711HphHVpDa2pha279Zv1u/dH6s/V36581Jb19Sy3uWyvP2p1/Aaxh66I= 2 S) ∈ R(G,B), and thus the deﬁnition of the equiva-

⇢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 04 lence relation ∼ρ0 involves exclusively B-regions.

⇢AAAIVXicdZVbb9s2FMfVrmu8dFvb7XEvbNUCHRAYlpP1AgRBY8e5NL34mtipgoCiKEfQNSRdxGD1Kfq6fa5iH2bAKJEU660VYFjnd/7nwiOR8vI4pKzV+vvGze9ufX97rfHD+p0ff/r57r37v5zQbEEQnqAszsjUgxTHYYonLGQxnuYEw8SL8akXdUv/6QdMaJilY7bM8XkC52kYhAgygWYuucwueOtFcXHPbjU32388dzaBvHm6JW+evdgCTrNVXbalrv7F/bWnrp+hRYJThmJI6XunlbNzDgkLUYyLdXdBcQ5RBOeYV30W4LFAPggyIn4pAxVd0cGEJpBdboh/uky8DcoSSJbE3/ASEV266Iq+JIQGtPhalq9FvF+w4Pk5D9N8wXCKZEvBIgYsA+V4gB8SjFi8FDcQkVCsBaBLSCBiYogrVeYE5pchui7WH6+0tPRx1VBVA1wvgXu9zPJy2hzGsXYI3QCFBC1CJpAIcWmObIdzt+o5ALZTaMchL53gUNl9kmWB0GGfRmHu5pC4aRamvngQ3PUCUPmbQIdfYZ8/cjdcLxYd0qsFJPhRGVQA6e8meXwty3oe6KqoHe4SmM6rB1na29yNpV2ZUcTdHfcBcHek7Xnc3XYfuNsFqGxGUhqQRCxIrgeRckGVK07/4yRJ6ZNxZFdMQIDdQopJR9odvRrSlaCrBXvS3iv0ekhPkl6hc+5LsG8kB5Ic1JIjCY7qMkgCpMsMpD0wOYaSDOscIwlGRjKVZFpLZhLMjORMkrOasF2upyemIFHHoI5qkHUN6yrZnkF7dbqRgSOlOzDoQKd7bdhrJXtr0FstOzbsWD/qI8OOtG5s2Filmxg0qbs7MfBE6U4NOtXp3hj2RsmmBk21bGbYTMnODDrTsn3DxEsh2TvD3mldz7CeXuyrL2JfaeHQwKHK1zeoD4pvP4uVQekdmTKKWX0U8F4QFPX7MVr1jZhxjUWiL31jsYNpGarKD0ytgWJzkleHjpfFfnniZjEvDx612lEOU1NJGDpVxETUR9sR2x9IqUfK7bxtOx+VxI+UJIq0xvdKjecZ0bHU/K4VHWE/Me5R13bsdkXsdi0aE7lnxqQ+IRGshFV9u226+r+0x2BJ+KHa2GJg1QBWTtwhhjHl9Zk41Nz7gOUZ7QVyTAqLIZUukGoQKRBpcKXAVbEuvrT6cwq+fXPSbjqbzfZgy37ZUd/chvWb9dB6YjnWM+uldWj1rYmFrMT6ZP1p/bX2ee2fxq3GbSm9eUPF/GqtXI27/wLwyu1x 09 Moreover, all the regions involved in the deﬁni-

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 1

⇢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 07 (B) 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 3

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 8 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 7 Lemma 72. Let σ ∈ Pre[[A ]] . Then for ⇢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 06 S G R (G,B)

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 11 every R ∈ R such that R∩S = ∅ one has (σ⊗ (B) 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 6 ρ ) ∈ Pre[[A ]] and (σ ⊗ ρ ) ∼ ⇢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 010 0|R S∪R G R 0|R S∪R ρ0

⇢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 10 03 σS. (a) Proof. First of all, as R(G,B) is closed under union, clearly S ∪ R ∈ R(G,B), and thus (σ ⊗ (B)

