Faithful orthogonal representations of graphs from partition logics
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Abstract
Partition logics often allow a dual probabilistic interpretation: a classical one for which probabilities lie on the convex hull of the dispersionfree 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 logicoempirical structure of observables does not determine the type of probabilities alone and that complementarity does not imply contextuality.
Keywords
Quantum mechanics Gleason theorem Kochen–Specker theorem Born rule Partition logic Grötschel–Lovász–Schrijver set1 Partition logics as nonboolean structures pasted from Boolean subalgebras
Partitions provide ways to distinguish between elements of a given finite set \({{\mathscr {S}}}_n=\{1,2,\ldots ,n\}\). The Bell number \(B_n\) (after Eric Temple Bell) is the number of such partitions (Sloane 2018). (Obvious generalizations are infinite denumerable sets or continua.) We shall restrict our attention to partitions with an equal number \(1\le m\le n\) 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 (Svozil 1993; Schaller and Svozil 1994, 1996; Dvurečenskij et al. 1995; Svozil 2005) is the logic obtained (i) from collections of such partitions, each partition being identified with an matomic 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.
Partitions represent classical miniuniverses which satisfy compatible orthogonality, or Specker’s exclusivity principle (Specker 1960, 2009; Cabello 2013; Fritz et al. 2013; Henson 2012; Cabello 2012; Cabello et al. 2013, 2014): if any two observables corresponding to two elements of a partition are comeasurable, the entire set of observables corresponding to all elements of that partition are simultaneously measurable. (For Hilbert spaces, this is a wellknown theorem; see, for instance, (von Neumann 1931, Satz 8, p. 221) and (von Neumann 1955, p. 173), or (Halmos 1958, § 84, Theorem 1, p. 171).)
Many partition logics, such as the pentagon logic, have quantum doubles. One of the (necessary and sufficient) criteria for quantum logics to be representable by a partition logic is the separability of pairs of atoms of the logic by dispersionfree (aka twovalued, \(\{0,1\}\)valued) weights/states (Kochen and Specker 1967, Theorem 0, p. 67), interpretable as classical truth assignments.
2 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 comeasurable propositions (satisfying Specker’s exclusivity principle) should obey Kolmogorov’s axioms, in particular nonnegativity and additivity. Nonnegativity implies that all probabilities are nonnegative: \(P(E_1),\ldots ,P(E_k)\ge 0\). Additivity among (pairwise) mutually exclusive outcomes \(E_1,\ldots , E_k\) means that the probabilities of joint outcomes are equal to the sum of probabilities of these outcomes, that is, within cliques/contexts, for \(k\le m\): \(P(E_1\vee \cdots \vee E_k) =P(E_1)+\cdots + P(E_k) \le 1\). In particular, probabilities add to 1 on each of the cliques/contexts. Furthermore, Kolmogorov’s axioms can be extended to configurations of more than one (classical) context by assuming that, relative to any atomic element of some context, the sum of the conditional probabilities of all atomic elements in any other context adds up to one (Svozil 2018).
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 \(\frac{1}{2}\) on the vertices of the pentagon (Ron 1978, \(\omega _0\), p. 68) and on the triangle vertices (Wright 1990, pp. 899–902) (the latter logic is representable as partition logic (Dvurečenskij et al. 1995, Example 8.2, pp. 420,421), but not in two or threedimensional 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 (Ron 1978; Wright 1990; Svozil 2006, 2014) or the initial state identification problem for finite deterministic automaton (Moore 1956; Svozil 1993; Schaller and Svozil 1995, 1996)—both are equivalent models of partition logics (Svozil 2005) featuring complementarity without value indefiniteness.
Specker’s parable of the overprotective seer (Specker 1960; Liang et al. 2011, 2017; Svozil 2016) involving three boxes is an example for which the exclusivity principle does not hold (Tarrida 2014, 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.
2.1 Probabilities from the convex hull of dispersionfree 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 (Kochen and Specker 1967) states that separability by dispersionfree states (of image \(2^1={0,1}\)) for every pair of atoms of the lattice is a necessary and sufficient criterion for a homomorphic embedding into some “larger” Boolean algebra. In 1978, Wright explicitly stated (Ron 1978, p. 272) “that every urn weight is “classical,” i.e., in the convex hull of the dispersionfree weights.” In the graph theoretical context Grötschel, Lovász and Schrijver have discussed the vertex packing polytopeVP(G) of a graph G, defined as the convex hull of incidence vectors of independent sets of nodes (Grötschel et al. 1986). This author has employed dispersionfree weights for hull computations on the Specker bug (Svozil 2001) and other (partition) logics supporting a separating set of twovalued states.
2.2 Born–Gleason–Grötschel–Lovász–Schrijver type probabilities
Motivated by cryptographic issues outside quantum theory, (Lovász 1979) has proposed an “indexing” of vertices of a graph by vectors reflecting their adjacency: the graphtheoretic 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 (Lovász et al. 1989; Parsons and Pisanski 1989). For physical applications (Cabello et al. 2010; SolísEncina and Portillo 2015) and others have used 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.
