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arXiv:quant-ph/0210082 · Physical Review Letters

Kochen-Specker Theorem for a Single Qubit using Positive Operator-Valued Measures

Adán Cabello

Published May 12, 2003• 62 citations
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Abstract

A proof of the Kochen-Specker theorem for a single two-level system is presented. It employs five eight-element positive operator-valued measures and a simple algebraic reasoning based on the geometry of the dodecahedron.
DOI
10.1103/physrevlett.90.190401

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Kochen-Specker theorem for a single qubit

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A proof of the Kochen-Specker theorem for a single two-level system is presented. It employs five eight-element POVMs and a simple algebraic reasoning based on the geometry of the dodecahedron.

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Kochen-Specker Theorem for a Single Qubit

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The Kochen-Specker theorem (KS) states that Ilonconrextual hidden variable theories are incompatible with quantum mechanics. Since the first proof of the theorem, von Neumann projection measurement has been used and the quantum system considered has dimensionality of at least three. However, recently generalized measurements represented by positive-operator-va!ued measures (POVM) have been applied to extend the proof to two dimensional quantum systems. This note gives numerical calculations on the first KS proof for a single qubit.

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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.
The Kochen-Specker theorem (KS) states that Ilonconrextual hidden variable theories are incompatible with quantum mechanics. Since the first proof of the theorem, von Neumann projection measurement has been used and the quantum system considered has dimensionality of at least three. However, recently generalized measurements represented by positive-operator-va!ued measures (POVM) have been applied to extend the proof to two dimensional quantum systems. This note gives numerical calculations on the first KS proof for a single qubit.
Cabello and Nakamura have shown [A. Cabello, Phys. Rev. Lett. 90, 190401 (2003)10.1103/PhysRevLett.90.190401] that the Kochen-Specker theorem can be applied to two-dimensional systems if one uses positive operator-valued measures (POVM). We show that the contextuality in their models is not of the Kochen-Specker type, but it is rather a result of not keeping track of the whole system on which the measurement is performed. This is connected to the fact that there is no one-to-one correspondence between the POVM elements and projectors on the extended Hilbert space, and the same POVM element has to originate from two different projectors when used in Cabello-Nakamura models. Moreover, we propose a hidden-variable formulation of the above models.