Degrees, Levels, and Profiles of Contextuality

Degrees, Lev els, and Profiles of Con textualit y Eh tibar N. Dzhafaro v 1 & Víctor H. Cerv an tes 2 1 Purdue Univ ersity , USA, eh tibar@purdue.edu 2 Univ ersity of Illinois at Urbana-Champaign, USA, victorhc@illinois.edu Abstract W e in tro duce a new notion, that of a contextuality pr ofile of a system. Rather than characterizing a system’s contextualit y by a single num b er, its ov erall de gr e e of c ontextuality , w e show ho w it can b e characterized b y a curve relating degree of contextualit y to level at whic h the system is considered, lev el 1 · · · n − 1 n > 1 n + 1 · · · N degree 0 · · · 0 d n > 0 d n +1 ≥ d n · · · d N ≥ d N − 1 , where N is the maxim um num b er of v ariables per system’s context. A system is represented at level n if one only considers the joint distribu- tions with k ≤ n v ariables, ignoring higher-order joint distributions. W e sho w that the level-wise con textuality analysis can be used in conjunction with an y well-constructed measure of con textuality . W e present a metho d of concatenated sy stems to explore contextualit y profiles systematically , and we apply it to the contextualit y profiles for three ma jor measures of con textuality prop osed in the literature. Keyw ords : con textuality , contextualit y profile, concatenated sys- tems, degree of contextualit y , disturbed systems, lev el of con textuality , measure of contextualit y , undisturbed systems There is a consensus that it is not sufficien tly informativ e to merely ascertain whether a system is contextual. It is also desirable and useful to measure its de gr e e of c ontextuality . W e recently prop osed that it is als o of in terest to de- termine a system’s first level of c ontextuality . The meaning of this term is as follo ws. Let the join t distribution of any k v ariables in the system b e called a k - mar ginal . F or an y n , if one c haracterizes the system b y all its k -marginals with k ≤ n , ignoring the higher-lev el marginals, we say that the system is r epr esente d at level n . A con textual system is alwa ys noncontextual at level 1, and there is a lo west lev el n > 1 at which it becomes contextual. This n is the system’s first level of c ontextuality , and at this level the degree of contextualit y , d n , can b e measured in sev eral kno wn w ays. In Ref. [1], this procedure is describ ed in detail for the measure based on the L 1 -distance b etw een a p oin t represen ting a system at a giv en level and the corresponding noncontextualit y p olytope. 1 W e no w sho w that the level-wise analysis can b e used in conjunction with an y other measure of contextualit y (notably , the contextual fraction, and the measure inv olving “negativ e probabilities”, b oth extended to apply to systems with disturbance). Moreo ver, for a system determined to b e contextual at level n , one can con tin ue to measure its contextualit y degree at levels n + 1 , n + 2 , . . . , N (where N is the maximum num b er of v ariables p er system’s con text). In this w ay one c haracterizes a system by a ve ctor of c ontextuality values lev el 1 · · · n − 1 n n + 1 · · · N degree 0 · · · 0 d n d n +1 · · · d N (1) that can b e called the system’s c ontextuality pr ofile . W e discuss w ays of explor- ing the patterns of contextualit y systematically . F or well-constructed measures of con textuality , d k is nondecreasing in k . This is true for the three measures men tioned ab o ve, none of them b eing a function of an y other. The tendency of d k to increase with k is minimal for the “negativ e probabilities” measure and maximal for the L 1 distance measure, with the contextual fraction falling in b et w een. The structure of the pap er is as follo ws. In the next section w e in tro duce the notion of a level- n representation of a system of random v ariables. Section 2 presen ts the main idea of the paper: given a measure of contextualit y , to define a system’s contextualit y profile as the sequence of its contextualit y v alues at differen t lev els. In Sections 3 and 4 w e remind the reader of the definition of con textuality and discuss three sp ecific measures of contextualit y that hav e b een prop osed in the contextualit y literature. In Section 5 w e prop ose a metho d of concatenated systems for exploring how fast contextualit y profiles grow from one level to another. In Section 6 we present the results of applying this method to the three measures of contextualit y just men tioned. In Section 7 we discuss con textuality profiles for a selection of systems of interest. Section 8 offers a summary and some questions for future w ork. 1 Lev el-wise represen tations Consider a generic example of a system: R 1 2 R 1 3 c = 1 R 2 1 R 2 2 R 2 4 R 2 5 2 R 3 1 R 3 2 R 3 3 R 3 4 3 R 4 1 R 4 2 R 4 5 4 q = 1 2 3 4 5 R . (2) W e use our usual notation here. R c q is a random v ariable recorded in context c and answering question q . If R c q is defined for a given ( q , c ) , i.e., if the cell is not empt y , we write q  c (question q is answered in context c ). The main property of a system of random v ariables is that all the v ariables in each row (sharing 2 a context) are jointly distribute d , whereas no tw o v ariables from differen t rows are join tly distributed (they are sto chastic al ly unr elate d ). A system is c onsistently c onne cte d if, for any q  c, c ′ , R c q d = R c ′ q , (3) that is, an y tw o v ariables answering the same question are identically dis- tributed. A system is undisturb e d (or str ongly c onsistently c onne cte d ) if, for an y subset of questions q 1 , . . . , q k  c, c ′ ,  R c q 1 , . . . , R c q k  d =  R c ′ q 1 , . . . , R c ′ q k  . (4) F or instance, if the system R ab o ve is undisturb ed, then  R 2 1 , R 2 2 , R 2 4  d =  R 3 1 , R 3 2 , R 3 4  ,  R 1 2 , R 1 3  d =  R 3 2 , R 3 3  , etc . (5) T o define the represen tations of the system R on v arious levels, let us first define them for a single ro w of this system, say , R 2 1 R 2 2 R 2 4 R 2 5 c = 2 q = 1 2 3 4 5 R 2 . (6) Skipping lev el 1, whic h will b e discussed later, the representation of this row on lev el 2 is 1 R 2 . 