Classical Predictions for Intertwined Quantum Observables Are Contingent and Thus Inconclusive
Abstract
:1. Quantum Clouds as Collections of Intertwined Contexts and Their Classical Doubles
- (i)
- a quantum mechanical realization in terms of intertwining orthonormal bases, as mentioned earlier;
- (ii)
- a pseudo-classical realization in terms of partition logic, which in turn have automaton logic or generalized urn models;
- (iii)
- a classical realization if there is only a single context involved;
- (iv)
- none of the above (such as a tightly interlinked “triangle” configuration of three contexts with two vertices per context).
2. Enforcing Classical Two-Valued States
- (I)
- The “measures” or value assignments employed in so-called “contextuality inequalities” merely assume that every proposition is either true or false, regardless of the other propositions in that context, which are simultaneously measurable [32]. This allows all possible possibilities of value assignments in a d-dimensional context with d vertices, thereby vastly expanding the multitude of possible value assignments. With this expansion, all Kochen–Specker sets trivially allow value assignments.
- (II)
- The prevalent assumption of two-valued states or value assignments, also used by Kochen and Specker [19], as well as Pitowsky [33], is that only a single one of the d vertices within a d-dimensional context is true, and all the others are false; therefore, any isolated d-dimensional context can have only d such standard two-valued value assignments.
- (III)
- An even more restricted rule of value assignment abandons uniform definiteness and supposes [11,12,34] that, if all but one vertex in a d-dimensional context are false, the remaining one is true, and if one vertex within a d-dimensional context is true, all remaining vertices are false. These latter value assignments allow for partial functions, which can be undefined.
- (u)
- unital, if for every , there is a two-valued state such that ;
- (s)
- separating, if for every distinct pair of vertices with , there is an such that ;
- (f)
- full, if for every nonadjacent pair of vertices , there is an such that .
3. Chromatic Separability
4. Formation of Gadgets as Useful Subgraphs for the Construction of Clouds
- Zeroth order gadget: a single context (also known as clique/block/Boolean (sub)algebra/ maximal observable/orthonormal basis). This can be perceived as the most elementary form of a true-implies-false (TIFS/10) [14]/01-(maybe better 10)-gadget [50,52] configuration, because a truth/value one assignment of one of the vertices implies falsity/value zero assignments of all the others;
- First order “firefly” gadget: two contexts connected in a single intertwining vertex;
- Second order gadget: two first order firefly gadgets connected in a single intertwining vertex;
- Third order house/pentagon/pentagram gadget: one firefly and one second order gadget connected in two intertwining vertices to form a cyclic orthogonality hypergraph;
5. Quantum Clouds Enforcing Particular Features When Interpreted Classically
- (a)
- (b)
- Already, Kochen and Specker utilized quantum clouds enforcing classical true-implies- false predictions and their compositions in the construction of a configuration that does not allow a uniform truth assignment (of Type (II)). Stairs ([54], pp. 588–589) has pointed out that the Specker bug ([9], Figure 1, p. 182) is a quantum cloud configuration that classically enforces true-implies- false: if a quantum system is prepared in such a way that is true—that is, if it is in the state –and measured along , and and are not orthogonal or collinear, then any observation of given amounts to a probabilistic proof of nonclassicality: because although quantum probabilities do not vanish, classical value assignments predict that never occurs. Minimal quantum cloud configurations for classical true-implies- false, as well as true-implies- true value assignments (of Type (II)) can be found in [14].As Cabello has pointed out [69,70], the original Specker bug configuration cannot go beyond the quantum prediction probability threshold because the angle between and cannot be smaller than radians (). A configuration ([71], Figure 5a) allowing Type (III) TIFS truth assignments with “maximally unbiased” quantum prediction probability is a sublogic of a quantum logic whose realization was enumerated in [12], Table. 1, p. 102201-7. This is depicted in Figure 3. A proof of Theorem 2 in [50] contains an explicit parametrization of a single TIFS/10 cloud allowing the full range of angles .
- (c)
- Clifton (note added in the proof to Stairs ([54], pp. 588–589)) presented a true-implies- true (TITS) cloud ([56,72,73], Sects. II, III, Figure 1) inspired by Bell ([57], Figure C.l. p. 67) (cf. also Pitowsky ([58], p. 394)), as well as by the Specker bug logic ([56], Sects. IV, Figure 2). Hardy [59,60,61], as well as Cabello, among others [50,63,64,65,66,67,69,70], utilized similar scenarios for the demonstration of nonclassicality ([74], Chapter 14). Figure 4 depicts an 11-gadget ([71], Figure 5b) with identical endpoints as the 10-gadget discussed earlier and depicted in Figure 3.