⇢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 011 ρ0|R)S∪R ∈ Pre[[AG]] . Now, S ∩ (S ∪ R) = S, BAAAIa3icdZXrbts2FMfVdlu87Jau37oNYKcGaIHAsJy0yYAgaOw4l6YXXxMnVWBQFOUIuoakihiqPu1p9nV7mj3E3mGUSIr11gowxPM7/3MOeWhSThr6lLVaf9+5e++LL79aaXy9+s23333/w9r9H89okhGEJygJEzJ1IMWhH+MJ81mIpynBMHJCfO4E3dJ//h4T6ifxmC1SfBXBeex7PoKMo9naL51ZvlXYKEtBZ/ZMvrfle2e2Zraam+1nO9YmEIPnW2Kw/dsWsJqt6jEN+fRn91ee226CsgjHDIWQ0ndWK2VXOSTMRyEuVu2M4hSiAM5xXs2+AOscucBLCP/FDFR0SQcjGkF2vcHfdBE5G5RFkCyIu+FEPLp00SV9SQj1aPGpLJ+KeJcxb+cq9+M0YzhGYkpeFgKWgLJpwPUJRixc8AFExOdrAegaEogYb+1SlTmB6bWPbovV9aUpLVxcTaiqAW4XwL5dJGm5BzkMQ+XgugHyCcp8xhEPsWmKTCvP7WrOHjCtQjmO89IJjqXdJ0nicR12aeCndgqJHSd+7PKNyG3HA5W/CVT4DXbzx/aG7YR8hvQmgwQ/LoMKIPzdKA1vRVnHAV0ZtZfbBMbzaiNLeze3Q2FXZhDk9p79CNh7wnac3N61H9m7BahsRmLqkYgvSKwHkXJBlSuM/+MkUekTcWSfd4CD/UKISUfYHbUa0hWgqwQHwj4o1HpIT5BeoXIeCnCoJUeCHNWSEwFO6jJIAKTKDIQ90DmGggzrHCMBRloyFWRaSy4EuNCSS0Eua8L2c9U93gWBOhp15ARZV7OulB1odFCnG2k4krojjY5UuleavZKyNxq9UbJTzU7VVp9odqJ0Y83GMt1Eo0k9uzMNz6TuXKNzle61Zq+lbKrRVMkuNLuQskuNLpXsUDP+pxDsrWZvla6nWU8t9uVHsS+VcKjhUObra9QHxef3YqlR6kTGjGJWXwV5z/OK+v8xWvaNmHaNeaKPfWN+gmkZKssPdK2BZHOSVpeOk4RueeMmYV5ePHK1oxTGuhI3VKqA8agPpsWPPxBSh5THede0PkiJG0hJECiN65Qax9GiU6F5qhQdbj/R7lHXtMx2Rcx2LRoTcWbGpL4hEayEVX2zrWf1f2mPwZLkx/Jg84ZVDVi6cYcYhjSv78Sh4s57LO5oxxNtkpg3qXSBWIFAgkCBGwluilX+pVWfU/D5wVm7aW0224Mt80VHfnMbxk/Gr8YTwzK2jRfGsdE3JgYyfjf+MP40/lr5p/Gg8bDxs5DevSNjHhhLT2P9Xzk886Y= 4 B5 B7 B8 ⇢AAAIVXicdZVpb9s2GMfVY42bHT32sm/YqQU6IDAsJz0GBEFjxzmaHj4TO2UQUBTlCDpD0kUMVZ9ib7fPNezDDBglkmK9tQIM6/k9/+fgI5FysyhgvNX6+8bNW7e/u7PWuLv+/Q8//nTv/oOHJyxdUEwmOI1SOnURI1GQkAkPeESmGSUodiNy6obd0n/6iVAWpMmYLzNyHqN5EvgBRlygGaSX6UXecoqL+3arudl+/srZBPLmxZa8efnbFnCareqyLXX1Lx6svYBeihcxSTiOEGMfnVbGz3NEeYAjUqzDBSMZwiGak7zqswBPBfKAn1LxSzio6IoOxSxG/HJD/LNl7G4wHiO6pN6GG4vo0sVW9CWhzGfF17J8LeLjgvuvzvMgyRacJFi25C8iwFNQjgd4ASWYR0txgzANxFoAvkQUYS6GuFJlTlF2GeDrYv3pSktLj1QNVTXA9RLA62WaldPOURRph9ANcEDxIuACiRDIMmw7eQ6rnn1gO4V2HOalExwqu0/T1Bc64rEwyGCGKEzSIPHEg8ih64PK3wQ6/Ip4+RO4Ad1IdMiuFoiSJ2VQAaS/G2fRtSzruqCronZySFEyrx5kaW/nMJJ2ZYZhDnfgYwB3pO26OdyGj+F2ASqb04T5NBYLkuvBtFxQ5YqS/zhpXPpkHN0VExBgt5Bi2pF2R6+GdiXoasGetPcKvR7ak6RX6Jz7EuwbyYEkB7XkSIKjugyWAOsyA2kPTI6hJMM6x0iCkZFMJZnWkpkEMyM5k+SsJnw319MTU5CoY1BHNci7hnWVbM+gvTrdyMCR0h0YdKDTvTXsrZK9N+i9lh0bdqwf9ZFhR1o3Nmys0k0MmtTdnRh4onSnBp3qdO8Me6dkU4OmWjYzbKZkZwadadm+YeKlkOyDYR+0rmdYTy/2zRexb7RwaOBQ5esb1AfFt5/FyqD0jkw4I7w+CvKe7xf1+zFa9Y24cY1Foi99Y7GDWRmqyg9MrYFic5pVh46bRl554qZRXh48arWjDCWmkjB0qpCLqM+2I7Y/kFKXltt523Y+K4kXKkkYao3nlhrXNaJjqflVKzrCfmbco67t2O2K2O1aNKZyz4xpfUJiVAmr+nbbdPV/aY+jkuSHamOLgVUDWDlxhwRFLK/PxKHm7iciz2jXl2NSWAypdIFEg1CBUIMrBa6KdfGl1Z9T8O2bk3bT2Wy2B1v264765jasR9Yv1jPLsV5ar61Dq29NLGzF1u/WH9afa3+t/dO43bgjpTdvqJifrZWrce9fsXrtaQ== 01 R [ [ [ and thus the condition in Eq. (123) is satisﬁed

⇢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 02 for T = S ∪ R, T \ S = R, S ∩ T = S, ν = σ,

BAAAIT3icdZVbb9s2FMfV7uI0u7Xr417YqQU6IDAsJ+sFCILGjnNperEtO3FSBQFJUY6ga0i6iKHqI+x1+1x73CfZ2zBKJMV6awUE4fmd/7nwyKRQHoeMdzp/3br9xZdffd1au7P+zbffff/D3Xs/nrBsQTGZ4izO6AxBRuIwJVMe8pjMckpggmJyiqJ+5T99TygLs3TClzm5SOA8DYMQQy6Q27vsXt61O+3N7q/PnE0gF0+25OLp8y3gtDv1Y1vqGV7eaz3x/AwvEpJyHEPG3jmdnF8UkPIQx6Rc9xaM5BBHcE6KusMSPBLIB0FGxV/KQU1XdDBhCeRXG+I/WyZog/EE0iX1N1AioisXW9FXhLKAlZ/K8qmIdwsePLsowjRfcJJi2VKwiAHPQDUY4IeUYB4vxQJiGoq9AHwFKcRcjG+lypzC/CrEN+X6o5WWlj6pG6prgJsl8G6WWV7NuYBxrB1CN8IhxYuQCyRCPJZj2ykKr+45ALZTasdhUTnBobKHNMsCoSM+i8LcyyH10ixMffEiCg8FoPa3gQ6/Jn7x0NvwUCw6ZNcLSMnDKqgE0t9P8vhGlkUI9FXUTuFRmM7rF1nZ24UXS7s2o6jwdrwHwNuRNkKFt+098LZLUNucpiygidiQ3A+m1YZqV5z+x0mTyifj6K6YgAC7pRTTnrR7eje0L0FfC/akvVfq/dCBJINS59yXYN9IDiQ5aCRHEhw1ZbAEWJcZSXtkcowlGTc5XAlcI5lJMmskZxKcGcm5JOcN4buFnp6YgkQ9g3qqQd43rK9kewbtNelcA12lOzDoQKd7ZdgrJXtj0BstOzbsWL/qI8OOtG5i2ESlmxo0bbo7MfBE6U4NOtXpXhv2WslmBs207MywMyU7N+hcy/YNEz8Kyd4a9lbrBoYN9GZffhT7UgvHBo5VvqFBQ1B+/l2sDEqfyJQzwpuroBgEQdn8PtxVn8uNayISfeybiBPMqlBVfmRqjRSb07y+dFAW+9WNm8VFdfGo3bo5TE0lYehUERdRH2xHHH8gpYhWx3nbdj4oiR8pSRRpjY8qDUJGdCw1v2hFT9iPjdvt247drYndbUQTKs/MhDY3JIa1sK5vd01X/5cOOKxIcagOthhYPYCVG3dMYMyK5k4ca47eE3lHo0COSWExpMoFUg0iBSINrhW4LtfFl1Z/TsHnFyfdtrPZ7o627Bc99c1ds36yfrYeW4711HphHVpDa2pha279Zv1u/dH6s/V36581Jb19Sy3uWyvP2p1/Aaxh66I= 2 ρ0|(T \S) = ρ0|R, and ρ0|(S\T ) = 1.

