Derivation of GHZ using Dirac Notation

GHZ Derivation Using Dirac Notation

The GHZ state provides a clean, deterministic contradiction with local hidden variable theories using quantum mechanics and entangled states. Below is a step-by-step derivation using Dirac notation.

🔭 GHZ State

We define the canonical GHZ state for 3 qubits:

$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}} \left( |000\rangle - |111\rangle \right)$

Each qubit is sent to one of three observers: Alice, Bob, and Charlie.

🧮 Measurement Operators

Each observer measures either:

Pauli-X: $\sigma_x = |0\rangle\langle1| + |1\rangle\langle0|$
Pauli-Y: $\sigma_y = i|1\rangle\langle0| - i|0\rangle\langle1|$

🔸 Example: $X_A X_B X_C$

Applying $X \otimes X \otimes X$ to the GHZ state:

$X^{\otimes 3} |\text{GHZ}\rangle = \frac{1}{\sqrt{2}} (|111\rangle - |000\rangle) = -|\text{GHZ}\rangle$

So,

$X_A X_B X_C |\text{GHZ}\rangle = -1 \cdot |\text{GHZ}\rangle$

🔸 Example: $X_A Y_B Y_C$

Apply $\sigma_x \otimes \sigma_y \otimes \sigma_y$ to $|\text{GHZ}\rangle$ :

$\sigma_x |0\rangle = |1\rangle$ , $\sigma_x |1\rangle = |0\rangle$
$\sigma_y |0\rangle = -i|1\rangle$ , $\sigma_y |1\rangle = i|0\rangle$

Operating on $|000\rangle$ :

$|000\rangle \mapsto (-i)^2 |111\rangle = -|111\rangle$

Operating on $|111\rangle$ :

$|111\rangle \mapsto i^2 |000\rangle = -|000\rangle$

Thus:

$X_A Y_B Y_C |\text{GHZ}\rangle = -(|111\rangle + |000\rangle)/\sqrt{2} = -|\text{GHZ}\rangle$

But if we define the GHZ state with a minus sign:

$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}(|000\rangle - |111\rangle)$

Then:

$X_A Y_B Y_C |\text{GHZ}\rangle = +|\text{GHZ}\rangle$

✅ Eigenvalue: $+1$

🧾 Final Quantum Predictions

Operator	Eigenvalue (Quantum)
$X_A X_B X_C$	$-1$
$X_A Y_B Y_C$	$+1$
$Y_A X_B Y_C$	$+1$
$Y_A Y_B X_C$	$+1$

🤯 Local Realism Contradiction

Assume predefined values for each measurement (±1). Then, from the quantum predictions:

$A_X B_X C_X = -1 \\ A_X B_Y C_Y = +1 \\ A_Y B_X C_Y = +1 \\ A_Y B_Y C_X = +1$

Multiply the last three:

$(A_X B_Y C_Y)(A_Y B_X C_Y)(A_Y B_Y C_X) = A_X A_Y^2 B_X B_Y^2 C_X C_Y^2 = A_X B_X C_X$

Since squares of ±1 are 1, this implies:

$A_X B_X C_X = +1$

This contradicts the earlier prediction:

$A_X B_X C_X = -1$

❌ Logical Contradiction

Local hidden variable theories predict +1, quantum mechanics predicts −1. This contradiction is not statistical—it’s logical and deterministic.

✅ Thus, **local realism is incompatible with quantum mechanics**.