Appendix A State-Independent Proofs of the Bell-Kochen-Specker Theorem In this chapter we present a brief review of the state-independent proofs of the Bell- Kochen-Specker Theorem. One of the first to challenge the idea of completing quantum theory was von Neumann, and his argument appears in his pioneer rigorous mathematical formulation of quantum theory [vN55]. Although one of his assumptions was too strong, his result was a landmark in foundations of physics. We then present Gleason’s lemma. Although his main result is motivated by a problem in foundations of quantum theory, Gleason’s goal had in principle nothing to do with completions of quantum theory, which are not even mentioned in his paper. Nevertheless, one of his lemmas provides an elegant proof of the Bell- Kochen-Specker theorem. This fact was noticed only after Kochen and Specker had proven this result by other means. The advantage of Kochen and Specker’s proof is that, contrary to Gleason’s argument, it requires only a finite set of one-dimensional projectors. The Kochen- Specker original proof is presented, along with other simple additive and multiplica- tive proofs. We finish this chapter presenting a contextual completion of a quantum probabilistic model, showing that the noncontextualitiy hypothesis is indeed crucial to discard the possibility of completions. A.1 von Neumann von Neumann was the first to rigorously establish a mathematical formulation for quantum theory, published in his 1932 work Mathematische Grundlagen der Quantenmechanik, and later translated to English in 1955 [vN55]. His rigorous approach allowed him to challenge the ideas of completion of quantum theory. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018 101 B. Amaral, M. Terra Cunha, On Graph Approaches to Contextuality and their Role in Quantum Theory, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-319-93827-1 102 A State-Independent Proofs of the Bell-Kochen-Specker Theorem He derived the quantum formula O = Tr (ρO) (A.1) for the expectation value of a measurement from a few general assumptions about the expectation-value function. Then, we can prove that there is no deterministic state consistent with this formula, and hence that completions of quantum theory are impossible. Although one of his assumptions was too strong, as we explain later, his result was breakthrough that preceded a series of elegant proofs of the Bell-Kochen- Specker theorem. A.1.1 von Neumann’s Assumptions Given a specific type of system in a probability theory, every state defines an expectation-value function: :M −→ R (A.2) where M stands for the set of measurements in the completion. Instead of focusing on the possible states, von Neumann was interested in the properties of these functions, and stated a number of requirements he believed to be natural impositions on them. Definition A.1 An expectation-value function :M −→ R is deterministic if M2=M2 (A.3) for every measurement M allowed in the model. Deterministic functions are the ones that come from deterministic states, that is, those for which one of the outcomes has probability one for every measurement M ∈ M . Definition A.2 An expectation-value function :M −→ R is called pure if =p + (1 − p) , 0 <p<1, (A.4) implies = =. Pure functions are the ones that cannot be written as a convex sum of others. A function is pure if, and only if, the state that defines it is a pure state of the system. Every deterministic function is pure, but the converse is not always true. For example, in quantum theory, pure functions are the ones defined by one-dimensional projectors, while there is no deterministic function. In a noncontextual completion of a probabilistic model the two notions coincide. A.1 von Neumann 103 In quantum theory, every measurement M can be associated to an observable, a Hermitian operator O acting on the Hilbert space of the system. von Neumann’s first assumption is that this correspondence is one-to-one and onto. Assumption A.1 There is a bijective correspondence between measurements in a quantum system and Hermitian operators acting on the Hilbert space of the system. This is not always the case, since some systems are subjected to certain superselection rules, which forbid some Hermitian operators [Wikd]. Although this is not a general assumption, there are other formulations of von Neumann’s result that circumvent this difficulty (see [CFS70] and the references therein). Suppose a completion of quantum theory is given. The states of the system, now corresponding to a quantum state plus an extra variable provided by the completion, define expectation-value functions acting on the set of measurements in the