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 9 representative of the equivalence class σRρ0 de-

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 1 ﬁned as follows.

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⇢AAAIVXicdZVpb9s2GMfVY42bHT32sm/YqQU6IDAsJz0GBEFjxzmaHj4TO2UQUBTlCDpD0kUMVZ9ib7fPNezDDBglkmK9tQIM6/k9/+fgI5FysyhgvNX6+8bNW7e/u7PWuLv+/Q8//nTv/oOHJyxdUEwmOI1SOnURI1GQkAkPeESmGSUodiNy6obd0n/6iVAWpMmYLzNyHqN5EvgBRlygGaSX6UXe2iwu7tut5mb7+StnE8ibF1vy5uVvW8BptqrLttTVv3iw9gJ6KV7EJOE4Qox9dFoZP88R5QGOSLEOF4xkCIdoTvKqzwI8FcgDfkrFL+Ggois6FLMY8csN8c+WsbvBeIzoknobbiyiSxdb0ZeEMp8VX8vytYiPC+6/Os+DJFtwkmDZkr+IAE9BOR7gBZRgHi3FDcI0EGsB+BJRhLkY4kqVOUXZZYCvi/WnKy0tPVI1VNUA10sAr5dpVk47R1GkHUI3wAHFi4ALJEIgy7Dt5DmsevaB7RTacZiXTnCo7D5NU1/oiMfCIIMZojBJg8QTDyKHrg8qfxPo8Cvi5U/gBnQj0SG7WiBKnpRBBZD+bpxF17Ks64KuitrJIUXJvHqQpb2dw0jalRmGOdyBjwHckbbr5nAbPobbBahsThPm01gsSK4H03JBlStK/uOkcemTcXRXTECA3UKKaUfaHb0a2pWgqwV70t4r9HpoT5JeoXPuS7BvJAeSHNSSIwmO6jJYAqzLDKQ9MDmGkgzrHCMJRkYylWRaS2YSzIzkTJKzmvDdXE9PTEGijkEd1SDvGtZVsj2D9up0IwNHSndg0IFO99awt0r23qD3WnZs2LF+1EeGHWnd2LCxSjcxaFJ3d2LgidKdGnSq070z7J2STQ2aatnMsJmSnRl0pmX7homXQrIPhn3Qup5hPb3YN1/EvtHCoYFDla9vUB8U334WK4PSOzLhjPD6KMh7vl/U78do1TfixjUWib70jcUOZmWoKj8wtQaKzWlWHTpuGnnliZtGeXnwqNWOMpSYSsLQqUIuoj7bjtj+QEpdWm7nbdv5rCReqCRhqDWeW2pc14iOpeZXregI+5lxj7q2Y7crYrdr0ZjKPTOm9QmJUSWs6ttt09X/pT2OSpIfqo0tBlYNYOXEHRIUsbw+E4eau5+IPKNdX45JYTGk0gUSDUIFQg2uFLgq1sWXVn9OwbdvTtpNZ7PZHmzZrzvqm9uwHlm/WM8sx3ppvbYOrb41sbAVW79bf1h/rv219k/jduOOlN68oWJ+tlauxr1/AcFO7Ws= \ 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 10 03 Rσ := S, σ˜Rσ ∼ρ0 σR, (124) S∈R(σ,R) (b) where Figure 3: An illustrative example of a reference local (G,B) state ρ0 (a) for a partition of a ﬁnite set G into 11 R(σ,R) := {S ∈ R | ∃τ ∈ [[AS]]R : τS ∼ρ0 σR}. regions, along with a primitive local state σR (b) with R = B4 ∪B5 ∪B7 ∪B8. The dots represent the elements Lemma 73. The minimal representative exists of G. and is unique. For the proof, see the analogous lemma9. Deﬁnition 58. The set of generalised local states over B upon ρ0 is

[[A ]](B) := Pre[[A ]](B)/ ∼ . (125) G ρ0LR G R ρ0

Its elements will be denoted by symbols like ρSρ0 . We now deﬁne linear combinations of local states. Deﬁnition 59. Let σ , τ ∈ [[A ]](B) , and Sρ0 T ρ0 G ρ0LR a ∈ R. Then we have ( (aσ)Sρ0 a 6= 0 a(σSρ0 ) := 0∅ρ0 a = 0,

σSρ0 + τT ρ0 := νS∪T ρ0 ,

ν := σ ⊗ ρ0|(T \S) + ρ0|(S\T ) ⊗ τ.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 59 We will denote by Rσ+τ the region where the Proof. By hypothesis, there exists ν ∈ [[AS∩T ]]R minimal representative of the class σSρ0 + τT ρ0 such that diﬀers from ρ0. As in the case of eﬀects, it is not always true that Rσ+τ = Rσ∪Rτ . As an example, σ = ν ⊗ ρ0S\T , τ = ν ⊗ ρ0T \S. consider σ = αi1 ⊗ βi2 and τ = αi1 ⊗ (ρ0i2 − βi2 ), Then we have with αi1 6= ρ0i1 , βi2 6= ρ0i2 , and Rσ = Rτ =

Bi1 ∪ Bi2 . Then σ + τ = αi1 ⊗ ρ0i2 , and clearly (aR|σSρ0 ) =(a|ν|(S∩T )∩R ⊗ ρ0|[(S\T )∩R]∪(R\S)) Rσ+τ = Bi1 , which is strictly included in Rσ =

Rτ = Rσ ∪ Rτ . (aR|τT ρ0 ) =(a|ν|(S∩T )∩R ⊗ ρ0|[(T \S)∩R]∪(R\T )).

Definition 60. The set of local states over B A straightforward set-theoretic calculation gives upon ρ0 is that [(S\T )∩R]∪(R\S) = [(T \S)∩R]∪(R\T ) = (B) (B) R \ (S ∩ T ), and thus (a |σ ) = (a |τ ). [[A ]] := {σ ∈ [[A ]] | σ ∈ [[A ]]}. R Sρ0 R T ρ0 G ρ0L Sρ0 G ρ0LR S On similar lines, we have the following result. An element σ ∈ [[A ]](B) is a local state over Sρ0 G ρ0LR B upon ρ0. A subset of special interest is that of Lemma 76. For every partition B ∈ B, every deterministic local states over B upon ρ0, defined reference local state ρ0 over B, and every local as state τ ∈ [[A ]](B) , the result of Eq. (127) is T ρ0 G ρ0LR [[A ]](B) := {σ ∈ [[A ]](B) | σ ∈ [[A ]] }. independent of the choice of representative bS in G ρ0L1 Sρ0 G ρ0L S 1 the class aR. We now characterise the vector space of generalised states as follows. Proof. If c˜Rc is the minimal representative of aR = bS, by Eq. (10), one has Lemma 74. The set of generalized local states in B upon ρ0 equipped with the operations defined a =c ˜ ⊗ eR\Rc , in Def. 59 is a real vector space. It is the space b =c ˜ ⊗ e . of finite real combinations of deterministic local S\Rc states in B upon the same state ρ : 0 Now, by the defining equation 127, we have that [[A ]](B) = Span ([[A ]](B) ), (126) G ρ0LR R G ρ0L1 (aR|τT ρ0 ) = (a|τ|R∩T ⊗ ρ0|(R\T )) The proof is trivial and we omit it. =(˜c ⊗ eR\Rc |τ|R∩T ⊗ ρ0|(R\T )) We now show that quasi-local states are =(˜c|τ ⊗ ρ ), ¯ |Rc∩T 0|Rc\T bounded linear functionals on [[AG]]QR. We start considering the pairing of local states [[A ]](B) G ρ0LR and similarly we obtain ¯ with local effects [[AG]]LR, defined as follows.

