Faithful orthogonal representations of graphs from partition logics

Partition logics often allow a dual probabilistic interpretation: a classical one for which probabilities lie on the convex hull of the dispersion-free weights and another one, suggested independently from the quantum Born rule, in which probabilities are formed by the (absolute) square of the inner product of state vectors with the faithful orthogonal representations of the respective graph. Two immediate consequences are the demonstration that the logico-empirical structure of observables does not determine the type of probabilities alone and that complementarity does not imply contextuality.


I. PARTITION LOGICS AS NONBOOLEAN STRUCTURES PASTED FROM BOOLEAN SUBALGEBRAS
Partitions provide ways to distinguish between elements of a given finite set S n = {1, 2, . . ., n}.The Bell number B n (after Eric Temple Bell) is the number of such partitions [1].(Obvious generalizations are infinite denumerable sets or continua.)We shall restrict our attention to partitions with an equal number 1 ≤ m ≤ n of elements.Every partition can be identified with some Boolean subalgebra 2 m -in graph theoretical terms a clique -of 2 n whose atoms are the elements of that partition.
A partition logic [2][3][4][5][6] is the logic obtained (i) from collections of such partitions, each partition being identified with an m-atomic Boolean subalgebra of 2 n , and (ii) by "stitching" or pasting these subalgebras through identifying identical intertwining elements.In quantum logic this is referred to as pasting construction; and the partitions are identified with, or are synonymously denoted by, blocks, subalgebras or cliques, which are representable by orthonormal bases or maximal operators.

II. PROBABILITIES ON PARTITION LOGICS
The following hypothesis or principle is taken for granted: probabilities and expectations on classical substructures of an empirical logic should be classical.That is, mutually exclusive co-measurable propositions (satisfying Specker's exclusivity principle) should obey the Kolmogorovian axioms; in particular, nonnegativity and additivity.Nonnegativity implies that all probabilities are nonnegative: P(E 1 ), . . ., P(E k ) ≥ 0. Additivity among (pairwise) mutually exclusive outcomes E 1 , . . ., E k means that the probabilities of joint outcomes are equal to the sum of probabilities of these outcomes; that is, within cliques/contexts, for k ≤ m: In particular, probabilities add to 1 on each of the cliques/contexts.At the moment at least three such types of probabilities are known to satisfy Specker's exclusivity principle, corresponding to classical, quantum and Wright's "exotic"pure weights, such as the weight 1  2 on the vertices of the pentagon [22, ω 0 , p. 68] and on the triangle vertices [19, pp. 899-902] (the latter logic is representable as partition logic [4, Example 8.2, pp.420,421], but not in 3-dimensional Hilbert space).The former two "nonexotic" types, based on representations of mutually disjoint sets, and on mutually orthogonal vectors, will be discussed later.
It is not too difficult to imagine boxes allowing input/output analysis "containing" classical or quantum algorithms, agents or mechanisms rendering the desired properties.For instance, a model realization of a classical box rendering classical probabilities is Wright's generalized urn model [19,[22][23][24] or the initial state identification problem for finite deterministic automaton [2,5,25,26] -both are equivalent models of partition logics [6] featuring complementarity without value indefiniteness.
Specker's parable of the overprotective seer [7,[27][28][29] involving three boxes is an example for which the exclusivity principle does not hold [30,Section 116,p. 40].It is an interesting problem to find other potential probability measures based on different approaches which are also linear in mutually exclusive events.

A. Probabilities from the convex hull of dispersion-free states
For nonboolean logics, it is not immediately evident which probability measures should be chosen.The answer is already implicit in Zierler and Schlessinger's 1965 paper on "Boolean embeddings of orthomodular sets and quantum logic".Theorem 0 of Kochen and Specker's 1967 paper [20] states that separability by dispersion free states (of image 2 1 = 0, 1) for every pair of atoms of the lattice is a necessary and sufficient criterium for a homomorphic embedding into some "larger" Boolean algebra.In 1978 Wright explicitly stated [22, p. 272] "that every urn weight is "classical", i.e., in the convex hull of the dispersion free weights".In the graph theoretical context Grötschel, Lovász and Schrijver have discussed the vertex packing polytope V P(G) of a graph G, defined as the convex hull of incidence vectors of independent sets of nodes [31].This author has employed dispersion free weights for hull computations on the Specker bug [32] and other (partition) logics supporting a separating set of two-value states.