system, which is, by Assumption A.1,thesetO(H) of Hermitian operators acting on the Hilbert space H of the system. Then, every state in the completion is associated with an expectation-value function :O (H) −→ R. (A.5) The next step in von Neumann’s approach was to impose a few assumptions on the functions that he believed to be valid if these functions came from expectation values of a real physical system. Assumption A.2 1. If M is by nature non-negative, then M ≥ 0; 2. If measurement M1 is associated to observable O1 and M2 is associated to observable O2, we can define measurement M1 + M2 and it is associated to observable O1 + O2; 3. If M1, M2,...are arbitrary measurements, then a1M1 + a2M2 + ... = a1 M1 + a2 M2 + ..., (A.6) that is, all expectation-value functions are linear; 4. If measurement M is associated to observable O and f : R −→ R is any real function,1 the measurement f (M) is associated to observable f(O). Theorem A.1 Under Assumptions A.1 and A.2, the expectation-value functions in any theory completing quantum theory are given by M = Tr (Oρ) , (A.7) 1Measurement f(M) is defined using the following rule: measure M and apply f to the outcome obtained. Observable f(O) can be defined easily if we write O in spectral decomposition. Let = | | {| } O i ai vi vi ,where vi is an orthonormal basis for the corresponding vector space. Then = | | f(O) i f(ai ) vi vi . 104 A State-Independent Proofs of the Bell-Kochen-Specker Theorem where O is the observable corresponding to measurement M and ρ is a density operator that depends only on the function (and not on the particular measure- ment M). This result implies that, as long as we impose Assumptions A.1 and A.2,we cannot circumvent the quantum rule for expectation values. As we already know, the pure functions of this form are the ones for which the associated density operator is a one-dimensional projector P and these functions only give deterministic outcomes for a small subset of measurements, namely, the ones for which the subspace in which P projects is an eigenspace of the associated observable. This in turn implies that there is no deterministic function, proving the impossibility of completions compatible with quantum theory. For a long time, it was generally believed that von Neumann’s theorem demon- strated that no deterministic theory reproducing the statistical quantum predictions was possible. In 1966, however, J. Bell published a paper with some serious criticism to one of the requirements made for the expectation-value functions [Bel66]. In von Neumann’s argument, he requires expectation-value functions to be linear, which is the case for quantum theory, but there is no physical reason to impose this property for more general theories. In fact, as von Neumann pointed out himself in Ref. [vN55], the sum of measurements a1M1 + a2M2 + ...is completely meaningless when the measurements involved are not compatible, since there is no way of constructing, in general, the corresponding experimental set-up to implement it. Bell argued that in the case of incompatible measurements, it is not reasonable to require that the expectation values necessarily reflect the observables’ algebraic relationships. It is a special property of quantum theory that the sum of the corresponding observables corresponds to another allowed measurement (as long as Assump- tion A.1 is valid), and the fact that the expectation value is linear is a consequence of the mathematical rules of quantum theory and is not enforced by any general physical law. In fact, it is not difficult to provide a completion of quantum theory for a qubit, a quantum system with two-dimensional Hilbert space, which does not satisfy linearity of expectation values. Example A.1 (An Example of Completion) In Ref. [Bel66], Bell showed an example of a completion for a qubit agreeing with quantum theory but violating von Neumann’s assumption of linearity. Let O be an operator acting on C2. Since the Pauli matrices 01 0 −i 10 σ = ,σ = ,σ = (A.8) x 10 y i 0 z 0 −1 and the identity I form a basis to the real vector space of 4 × 4 Hermitian operators we can always write O in the form O = a0I + a1σx + a2σy + a3σz, (A.9) where ai ∈ R. A.2 Gleason’s Lemma 105 Denote by v(O) the value associated to measurement O in a completion. If we set |a = (a1,a2,a3), the eigenvalues of O, and hence the possible values of v(O), can be written as v(O) = a0 ± a . (A.10) Let |φ ∈ C2 and |n be the point on the Bloch sphere [NC00] corresponding to |φ. Then, we have O = φ| O |φ = a0 + a | n . (A.11) Together with the quantum state |φ, we will use another vector |m in the Bloch sphere to represent the state of the system.