Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes occurring in quantum mechanics, which Kochen and Specker in [5] showed were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker [5]), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker [5], and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.
Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes occurring in quantum mechanics, which Kochen and Specker in [5] showed were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker [5]), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker [5], and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.
Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes occurring in quantum mechanics, which Kochen and Specker in 5 showed were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker 5), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker 5, and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.
Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes (and inconsistencies of measured values), occurring in quantum mechanics, where Kochen and Specker in [5] showed these were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker [5]), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker [5], and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.
Abstract In chapter 12 we give an overview of the historical development of hidden-variable theories aiming to attribute ‘hidden’ definite outcomes to any potential measurement on a quantum system. We start with a discussion of von Neumann’s assumption on hidden-variable models, which was later realized not to hold by Hermann, Kochen, Specker, and Bell, going into detail on Bell’s refutation of von Neumann’s arguments. We then turn to the notion of contextuality, which severely restricts hidden-variable models: we discuss Gleason’s theorem before stating and proving the famous Kochen-Specker theorem, as well as the simpler versions of Peres’ nonet and Mermin’s pentagram. These results show that quantum mechanics is not compatible with the assignment of definite measurement outcomes to all possible measurements, and such assignments can hence only be made once a restricted set of observables, a context, is selected
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use:
We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some characteristics that any model (and hence, any formulation of quantum mechanics compatible with its predictions and relying on a notion of measurement) should verify. The main results we obtain are that in the case of a Hilbert space of dimension at least 3, using a strengthening of the Kochen-Specker theorem, we show that no model can lead to the certain prediction of more than one atomic outcome. Moreover, if the Hilbert space is finite dimensional, then we are able to precisely describe the structure of the predictions of any model of our logic. In particular, we show that all the models of our logic do exactly make the same predictions regarding whether a given sequence of outcomes is possible or not, so that quantum mechanics can be considered complete as long as the possibility of outcomes is considered.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.