B. Born-Gleason-Grötschel-Lovász-Schrijver type probabilities
Motivated by cryptographic issues outside quantum theory Lovász [38] has proposed an "indexing" of vertices of a graph by vectors reflecting their adjacency: the graph-theoretic definition of a faithful orthogonal representation of a graph is by identifying vertices with vectors (of some Hilbert space of dimension d) such that any pair of vectors are orthogonal if and only if their vertices are not orthogonal [39,40].For physical applications, Cabello [41], Portillo [42] and others have uses an "inverse" notation, in which vectors are required to be mutually orthogonal whenever they are adjacent.Both notations are equivalent by exchanging graphs with their complements or inverses.
There is no systematic algorithm to compute the minimal dimension for a faithful orthogonal representation of a graph.Lovász [13,38] gave a (relative to entropy measures [43]) "optimal" vector representation of the pentagon graph depicted in Figure 1 which, by preparing the "(umbrella) handle" state vector 1, 0, 0 , turns out to render the maximal [34,36] quantum bound ∑ 5 j=1 c|u l 2 = √ 5, which exceeds the "classical" bound (3) of 2 from the computation of the convex hull of the dispersion free weights.
Based on Lovász's vector representation by graphs Grötschel, Lovász and Schrijver have proposed [31, Section 3] a Gleason-Born type probability measure [44] which results in convex sets different from polyhedra defined via convex hulls of vectors discussed earlier in Section II A. Essentially their probability measure is based upon m-dimensional faithful orthogonal representations of a graph G whose vertices v i are represented by unit vectors |v i which are orthogonal within, and nonorthogonal outside, of cliques/contexts.Every vertex v i of the graph G, represented by the unit vector |v i , can then be associated with a "probability" with respect to some unit "preparation" (state) vector |c by defining this "probability" to be the square of the inner product of |v i and |c ; that is, by Iff the vector representation (in the sense of Cabello-Portillo) of G is faithful the Pythagorean theorem assures that, within every clique/context of G, probabilities are positive, additive, and (as both |v i and |c are normalized) the sum of probabilities on that context adds up to exactly one; that is, ∑ i∈clique/context P(c, v i ) = 1.Thereby, probabilities and expectations of simultaneously comeasurable observables, represented by graph vertices within cliques or contexts, obey Specker's exclusivity principle and "behave classically".It might be challenging to motivate "quantum type" probabilities and their convex expansion, the theta body [31], by the very assumptions such as exclusivity [14,44].
A very similar measure on the closed subspaces of Hilbert space, satisfying Specker's exclusivity principle and additivity, had been proposed by Gleason [45, first and second paragraphs, p. 885]: "A measure on the closed subspaces means a function µ which assigns to every closed subspace a nonnegative real number such that if {A i } is a countable collection of mutually orthogonal subspaces having closed linear span B, then µ(B) = ∑ i µ(A i ).It is easy to see that such a measure can be obtained by selecting a vector v and, for each closed subspace A, taking µ(A) as the square of the norm of the projection of v on A." Gleason's derivation of the quantum mechanical Born rule [46, Footnote 1, Anmerkung bei der Korrektur, p. 865] operates in dimensions higher than two, and allows also mixed states; that is, outcomes of nonideal measurements.However, mixed states can always be "completed" or "purified" [47, Section 2.5,pp.109-111] (and thus outcomes of nonideal measurements made ideal [44]) by the inclusion of auxiliary dimensions.

III. QUASICLASSICAL ANALOGUES OF ENTANGLEMENT
In what follows classical analogs to entangled states will be discussed.These examples are local.They are based on Schrödinger's observation that entanglement among pairs of particles is associated with, or at least accompanied by, joint or relational [48] properties of the constituents, whereas nonentangled states feature individual separate properties of the pair constituents [49][50][51].(For early similar discussions in the measurement context, see von Neumann [16, Section VI.2, p 426, pp 436-437] and London and Bauer [52,53].)

A. Partitioning of state space
Wrights generalized urn model [19,22], in a nutshell, is the observation of black balls, on which multiple colored symbols are painted, with monochromatic filters in only one of those colors.Complementarity manifests itself in the necessity of choice of the particular color one observes: one may thereby obtain knowledge of the information encoded in this color; but thereby invariable loses messages encoded in different colors.A typical example is the logic L 12 encoded by the partition logic enumerated in (1) and depicted in Figure 1(a): suppose that there are 4 ball types and two colors on black backgrounds: • ball type 1 is colored with orange a and blue a; • ball type 2 is colored with orange b and blue c; • ball type 3 is colored with orange c and blue b; • ball type 4 is colored with orange c and blue c.Suppose an urn is loaded with balls of all four types.Suppose further that an agent's task is to draw one ball from the urn and, by observing this ball, to find which type it is.Of course, if the observer is allowed to look at both colors simultaneously, this would allow to single out exactly one ball type.But that maximal resolution can no longer be maintained in experiments restricted to one of the two colors.Any one of such two experiment varieties could resolve different, complementary properties: looking at the drawn ball with orange glasses the agent is able to resolve between balls (i) of type 1 associated with the symbol a; (ii) of type 2 associated with the symbol b; and (ii) of type 3 or 4 associated with the symbol c.The resolution between type 3 and 4 balls is lost.Alternatively, by looking at the drawn ball with blue glasses the agent is able to resolve between balls (i) of type 1 associated with the symbol a; (ii) of type 3 associated with the symbol b; and (ii) of type 2 or 4 associated with the symbol c.That is, for the color blue the resolution between type 2 and 4 balls is lost.In any case, the state of ball type is partitioned in different ways, depending on the color of the filter.(Similar considerations apply for initial state identification problems on finite automata.)

