On Counterfactuals and Contextuality

Abstract

Abstract. Counterfactual reasoning and contextuality is defined and critically evaluated with regard to its nonempirical content. To this end, a uniqueness property of states, explosion views and link observables are introduced. If only a single context associated with a particular maximum set of observables can be operationalized, then a context translation principle resolves measurements of different contexts. COUNTERFACTUALSWith the rise of quantum mechanics [1,2,3,4] physics proper entered an ancient and sometimes fierce debate in theology and philosophy: the controversy between realism versus idealism. Whereas realism has been subsumed by the proposition that [5] "some entities sometimes exist without being experienced by any finite mind," idealism put forward that "we have not the faintest reason for believing in the existence of unexperienced entities. [[Realism]] has been adopted . . . solely because it simplifies our view of the universe." And whereas these issues can be considered nonoperational and thus metaphysical or even ideological, it is also true that they have inspired a great number of minds, to the effect of stimulating new approaches to quantum mechanics, revealing many theoretical details, quantum phenomena and quantum technologies.The Kochen-Specker theorem [6], for example, was motivated from the onset by scholasticism, as in an early programmatic article [7] Ernst Specker related the discussion on the foundations of quantum mechanics to scholastic speculations about the existence of infuturabilities. The scholastic issue was whether or not the omniscience (comprehensive knowledge) of God extends to what nowadays are called counterfactuals. And if so, can all events be pasted together to form a consistent whole?Informally, counterfactuals will be defined as follows. By counterfactual events we mean events which would have occurred if something had happened which did not happen. The associated counterfactual proposition is henceforth regarded "true" if it states the occurrence of an event which would have occurred if something had happened which did not happen.Classically, at least in principle, it makes no difference whether or not a particular observable is measured. It is assumed to possess a definite value, irrespective of any measurement. Thus classical counterfactuals do not present a conceptual challenge.Quantum mechanically, the situation appears to be very different, and the use of counterfactuals for quantized systems is problematic and unresolved. Let us briefly mention two novel, nonrealistic features of quantum mechanics which challenge the sort of realism suggested by classical physics. Note, however, that also in classical times a certain uneasiness with the prevailing realistic perception remained; e.g., in Hertz's perception of the formalisms of classical mechanics Ref. [8].Complementarity and the uncertainty principle limit the precision of co-measurements of certain entities. However, nondistributive propositional structures characteristic for complementarity not necessa...

The Principle of Supplementarity: A Contextual Probabilistic Viewpoint to Complementarity, the Interference of Probabilities and Incompatibility of Variables in Quantum Mechanics

“…Before having a closer look at our model, it is perhaps necessary to discuss the meaning of the term contextuality, as it can obviously be interpreted in many different ways, see Ref. 31. The most common meaning (especially in the literature on quantum logic (33) ) is that the outcome for a measurement of an observable u under a contextual model is calculated using a different (albeit hidden) measure space, depending on whether or not compatible observables v, w, .…”

Noncontextuality in multipartite entanglement

“…Multiport interferometric analogues of multi-particle entanglement have been developed with quantum noncontextuality in mind [35]. Although there is no principal limit to the number of entangled particles involved, the complexity of the interferometric setup associated with certain tasks, as for example the encoding of "explosion views" of Kochen-Specker configurations, still appears to represent an insurmountable challenge.…”

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“…For the sake of the argument toward quantum (non)contextuality [35], rotations in the e 1 − e 2 plane along e 3 are considered; the corresponding matrix being…”

Characterization of quantum computable decision problems by state discrimination

“…A context can formally be defined [26] as a single (nondegenerate) "maximal" self-adjoint operator C. It has a spectral decomposition into some complete set of orthogonal projectors E i which correspond to propositions in the von Neumann-Birkhoff type sense [23,27]. That is, C = ∑ …”