1 1 R 2 . 1 2 2 . 1 R 2 . 2 1 R 2 . 2 4 2 . 2 R 2 . 3 1 R 2 . 3 5 2 . 3 R 2 . 4 2 R 2 . 4 4 2 . 4 R 2 . 5 2 R 2 . 5 5 2 . 5 R 2 . 6 4 R 2 . 6 5 2 . 6 q = 1 2 3 4 5 R 2 [2] , (7) where, for an y q , q ′  ( c = 2) ,  R 2 .x q , R 2 .x q ′  d =  R 2 q , R 2 q ′  . (8) In other w ords, each of the  4 2  = 6 rows of the new matrix is obtained by pic king from the row R 2 a subset of tw o v ariables, and creating their distributional copy . Clearly , the system R 2 [2] represen ting the row R 2 on lev el 2 is an undisturb ed system: e.g., R 2 . 1 2 d = R 2 . 4 2 d = R 2 . 5 2 d = R 2 2 , R 2 . 2 4 d = R 2 . 4 4 d = R 2 . 6 4 d = R 2 4 , etc . (9) 1 The reason we denote the row R 2 (italics) but denote the system below R 2 (script) is that we use italics for sets of v ariables when they are jointly distributed and script letters when they are not (or are not necessarily). W e use this notation conven tion throughout this pap er. 3 The system R 2 [2] has the same individual and pairwise distributions as the row R 2 , but R 2 [2] contains no higher-order distributions (no triples, quadruples, etc.) b ecause join t distributions only exist within but not across the contexts. The represen tation R 2 [3] of the row R 2 on lev el 3 is obtained analogously , b y picking from this ro w  4 3  = 4 p ossible triples of v ariables and creating their distributional copies: R 2 . 1 1 R 2 . 1 2 R 2 . 1 4 2 . 1 R 2 . 2 1 R 2 . 2 2 R 2 . 2 5 2 . 2 R 2 . 3 1 R 2 . 3 4 R 2 . 3 5 2 . 3 R 2 . 4 2 R 2 . 4 4 R 2 . 4 5 2 . 4 q = 1 2 3 4 5 R 2 [3] . (10) This system, to o, is undisturb ed b y construction. Finally , the level 4 representation of the row R 2 simply coincides with it, b ecause there is only one quadruple we can select from R 2 : R 2 1 R 2 2 R 2 4 R 2 5 c = 2 q = 1 2 3 4 5 R 2 [4] = R 2 [4] = R 2 1 R 2 2 R 2 4 R 2 5 c = 2 q = 1 2 3 4 5 R 2 . (11) This system is trivially undisturb ed. The higher-level representations R 2 [5] , R 2 [6] , etc. of R 2 also simply coincide with R 2 , which is explained as follows. A represen tation on level n > 1 should alw ays b e taken as cum ulative, to include not only n -tuples but also all low er- lev el tuples. How ever, if n -tuples exist (the original row con tains no less than n v ariables), inclusion or exclusion of the low er-level tuples nev er influences the contextualit y status of the represen ting system (i.e., whether the system is con textual or noncontextual) or an y reasonable measure of the degree of con textuality if it is contextual. So, e.g., the system R 2 [3] with an added ro w c = 2 . 5 that contains a pair, R 2 . 1 1 R 2 . 1 2 R 2 . 1 4 2 . 1 R 2 . 2 1 R 2 . 2 2 R 2 . 2 5 2 . 2 R 2 . 3 1 R 2 . 3 4 R 2 . 3 5 2 . 3 R 2 . 4 2 R 2 . 4 4 R 2 . 4 5 2 . 4 R 2 . 2 1 R 2 . 2 2 2 . 5 q = 1 2 3 4 5 R 2+ [3] , (12) is equiv alen t to R 2 [3] in any considerations of contextualit y . This is a general prop ert y of undisturb ed systems. That is wh y we defined R 2 [3] as con taining only triples rather than also pairs and singles, and R 2 [4] as containing only quadruples, ignoring triples, pairs, and singles. Ho wev er, on the next lev el, 5 , 4 no quintuples of v ariables exist, so w e ha ve to include the highest existing tuple, whic h in this case is the quadruple: R 2 1 R 2 2 R 2 4 R 2 5 c = 2 q = 1 2 3 4 5 R 2 [5] = R 2 [5] = R 2 1 R 2 2 R 2 4 R 2 5 c = 2 q = 1 2 3 4 5 R 2 . (13) By the same logic, the represen tation of the first row of the system R , R 1 2 R 1 3 c = 1 q = 1 2 3 4 5 R 1 , (14) is one and the same at all lev els n > 1 : R 1 = R 1 [2] = R 1 [3] = · · · (15) Let us now explain why we do not consider lev el 1 represen tations. In fact, w e do include them in the definition of a contextualit y p rofile (Section 2), but they are alw ays trivially noncontextual, requiring no separate analysis. F or our example, the ro w R 2 , the lev el 1 representation is R 2 . 1 1 2 . 1 R 2 . 2 2 2 . 2 R 2 . 3 4 2 . 3 R 2 . 4 5 2 . 4 q = 1 2 3 4 5 R 2 [1] . (16) A system with only one v ariable in each column is alw ays noncon textual, i.e., its con textuality degree is zero for an y measure of contextualit y . With the algorithm sp ecified, the level 4 representation of the en tire system R is R 1 2 R 1 3 c = 1 R 2 1 R 2 2 R 2 4 R 2 5 2 R 3 1 R 3 2 R 3 3 R 3 4 3 R 4 1 R 4 2 R 4 5 4 q = 1 2 3 4 5 R [4] = R , 5 and its lev el 3 and level 2 represen tations are, resp ectiv ely , R 1 2 R 1 3 c = 1 R 2 . 1 1 R 2 . 1 2 R 2 . 1 4 2 . 1 R 2 . 2 1 R 2 . 2 2 R 2 . 2 5 2 . 2 R 2 . 3 1 R 2 . 3 4 R 2 . 3 5 2 . 3 R 2 . 4 2 R 2 . 4 4 R 2 . 4 5 2 . 4 R 3 . 1 1 R 3 . 1 2 R 3 . 1 3 3 . 1 R 3 . 2 1 R 3 . 2 2 R 3 . 2 4 3 . 2 R 3 . 3 1 R 3 . 3 3 R 3 . 3 4 3 . 3 R 3 . 4 2 R 3 . 4 3 R 3 . 4 4 3 . 4 R 4 1 R 4 2 R 4 5 4 q = 1 2 3 4 5 R [3] . (17) and R 1 2 R 1 3 c = 1 R 2 . 1 1 R 2 . 1 2 2 . 1 R 2 . 2 1 R 2 . 2 4 2 . 2 R 2 . 3 1 R 2 . 3 5 2 . 3 R 2 . 4 2 R 2 . 4 4 2 . 4 R 2 . 5 2 R 2 . 5 5 2 . 5 R 2 . 6 4 R 2 . 6 5 2 . 6 R 3 . 1 1 R 3 . 1 2 3 . 1 R 3 . 2 1 R 3 . 2 3 3 . 2 R 3 . 3 1 R 3 . 3 4 3 . 3 R 3 . 4 2 R 3 . 4 3 3 . 4 R 3 . 5 2 R 3 . 5 4 3 . 5 R 3 . 6 3 R 3 . 6 4 3 . 6 R 4 . 1 1 R 4 . 1 2 4 . 1 R 4 . 2 1 R 4 . 2 5 4 . 2 R 4 . 3 2 R 4 . 3 5 4 . 