Based on Lovász’s vector representation by graphs, Grötschel, Lovász and Schrijver have proposed (Grötschel et al. 1986, Section 3) a Gleason–Born type probability measure (Cabello 2019) which results in convex sets different from polyhedra defined via convex hulls of vectors discussed earlier in Sect. 2.1. Essentially their probability measure is based upon mdimensional faithful orthogonal representations of a graph G whose vertices \(v_i\) are represented by unit vectors \(\vert v_i\rangle \) which are orthogonal within, and nonorthogonal outside, of cliques/contexts. Every vertex \(v_i\) of the graph G, represented by the unit vector \(\vert v_i\rangle \), can then be associated with a “probability” with respect to some unit “preparation” (state) vector \(\vert c\rangle \) by defining this “probability” to be the absolute square of the inner product of \(\vert v_i\rangle \) and \(\vert c\rangle \), that is, by \(P(c,v_i)=\left \langle c \vert v_i\rangle \right ^2\). 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 and additive, and (as both \(\vert v_i\rangle \) and \(\vert c\rangle \) are normalized) the sum of probabilities on that context adds up to exactly one, that is, \(\sum _{i \in \text {clique/context}} P(c,v_i)=1\). 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 (Grötschel et al. 1986), by the very assumptions such as exclusivity (Cabello et al. 2014; Cabello 2019).
A very similar measure on the closed subspaces of Hilbert space, satisfying Specker’s exclusivity principle and additivity, had been proposed by Gleason Gleason (1957), first and second paragraphs, p. 885: “A measure on the closed subspaces means a function\(\mu \)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 spanB, then\(\mu (B) = \sum _i \mu (A_i)\). It is easy to see that such a measure can be obtained by selecting a vectorvand, for each closed subspaceA, taking\(\mu (A)\)as the square of the norm of the projection ofvonA.” Gleason’s derivation of the quantum mechanical Born rule (Born 1926, 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” (Nielsen and Chuang 2010, Section 2.5, pp. 109–111) (and thus outcomes of nonideal measurements made ideal Cabello 2019) by the inclusion of auxiliary dimensions.
3 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 (Zeilinger 1999) properties of the constituents, whereas nonentangled states feature individual separate properties of the pair constituents (Schrödinger 1935a, b, 1936). (For early similar discussions in the measurement context, see von Neumann 1955, Section VI.2, p 426, pp 436–437 and London and Bauer 1939; London and Edmond 1983.)
3.1 Partitioning of state space

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.
Six subensembles \(E_1\)–\(E_6\) of the set Open image in new window with the following properties: Open image in new window encodes the first digit being 0; Open image in new window encodes the first digit being 1; Open image in new window encodes the second digit being 0; Open image in new window encodes the second digit being 1; Open image in new window encodes the first and the second digit being equal; Open image in new window encodes the first and the second digit being different
Subensembles \((E_5)^2\) and \((E_6)^2\) of the set { Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window } with the following properties: Open image in new window encodes the first and the second pair, as well as the third and the fourth pair of digits being equal; Open image in new window encodes the first and the second pair, as well as the third and the fourth pair of digits being different
3.2 Relational encoding
Tables 1 and 2 enumerate a relational encoding among two or more colors not dissimilar to Peres’ detonating bomb model (Peres 1978). Suppose that an urn is loaded with balls of the type occurring in subensemble \(E_6\) of Table 1. The observation of some symbol \(s\in \{0,1\}\) in green implies the (counterfactual) observation of the same symbol s in red, and vice versa. Table 2 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 dispersionfree 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.
4 Partition logic freak show
Let us, for a moment, consider partition logics not restraint by lowdimensional faithful orthogonal representability, but with a separable set of twovalued states (with the exception of the logic depicted in Fig. 2f). These have no quantum realization. Yet, due to the automaton logic, they are intrinsically realized in Turing universal environments.
5 Identical graphs realizable by different physical resources require different probability types
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., Figure 1 in both References Fritz et al. 2013; Cabello et al. 2014). The difference is the representation of these graphs: the quantum case has a faithful orthogonal representation in some finitedimensional 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 inbetween classical Boolean and quantum realms.
It should be stressed that the algebraic structure of empirical logics, or graphs, does 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 (Kochen and Specker 1967, Theorem 0) sets of twovalued states and thus are not “contextual” in the Specker sense (Specker 1960).
Notes
Acknowledgements
Open access funding provided by TU Wien (TUW). This work was greatly inspired by Adán Cabello’s insistence on graph theoretical importance for quantum mechanics and for his challenge to come up with a classical Einstein–Podolsky–Rosen type scenario.
Compliance with ethical standards
Conflicts of interest
The author declares that he has no conflict of interest.
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