(bS|τT ρ0 ) = (˜c|τ|Rc∩T ⊗ ρ0|(Rc\T )). Definition 61. Every σSρ0 ∈ [[AG]]ρ0LR iden- ¯ tifies a functional on [[AG]]LR as follows. For The above results give us a hint that for ev- a ∈ [[A¯ ]] let (B) R G LR ery B ∈ B and every ρ , the space [[A ]] is a 0 G ρ0LR R∩S σ|R∩S submanifold of the space [[AG]]R. We now com- (B) (aR|σSρ0 ) := a . (127) plete the real vector spaces [[AG]] to Banach ρ R\S ρ0LR 0|R\S spaces. For this purpose, we start equipping all spaces [[A ]](B) with a norm. We now prove that the above pairing is well- G ρ0LR defined, namely it does not depend on the repre- Definition 62. Given an element σ of the space sentative in the class σSρ0 , nor on the representa- (B) [[AG]] , its norm is defined as tive in the class aR. This is done in the next two ρ0LR lemmas. kσkB,ρ := kσ˜kop, (128) ¯ 0 Lemma 75. For every aR ∈ [[AG]]LR the result of Eq. (127) is independent of the choice of rep- where k·kop denotes the operational norm on resentative τT ρ0 in the class σSρ0 . [[ARσ ]]R.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 60 (B) Lemma 77. Given an element σ of [[A ]] , and thus actually (anR |ρSρ ) is a Cauchy se- G ρ0LR n 0 for τT in the corresponding equivalence class, one quence. Finally, since for every n ∈ N one has has

|(anRn |ρSρ0 )| ≤ kanksupkρSkB,ρ0 ,

kσkB,ρ0 = kτkop, taking the limit for n → ∞ on both sides we have where k·kop denotes the operational norm on that [[AT ]]R.

|(a|ρSρ0 )| ≤ kaksupkρSρ0 kB,ρ0 . Proof. By deﬁnition of the relation ∼ρ0 , and of minimal representative σ˜ of σ, one has Rσ We now show that the deﬁnition in Eq. (129) is independent of the sequence a in the class τ =σ ˜ ⊗ ρ . nRn 0(T \Rσ) of a.

Thus, kτkop = kσ˜ ⊗ ρ0(T \Rσ)kop. Now, by Lemma 79. Let a = [an ] = [bn 0 ] ∈ [[A¯ G]]Q , lemma3 Rn Rn R and ρSρ0 ∈ [[AG]]ρ0LR. Then

kτkop = kσ˜kop = kσkB,ρ0 . lim (an|ρSρ ) = lim (bn|ρSρ ). (131) n→∞ 0 n→∞ 0 We can now show that vectors in [[A ]](B) are G ρ0LR ¯ Proof. Remind that by definition, for every ε bounded linear functionals on [[AG]]QR. there exists n0 such that for n ≥ n0, it is ¯ ka − b 0 k ≤ ε. Now, by lemma 78, this Definition 63. Let a = [anRn ] ∈ [[AG]]QR, and nRn nRn sup implies that ρ = ρSρ0 ∈ [[AG]]ρ0LR. We define

(a|ρ) := lim (an |ρSρ ). (129) |(an − bn 0 |ρSρ )| ≤ εkρSρ kB,ρ , n→∞ Rn 0 Rn Rn 0 0 0 With this definition, we can now prove that lo- and thus the thesis follows. cal states are actually bounded linear functionals (B) ¯ We finally close the vector space [[AG]] by on [[AG]]QR. ρ0LR adding the limits of Cauchy sequences in the ¯ Lemma 78. For a = [anR ] ∈ [[AG]]Q , and n R norm k·kB,ρ0 , thus obtaining a Banach space ρ ∈ [[A ]] , one has (B) Sρ0 G ρ0LR that we denote [[A ]] . We start considering G ρ0QR Cauchy sequences |(a|ρ)| ≤ kaksupkρkB,ρ0 . (130) σ : → [[A ]](B) :: n 7→ σ ∈ [[A ]] , Proof. We start considering the case of a local N G ρ0R nRnρ0 G ρ0LR effect aR. In this case, us define (B) which make a real vector space [[AG]] , con- −1 ρ0CR a0 := (2kaksup) (kaksupeR + a) 0, (B) taining [[AG]]ρ L as the subspace of constant a := (2kak )−1(kak e − a) 0, 0 R 1 sup sup R sequences. We then quotient [[A ]](B) by the G ρ0CR a0 + a1 = eR, kaksup(a0 − a1) = a, usual equivalence relation =∼, obtaining complete normed real vector space. we have that Definition 64. The space of quasi-local states |(aR|ρSρ )| = |(aR|ρ ⊗ ρ )| 0 |R∩S 0|R\S in B upon ρ is defined as the space [[A ]](B) := 0 G ρ0QR = kaksup|({a¯0 − a¯1} ⊗ eS\R|ρ ⊗ ρ0|R\S)| [[A ]](B) / =∼. G ρ0CR ≤ kaksupkρSρ0 kB,ρ0 , Inside the new Banach space that we defined, where (¯ai| := (ai|IR∩S ⊗ |ρ0|R\S). Let now a = we can single out a convex set of quasi-local [a ]. Then we have nRn states, corresponding to preparations that can be arbitrarily approximated by local protocols start- |(anRn − amRm |ρSρ0 )| ing from ρ0, along with the cone that they span, ≤ ka − a k kρ k , nRn mRm sup Sρ0 B,ρ0 and the convex set of deterministic states.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 61 Deﬁnition 65. An element σ of [[A ]](B) is a The above deﬁnition is well-posed, as shown by G ρ0QR the following lemma, whose proof is a straightfor- quasi-local state if there exists σnRnρ0 in the class ward consequence of lemma 78. σ, and n0 ∈ N, such that, for n ≥ n0, σnRnρ0 ∈ (B) [[AG]] . The set of quasi-local states in B upon ¯ ρ0L Lemma 80. Let a ∈ [[AG]]QR, and σ = (B) 0 (B) ρ0 is denoted by [[AG]] . [σn ] = [σ 0 ] ∈ [[AG]] . Then ρ0Q Snρ0 nSnρ0 ρ0QR (B) A quasi-local state σ in [[AG]]ρ Q is determinis- 0 0 lim (a|σmS ρ ) = lim (a|σ 0 ). (133) n→∞ m 0 n→∞ mSmρ0 tic, if there exists a sequence σnRn in the class of