3 q = 1 2 3 4 5 R [2] . (18) 2 Profiles of contextualit y Let us assume that w e hav e a measure of con textualit y that applies to an y system R (from a sufficien tly broad class of systems). Let us denote its v alue by deg R . The main idea of this pap er is this: contextualit y v alues of the lev el-wise represen tations of R , deg R [1] = 0 , deg R [2] = d 1 , . . . , deg R [ N ] = d N , (19) can b e considered the c ontextuality pr ofile of the system R . Here, N is the maximal n umber of v ariables in a row of R . The v alues of d N +1 , d N +2 , etc., need 6 Figure 1: F our p ossible contextualit y profiles with the same final degree of contextu- alit y at level 5. not b e considered b ecause they alw ays equal d N . W e include the uninformativ e deg R [1] = 0 as the “anchoring p oin t” of a profile, primarily for aesthetic reasons. Figure 1 presen ts hypothetical contextualit y profiles for four systems with N = 5 . Observ e that on level 5 all four profiles hav e the same v alue. This common v alue is the con textuality degree that our measure deg will sho w for all four systems, b ecause lev el 5 represen tations of these systems coincide with the systems themselves. The existing w ays of con textuality analysis therefore w ould treat these four systems as essentially indistinguishable. In Ref. [1], this idea is partially implemented for the measure of contextualit y that we called “hierarc hical.” The implementation inv olves lev els of consider- ation, but the pro cess describ ed there stops at the first contextual lev el (the smallest n with d n > 0 ). In essence, for a con textual system, this merely re- places a p oin t measure of contextualit y (the single num b er d N ) with a tw o-p oin t one: ( n min , d n min ) . (20) What w e prop ose no w is that (A) there is no reason to stop at the smallest n with d n > 0 , one can compute an en tire function n 7→ d n ( n = 1 , 2 , . . . , N ), the system’s con textuality profile; (B) one can do this for an y well-constructed measure of con textuality; (C) computing contextualit y profiles for different measures can b e an informativ e wa y of comparing them. W e prop ose that a w ell-constructed contextualit y measure should hav e the fol- lo wing three prop erties: 1. for any noncontextual system its final-level degree of contextualit y is zero; 7 2. for any contextual system its final-level degree of con textuality is p ositive; 3. its con textuality profile is a nondecreasing function of lev el. These requirements are obviously satisfied for the three measures w e are going to explore in the next section. Ho wev er, it is w orth men tioning that some seemingly reasonable measures of con textuality may fail them. Thus, in Ref. [2] w e describ e a measure abbreviated CNT 1 , which, as it turns out, may produce decreasing contextualit y profiles. One therefore should consider this measure not w ell-constructed, and this is the reason w e do not include it in the analysis b elo w. 3 Definition of contextualit y The notion of contextualit y , extended to include disturb ed systems, is describ ed in detail in our previous publications (e.g., Refs. [1, 2]). Put briefly , and using the same example as b efore, given a system R , R 1 2 R 1 3 c = 1 R 2 1 R 2 2 R 2 4 R 2 5 2 R 3 1 R 3 2 R 3 3 R 3 4 3 R 4 1 R 4 2 R 4 5 4 q = 1 2 3 4 5 R , (21) w e attempt to construct its pr ob abilistic c oupling S 1 2 S 1 3 c = 1 S 2 1 S 2 2 S 2 4 S 2 5 2 S 3 1 S 3 2 S 3 3 S 3 4 3 S 4 1 S 4 2 S 4 5 4 q = 1 2 3 4 5 S , (22) in which all v ariables are join tly distributed (not just within contexts but o ver- all), sub ject to the following constraints: (a) The v ariables in the rows of S are jointly distributed as the v ariables in the corresp onding row of R , e.g.,  S 1 2 , S 1 3  d =  R 1 2 , R 1 3  ,  S 2 1 , S 2 2 , S 2 4 , S 2 5  d =  R 2 1 , R 2 2 , R 2 4 , R 2 5  , etc . (b) An y tw o v ariables answering the same question (e.g., S 1 2 and S 3 2 ) coincide with the maximal p ossible probability . F or consisten tly connected systems, where the v ariables answ ering the same question are identically distributed, the maximal probability of coinciding, e.g., of  S 1 2 = S 3 2  , equals 1. That is, the requirement (b) then simply makes all v ariables answering the same question iden tical. 8 If a coupling of R satisfying (a) and (b) exists, the system R is nonc ontextual , otherwise it is c ontextual . Let v be a v ector of probabilities 2 that represents the constrain ts imp osed on the distribution of the v ariables in the system’s coupling. F or now, it is not imp ortan t precisely ho w it is constructed. Suffice it to stipulate that v uniquely determines the join t distributions of the v ariables in eac h context, and also ensures that all same-question pairs of v ariables coincide with maximal probabilities. The existence of the coupling with prop erties (a)-(b) then means that there is a v ector x satisfying the following matrix equation: M x = v , x ≥ 0 ( 1 ⊺ x = 1) . (23) Here, x ≥ 0 means that every component of x is nonnegative; 1 is a v ector of 1 s (so that 1 ⊺ x is the sum of the comp onen ts of x ). The comp onents of x are probabilities of all p ossible v alues of the coupling S . Thus, in our example (22), if all v ariables are dic hotomous, there are 2 13 p ossible v alues, such as  S 1 2 = 1 , S 1 3 = 0 , S 2 1 = 0 , . . . , S 4 5 = 1  , and x contains the same num b er of probabilities. M is a Bo olean incidence matrix which tells us which comp onents of x sum to a given probabilit y in v . The reason 1 ⊺ x = 1 is sho wn in (23) paren thetically is that this condition is satisfied automatically due to the fact that v represents probabilit y distributions. 4 Three measures of con textualit y The three measures of contextualit y we are interested in are describ ed in detail in Ref. [2]. Here w e present their brief c haracterization. The first measure is the already-men tioned “hierarc hical” measure. In this pap er we call it the distanc e me asur e . It is based on the notion of distance b et w een a system and a nonc ontextuality p olytop e . Consider all p ossible matrices of a giv en format . The latter is defined by the set of all questions q , the set of all contexts c , and the relation q  c . 3 T wo systems of the same format differ in the join t distributions within their corresp onding con texts. Let V be the set of all vectors v suc h that the equation (23) has a solution: V = { v : M x = v for some x ≥ 0 } . (24) It is known that in the space of v -v ectors V forms a p olytop e, which we call the noncontextualit y p olytop e for systems of a given format (and with fixed 2 W e use standard matrix notation, with all v ectors (such as v ) treated as columns unless shown as transp osed ( v ⊺ ). 