- (d)
- Various parallel and serial compositions of 10- and 11-gadgets serve as a “gadget toolbox” to obtain clouds, which, if they are interpreted classically, exhibit other interesting relational properties. For instance, the parallel composition (pasting) of two quantum clouds of the 10-gadget type: one 10-gadget classically demanding true-implies- false and the other 10-gadget classically demanding true-implies- false, results in a quantum cloud that has two observables and , which are classically always “opposite”: if one is true, the other one is false, and vice versa.
- (e)
- The parallel composition (pasting) of two quantum clouds of the TITS type, with one TITS, classically demanding true-implies- true and the other TITS classically demanding true-implies- true, results in a quantum cloud that has two observables and , which are classically nonseparable, which is a sufficient criterion for nonclassicality ([19], Theorem 0, p. 67). As pointed out by Portillo [75], this is equivalent to is true if and only if is true (TIFFTS). Figure 5 depicts a historic example of such a construction. The serial composition of suitable TITS of the form true-implies- true-implies- true eventually yields two or more vectors and that are mutually orthogonal; a technique employed by Kochen and Specker for the construction of a quantum cloud admitting no Type (II) truth assignment ([19], , p. 69).
- (f)
- The parallel composition (pasting) of the two quantum clouds that respectively represent a 10-gadget and an 11-gadget and identical endpoints and yields a true-implies- value indefinite cloud discussed in [12].
6. Some Technical Issues of Gadget Construction
7. Discussion
- if the quantum cloud allows both values, then the claim is that there is no determination of the outcome; the event “popped up” from nowhere, ex nihilo, or, theologically speaking, has come about by creatio continua (cf. Kelly James Clark’s God-as-Curler metaphor [76]);
- in the case of a 10-gadget, the system is truly quantum and cannot be classical;
- in the case of an 11-gadget, the system could be classical;
- in the case of a cloud inducing value indefiniteness, the claim can be justified that the system cannot be classical, as no such event (not even its absence) should be recorded. Indeed, relative to the assumptions made, the (non)occurrence of any event at all is in contradiction to the classical predictions.
- as mentioned earlier, if the quantum cloud allows both values, then there exists creatio continua (currently, this appears to be the orthodox majority position);
- in the case of a 10-gadget, the system could be classical;
- in the case of an 11-gadget, the system is truly quantum and cannot be classical;
- just as mentioned earlier, in the case of a cloud inducing value indefiniteness, the claim can be justified that the system cannot be classical, as no such event (not even its absence) should be recorded.
Funding
Acknowledgments
Conflicts of Interest
References
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If A is True Classical Value Assignments | Anecdotal, Historic Quantum Realization | Reference to Utility or Relational Properties |
---|---|---|
imply is independent (arbitrary) | firefly logic ([53], pp. 21, 22) | |
imply false (TIFS/10) | Specker bug logic ([9], Figure 1, p. 182) | ([54], pp. 588–589), [14,55] |
imply true (TITS) | extended Specker bug logic | ([19], , p. 68), |
([56], Sects. II, III, Figure 1), | ||
([57], Figure C.l. p. 67), | ||
([58], p. 394), [59,60,61], | ||
[14,62,63,64,65,66,67] | ||
iff true (nonseparability) | combo of intertwined Specker bugs | ([19], , p. 70) |
imply value indefiniteness of | depending on Type (II), (III) assignments | [12,33] |
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Svozil, K. Classical Predictions for Intertwined Quantum Observables Are Contingent and Thus Inconclusive. Quantum Rep. 2020, 2, 278-292. https://doi.org/10.3390/quantum2020018
Svozil K. Classical Predictions for Intertwined Quantum Observables Are Contingent and Thus Inconclusive. Quantum Reports. 2020; 2(2):278-292. https://doi.org/10.3390/quantum2020018
Chicago/Turabian StyleSvozil, Karl. 2020. "Classical Predictions for Intertwined Quantum Observables Are Contingent and Thus Inconclusive" Quantum Reports 2, no. 2: 278-292. https://doi.org/10.3390/quantum2020018