Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure

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 i...

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Main Author:	Toh, Sing Poh
Format:	Thesis
Language:	English English
Published:	2008
Subjects:	Algorithms Mathematical analysis
Online Access:	http://psasir.upm.edu.my/id/eprint/5419/1/IPM_2008_3a.pdf
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spelling	my-upm-ir.54192013-05-27T07:22:41Z Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure 2008 Toh, Sing Poh 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. Algorithms Mathematical analysis 2008 Thesis http://psasir.upm.edu.my/id/eprint/5419/ http://psasir.upm.edu.my/id/eprint/5419/1/IPM_2008_3a.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Algorithms Mathematical analysis Institute Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Algorithms Mathematical analysis
spellingShingle	Algorithms Mathematical analysis Toh, Sing Poh Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
description	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.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Toh, Sing Poh
author_facet	Toh, Sing Poh
author_sort	Toh, Sing Poh
title	Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
title_short	Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
title_full	Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
title_fullStr	Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
title_full_unstemmed	Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure
title_sort	proving kochen-specker theorem using projection measurement and positive operator-valued measure
granting_institution	Universiti Putra Malaysia
granting_department	Institute Mathematical Research
publishDate	2008
url	http://psasir.upm.edu.my/id/eprint/5419/1/IPM_2008_3a.pdf
_version_	1747810420912029696

Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure

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