3 F or some measures of contextualit y one should also include in the notion of a format the individual distributions of the v ariables in the system. W e do not do this in this pap er, but it should be noted that all systems of a given format should ha ve the same sets of possible v alues for the corresp onding v ariables (if we consider only categorical v ariables; more generally , they should hav e the same sigma-algebras). 9 Figure 2: A tw o-dimensional pro jection of a v ector v ∗ and a noncontextualit y p olytope V , with the L 1 -distance b et ween them. individual distributions of the v ariables, see fo otnote 3). If the system we study (let’s denote its vectorial representation b y v ∗ ) is contextual, then it falls outside V , and its distance from V can b e viewed as a measure of con textuality . When dealing with probabilit y distributions, the distance measure of c hoice is L 1 , defined b y L 1 ( a , b ) = X | a i − b i | , (25) where the summation is ov er all dimensions of the vector space. Our distance measure is deg = L 1 ( v ∗ , V ) , (26) the L 1 -distance b et ween the vector v ∗ and the p olytope V , as sho wn in Figure 2. W e call this measure CNT 2 , follo wing the nomenclature adopted in several previous publications, e.g. [2]. The next measure of con textuality is based on quasi-pr ob abilities (p ositiv e and negativ e num bers that sum to 1). It is obtained by dropping in the definition of noncon textuality (23) the nonn egativit y constrain t x ≥ 0 : M x = v ( 1 ⊺ x = 1) . (27) This matrix equation is alw ays solv able for x , but some of the comp onen ts of a solution ma y b e negative. Let | x | denote the v ector of absolute v alues of the comp onen ts of x . It is easy to see that 1 ⊺ | x | ≥ 1 , (28) and the equalit y is ac hieved if and only if x con tains no negative comp onen ts. Among all solutions x one can alwa ys find some for which 1 ⊺ | x | has the smallest 10 p ossible v alue, and then deg = 1 ⊺ | x | − 1 (29) is a measure of contextualit y . W e refer to it as the quasi-pr ob ability me asur e of con textuality , and denote it CNT 3 , as in our previous publications. F or the third measure of contextualit y , replace the equalit y in the definition of noncon textuality (23) with M x ≤ v , x ≥ 0 ( 1 ⊺ x ≤ 1) , (30) where the inequalities are taken to hold comp onen t-wise. This inequality alw ays has solutions, and among them there are some with the maximal v alue of 1 ⊺ x . Then deg = 1 − ( 1 ⊺ x ) max , (31) is a measure of con textuality , and it is called c ontextual fr action, CNTF . The con textual fraction and the quasi-probability measures w ere first pro- p osed b y Abramsky and Brandenburger [3]. W e later extended them to also apply to disturb ed systems (see Ref. [1] for details). Eac h of the three measures, CNT 2 , CNT 3 , and CNTF , can b e, at least in principle, applied to systems of any format. In particular, given a system R , eac h of them can b e applied to all its level representations, R [1] , R [2] , . . . , R [ N ] , to form their resp ectiv e profiles. 5 Concatenated systems Con textuality profiles can b e studied in man y w ays, b ecause, as all functions , they can b e c haracterized in many wa ys. Moreov er, as should b e exp ected, their prop erties dep end on the format of the systems we c ho ose. This pap er b eing in tro ductory , w e fo cus here on one asp ect of contextualit y profiles only: on comparing our three measures of con textuality , CNT 2 , CNT 3 , and CNTF , on ho w fast the degree of contextualit y tends to increase with its lev el. Let us explain what w e mean by this. Supp ose a measure deg pro duces a profile that c hanges its v alue from d n to d n +1 as one mov es from lev el n to level n +1 ; and supp ose that the corresp onding v alues for another measure, deg ′ , are d ′ n and d ′ n +1 . Both measures are w ell- constructed, so d n +1 ≥ d n , d ′ n +1 ≥ d ′ n . (32) Ho wev er, it would not b e informative to directly compare the numerical v alues of d n +1 − d n and d ′ n +1 − d ′ n (unless one of these differences is zero). The t wo measures are on completely differen t scales, so we may b e comparing meters to grams, or ev en worse, meters to decib els. The same reasoning applies, of course, to their ratios, differences of their cub es, or other measures of incremen tation. 11 W e need to find a wa y to consider the increase from d n to d n +1 and from d ′ n to d ′ n +1 in trinsically , within their resp ectiv e scales. Ho w can this b e done? The increase from d n to d n +1 o ccurs because the system R [ n + 1] contains ( n + 1) -tuples of v ariables, in addition to the k -tuples of v ariables with k ≤ n contained in R [ n ] . The degree of contextualit y brought in by these ( n + 1) -tuples somehow combines with the contextualit y present in R [ n ] to pro duce d n +1 . If we had a wa y of measuring the contextualit y ∆ n +1 brough t in b y these ( n + 1) -tuples only , then we would b e able to compare d n + ∆ n +1 to d n +1 : (1) d n + ∆ n +1 < d n +1 (sup eradditiv e incremen t) , (2) d n + ∆ n +1 = d n +1 (additiv e increment), (3) d n + ∆ n +1 > d n +1 (subadditiv e increment), (4) d n = d n +1 (plateau). But is there a wa y to find ∆ n +1 indep enden tly of d n +1 ? W e prop ose one such w ay as follows. Consider t wo systems, A 1 1 A 1 2 · · · A 1 n c = 1 A 2 1 A 2 2 · · · A 2 n 2 . . . . . . . . . . . . . . . A s 1 A s 2 · · · A s n s q = 1 2 · · · n A (33) and B 1 1 B 1 2 · · · B 1 n B 1 n +1 c = 1 ′ B 2 1 B 2 2 · · · B 2 n B 2 n +1 2 ′ . . . . . . . . . . . . . . . . . . B t 1 B t 2 · · · B t n B t n +1 t ′ q = 1 ′ 2 ′ · · · n ′ ( n + 1) ′ B , where some of the v ariables sho wn can b e constants or empt y cells. Let system A ha ve a contextualit y profile k 1 2 · · · n deg A [ k ] 0 d 2 · · · d n ≥ 0 . (34) F or system B , let us assume that its con textuality profile is k 1 2 · · · n n + 1 deg B [ k ] 0 0 · · · 0 ∆ n +1 ≥ 0 . (35) 12 That is, this system is noncontextual at all levels except for the last one. Let us concatenate these t wo systems in to a larger system as shown: A 1 1 A 1 2 · · · A 1 n c = 1 A 2 1 A 2 2 · · · A 2 n 2 . . . . . . . . . . . . . . . A s 1 A s 2 · · · A s n s B 1 1 B 1 2 · · · B 1 