σ and n0 ∈ N such that, for n ≥ n0, σnRn ∈ Finally, we use the definition of sup-norm of ef- (B) (B) [[AG]] . The set of deterministic quasi-local fects to prove that vectors in the spaces [[A ]] ρ0L1 G ρ0QR (B) ¯ states is denoted by [[A ]] . are bounded linear functionals on [[AG]]Q . Also G ρ0Q1 R (B) in this case the proof follows straightforwardly We denote the subset of [[AG]] of elements ρ0QR form lemma 78. (B) σ = λρ, for λ ≥ 0 and ρ ∈ [[AG]] , as ρ0Q Lemma 81. Let a ∈ [[A¯ ]] and ρ ∈ [[A ]](B) . (B) G QR G ρ0QR [[AG]] . We will equivalently write σ 0 for ρ0Q+ Then one has σ ∈ [[A ]](B) G ρ0Q+ |(a|ρ)| ≤ kaksupkρkB,ρ0 . (134) Analogously to the case of effects, the ele- Lemma 82. The spaces [[A ]](B) are closed (B) G ρ0QR ments of [[AG]] will be denoted by ρ = ρ0QR subspaces of the extended space of states [[A ]] . [ρ ], with the convention that for a con- G R nRnρ0 On [[A ]](B) it holds that k·k ≡ k·k . G ρ0QR B,ρ0 ∗ stant Cauchy sequence ρnRnρ0 = ρRρ0 we will : : set ρRρ0 = [ρnRnρ0 ]. We define kρkB,ρ0 = Proof. We only need to prove that k·k∗ ≡ k·kQ, (B) limn→∞ kρn kB,ρ . One can straightforwardly Rnρ0 0 since we already proved that [[AG]]ρ L is a lin- (B) 0 R check that the space [[AG]]ρ Q of quasi-local ear manifold in the space of bounded linear func- 0 R ¯ states in B upon ρ0 is a Banach space, the sets tionals on [[AG]]QR. First of all, by eq. (134) (B) (B) [[A ]] and [[A ]] are convex subsets, and one clearly has kρk∗ ≤ kρkB,ρ0 for all ρ ∈ G ρ0Q G ρ0Q1 (B) (B) [[AG]]QR. For the converse, one can use the [[AG]] is a convex cone. The space [[AG]] ρ0Q+ ρ0LR same technique as for lemma 16, to prove that can be identified with the dense submanifold con- kρkB,ρ0 ≤ kρk∗ for the case of a local state taining constant sequences σSρ0 . ρ ∈ [[A ]](B) . Thus kρk = kρk for local For every partition B and every reference local G ρ0LR ∗ B,ρ0 (B) (B) state ρ , the Banach space [[A ]] is separa- states ρ ∈ [[AG]]ρ L . It is then straightforward 0 G ρ0QR 0 R to conclude that equality holds for every quasi ble. Indeed, one can choose a basis in [[ASn ]]R for each region S := Sn B in the increasing local state ρ ∈ [[A ]](B) . n j=1 j G ρ0QR sequence {Sn}n∈N of regions of B. Now, for ev- 0 (B) Notice that two partitions B and B along with ery ρ ∈ [[AG]]ρ Q , and for every n ∈ N, given 0 0 R two local reference states ρ0 and ρ0 might define 0 the class ρnRnρ0 defining ρ, one has Rn ⊆ Sm for (B) (B ) the same space [[AG]] ≡ [[AG]] 0 . This is sufficiently large m. Thus, linear combinations of ρ0QR ρ0QR (B) the countable collection of bases for Sm are dense the case if for every σ ∈ [[A ]] there exists a G ρ0QR (B) 0 in [[AG]] . 0 (B ) 0 ρ0QR state σ ∈ [[AG]] 0 such that, σ = σ , as func- ρ0QR We now show that actually, for every pair ¯ (B) tionals on [[AG]]QR. We then provide the following (B, ρ0), the space [[AG]] is a subspace of ρ0QR definition. [[A ]] , and k·k ≡ k·k on [[A ]](B) . First G R ∗ B,ρ0 G ρ0QR ¯ Definition 67. We say that the two reference of all, we extend the pairing of [[AG]]QR and 0 0 local states (B, ρ0) and (B , ρ0) are compatible, [[A ]](B) to [[A¯ ]] and [[A ]](B) . G ρ0LR G QR G ρ0QR and denote this relation by the symbol (B, ρ0) ∼ (B0, ρ0 ), if for all σ ∈ [[A ]](B) there exists a 0 G ρ0QR Definition 66. Let a = [a ] ∈ [[A¯ ]] and 0 nRn G QR 0 (B ) (B) state σ ∈ [[AG]]ρ0 Q such that, for every a ∈ σ = [σn ] ∈ [[AG]] . Then we define 0 R Snρ0 ρ0QR ¯ [[AG]]QR, one has

(a|σ) := lim (a|σn ). (132) 0 n→∞ Smρ0 (a|σ) = (a|σ ).

