8.21: Quantum v. Realism

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This tutorial demonstrates the conflict between quantum theory and realism in an experiment described in Physical Review Letters on October 31, 2003 by Zhao, et al. titled ʺExperimental Violation of Local Realism by Four‐Photon GHZ Entanglement.ʺ It draws on the methodology outlined by N. David Mermin in two articles in the general physics literature: Physics Today, June 1990; American Journal of Physics, August 1990.

The experiment involves the measurement of the diagonal and circular polarization states of a four‐photon entangled state using the following measurement protocols.

\[ \begin{matrix} \sigma_d^1 \otimes \sigma_d^2 \otimes \sigma_d^3 \otimes \sigma_d^4 & \sigma_d^1 \otimes \sigma_c^2 \otimes \sigma_d^3 \otimes \sigma_c^4 & \sigma_d^1 \otimes \sigma_d^2 \otimes \sigma_c^3 \otimes \sigma_c^4 & \sigma_d^1 \otimes \sigma_c^2 \otimes \sigma_c^3 \otimes \sigma_d^4 \end{matrix} \nonumber \]

The individual polarization operators and their eigenvalues are:

\[ \begin{matrix} D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} & \text{eigenvals}(D) = \begin{pmatrix} 1 \\ -1 \end{pmatrix} & C = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} & \text{eigenvals}(C) = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \end{matrix} \nonumber \]

The composite operators are formed by tensor matrix multiplication, where kronecker is Mathcadʹs command for tensor multiplication.

\[ \begin{matrix} \sigma_{dddd} = \text{kronecker(D, kronecker(D, kronecker(D, D)))} & \sigma_{dcdc} = \text{kronecker(D, kronecker(C, kronecker(D, C)))} \\ \sigma_{ddcc} = \text{kronecker(D, kronecker(D, kronecker(C, C)))} & \sigma_{dccd} = \text{kronecker(D, kronecker(C, kronecker(C, D)))} \end{matrix} \nonumber \]

These operators commute with each other allowing them to have simultaneous eigenvalues.

\[ \begin{matrix} \sigma_{dddd} \sigma_{dcdc} - \sigma_{dcdc} \sigma_{dddd} \rightarrow 0 & \sigma_{dddd} \sigma_{dccd} - \sigma_{dccd} \sigma_{dddd} \rightarrow 0 & \sigma_{dddd} \sigma_{ddcc} - \sigma_{ddcc} \sigma_{dddd} \rightarrow 0 \\ \sigma_{ddcc} \sigma_{dcdc} - \sigma_{dcdc} \sigma_{ddcc} \rightarrow 0 & \sigma_{ddcc} \sigma_{dccd} - \sigma_{dccd} \sigma_{ddcc} \rightarrow 0 & \sigma_{dcdc} \sigma_{dccd} - \sigma_{dccd} \sigma_{dcdc} \rightarrow 0 \end{matrix} \nonumber \]

Next we show that the null matrix results when the fourth operator is added to the product of the first three.

\[ \sigma_{dddd} \sigma_{dcdc} \sigma_{ddcc} + \sigma_{dccd} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \nonumber \]

To facilitate further analysis, the null result is written as follows.

\[ \sigma_{dddd} \sigma_{dcdc} \sigma_{ddcc} = - \sigma_{dccd} \nonumber \]

Realism maintains that objects have values for observable properties that exist prior to measurement and independent of the choice of measurement (noncontextual). If this assumption is valid, then the operators highlighted with the same color must have the same eigenvalues (+1 or ‐1) and therefore the product of their eigenvalues must be unity.

\[ \left( \textcolor{red}{ \sigma_d^1} \otimes \textcolor{blue}{ \sigma_d^2} \otimes \textcolor{green}{ \sigma_d^3} \otimes \sigma_d^4 \right) \left( \textcolor{red}{ \sigma_d^1} \otimes \sigma_d^2 \otimes \textcolor{green}{ \sigma_d^3} \otimes \textcolor{yellow}{ \sigma_d^4} \right) \left( \sigma_d^1 \otimes \textcolor{blue}{ \sigma_d^2} \otimes \sigma_d^3 \otimes \textcolor{yellow}{ \sigma_d^4} \right) = - \left( \sigma_d^1 \sigma_c^2 \sigma_c^3 \sigma_d^4 \right) \nonumber \]

Thus, applying a classical concept (noncontextual realism) to the above quantum mechanical equation leads to the following contradiction.

\[ \sigma_d^1 \otimes \sigma_c^2 \otimes \sigma_c^3 \otimes \sigma_d^4 = - \sigma_d^1 \otimes \sigma_c^2 \otimes \sigma_c^3 \otimes \sigma_d^4 \nonumber \]

The experimental results reported by Zhao, et al. validate the quantum mechanical analysis, and contradict the realistic interpretation. See the preceding tutorial for a summary of their experimental results.