n B 1 n +1 1 ′ B 2 1 B 2 2 · · · B 2 n B 2 n +1 2 ′ . . . . . . . . . . . . . . . . . . B t 1 B t 2 · · · B t n B t n +1 t ′ q = 1 2 · · · n 1 ′ 2 ′ · · · n ′ ( n + 1) ′ A ∗ B . (36) Note that the sets of b oth questions and contexts of the tw o subsystems are completely disjoin t. This means that a coupling of A ∗ B can b e constructed as separate couplings for A and for B , with their join t distribution defined arbitrarily (in particular, they can alw ays b e treated as indep enden t even ts). In other words, for all k ≤ n , since B [ k ] is noncontextual, the contextualit y of ( A ∗ B ) [ k ] is determined b y A [ k ] alone. It is natural to expect then that, for all k ≤ n , deg ( A ∗ B ) [ k ] = deg ( A ) [ k ] . (37) This can ev en b e added as a fourth requirement for a w ell-constructed measure of con textuality , in addition to the three requiremen ts listed at the end of Section 2. Whether we do this or not, this prop ert y holds for all three measures CNT 2 , CNT 3 , and CNTF . Consequently , for all of them w e hav e k 1 2 · · · n deg A [ k ] 0 d 2 · · · d n deg B [ k ] 0 0 · · · 0 deg ( A ∗ B ) [ k ] 0 d 2 · · · d n . (38) A t the next, ( n + 1) st level w e get k 1 2 · · · n n + 1 deg A [ k ] 0 d 2 · · · d n d n deg B [ k ] 0 0 · · · 0 ∆ n +1 deg ( A ∗ B ) [ k ] 0 d · · · d n d n +1 . (39) The reason d n rep eats at level n + 1 for deg A [ k ] is that A [ n + 1] = A [ n ] . Clearly , we no w hav e what we hav e aimed at: the p ossibilit y to compare d n +1 and d n + ∆ n +1 , in order to determine if the com bination of d n and ∆ n +1 b y the measure deg is additive, superadditive, or subadditive (including the plateau case, d n +1 = d n ). 13 Figure 3: Con textuality profiles for the metho d of concatenated systems, n = 2 . The b o xes represent the systems b eing concatenated, with the num b ers in them indicating their final level of contextualit y . Symbols attached to the curves indicate contextualit y v alues. 6 Con textualit y profiles for the three measures W e implemen t the method presented in the previous section using its simplest sp ecial case: with n = 2 . Figures 3 and 4 illustrate the logic and the p ossible t yp es of con textuality profiles for this special case. W e chose the formats for systems A and B as shown, A 1 1 A 1 2 1 A 2 2 A 2 3 2 A 3 1 A 3 3 3 1 2 3 A , B 1 1 B 1 2 B 1 3 1 ′ B 2 2 B 2 3 B 2 4 2 ′ B 3 1 B 3 3 B 3 4 3 ′ 1 ′ 2 ′ 3 ′ 4 ′ B , (40) with all v ariables b eing dichotomous (say , ± 1 ). The format of the concatenated system then acquires the form A 1 1 A 1 2 1 A 2 2 A 2 3 2 A 3 1 A 3 3 3 B 1 1 B 1 2 B 1 3 1 ′ B 2 2 B 2 3 B 2 4 2 ′ B 3 1 B 3 3 B 3 4 3 ′ 1 2 3 1 ′ 2 ′ 3 ′ 4 ′ A ∗ B . (41) The systems we explored were obtained b y sp ecifying the joint distribution of the v ariables in the systems A and B . Each contextualit y profile sho wn b elo w 14 Figure 4: F our possible types of the contextualit y profiles for concatenated systems ( n = 2 ): sup eradditiv e (top left panel), additive (top right), subadditive (b ottom left), and, as the extreme case of subadditivity , plateau (b ottom righ t). has symbols attached to it, referring to the systems whose detailed sp ecifications are giv en in App endix. Figures 5-6 sho w the contextualit y profiles for a selection of undisturb ed concatenated systems. W e see that the measure CNT 2 sho ws precise additivity , while b oth CNT 3 and CNTF are subadditive. The subadditivity in these mea- sures, especially in CNT 3 , is often extreme, resulting in a plateau in most cases sho wn. There is no qualitative difference b etw een the profiles of the undisturb ed and disturb ed concatenated systems. Figures 7-8 for a selection of disturb ed systems exhibit the same pattern as in Figures 5-6. No w that the subadditivit y of the contextualit y profiles for CNT 3 and CNTF has b een observ ed, can we determine its cause? It turns out w e can. Figures 9 and 10 exhibit the con textuality profiles for CNT 3 and CNTF with the sup er- imp osed profiles of the individual A - and B -subsystems. One can see that d 3 coincides with ∆ 3 if the latter exceeds d 2 ; otherwise d 3 remains on the lev el of d 2 . In other words, for b oth CNT 3 and CNTF profiles shown, we hav e the rule of maxim um: d 3 = max ( d 2 , ∆ 3 ) . (42) The subadditivit y therefore is the consequence of max ( d 2 , ∆ 3 ) ≤ d 2 + ∆ 3 . T ables 1 and 2 provides an illustration of the addition rule for CNT 2 and the rule of maximum for CNT 3 and CNTF using larger selections of subsystems A and B than in our figures. The results presented here are just a fraction of the systems we explored for this w ork: 6 × 49 undisturb ed A - B pairs and 125 × 49 disturb ed A - B pairs 15 A 0 ∗ B 2 A 2 ∗ B 2 A 4 ∗ B 2 0 . 1 2 0 . 3 3 0 . 5 3 1 2 3 L e v e l C N T 2 A 0 ∗ B 2 A 2 ∗ B 2 A 4 ∗ B 2 0 . 2 0 0 . 4 0 1 2 3 L e v e l C N T 3 A 0 ∗ B 2 A 2 ∗ B 2 A 4 ∗ B 2 0 . 5 0 0 . 8 0 1 2 3 L e v e l C N T F Figure 5: Con textuality profiles for a selection of undisturb ed concatenated systems. Sym b ols A and B with indices refer to A - and B -subsystems, respectively (as sp ecified in App endix). The dashed lines attached to each profile show the incremen t from d 2 to d 2 + ∆ 3 : if it is ab o ve the corresp onding segment of the profile we hav e subadditivity , and when the dashed line is not seen (coincides with the segmen t) we hav e additivit y . A 1 ∗ B 2 A 3 ∗ B 2 A 5 ∗ B 2 0 . 2 3 0 . 4 2 0 . 6 2 1 2 3 L e v e l C N T 2 A 1 ∗ B 2 A 3 ∗ B 2 A 5 ∗ B 2 0 . 1 7 0 . 3 0 0 . 5 0 1 2 3 L e v e l C N T 3 A 1 ∗ B 2 A 3 ∗ B 2 A 5 ∗ B 2 0 . 5 0 1 . 0 0 1 2 3 L e v e l C N T F Figure 6: The same as in Figure 5, for another selection selection of the A -subsystems. 16 A 0 ' ∗ B 3 ' A 2 ' ∗ B 3 ' A 4 ' ∗ B 3 ' A 6 ' ∗ B 3 ' 0 . 0 9 0 . 2 0 0 . 3 1 0 . 4 2 1 2 3 L e v e l C N T 2 A 0 ' ∗ B 3 ' A 2 ' ∗ B 3 ' A 4 ' ∗ B 3 ' A 6 ' ∗ B 3 ' 0 . 1 2 0 . 2 2 0 . 3 3 1 2 3 L e v e l C N T 3 A 0 ' ∗ B 3 ' A 2 ' ∗ B 3 ' A 4 ' ∗ B 3 ' A 6 ' ∗ B 3 ' 0 . 3 6 0 . 4 4 0 . 