Quantum Measurements Generating Structures of Numerical Events

Let S be a set of states of a physical system and ( ) p s the probability of an occurrence of an event when the system is in state s S ∈ . The function p from S to [ ] 0,1 is called a numerical event, multidimensional probability or, more precisely, S-probability. If a set of numerical events is ordered by the order of real functions one obtains a partial ordered set P in which the sum and difference of S-probabilities are related to their order within P. According to the structure that arises, this further opens up the opportunity to decide whether one deals with a quantum mechanical situation or a classical one. In this paper we focus on the situation that P is generated by a given set of measurements, i.e. S-probabilities, without assuming that these S-probabilities can be complemented by further measurements or are embeddable into Boolean algebras, assumptions that were made in most of the preceding papers. In par-ticular, we study the generation by S-probabilities that can only assume the values 0 and 1, thus dealing with so called concrete logics. We characterize these logics under several suppositions that might occur with measurements and generalize our findings to arbitrary S-probabilities, this way providing a possibility to distinguish between potential classical and quantum situations and the fact that an obtained structure might not be sufficient for an appropriate decision. Moreover, we provide some explanatory examples from physics.


Introduction
Let S be a set of states of a physical system and ( ) p s the probability of the oc-S-probabilities for the functions ( ) p s for arbitrary S. This notion was taken up later and will be used in this paper. Given S, a set P of numerical events can be ordered by the order ≤ of functions, similar to the ordering of events of a classical event space (σ-algebra) by the set-theoretic inclusion ⊆ . We will require some of the characteristic features of classical event spaces to belong to the partially ordered set ( ) A3) if , p q P ∈ and p q ⊥ then p q P + ∈ ; A4) if , , p q r P ∈ and p q r p ⊥ ⊥ ⊥ then p q r P + + ∈ .
Further, we point out that (A3) is a special case of (A4). It should also be remarked that under the assumption of (A2), Axiom (A3) is equivalent to the following axiom: A5) If , p q P ∈ and p q ≤ then q p P − ∈ .
Let ' also denote the complementation of sets and  and  stand for the union and intersection of sets, respectively. Thinking of classical event systems, (A2) can be considered as a translation of A B′ ⊆ into 1 p q ≤ − , and (A4) is motivated by the fact that pairwise orthogonality of a triple , , A B C of events implies ( ) , which in terms of functions means ( ) An algebra of S-probabilities is an orthomodular poset with a full set of states, and algebras of S-probabilities and orthomodular posets with a full sets of states are in one-to-one correspondence (cf. [4]). Boolean algebras are a special class of algebras of S-probabilities and in general represent the situation that one deals with a classical physical system. Given a set of measurements represented by S-probabilities, our goal is to find out to which kind of structure the obtained measurements belong or give rise to.

D. Dorninger, H. Länger Journal of Applied Mathematics and Physics
If one deals with an algebra of S-probabilities that is not a Boolean algebra one may assume to be concerned with a quantum mechanical phenomenon. GFEs which are not algebras of S-probabilities can also give a clue to the lack of further information or else, they might also not be appropriate to determine the structure of events underlying an experiment.
Several papers have been published on the subject of assessing the situation whether the obtained measurements give rise to or can be embedded into a Boolean algebra by adding further measurements (cf. [2] [3] and [5] [6] [7] [8] [9]). In this paper we will focus on the question which kind of structure will be generated by already achieved measurements without taking into account the possibility to obtain further relevant data.
We will study GFEs and algebras of S-probabilities generated by given measurements dependent on the structure of these quantities. First, we will assume that there will only be two outcomes to measurements, namely that an S-probability might be either 0 or 1, and then we will generalize some of the obtained results to arbitrary numerical events. As for the structure of the generating sets of S-probabilities we will focus on the situations that the S-probabilities and their complements are pairwise incomparable or else orthogonal or that they form chains. In particular, we will show when under these assumptions the resulting GFE or algebra of S-probabilities is a Boolean algebra, which means that one deals with a classical phenomenon, or when there is not enough information available to decide about classicality or non-classicality. To explain our results we provide some examples of physical experiments.

