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 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 three-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 (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.
2.1 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 (Kochen and Specker
1967) 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 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 dispersion-free 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 dispersion-free weights for hull computations on the Specker bug (Svozil
2001) and other (partition) logics supporting a separating set of two-valued states.
Hull computations based on the pentagon (modulo pentagon/pentagram graph isomorphisms) can be found in Refs. Klyachko et al. (
2008), Bub and Stairs (
2009,
2010), Badzia̧g et al. (
2011) (for a survey see Svozil
2018, Section 12.9.8.3). The Bub and Stairs inequality (Bub and Stairs
2009, Equation (10), p. 697) can be directly read off from the partition logic (
2), as depicted in Fig.
1b, which in turn are the cumulated indices of the nonzero dispersion-free weights on the atoms: the sum of the convex hull of the dispersion-free weights on the 5 intertwining atoms (the “vertices” of the pentagon diagram) represented by the subsets
\(\{1,2,3\} \),
\(\{ 6,8,10\}\),
\(\{ 3,5,9\} \),
\(\{ 2,7,8\} \),
\(\{ 4,5,6\} \) of
\({{\mathscr {S}}}_{11}\) is
$$\begin{aligned} \begin{aligned}&(\lambda _1+\lambda _2+\lambda _3 )+ (\lambda _6+\lambda _8+\lambda _{10} )+ (\lambda _3+\lambda _5+\lambda _9 ) \\&\quad + (\lambda _2+\lambda _7+\lambda _8 )+ (\lambda _4+\lambda _5+\lambda _6 ) \le 2 \sum _{i=1}^{11} \lambda _i =2. \end{aligned} \nonumber \\ \end{aligned}$$
(3)
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 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 (Lovász et al.
1989; Parsons and Pisanski
1989). For physical applications (Cabello et al.
2010; Solís-Encina 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.
There is no systematic algorithm to compute the minimal dimension for a faithful orthogonal representation of a graph. Lovász (
1979), Cabello et al. (
2013) gave a (relative to entropy measures (Haemers
1979) “optimal” vector representation of the pentagon graph depicted in Fig.
1b in three dimensions [
\(L_{12}\) depicted in Fig.
1a is a sublogic thereof]: modulo pentagon/pentagram graph isomorphisms which in two-line notation is
\(\begin{pmatrix} 1&{}2&{}3&{}4&{}5\\ 1&{}4&{}2&{}5&{}3 \end{pmatrix}\) and in cycle notation is (1)(2453); its set of five intertwining vertices
\(\{v_1,\ldots ,v_5\}=\{u_1,u_3,u_5,u_2,u_4\}\) are represented by the three-dimensional unit vectors (the five vectors corresponding to the “inner” vertices/atoms can be found by a Gram–Schmidt process)
$$\begin{aligned} \vert u_l \rangle = 5^{-\frac{1}{4}} \begin{pmatrix}1, \left[ 5^\frac{1}{2}-1\right] ^\frac{1}{2} \cos \frac{2 \pi l}{5} , \left[ 5^\frac{1}{2}-1\right] ^\frac{1}{2} \sin \frac{2 \pi l}{5} \end{pmatrix},\nonumber \\ \end{aligned}$$
(4)
which, by preparing the “(umbrella) handle” state vector
\(\begin{pmatrix} 1,0,0 \end{pmatrix} \), turns out to render the maximal (Bub and Stairs
2009; Badzia̧g et al.
2011) quantum-bound
\(\sum _{j=1}^5 \vert \langle c\vert u_l \rangle \vert ^2 =\sqrt{5}\), 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 (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
m-dimensional 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 co-measurable 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.