6 7 1 2 3 L e v e l C N T F Figure 7: Contextualit y profiles for a selection of disturb ed A -subsystems concatenated with a disturb ed subsystem B 3 . The rest is the same as in Figure 5. A 1 ' ∗ B 3 ' A 3 ' ∗ B 3 ' A 5 ' ∗ B 3 ' 0 . 1 5 0 . 2 6 0 . 3 7 1 2 3 L e v e l C N T 2 A 1 ' ∗ B 3 ' A 3 ' ∗ B 3 ' A 5 ' ∗ B 3 ' 0 . 1 2 0 . 1 7 0 . 2 8 1 2 3 L e v e l C N T 3 A 1 ' ∗ B 3 ' A 3 ' ∗ B 3 ' A 5 ' ∗ B 3 ' 0 . 3 6 0 . 5 6 1 2 3 L e v e l C N T F Figure 8: The same as in Figure 7 but for another selection of A -subsystems concate- nated with a disturbed subsystem B 3 . 17 d 2 Δ 3 0 1 / 5 1 / 2 1 2 3 L e v e l C N T F d 2 Δ 3 0 1 / 2 3 / 5 1 2 3 L e v e l C N T F Figure 9: CNTF profiles for a selection of concatenated systems ( A 1 ∗ B 2 left and A 3 ∗ B 2 righ t). The dashed lines represent the CNTF profiles for the systems’ B -parts. The dotted lines represent the CNTF profiles for the system’s A -parts (invisible if it coincides with a systems’ profile). d 2 Δ 3 0 1 / 9 1 / 6 1 2 3 L e v e l C N T 3 d 2 Δ 3 0 1 / 6 3 / 1 0 1 2 3 L e v e l C N T 3 Figure 10: The same as in Figure 9 but for CNT 3 . 18 T able 1: Con textuality v alue d 3 of undisturb ed concatenated systems for the measures CNT 2 , CNT 3 , and CNTF . The v alue of d 3 for A i ∗ B j is in the intersection of column A i and row B j . The corresp onding v alues of d 2 for A i and ∆ 3 for B j are shown, resp ectiv ely , in the ro w and the column con taining the measure’s name. Observ e that for CNT 2 , d 3 is the sum of the corresp onding v alues of d 2 and ∆ 3 ; and for both CNT 3 and CNTF , d 3 is the larger of the corresponding v alues of d 2 and ∆ 3 . (Note that for the A -subsystems, CNT 2 = CNT 3 = 1 2 CNTF , as it was previously established for all cyclic systems [4]). A 0 A 1 A 2 A 3 A 4 A 5 CNT 2 0 1 / 10 1 / 5 3 / 10 2 / 5 1 / 2 B 1 1 / 24 1 / 24 17 / 120 29 / 120 41 / 120 53 / 120 13 / 24 B 2 1 / 8 1 / 8 9 / 40 13 / 40 17 / 40 21 / 40 5 / 8 CNT 3 0 1 / 10 1 / 5 3 / 10 2 / 5 1 / 2 B 1 1 / 18 1 / 18 1 / 10 1 / 5 3 / 10 2 / 5 1 / 2 B 2 1 / 6 1 / 6 1 / 6 1 / 5 3 / 10 2 / 5 1 / 2 CNTF 0 1 / 5 2 / 5 3 / 5 4 / 5 1 B 1 1 / 6 1 / 6 1 / 5 2 / 5 3 / 5 4 / 5 1 B 2 1 / 2 1 / 2 1 / 2 1 / 2 3 / 5 4 / 5 1 (with many different subsystems pro ducing iden tical profiles). The contextualit y curv es w e had to leav e out in order not to clutter the graphs and tables or m ultiply their num b er conform to the same pattern: CNT 2 is alwa ys additiv e, and the measures CNT 3 and CNTF are subadditive b ecause they conform to the rule of maxim um. With these regularities b eing established as inductive generalizations, we can lo ok for their analytic justification. Although this is not essential for this pap er, whose purp ose is to in tro duce and demonstrate the usefulness of the concept of a con textuality profile, we outline these analytic argumen ts b elo w. F or CNT 2 , since B is noncontextual at level 2, the v alue of d 2 is the L 1 - distance b etw een the system A and the lev el-2 noncon textuality p olytope. The system and the p olytope are defined in the space spanned by the axes repre- sen ting all pairwise probabilities. The v alue of ∆ 3 is the L 1 -distance b et ween the system B and the lev el-3 noncon textuality p olytope. Because B is noncon- textual at level 2, this distance is entirely within the space spanned by the axes represen ting all triple probabilities. In the system A ∗ B the axes of these tw o spaces, of the pairwise and of the triple probabilities, are com bined as m utually orthogonal subspaces. By the nature of L 1 , therefore, the ov erall distance in this com bined space is the sum of the tw o subspace distances. If instead of the L 1 -distance we chose an L p -distance with p > 1 , the ov erall distance would hav e satisfied d p 3 = d p 2 + ∆ p 3 . (43) The argument establishing the rule of maximum is essentially the same for CNT 3 and CNTF . Let us present the details for the latter. With reference to (30) and (31), let x A = ( α 1 , . . . , α K ) b e a vector of probabilities assigned to 19 T able 2: Con textuality v alue d 3 of disturb ed concatenated systems for the measures CNT 2 , CNT 3 , and CNTF . The rest as in T able 1. A ′ 0 A ′ 1 A ′ 2 A ′ 3 A ′ 4 A ′ 5 A ′ 6 CNT 2 0 1 / 18 1 / 9 1 / 6 2 / 9 5 / 18 3 / 9 B ′ 1 1 / 100 1 / 100 59 / 900 109 / 900 159 / 900 209 / 900 259 / 900 309 / 900 B ′ 2 5 / 100 5 / 100 19 / 180 29 / 180 13 / 60 49 / 180 59 / 180 23 / 180 B ′ 3 9 / 100 9 / 100 131 / 900 181 / 900 77 / 300 281 / 900 331 / 900 127 / 300 CNT 3 0 1 / 18 1 / 9 1 / 6 2 / 9 5 / 18 3 / 9 B ′ 1 2 / 150 2 / 150 1 / 18 1 / 9 1 / 6 2 / 9 5 / 18 3 / 9 B ′ 2 1 / 15 1 / 15 1 / 15 1 / 9 1 / 6 2 / 9 5 / 18 3 / 9 B ′ 3 3 / 25 3 / 25 3 / 25 3 / 25 1 / 6 2 / 9 5 / 18 3 / 9 CNTF 0 1 / 9 2 / 9 1 / 3 4 / 9 5 / 9 2 / 3 B ′ 1 1 / 25 1 / 25 1 / 9 2 / 9 1 / 3 4 / 9 5 / 9 2 / 3 B ′ 2 1 / 5 1 / 5 1 / 5 2 / 9 1 / 3 4 / 9 5 / 9 2 / 3 B ′ 3 9 / 25 9 / 25 9 / 25 9 / 25 9 / 25 4 / 9 5 / 9 2 / 3 the K combinations of v alues of the v ariables in A . Let x B = ( β 1 , . . . , β L ) b e defined analogously for B , and x AB = ( γ 11 , . . . , γ K L ) for A ∗ B , where γ ij is assigned to the concatenation of the i th combination of v alues in A and the j th com bination of v alues in B . Then we hav e ( 1 ⊺ x AB ) max ≤ min (( 1 ⊺ x A ) max , ( 1 ⊺ x B ) max ) . (44) Indeed, if we had, e.g., ( 1 ⊺ x AB ) max > ( 1 ⊺ x B ) max , then we could redefine the v alues of x B as β j = K X i =1 γ ij , j = 1 , . . . , L, (45) and obtain thereb y a coupling of B with a greater v alue of 1 ⊺ x B than ( 1 ⊺ x B ) max . It is also true that ( 1 ⊺ x AB ) max ≥ min (( 1 ⊺ x A ) max , ( 1 ⊺ x B ) max ) , (46) b ecause if w e had, e.g., ( 1 ⊺ x AB ) max < ( 1 ⊺ x B ) max ≤ ( 1 ⊺ x A ) max , then we could redefine the v alues of x AB as γ ij =  β j if i = 1 , 0 if i > 1 , (47) and obtain thereb y a coupling of A ∗ B with a greater v alue of 1 ⊺ x AB than ( 1 ⊺ x B ) max . The conjunction of (44) and (46) yields the rule of maximum, (42), b ecause CNTF = 1 − ( 1 ⊺ x ) max . F or CNT 3 , with reference to (27) and (29), w e show that 20 ( 1 ⊺ | x AB | ) max ≥ min (( 1 ⊺ | x A | ) max , ( 1 ⊺ | x B | ) max ) , (48) b ecause if, e.g., ( 1 ⊺ | x AB | ) max < ( 1 ⊺ | x B | ) max , one could redefine the v alues of x AB as in (47), and ac hieve an increase in 1 ⊺ | x AB | . Also, ( 1 ⊺ | x AB | ) max ≤ min (( 1 ⊺ | x A | ) max , ( 1 ⊺ | x B | ) max ) , (49) b ecause if, e.g., ( 1 ⊺ | x AB | ) max > ( 1 ⊺ | x B | ) max ≥ ( 1 ⊺ | x A | ) max , one could redefine the v alues of x B as in (45), and ac hieve an increase in 1 ⊺ | x B | . 