S-Probabilities Which Can Only Assume the Values 0 and 1
To illustrate the structure of GFEs and algebras of S-probabilities we listen here some basic facts concerning GFEs, writing p q ∧ and p q ∨ for the infimum and supremum of two S-probabilities p and q, respectively (if these exist).
Proposition 2. (cf. [3]) For a GFE P the following hold: 1) If p q ≤ for , p q P ∈ then q p P − ∈ , and if P is an algebra of S-probabilities then q p q p′ − = ∧ ; 2) P is an algebra of S-probabilities if and only if for all , exists for all , p q P ∈ then P is a lattice; 4) If P is an algebra of S-probabilities which is a lattice then P is an orthomodular lattice; 5) A lattice-ordered algebra P of S-probabilities is a Boolean algebra, if and only if p q p q ∨ ≤ + for all , p q P ∈ .
All forthcoming theorems in this section will be exclusively about S-probabilities which only take on the values 0 and 1.
As one can see immediately, a GFE P of numerical events assuming only the values 0 and 1 is an algebra of S-probabilities. We further point out that such an algebra of S-probabilities can be represented by sets (see below) and is therefore S-probabilities can also be considered as quantum logics can be concluded from the fact that they are orthomodular posets which do serve as quantum logics.) Definition 3. Let S be a set and M a set of subsets of S (we will link to GFEs further on). We call M a G-system on S if it satisfies the following axioms: It should be remarked that in general M is not an algebra of sets. Moreover, (G3) implies the following axiom: We further observe that under the assumption of (G2), Axiom (G3) is equivalent to the following axiom: For every set S and every subset A of S let A I denote the mapping from S to 3) The correspondence described in (i) and (ii) is one-to-one.
Due to Theorem 4 the study of GFEs P of S-probabilities assuming only the values 0 and 1 can be reduced to arguments about G-systems and indicator functions.
For every set S and any set Q of subsets of S let Q denote the G -system on S generated by Q, i.e. the smallest G-system on S including Q or the intersection of all G-systems on S including Q or the smallest set of subsets of S including Q and satisfying (G1)-(G3).
Analogously, for every set S and any set Q of functions from S to [ ] 0,1 let likewise Q denote the GFE generated by Q, i.e. the smallest GFE including Q or the intersection of all GFEs including Q or the smallest set of functions from S to [ ] 0,1 including Q and satisfying (A1)-(A3).
As for arbitrary algebras P of S-probabilities, if for a p P  1) If the four elements , , ,  is a Boolean algebra having at most eight elements.
Proof. Put 1) This is clear.
2) If e.g. A and A' are comparable then consists of all unions of some of the pairwise disjoint sets , \ , A B A B′ .
 Proposition 5 will be the initial point for generalizations (see Section 3). Throughout this and the next section of the paper let 2 n ≥ be an arbitrary integer and put is a Boolean algebra having at most 2 n elements.
consists of all unions of some of the pairwise disjoint sets ( ) is a Boolean algebra having at most 2 n elements. Proof.
consists of all unions of some of the pairwise disjoint sets is a Boolean algebra having at most 2 2 n elements.

Generating Algebras of Arbitrary S-Probabilities
, and hence the latter is a 2 n -element Boolean algebra, too. Now the following are equivalent: are pairwise orthogonal then , , J K L will be pairwise disjoint and hence . This shows that A is an algebra of S-probabilities. If is a 2 n -element Boolean algebra.   Now, let p be the probability that the beam consists of electrons with x ↑ and y ↓ and q the probability that this is the case for electrons with x ↓ and y ↓ . According to our arrangement we then obtain The second example is based on a simple experiment described in [1] and serves to demonstrate the difference between generating GFEs and algebras of S-probabilities: Consider a coin with faces H (heads) and T (tails) in a box with a window in the upper side such that one can look in. The box is carried from one table (table   I)-its state there should be s 1 -to a second table (table II), where the box is assumed to be in state s 2 . During the process of transporting the box the face of the coin may change. Let p be the relative frequency that the face changes from a given position on table I, let us say it should always be H, to T. If measurements show that due to a certain asymmetry of the coin the relative frequency b that the coin shows T on table II is very high, certainly strictly above 1/2, we will obtain ( ) ( ) ( ) ( ) As one can easily verify, P is a lattice with six elements and not a Boolean algebra. Whether there is any indication for a quantum process cannot be said, what we can only derive from this result is that more information would be necessary or that the process can't be properly described by the structure of observed numerical events.
As also already discussed in [1], but from a different point of view, we next consider the states s 1 , s 2 of a photon linearly polarized along two orthogonal axes (and propagating in a direction orthogonal to the plane E spanned by these axes). Fixing an orthogonal x,y-coordinate system and introducing a new x α , y α -coordinate system in E by rotating the x-and y-axes by an angle α one obtains that the transmission probability in the direction of the x α -coordinate will be If we pick only one angle π 4 α ≠ for a measurement besides considering the S-probability ( ) 0, 0 which can be interpreted as observing nothing and ( ) p p ′ = accounting for a GFE which is a three-element chain and no Boolean algebra indeed. But still, with a practical experiment the position of the polarizer cannot be assumed to be π/4 with 100% security so one has to take into account at least two angles α different from 0, π/2 and π/4, which give rise to M n (or theoretically) M ∞ . If one furthermore adds π 4 p to these sets of numerical events one obtains GFEs which will suggest the non-classicality of the experiment at hand.

Conclusions and Suggestions
In 1991 the physicist E. Beltrametti and the mathematician M. Maczyński introduced the notion of a numerical event, i.e. S-probability. Their goal was to provide an approach to quantum mechanics devoid of the necessity to know something about the structure of events pertaining to a physical experiment (cf. [1]).
(If one first described events and next measured them in various states then one would have to know the logical structure of events from the beginning or assumed axiomatically.) In the years that followed, properties of algebras of S-probabilities and generalizations of them have been thoroughly studied, mainly from the algebraic point of view (cf. [2] [3] and [5] [6] [7] [8] [9]). In particular, many characterizations were found for an algebra of S-probabilities to be a Boolean algebra or to be embeddable into a Boolean algebra, both cases in which classicality of the system can be assumed. However, for those characterizations the existence of certain S-probabilities has to be secured which in general means that further measurements have to be carried out.
In our approach we refrained from possibly requiring further information by measurements and investigated the structures that arise when generated by the S-probabilities on hand. We studied various relations between the given S-probabilities like incompatibility, orthogonality or a special order. Beginning with numerical events that can only assume the values 0 and 1 and then generalizing our findings to arbitrary S-probabilities we gained some insight into the logical structure induced by a given set of measurements and this way obtained a clue for classicality or non-classicality.
What remains to be investigated is the influence of the cardinality of S, the number of the possible states of a system, the assumption of a more complex relationship between the given S-probabilities, like certain correlations between D. Dorninger, H. Länger Journal of Applied Mathematics and Physics the multidimensional probabilities or more general kinds of their partial order, and the application of the prospective results to concrete physical problems.