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 62 The equivalence classes of pairs (B, ρ0) under We can now prove the following important re- the above relation will be denoted by [(B, ρ0)]. sult. The norm k·kB,ρ can be extended to the space 0 Lemma 84. The space [[A ]] is a closed sub- of finite linear combinations of generalised quasi- G QR space of the extended space of states [[A ]] . On local states. G R [[AG]]Q it holds that k·kQ ≡ k·k∗. (B(i)) R Definition 68. Let σi ∈ [[AG]] , for i = ρ(i)Q 0 R Proof. We only need to prove that k·k∗ ≡ k·kQ (i) (i) (j) (j) 1, . . . , k, with [(B , ρ0 )] 6= [(B , ρ0 )]. Then on finite sums of the form of eq. (136), since we we define already proved that [[AG]]QR is obtained closing k k a linear manifold in the space of bounded lin- X X ear functionals on [[A¯ G]]Q . The proof follows ex- aiσi := |ai|kσik (i) (i) . R B ,ρ0 i=1 Q i=1 actly the same argument as for lemma 82, using eq. (135) instead of eq. (134). The space of finite sums as from definition 68 is a real vector subspace of [[AG]]R. Moreover, Notice that all local states allow us to de- as a real vector space, it is equipped with the fine states of arbitrary finite regions S ⊆ G, by norm k·kQ. We can define the space of Cauchy Eq. (122). We will now extend the definition of sequences in this space, and complete it by the local state for all quasi-local states in a fixed sec- usual procedure. This leads us to the following tor [[A]](B) , as follows. First, if ρ is quasi local, ρ0Q definition of the space of quasi-local states. then kρn − ρmkB,ρ0 ≤ ε, and k(ρn − ρm)|Skop ≤ kρ − ρ k ≤ ε. Thus, being the set of states Definition 69. The space of generalised quasi- n m B,ρ0 [[A ]] compact, ρ := lim ρ | ∈ [[A ]]. local states of A , denoted as [[A ]] , is S S n→∞ n S S G G QR The same is true for finte linear combinations, [[A ]] := M [[A ]](B) , as well as limits of Cauchy sequences thereof. In- G QR G ρ0QR [B,ρ ] deed, given a finite combination of states ρ = 0 P i aiρi, one has where the direct sum denotes closure of the space X of finite linear combinations in the norm k·kQ. ρ|S := aiρi|S ∈ [[AS]]R. (138) i The norm introduced above allows us to prove that [[AG]]QR is a space of bounded linear func- Finally, if τ : N → [[A]]QR, and kτn − τmkQ ≤ ε, ¯ then clearly k(τ − τ ) k ≤ ε, and thus tionals on [[AG]]QR, namely a subspace of [[AG]]R. n m |S op

Lemma 83. Every quasi-local state ρ ∈ [[AG]]Q R τ|S := lim (τn)|S ∈ [[AS]]R. (139) is a continuous functional on the space of quasi- n→∞ ¯ local effects a ∈ [[AG]]QR, and In view of the above considerations, we can now define the set of states in [[AG]]QR. |(a|ρ)| ≤ kaksupkρkQ (135) Definition 70. The set [[A ]] of quasi-local Proof. Let ρ ∈ [[A ]] be a finite sum of the form G Q G R states is the (convex) set of generalised states ρ k (G) (i) for which, for every finite region S ∈ R , the ρ = X a σ , σ ∈ [[A ]](B ). (136) i i i G ρ0QR restriction ρ|S of ρ to S is a state in [[AS]]. The i=1 positive cone [[AG]]Q+ of AG is Then [[AG]]Q+ k X := {σ ∈ [[AG]]Q | ∃λ ≥ 0, ρ ∈ [[AG]]Q : σ = λρ}. (a|ρ) := ai(a|σi). (137) R i=1 The set of deterministic quasi-local states of AG, One can easily realise that |(a|ρ)| ≤ kaksupkρkQ. denoted as [[AG]]Q1, is Finally, by the above inequality, also the limit [[A ]] := {ρ ∈ [[A ]] | (e |ρ) = 1}. of a Cauchy sequence in [[AG]]QR is a continuous G Q1 G Q G linear functional on [[A¯ G]]Q , and satisfies R Lemma 85. Definition 70 is compatible with def- |(a|ρ)| ≤ kaksupkρkQ. inition 16

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 63 Proof. We need to prove that the definition of D Homogeneity and causal influence restriction to the finite region S provided in Eq. (33) is equivalent to that of Eq. (139). We This section is devoted to the proof that, for π ∈ start from the case of local states. In this case, ΠV , since effects are separating for states of finite 0 0 π(g) π(g ) ⇔ g g dimensional systems, one has (aS|ρT ) = (aS ⊗ By definition, f f 0 if there exists R 3 f and eT \S|ρ|T ⊗ρ0|S\T ), and thus ρ|S = ρ|(T ∩S)⊗ρ0|S\T , which is precisely the same as in Eq. (122). The (G) (G) ∃F ∈ [[Af C → Af C]]QR, statement in the case of a general state ρ ∈ ( ⊗ ) ( −1 ⊗ ) 6∈ [[A(G)C → A(G)C]] . [[AG]]Q is then straightforwardly proved, remind- V IC F V IC f¯0 f¯0 QR ing that |(aS|ρ − ρn)| ≤ kaSksupkρ − ρnkQ. (141) Let now f = π(g) for some π ∈ Π . Now, by Corollary 19. One has the following inclusions V homogeneity, and in particular by Eq. (104), one (G) (G) has that F 0 ∈ [[A C → A C]] if and only [[AG]]Q ⊆ [[AG]], [[AG]]Q1 ⊆ [[AG]]1. (140) π(g) π(g) (G) (G) if there exists F ∈ [[Ag C → Ag C]] such that 0 −1 F = (Tπ ⊗ I )F (T ⊗ I ). Then one has C Proof of identity 94 C π C 0 −1 (V ⊗ IC)F (V ⊗ IC) −1 −1 In order to keep the notation lighter, we do the = (VTπ ⊗ I )F (T V ⊗ I ) ∼ C π C calculation for the case of C = I. The proof can −1 −1 = (TπV ⊗ I )F (V T ⊗ I ). then be adapted to the general case by suitably C π C (142) padding all transformations with IC and extending the region S˜ to S˜C. As a consequence, again by eq. (104), we have 0 −1 (G) that (V ⊗ IC)F (V ⊗ IC) ∈ [[A ¯0 C → −1 † −1† † † f (VAV ⊗ IH ) a ˜ = W (A ⊗ IH ) W a ˜ (G) 1 S 1 S A C]], for f 0 = π(g0), if and only if ( ⊗ f¯0 V = −1†( ⊗ )† † 0† a W A IH1 SN +∗∪N + SN +∗∪S S˜ −1 (G) (G) S S S IC)F (V ⊗ IC) ∈ [[Ag¯0 C → Ag¯0 C]]. Fi- = 0† † ( ⊗ )† nally, comparing with Eq. (141), this implies that SP +∗∪N +∗∪RSN +∪N +∗∪R A IH1 S S S S π(g) π(g0) if and only if g g0. † 0† × S +∗ + S +∗ aS˜ NS ∪NS NS ∪S 0† † † =S +∗ +∗ S + +∗ (A ⊗ IH1 ) E Proof of right and left invertibility of PS ∪NS ∪R NS ∪NS ∪R † 0† a locally defined GUR × S + +∗ S +∗ +∗ aS˜ NS ∪NS ∪R PS ∪NS ∪R = 0† † ( ⊗ )† † 0† a Here we prove right and left invertibility of V SP +∗∪N +∗∪RSR A IH1 SRSP +∗∪N +∗∪R S˜ S S S S defined starting from Eq. (109). First, notice that 0† † 0† = ( ⊗ ) a − − − − S +∗ +∗ IH0 A S +∗ +∗ S˜ PR PR \R PR PR \R PS ∪NS ∪R PS ∪NS ∪R ψ e ψ e 0† 0† † R R R e =SR (IH0 ⊗ A ) SR [aS˜] 0 = 0 SN − a SN − R C R C A C = 0† † ( ⊗ )† † 0†[a ], e SR SR A IH0 SRSR S˜ N − N − N − N − R R e R R e where in the third identity we used Eq. (91), ob-