7 Hyp ercyclic systems Hyp ercyclic systems were in tro duced in Ref. [5] as a set of systems that are b oth highly structured and sufficiently div erse to form testing grounds for contextu- alit y research. The subsystem A in our concatenated systems is a cyclic system of rank 3 (a sp ecial case of a hypercyclic system), and the subsystem B was a h yp ercyclic system of order 3 and rank 4 but without a last ro w (so abridged to sp eed up execution of the linear programs). The order of a hypercyclic system is the num b er of v ariables in each context; the rank is the num b er of the system’s con texts (which is the same as the num b er of questions). The v ariables in each ro w are cyclically shifted clo c kwise with resp ect to the previous row. Figures 11 and 12 presen t contextualit y profiles for a selection of h yp ercyclic systems of order 3 and rank 4: R 1 1 R 1 2 R 1 3 c = 1 R 2 2 R 2 3 R 2 4 2 R 3 1 R 3 3 R 3 4 3 R 4 1 R 4 2 R 4 4 4 q = 1 2 3 4 . (50) Figure 11 has the prop erty common to all undisturb ed hypercyclic systems: they are noncontextual at all but the final level (in our case, lev el 3). Some disturbed h yp ercyclic systems ha ve th is prop ert y to o, but it do es not hold generally . Ho wev er, this prop ert y is not why we consider the h yp ercyclic systems here. In this pap er they serve another purp ose. W e kno w from Ref. [5] that none of the three measure s of con textuality we studied, CNT 2 , CNT 3 , and CNTF , is a function of any other of them when considered across differen t systems. Sp ecifically , it was shown that, for any ordered pair of these contextualit y mea- sures, e.g., ( CNTF , CNT 2 ) , one can find tw o h yp ercyclic systems suc h that the first measure changes from one of them to another while the second measure remains constant. F or our selection of undisturbed h ypercyclic systems this is shown in Figure 13. The interpretation of this observ ation is that, unlik e, sa y , CNTF and log ( CNTF ) , the three measures CNT 2 , CNTF , and CNT 3 reflect pairwise distinct asp ects of contextualit y . The question w e pose in this paper is whether the same is true for the con textual profiles generated by the three measures for one and the same system. 21 H 0 H 1 & H 2 H 3 0 . 0 0 0 . 1 0 0 . 1 3 1 2 3 L e v e l C N T 2 H 0 H 1 H 2 & H 3 0 . 0 0 0 . 0 9 0 . 1 0 1 2 3 L e v e l C N T 3 H 0 H 2 H 1 & H 3 0 . 0 0 0 . 3 0 0 . 3 5 1 2 3 L e v e l C N T F Figure 11: Contextualit y profiles for a selection of undisturb ed hypercyclic systems of order 3 and rank 4. H 0 ' H 1 ' H 2 ' H 3 ' H 4 ' 0 . 0 0 0 . 1 7 0 . 2 5 0 . 4 0 0 . 6 0 1 2 3 L e v e l C N T 2 H 0 ' H 1 ' H 2 ' H 3 ' & H 4 ' 0 . 0 0 . 1 0 . 2 0 . 4 1 2 3 L e v e l C N T 3 H 0 ' H 1 ' H 2 ' H 3 ' H 4 ' 0 . 0 0 0 . 1 7 0 . 4 0 0 . 5 0 1 2 3 L e v e l C N T F Figure 12: Contextualit y profiles for a selection of disturb ed h yp ercyclic systems of order 3 and rank 4. 22 H 0 H 1 H 2 H 3 0 . 0 0 0 . 3 0 0 . 3 5 0 . 0 0 0 . 1 0 0 . 1 3 C N T 2 C N T F H 0 H 1 H 2 H 3 0 . 0 0 0 . 3 0 0 . 3 5 0 . 0 0 . 1 C N T 3 C N T F H 0 H 1 H 2 H 3 0 . 0 0 0 . 0 9 0 . 1 0 0 . 0 0 0 . 1 0 0 . 1 3 C N T 2 C N T 3 Figure 13: Relationship b et ween the ov erall contextualit y degrees generated by t wo con textuality measures applied to our selection of undisturbed hypercyclic systems. The horizontal lines show that the abscissa measure cannot be a function of the or- dinate one; the v ertical lines show that the ordinate measure cannot b e a function of the abscissa one. H 0 ' H 1 ' H 2 ' H 3 ' H 4 ' 0 . 0 0 0 . 1 7 0 . 4 0 0 . 5 0 0 . 0 0 0 . 1 7 0 . 2 5 0 . 4 0 0 . 6 0 C N T 2 C N T F H 0 ' H 1 ' H 2 ' H 3 ' H 4 ' 0 . 0 0 0 . 1 7 0 . 4 0 0 . 5 0 0 . 0 0 0 . 0 9 0 . 2 0 0 . 4 0 C N T 3 C N T F H 0 ' H 1 ' H 2 ' H 3 ' H 4 ' 0 . 0 0 0 . 0 9 0 . 2 0 0 . 4 0 0 . 0 0 0 . 1 7 0 . 2 5 0 . 4 0 0 . 6 0 C N T 2 C N T 3 Figure 14: Relationship b etw een the segments of contextualit y profiles (b et ween levels 2 and 3, as indicated b y arro ws) generated b y t w o measures applied to the same disturb ed h ypercyclic system. The horizontal lines sho w that the abscissa profile cannot be a function of the ordinate one; the vertical lines sho w that the ordinate profile cannot b e a function of the abscissa one. 23 Figure 15: Contextualit y profiles for a triple-concatenation of systems. Figure 14 tells us that this is indeed the case: for an y ordered pair of our three measures one can find a system such that the first measure changes b et ween lev els 2 and 3 while the second measure remains constant. 8 Conclusion W e ha ve introduced a new notion, that of a contextualit y profile of a system, and inv estigated some of its basic prop erties. W e compared the con textuality profiles of three w ell-constructed measures, CNT 2 , CNTF , and CNT 3 using the metho d of concatenated systems. W e established that CNT 2 profiles are additive while CNTF and CNT 3 profiles are subadditive, b ecause they conform to the rule of maxim um. W e hav e also established that none of these three measures is a function of any other, not only across differen t systems (which has b een known previously), but also within a system tak en at different lev els. Note that concatenation can b e used recursively , creating combinations of three and more systems, as shown in Figure 15. F or concatenations of A 1 , . . . , A k , the arguments for the additivit y of CNT 2 and the maxim um rule for CNTF and CNT 3 can b e recursively applied to sho w that, CNT 2 = k X k =2 ∆ k , CNT 3 = max(∆ ′ 2 , . . . , ∆ ′ k ) , CNTF = max k (∆ ′′ 2 , . . . , ∆ ′′ k ) , (51) where w e replaced d 2 with ∆ 2 for uniformit y and added prime s to emphasize that the v alue of ∆ k is measure-sp ecific. 