# 8.57: The Kochen-Specker Theorem Illustrated Using a Three-Qubit GHZ System

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

In the '90s N. David Mermin published two articles in the general physics literature (Physics Today, June 1990; American Journal of Physics, August 1990) on the Greenberger-Horne-Zeilinger (GHZ) gedanken experiment (American Journal of Physics, December 1990; Nature, 3 February 2000) involving three spin-1/2 particles that illustrated the clash between local realism and the quantum view of reality for the quantum nonspecialist. The purpose of this tutorial is to use the GHZ example to illustrate the Kochen-Specker (KS) theorem by stripping away the use of the three-spin wave function in the analysis of the thought experiment.

The KS theorem asserts that no noncontextual (NC) hidden variable (HV) model (NCHV) can agree with the measurement predictions of quantum theory for Hilbert space dimensions greater than 2. The problem dealt with here is, of course, three-dimensional.

The three spin-1/2 particles are created in a single event and move apart in the horizontal y-z plane. It will be shown that a consideration of spin measurements (in units of h/4π in the x- and y-directions reveals the impossibility of assigning values to the spin observables independent of measurement.

The x- and y-direction spin operators are the Pauli matrices:

$\begin{matrix} \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} & \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \end{matrix} \nonumber$

The eigenvalues of the Pauli matrices are +/- 1:

$\begin{matrix} \text{eigenvals}( \sigma_x) = \begin{pmatrix} 1 \\ -1 \end{pmatrix} & \text{eigenvals}( \sigma_y) = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \end{matrix} \nonumber$

The following operators represent the measurement protocols for spins 1, 2, and 3.

$\begin{matrix} \sigma_x^1 \otimes \sigma_y^2 \otimes \sigma_y^3 & \sigma_y^1 \otimes \sigma_y^2 \otimes \sigma_x^3 & \sigma_x^1 \otimes \sigma_x^2 \otimes \sigma_x^3 \end{matrix} \nonumber$

The tensor matrix product, also known as the Kronecker product, is available in Mathcad. The operators in tensor format are formed as follows.

$\begin{matrix} \sigma_{xyy} \sigma_{yxy} - \sigma_{yxy} \sigma_{xyy} \rightarrow 0 & \sigma_{xyy} \sigma_{yyx} - \sigma_{yyx} \sigma_{xyy} \rightarrow 0 & \sigma_{xyy} \sigma_{xxx} - \sigma_{xxx} \sigma_{xyy} \rightarrow 0 \\ \sigma_{yxy} \sigma_{yyx} - \sigma_{yyx} \sigma_{yxy} \rightarrow 0 & \sigma_{yxy} \sigma_{xxx} - \sigma_{xxx} \sigma_{yxy} \rightarrow 0 & \sigma_{yyx} \sigma_{xxx} - \sigma_{xxx} \sigma_{yyx} \rightarrow 0 \end{matrix} \nonumber$

The next step is to compare the matrix for the product of the first three operators $$(\sigma_{xyy} \sigma_{yxy} \sigma_{yyx})$$ with that of the fourth $$( \sigma_{xxx})$$.

$\begin{matrix} \sigma_{xyy} \sigma_{yxy} \sigma_{yyx} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} & \sigma_{xxx} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \end{matrix} \nonumber$

This indicates the following relationship between the four operators and leads quickly to a refutation of the concept of noncontextual, hidden values for quantum mechanical observables.

$\begin{matrix} \left( \sigma_x^1 \otimes \sigma_y^2 \otimes \sigma_y^3 \right) \left( \sigma_y^1 \otimes \sigma_x^2 \otimes \sigma_y^3 \right) \left( \sigma_y^1 \otimes \sigma_y^2 \otimes \sigma_x^3 \right) = - \left( \sigma_x^1 \otimes \sigma_x^2 \otimes \sigma_x^3 \right) \end{matrix} \nonumber$

Local realism assumes that objects have definite properties independent of measurement. In this example it assumes that the x- and y-components of the spin have definite values prior to measurement. This position leads to a contradiction with the above result. There is no way to assign eigenvalues (+/-1) to the operators that is consistent with the above result.

Concentrating on the operator on the left side, we notice that there is a σy measurement on the first spin in the second and third term. If the spin state is well-defined before measurement those results have to be the same, either both +1 or both -1, so that the product of the two measurements is +1. There is a σy measurement on the second spin in terms one and three. By similar arguments those results will lead to a product of +1 also. Finally there is a σy measurement on the third spin in terms one and two. By similar arguments those results will lead to a product of +1 also. Incorporating these observations into the expression above leads to the following contradiction.

$\begin{matrix} \sigma_x^1 \otimes \sigma_x^2 \otimes \sigma_x^3 = - \sigma_x^1 \otimes \sigma_x^2 \otimes \sigma_x^3 \end{matrix} \nonumber$

A brute force method can be used to confirm this result by showing that the left and right sides of the equation are not equal for any legitimate set of values for the individual spins. This is shown for several such sets below.

$\begin{matrix} x1 = 1 & x2 = 1 & x3 = 1 & y1 = 1 & y2 = 1 & y3 = 1 \end{matrix} \nonumber$

$\begin{matrix} \text{(x1 y2 y3) (y1 x2 y3) (y1 y2 x3) = 1} & - \text{(x1 x2 x3)} = -1 \end{matrix} \nonumber$

$\begin{matrix} x1 = -1 & x2 = 1 & x3 = 1 & y1 = 1 & y2 = -1 & y3 = -1 \end{matrix} \nonumber$

$\begin{matrix} \text{(x1 y2 y3) (y1 x2 y3) (y1 y2 x3) = -1} & - \text{(x1 x2 x3)} = 1 \end{matrix} \nonumber$

$\begin{matrix} x1 = -1 & x2 = -1 & x3 = -1 & y1 = -1 & y2 = -1 & y3 = -1 \end{matrix} \nonumber$

$\begin{matrix} \text{(x1 y2 y3) (y1 x2 y3) (y1 y2 x3)} = -1 & - \text{(x1 x2 x3)} = 1 \end{matrix} \nonumber$

The Kochen-Specker theorem demonstrates that it is, in general, impossible to ascribe to an individual quantum system a definite value for each of a set of observables not all of which necessarily commute. N. David Mermin, Physical Review Letters, 65, 3373 (1990).

This page titled 8.57: The Kochen-Specker Theorem Illustrated Using a Three-Qubit GHZ System is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.