† † 0† P − P −\R P − serving that (A ⊗ IH1 ) S +∗ + S +∗ [aS˜] ∈ ψ R R R e NS ∪NS NS ∪S ¯ (G) + R R [[A +∗ ∪ (NS , 1)]]QR. In the fourth identity = 0 0 (NS ∪R,0 SN − A SN − R C C e R we used the fact that a ˜ can be padded with eX S − − − NR NR NR to obtain an effect a˜X on the region S˜ ∪ X, and e † † then W a˜X is an effect equivalent to W aS. In the fifth identity we used the fact that SX com- P − ψ R e mutes with YY ⊗ IH1 for Y ∩ X = ∅, and in 0 = C C e , the seventh identity we used the fact that SX 0 N − A N − R R e commutes with IH0 ⊗ YY for Y ∩ X = ∅.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 64 where we used Eq. (94). have P − † † N+ N + N + Let us now deﬁne by b := {( ⊗ P + P + P + W W SC WS R R R e † ψ )b} + , where IC N C C S 0 P − \N + + + + 0 S N P + + + P + + R S b P P PR NS NS R R R + e N e + + S P + P + + + S 0 P P P P \R WS := S . R R R R η S S S e η e

(143) P − N+ N + \N + P + P + R ψ R R e † † Then one can calculate W V aRC by the follow- N + N + R R e C ing diagram B00 C C b 0 e R − − − − = SP + = , PR \R PR PR PR \R P + e ψ e R − + P + \R P \N P + N+ P + R P + R e R R R e 0 0 + + S − S − a P + P + NR NR P P C R R e N − N − N − η R R R e where Eq. (97) was used in the last step. The arguments are straightforwardly generalised to (G) P −\R P − ¯ ¯ R R e b ∈ [[AS C]]QR. R 0 C C = SN − e R 0 N − N − A N − η R R R e F Proof of theorem 10

P −\R P − P −\R Proof. In the following, Gi := (G, i) for i = 0, 1. R R R R e − PR \R The argument in appendixE shows that R R R e e 0 0 R = SN − SN − A = . G0 R R C C a G0 G0 e e C e N − N − N − η R R R e W = , G1 G1 G1 G0 V −1 Finally, we observe that [(a ⊗ eP −\R)P −C] = R R −1† † † † where (G, A, V ) is a GUR. Moreover, since [aRC], then W V = IG. Similarly, one can cal- † † G0 G1 G culate V W bSC. First we observe that [bSC] = 0 e † † = , [(b ⊗ e + ) + ]. Then W bSC = {(W + (b ⊗ SG P \S P C P G1 G0 S S S e G1 eN + \N + )}N + , and P + S P +C S S one has

N + \N + N + N + \N + N + G0 G0 G0 G0 P + S P + P + S P + e e S S e S S e + + + + = −1 N P \S N P \S W S S S S G1 W G1 G1 0 e 0 e G1 SP + = SP + S C S C C e P + b P + B η S S η S S S e G0 G1 G1 G0 e

N + \N + N + N + P + S P + P + = SG W SG S S S e G1 G0 G0 G1 G1 + + −1 NS PS \S V 0 0 = SP + SP + P + S C C e S S e P + B P + η S S S S e G0 G1 G1 G0 V −1 = SG W , N + \N + P + S G1 G0 G0 S e e N + N + S S e namely = B00 , C C e G1 G1 G1 G0 P + −1 η S e V = W . G0 G G † † e 0 0 where we used Eq. (97). Now, for V W [bSC] we e

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 65 On the other hand, one can write the latter as that

C C G1 G1 G1 G1 C C V − VL (AR) G G AR G G = , (144) G0 G0 = 0 0 0 0 G W G G0 0 0 e W −1 W e G1 G1 G1 G1 or equivalently C C

G0 G G e 0 0 e AR = G0 −1 G0 G0 G0 , = W −1 . V V G1 G1 G1 G1 V G1 Now, let us consider and