24 This b eing only a concept pap er, it lea ves many questions unanswered. F rom a mathematical p oin t of view, m uch remains to b e in vestigated analytically re- garding the prop erties of the contextualit y profiles. Ho wev er, aside from the in trinsic mathematical interest, the main substantiv e question is that of applica- bilit y: do the contextualit y profiles tell us something ab out other, indep enden tly defined prop erties of the empirical entities described b y the systems? Could, e.g., the resource-theoretical asp ects of the systems [6] b e b etter understoo d if w e relate them to v arious asp ects of contextualit y profiles rather than to the o verall degree of contextualit y only? More w ork is needed. App endix T ables 3-8 show the distributions of all the systems men tioned in the main text. Disturb ed systems are indicated by primes ( A ′ 1 , B ′ 2 , H ′ 0 , etc.). T able 3: Distributions of the v ariables in the selection of the undisturb ed A -subsystems used. All v ariables are dichotomous ( ± 1 ). Each num b er shows the probability with whic h the corresp onding v ariable(s) equal 1. Th us, a | b c in the second row means that P  R 2 2 = 1  = a , P  R 2 3 = 1  = b , and P  R 2 2 = R 2 3 = 1  = c . A 0 A 1 A 2 A 3 A 4 A 5 R 1 1 R 1 2 R 1 1 = R 1 2 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 R 2 2 R 2 3 R 2 2 = R 2 3 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 0 . 5 0 . 5 0 R 3 3 R 3 4 R 3 3 = R 3 4 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 4 0 . 5 0 . 5 0 . 3 0 . 5 0 . 5 0 . 2 0 . 5 0 . 5 0 . 1 0 . 5 0 . 5 0 T able 4: The same as T able 3 but for the selection of the disturb ed A -subsystems used. A ′ 0 A ′ 1 A ′ 2 A ′ 3 A ′ 4 A ′ 5 A ′ 6 R 1 1 R 1 2 R 1 1 = R 1 2 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 R 2 2 R 2 3 R 2 2 = R 2 3 4 / 9 5 / 9 1 / 9 4 / 9 5 / 9 1 / 18 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 4 / 9 5 / 9 0 R 3 3 R 3 4 R 3 3 = R 3 4 4 / 9 5 / 9 2 / 9 4 / 9 5 / 9 2 / 9 4 / 9 5 / 9 2 / 9 4 / 9 5 / 9 3 / 18 4 / 9 5 / 9 1 / 9 4 / 9 5 / 9 1 / 18 4 / 9 5 / 9 0 25 T able 5: Distributions of the v ariables in the selection of the undisturb ed B -subsystems used. As in T able 3, each n umber sho ws the probabilit y with which the corresp ond- ing v ariable(s) equal 1. Thus, a | b | c d | e | f g in the second row means that P  R 2 2 = 1  = a , P  R 2 2 = R 2 3 = 1  = d , P  R 2 2 = R 2 3 = R 2 4 = 1  = g , etc. B 1 B 2 R 1 1 R 1 2 R 1 3 R 1 1 = R 1 2 R 1 2 = R 1 3 R 1 1 = R 1 3 R 1 1 = R 1 1 = R 1 3 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 8 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 8 R 2 2 R 2 3 R 2 4 R 2 2 = R 2 3 R 2 3 = R 2 4 R 2 2 = R 2 4 R 2 2 = R 2 3 = R 2 4 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 0 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 6 R 3 3 R 3 4 R 3 1 R 3 3 = R 3 4 R 3 4 = R 3 1 R 3 3 = R 3 1 R 3 3 = R 3 4 = R 3 1 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 4 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 0 T able 6: The same as T able 5, but for the selection of the disturb ed B -subsystems used. B ′ 1 B ′ 2 B ′ 3 R 1 1 R 1 2 R 1 3 R 1 1 = R 1 2 R 1 2 = R 1 3 R 1 1 = R 1 3 R 1 1 = R 1 1 = R 1 3 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 3 / 25 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 3 / 25 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 3 / 25 R 2 2 R 2 3 R 2 4 R 2 2 = R 2 3 R 2 3 = R 2 4 R 2 2 = R 2 4 R 2 2 = R 2 3 = R 2 4 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 0 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 0 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 0 R 3 3 R 3 4 R 3 1 R 3 3 = R 3 4 R 3 4 = R 3 1 R 3 3 = R 3 1 R 3 3 = R 3 4 = R 3 1 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 2 / 25 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 1 / 25 13 / 25 1 / 2 12 / 25 13 / 50 6 / 25 6 / 25 0 T able 7: Distributions in the selection of the undisturbed hypercyclic systems used. The format is the same as in T able 5. H 0 H 1 H 2 H 3 R 1 1 R 1 2 R 1 3 R 1 1 = R 1 2 R 1 2 = R 1 3 R 1 1 = R 1 3 R 1 1 = R 1 1 = R 1 3 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 4 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 R 2 2 R 2 3 R 2 4 R 2 2 = R 2 3 R 2 3 = R 2 4 R 2 2 = R 2 4 R 2 2 = R 2 3 = R 2 4 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 R 3 3 R 3 4 R 3 1 R 3 3 = R 3 4 R 3 4 = R 3 1 R 3 3 = R 3 1 R 3 3 = R 3 4 = R 3 1 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 R 4 4 R 4 1 R 4 2 R 4 4 = R 4 1 R 4 1 = R 4 2 R 4 4 = R 4 2 R 4 4 = R 4 1 = R 4 2 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 3 / 20 1 / 2 1 / 2 1 / 2 1 / 4 1 / 4 1 / 4 1 / 5 26 T able 8: Distributions in the selection of the disturb ed hypercyclic systems used. The format is the same as in T able 5. H ′ 0 H ′ 1 H ′ 2 H ′ 3 H ′ 4 R 1 1 R 1 2 R 1 3 R 1 1 = R 1 2 R 1 2 = R 1 3 R 1 1 = R 1 3 R 1 1 = R 1 1 = R 1 3 2 / 3 2 / 3 1 / 2 1 / 3 1 / 3 1 / 3 1 / 6 1 / 3 1 / 2 2 / 3 1 / 6 1 / 3 1 / 3 1 / 6 1 / 4 1 / 4 1 / 4 0 0 0 0 3 / 5 3 / 5 1 / 5 2 / 5 1 / 5 1 / 5 1 / 5 3 / 5 3 / 5 1 / 5 2 / 5 1 / 5 1 / 5 1 / 5 R 2 2 R 2 3 R 2 4 R 2 2 = R 2 3 R 2 3 = R 2 4 R 2 2 = R 2 4 R 2 2 = R 2 3 = R 2 4 1 / 2 1 / 2 1 / 3 1 / 6 1 / 3 1 / 3 1 / 6 2 / 3 2 / 3 1 / 2 1 / 3 1 / 3 1 / 3 1 / 6 1 / 2 1 / 4 1 / 4 1 / 4 0 0 0 3 / 5 2 / 5 2 / 5 1 / 5 1 / 5 2 / 5 1 / 5 3 / 5 2 / 5 3 / 5 1 / 5 1 / 5 2 / 5 1 / 5 R 3 3 R 3 4 R 3 1 R 3 3 = R 3 4 R 3 4 = R 3 1 R 3 3 = R 3 1 R 3 3 = R 3 4 = R 3 1 1 / 2 1 / 2 1 / 3 1 / 3 1 / 3 1 / 3 1 / 6 2 / 3 1 / 2 2 / 3 1 / 3 1 / 3 1 / 3 1 / 6 1 / 2 1 / 4 1 / 2 0 1 / 4 1 / 4 0 2 / 5 2 / 5 3 / 5 0 1 / 5 1 / 5 0 2 / 5 3 / 5 3 / 5 1 / 5 1 / 5 1 / 5 0 R 4 4 R 4 1 R 4 2 R 4 4 = R 4 1 R 4 1 = R 4 2 R 4 4 = R 4 2 R 4 4 = R 4 1 = R 4 2 1 / 2 1 / 2 1 / 3 1 / 3 1 / 3 1 / 3 1 / 6 1 / 2 2 / 3 2 / 3 1 / 3 1 / 3 1 / 3 1 / 6 3 / 4 1 / 4 1 / 2 1 / 4 1 / 4 1 / 2 1 / 4 3 / 5 2 / 5 3 / 5 1 / 5 1 / 5 2 / 5 1 / 5 3 / 5 3 / 5 3 / 5 2 / 5 1 / 5 2 / 5 1 / 5 Author Contributions: The initial idea: END. 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