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New Method to Get the Kochen-Specker Theorem

Koji Nagata, Tadao Nakamura

Published Jun 1, 2016• 0 citations
OpenAlex

Abstract

We derive new type of no-hidden-variables theorem based on the assumptions proposed by Kochen and Specker. We consider $N$ spin-1/2 systems. The hidden results of measurement are either $+1$ or $-1$ (in $\hbar/2$ unit). We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in $N$ spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. Also we discuss two-dimensional no-hidden-variables theorem of the KS type. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.

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Inequalities for experimental tests of the Kochen-Specker theorem

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We derive inequalities for n-partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than 1∕2 is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An n-partite uncorrelated state violates the n-partite inequality derived here by an amount that grows exponentially with n.

arXiv:quant-ph/0312052•Journal of Mathematical Physics•Oct 1, 2005•8 citations
We derive new type of no-hidden-variables theorem based on the assumptions proposed by Kochen and Specker. We consider NNN spin-1/2 systems. The hidden results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit). We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in NNN spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. Also we discuss two-dimensional no-hidden-variables theorem of the KS type. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.
Recently, [{arXiv:0810.3134}] is accepted and published. We present ultimate version of no-hidden-variables theorem. We derive a proposition concerning the quantum theory under the existence of the Bloch sphere in a single spin-1/2 system. The existence of a single classical probability space for measurement outcome within the formalism of von Neumann's projective measurement does not coexist with the proposition concerning the quantum theory. We have to give up the existence of such a classical probability space for measurement outcome in the two-dimensional Hilbert space formalism of the quantum theory. The quantum theory does not accept a hidden-variable interpretation in the two-dimensional space.
We study the relation between the possibility of describing quantum correlation with hidden variables and the existence of the Bloch sphere. We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in N spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. We show that the introduction of curved information and the continuity equation of probability are in agreement with classical quantum mechanics. So we give up the hidden variable theory as local theory and we accept the Bloch sphere as global theory connected with the information space.
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:
Meyer [Phys. Rev. Lett. 83, 3751 (1999)] recently showed that the Kochen-Specker theorem, which demonstrates the impossibility of a deterministic hidden variable description of ideal spin measurements on a spin 1 particle, can be effectively nullified if only finite precision measurements are considered. We generalize this result: one can ascribe consistent outcomes to a dense subset of the set of projection valued measurements, or to a dense subset of the set of positive operator valued measurements, on any finite dimensional system. Hence no Kochen-Specker-like contradiction can rule out hidden variable theories indistinguishable from quantum theory by finite precision measurements in either class.
One of the main theorems on the impossibility of hidden variables in quantum mechanics is Kochen-Specker theorem (KS). This theorem says that any hidden variable theory that satisfies quantum mechanics must be contextual. More specifically, it asserts that, in Hilbert space of dimension ≥ 3, it is impossible to associate definite numerical values, 1 or 0, with every projection operator Pm, in such a way that, if a set of commuting Pm satisfies 1=ΣmP, the corresponding values will also satisfy . Since the first proof of Kochen and Specker using 117 vectors in R3, there were many attempts to reduce the number of vector either via conceiving ingenious models or extending the system being considered to higher dimension. By considering eight dimensional three qubits system, we found a state dependent proof that requires only five vectors. The state that we assign value of 1 is the ray that arises from intersection of two planes. The recent advancements show that the KS theorem proof can be extended to two dimensional quantum system through generalized measurement represented by positive operator-valued measured (POVM). In POVMs the number of available outcomes of a measurement may be higher than the dimensionality of the Hilbert space and N-outcome generalized measurement is represented by N-element POVM which consists of N positive semidefinite operators {}dE that sum to identity. Each pair of elements is not mutually orthogonal if the number of outcome of measurements is bigger than the dimensionality. In terms of POVM, Kochen-Specker theorem asserts that and could not be satisfied for . We developed a general model that enables us to generate different sizes of the POVM for the proof of the Kochen-Specker theorem. We show that the current simplest Nakamura model is in fact a special case of our model. W also provide another model which is as simple as the Nakamura’s but consists of different sets of POVM.
We derive inequalities for n-partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than 1∕2 is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An n-partite uncorrelated state violates the n-partite inequality derived here by an amount that grows exponentially with n.