G0 G1 e G0 G0 G0 G0 ˜ −1 SG W W G1 G0 G1 G1 = G1 G1 G1 G1 + V (AR) AR C L C C C G2

G0 G0 G1 e SG G1 G1 G0 G0 G1 G1 G1 G1 := = V V −1 . −1 W G2 W G2 G2 AR C C

G G 0 0 G Proof of lemmas 66 and 67 W G2 G2 G2 = We start with the proof of lemma 66. W G1 G1 e Proof. (1) Notice that N 0± = φ(S ) = φ(N ± ). eH ± eG Then |N 0± | ≤ |N ± |, with strict inequality if and eH eG G0 G0 only if φ(h ) = φ(h ) for some h 6= h ∈ N ± . a b a b eG W However, this is impossible because it would be G2 G2 G2 −1 0 = V , equivalent to φ(hahb ) ∈ R , namely ha = hb, contrarily to the hypothesis. Thus, |N 0± | = eH G1 |N ± |. Invariance of neighbourhoods under left- e eG multiplication ﬁnally gives the thesis. from which one concludes that (2) Let now f ∈ P 0± = N 0∓ , namely there eH N 0± eH 0± 0∓ G1 exists h ∈ N such that f ∈ N , or equiva- G0 G1 e e eH h lently h ∈ N 0±. Then, f ∈ P 0± iﬀ there exists ˜ = , f eH SG 0± G1 G0 G0 G0 h ∈ N 0± ∩ N . Equivalently, there exists h ∈ H X eH f such that h = φ(ha) and h = fφ(hb), namely f = φ(h h−1). Then P 0± = {φ(h h−1) | h , h ∈ for some X . Thus a b eH a b a b N ± }. By a similar argument, we have that eG G0 G0 G0 G0 P ± = {h h−1 | h , h ∈ N ± }. Consequently, X eG a b a b eG = . |P 0±| ≤ |{h h−1 | h , h ∈ N + }| = |P ± |. Now, W eH a b a b eG eG G2 G2 G2 V G2 strict inequality in the last relation would require −1 −1 that φ(hahb ) = φ(hdhc ), contrarily to the hy- Finally, this implies that = ⊗ −1, and pothesis. We conclude that |P 0±| = |P ± |. Also W X V eH eG in particular, by Eq. (144), it must be X = V , in this case the thesis can be obtained by invari- namely W = V ⊗ V −1. Consequently, we have ance of neighbourhoods under left-multiplication.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 66 (3) Let N 0± ∩ N 0± = ∅. Then clearly for every φ(P ± ∩ P ±). Finally, by the same argument as f1 f2 g1 g2 −1 ± ± for the proof of item (2) of lemma 66, we get the gi ∈ φ (fi) it must be Ng1 ∩ Ng2 = ∅. Let then now h ∈ N 0± ∩ N 0±. This implies that thesis. f1 f2 ± (2) The hypothesis clearly implies that N ± ∩ h = f1φ(ha) = f2φ(hb) for ha, hb ∈ N . Conse- g1 eG ± ± ± −1 Ng = ∅. Suppose now that h ∈ Pg ∩ Pg . Then quently, f2 = f1φ(hahb ). Now, either f1 = f2, 2 1 2 h = g h h−1 = g h h−1 for h , h , h , h ∈ N ± . and then choosing g1 = g2 the thesis trivially fol- 1 a b 2 d c a b c d eG −1 0 −1 −1 −1 lows, or φ(hah ) 6∈ R , and then hah 6∈ R. This implies that g2 = g1hahb hchd , and con- b b −1 −1 −1 sequently φ(g2) = φ(g1)φ(hah hch ). On the One can then set g2 = g1hahb , and verify that b d −1 g2 ∈ φ (f2), as well as k = g1ha = g2hb ∈ other hand, we proved that for every he, hf ∈ ± ± N ± one has g 6= g h h−1, while by hypothesis Ng ∩ Ng . Thus, h = φ(k), and consequently eG 2 1 f e 1 2 −1 ± 0± 0± ± ± φ(g2) = φ(g1)φ(hf h ) for some he, hf ∈ N . N ∩ N ⊆ φ(Ng ∩ Ng ). This implies that e eG f1 f2 1 2 −1 −1 −1 0± 0± ± ± Thus, while hah hch heh 6∈ R, it must be |Nf ∩Nf | ≤ |Ng ∩Ng |, with a strict inequality b d f 1 2 1 2 −1 −1 −1 0 ± ± φ(hah hch heh ) 6∈ R . We reached a con- iff φ(k1) = φ(k2) for k1 6= k2 ∈ Ng ∩ Ng . How- b d f 1 2 ± ± ever, form the proof of item (1) we know that if tradiction, and thus it must be Pg1 ∩Pg2 = ∅. ± k1 6= k2 ∈ Ng it must be φ(k1) 6= φ(k2). Thus, |N 0± ∩ N 0±| = |φ(N ± ∩ N ±)|, and N 0± ∩ N 0± = f1 f2 g1 g2 f1 f2 References ± ± φ(Ng1 ∩ Ng2 ). (4) Let h ∈ P 0∓ ∩ N 0±. Then h = f φ(h−1h ) = [1] Lucien Hardy. Disentangling nonlocality f2 f1 2 c b f φ(h ), with h , h , h ∈ N ± . This implies and teleportation, 1999. arXiv:quant-ph/ 1 a a b c eG −1 9906123. that f2 = f1φ(hahb hc). We can now set g2 := −1 −1 g1hahb hc with g1 ∈ φ (f1). In this way, [2] Lucien Hardy. Quantum theory from five one has f2 = φ(g2). Let us set k := g1ha = reasonable axioms, 2001. arXiv:quant-ph/ −1 ∓ ± g2hc hb ∈ Pg2 ∩ Ng1 . Then, h = φ(k), and 0101012. P 0∓ ∩N 0± ⊆ φ(P ∓ ∩N ±). By the same argument f1 f1 g2 g1 [3] Christopher A. Fuchs. Quantum mechanics as for item (3), the thesis follows. as quantum information (and only a little (5) Let h ∈ P ± ∩ N ∓. Then h = g h h−1, h = g1 g2 1 a b more), 2002. arXiv:quant-ph/0205039. g h−1, for some h , h , h ∈ N ± , and thus g = 2 c a b c eG 2 −1 [4] Gilles Brassard. Is information the key? Na- g1hahb hc. On the other hand, by hypothesis, ± ture Physics, 1(1):2–4, 2005. URL: https: for every hx ∈ N one has g2 6= g1hx. Collecting eG . −1 −1 //doi.org/10.1038/nphys134 all the informations, we have hahb hchx 6∈ R, −1 −1 0 ± [5] Jonathan Barrett. Information processing in i.e. φ(hah hch ) 6∈ R , for every hx ∈ N . b x eG generalized probabilistic theories. Phys. Rev. Then one has φ(g ) = φ(g )φ(h h−1h ). By hy- 2 1 a b c A, 75:032304, Mar 2007. URL: https:// pothesis, it is also φ(g ) = φ(g )φ(h ) for some 2 1 d doi.org/10.1103/PhysRevA.75.032304. h ∈ N ± . This is clearly absurd, thus it must be d eG ± ∓ [6] Giacomo M. D’Ariano. Probabilistic theo- Pg1 ∩ Ng2 = ∅. ries: What is special about quantum me- Now, we provide the proof of lemma 67 chanics? In Alisa Bokulich and Gregg Jaeger, editors, Philosophy of Quantum In- Proof. (1) Let f ∈ P 0±, namely f = 2 f1 2 formation and Entanglement, chapter 5, f φ(h h−1) for h , h ∈ N ± . If f = f , it pages 85–126. Cambridge University Press., 1 a b a b eG 1 2 −1 is sufficient to choose g1 = g2 ∈ φ (f1). Let 2010. URL: https://doi.org/10.1017/ −1 now f1 6= f2. Then, let g1 ∈ φ (f1), and CBO9780511676550.007. −1 g2 := g1hahb . By construction, φ(g2) = f2 and [7] Giacomo Mauro D’Ariano and Alessandro g ∈ P ±. Now, let h ∈ P 0± ∩ P 0±, namely h = 2 g1 f1 f2 Tosini. Testing axioms for quantum the- −1 −1 −1 −1 f1φ(hf he ) = f2φ(hchd ) = f1φ(hahb hchd ). ory on probabilistic toy-theories. Quan- −1 −1 −1 0 Then we have φ(hahb hchd hehf ) ∈ R , and tum Information Processing, 9(2):95–141, −1 −1 −1 necessarily hahb hchd hehf ∈ R. This implies 2010. URL: https://doi.org/10.1007/ −1 −1 −1 −1 s11128-010-0172-3. that g1hf he = g1hahb hchd = g2hchd . Now, −1 −1 setting k := g1hf he , we have k = g2hchd ∈ [8] Giulio Chiribella, Giacomo Mauro D’Ariano, P ± ∩ P ±, and h = φ(k). Thus, P 0± ∩ P 0± ⊆ g1 g2 f1 f2 and Paolo Perinotti. Probabilistic theories

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