B. Relational encoding
Tables I and II enumerate a relational encodings among two or more colors not dissimilar to Peres' detonating bomb model [54].Suppose that an urn is loaded with balls of the type occurring in subensemble E 6 of Table I.The observation of some symbol s ∈ {0, 1} in green implies the (counterfactual) observation of the same symbol s in red, and vice versa.Table II is just an extension to two colors per observer, and an urn loaded with subensembles (E 6 ) 2 Agent Alice draws a ball from the urn and looks at it with her red (exclusive) or blue filters.Then Alice hands the ball over to Bob.Agent Bob looks at the ball with his green (exclusive) or orange filters.This latter scenario is similar to a Clauser-Horne-Shimony-Holt scenario of the Einstein-Podolsky-Rosen type, except that the former is totally local and its probabilities derived from the convex hull of the dispersion-free weights can never violate classical bounds; whereas the latter one may be (and hopefully is) nonlocal, and its performance with a quantum resource violates the classical bounds.

IV. PARTITION LOGIC FREAK SHOW
Let us, for a moment, consider partition logics not restraint by low-dimensional faithful orthogonal representability, but with a separable set of two-valued states [with the exception of the logic depicted in Figure 2(f)].These have no quantum realization.Yet, due to the automaton logic, they are intrinsically realized in Turing universal environments.
An example of such a structure is Wright's triangle logic [19, Figure 2, p. 900] depicted in Figure 2(a) (see also [4,Fig. 6, Example 8.2, pp.414,420,421]).As mentioned earlier, together with 4 dispersion free weights it allows another weight 1  2 on its intertwining vertices.Figures 2(b&c) depict a square logic, the latter one being formed by a pasting of two triangle logics along a common leg.Figures 2

(d&e) depict
It is important to emphasize that both scenarios -the classical generalized urn scenario as well as the quantized one -from a graph-theoretical point of view, operate with identical (exclusivity) graphs (e.g., Figures 1 in both References [10,14]).The difference is the representation of these graphs: the quantum case has a faithful orthogonal representation in some finite-dimensional Hilbert space, whereas the classical case in terms of a generalized urn model has a settheoretic representation in terms of partitions of some finite set.
Generalized urn models and automaton logics are models of partition logics which are capable of complementarity yet fail to render (quantum) value indefiniteness.They are important for an understanding of the "twilight zone" spanned by nonclassicality (nondistributivity, nonboolean logics) and yet full value definiteness -one may call this a "purgatory"floating in-between classical Boolean and quantum realms.
It should be stressed that the algebraic structure of empirical logics, or graphs, do in general not determine the types of probability measures on them.For instance, a generalized urn loaded with balls rendering the pentagon structure, as envisioned by Wright, has probabilities different from the scheme of Grötschel, Lovász and Schrijver, which is based on orthogonal representations of the pentagon.Likewise, a geometric resource such as a "vector contained in a box" and "measured along projections onto an orthonormal basis" will not conform to probabilities induced by the convex hull of the dispersionfree weights -even if these weights are separating.Therefore the particular physical resource -what is actually inside the black box -determines which type of probability theory is applicable.
Furthermore, partition logics which are not just a single Boolean algebra represent empirical configurations featuring complementarity.And yet they all have separating [20, Theorem 0] sets of two-valued states and thus are not "contextual" in the Specker sense [7].
267, Fig. 2] support 11 dispersion free states v 1 , . . ., v 11 .By constructing 5 contexts/cliques from the occurrences of the dispersion free value 1 on the respective 10 atoms results in the partition logic based on the set of indices of the dispersion free states S 11 = {1, . . ., 11}, as depicted in Fig. 1(b)

FIG. 1 .
FIG. 1. Greechie orthogonality diagrams of (a) the L 12 logic and (b) the pentagon (pentagram) logic, with two of their associated (quasi)classical partition logic representations.