8.15: GHZ Math Appendix

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This appendix shows another way of "doing the math" in the GHZ experiment.

\[ \Psi_{yyx} = \frac{1}{ \sqrt{2}} (H_1 H_2 H_3 + V_1 V_2 V_3)~ \begin{array}{|l} \text{substitute,}~ H_1 = \frac{1}{ \sqrt{2}} (R_1 + L_1) \\ \text{substitute,}~ H_2 = \frac{1}{ \sqrt{2}} (R_2 + L_2) \\ \text{substitute,}~ H_3 = \frac{1}{ \sqrt{2}} (H'_3 + V'_3) \\ \text{substitute,}~ V_1 = \frac{i}{ \sqrt{2}} (L_1 - R_1) \\ \text{substitute,}~ V_2 = \frac{i}{ \sqrt{2}} (L_2 - R_2) \\ \text{substitute,}~ V_3 = \frac{1}{ \sqrt{2}} (H'_3 - V'_3) \\ \text{simplify} \end{array} \rightarrow \Psi_{yyx} = \frac{1}{2} R_1 R_2 V'_3 + \frac{1}{2} R_1 L_2 H'_3 + \frac{1}{2} L_1 R_2 H'_3 + \frac{1}{2} L_1 L_2 V'_3 \nonumber \]

\[ \Psi_{yxy} = \frac{1}{ \sqrt{2}} (H_1 H_2 H_3 + V_1 V_2 V_3)~ \begin{array}{|l} \text{substitute,}~ H_1 = \frac{R_1 + L_1}{ \sqrt{2}} \\ \text{substitute,}~ H_2 = \frac{(H'_2 + V'_2)}{ \sqrt{2}} \\ \text{substitute,}~ H_3 = \frac{R_3 + L_3}{ \sqrt{2}} \\ \text{substitute,}~ V_1 = \frac{i}{ \sqrt{2}} (L_1 - R_1) \\ \text{substitute,}~ V_2 = \frac{H'_2 - V'_2}{ \sqrt{2}} \\ \text{substitute,}~ V_3 = \frac{i (L_3 - R_3)}{ \sqrt{2}} \\ \text{simplify} \end{array} \rightarrow \Psi_{yxy} = \frac{1}{2} R_1 H'_2 L_3 + \frac{1}{2} R_1 V'_2 R_3 + \frac{1}{2} L_1 H'_2 R_3 + \frac{1}{2} L_1 V'_2 L_3 \nonumber \]

\[ \Psi_{xyy} = \frac{1}{ \sqrt{2}} (H_1 H_2 H_3 + V_1 V_2 V_3)~ \begin{array}{|l} \text{substitute,}~ H_1 = \frac{H'_1 + V'_1}{ \sqrt{2}} \\ \text{substitute,}~ H_2 = \frac{R_2 + L_2}{ \sqrt{2}} \\ \text{substitute,}~ H_3 = \frac{R_3 + L_3}{ \sqrt{2}} \\ \text{substitute,}~ V_1 = \frac{H'_1 - V'_1}{ \sqrt{2}} \\ \text{substitute,}~ V_2 = \frac{i (L_2 - R_2)}{ \sqrt{2}} \\ \text{substitute,}~ V_3 = \frac{i (L_3 - R_3}{ \sqrt{2}} \\ \text{simplify} \end{array} \rightarrow \Psi_{xyy} = \frac{1}{2} H'_1 R_2 L_3 + \frac{1}{2} H'_1 L_2 R_3 + \frac{1}{2} V'_1 R_2 R_3 + \frac{1}{2} V'_1 L_2 L_3 \nonumber \]

\[ \Psi_{xxx} = \frac{1}{ \sqrt{2}} (H_1 H_2 H_3 + V_1 V_2 V_3)~ \begin{array}{|l} \text{substitute,}~ H_1 = \frac{H'_1 + V'_1}{ \sqrt{2}} \\ \text{substitute,}~ H_2 = \frac{H'_2 + V'_2}{ \sqrt{2}} \\ \text{substitute,}~ H_3 = \frac{H'_3 + V'_3}{ \sqrt{2}} \\ \text{substitute,}~ V_1 = \frac{H'_1 - V'_1}{ \sqrt{2}} \\ \text{substitute,}~ V_2 = \frac{H'_2 - V'_2)}{ \sqrt{2}} \\ \text{substitute,}~ V_3 = \frac{H'_3 - V'_3}{ \sqrt{2}} \\ \text{simplify} \end{array} \rightarrow \Psi_{xxx} = \frac{1}{2} H'_1 H'_2 H'_3 + \frac{1}{2} H'_1 V'_2 V'_3 + \frac{1}{2} V'_1 H'_2 V'_3 + \frac{1}{2} V'_1 V'_2 H'_3 \nonumber \]