This lesson introduces quantum entanglement through Bell states and shows how strong non-classical correlations emerge in multi-qubit systems.
Module 10 explains entanglement by constructing and analyzing Bell states. It shows how correlated quantum systems exhibit behavior that cannot be described by independent classical components.
Understand what entanglement means operationally, learn how Bell states are built in circuits, and interpret the measurement correlations that reveal genuinely quantum relationships.
Learners gain a concrete foundation in entanglement, stronger intuition for correlated quantum systems, and a clear pathway into more advanced quantum communication and algorithm topics.
Quantum entanglement is a phenomenon where multiple qubits become so strongly correlated that one cannot be fully described without the others.
$$|\psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$$
This state is factorizable:
$\frac{1}{\sqrt{2}}(|0\rangle_A + |1\rangle_A) \otimes \frac{1}{\sqrt{2}}(|0\rangle_B + |1\rangle_B)$
Measuring qubit $A$ does not force a unique value of qubit $B$.
$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$
This cannot be factored into independent single-qubit states. Measuring qubit $A$ immediately constrains qubit $B$ to matching correlated outcomes.
| System Type | Measured State of Qubit A | Resulting Subspace of Qubit B | Global Configuration Behavior |
|---|---|---|---|
| Product state | $|0\rangle$ or $|1\rangle$ | Can be either $|0\rangle$ or $|1\rangle$ | Independent collapse branches |
| Bell state $|\Phi^+\rangle$ | $|0\rangle$ | Forced to $|0\rangle$ | Strict correlation |
| Bell state $|\Phi^+\rangle$ | $|1\rangle$ | Forced to $|1\rangle$ | Strict correlation |
$$|\psi_1\rangle = (H \otimes I)|00\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)$$
$$|\psi_1\rangle = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \\ 0 \end{bmatrix}$$
$$|\psi_2\rangle = CNOT|\psi_1\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$
$$|\psi_2\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \\ 0 \end{bmatrix} = \begin{bmatrix} 1/\sqrt{2} \\ 0 \\ 0 \\ 1/\sqrt{2} \end{bmatrix}$$
| Circuit Input State | Bell State Notation | Expanded Dirac Vector | Final Column Vector | Readout Correlation Behavior |
|---|---|---|---|---|
| $|00\rangle$ | $|B_{00}\rangle$ | $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ | $\begin{bmatrix}1/\sqrt{2}\\0\\0\\1/\sqrt{2}\end{bmatrix}$ | Perfectly correlated bits |
| $|01\rangle$ | $|B_{01}\rangle$ | $\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ | $\begin{bmatrix}0\\1/\sqrt{2}\\1/\sqrt{2}\\0\end{bmatrix}$ | Odd-parity entangled structure |
| $|10\rangle$ | $|B_{10}\rangle$ | $\frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ | $\begin{bmatrix}1/\sqrt{2}\\0\\0\\-1/\sqrt{2}\end{bmatrix}$ | Even-parity entangled structure with relative phase inversion |
| $|11\rangle$ | $|B_{11}\rangle$ | $\frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ | $\begin{bmatrix}0\\1/\sqrt{2}\\-1/\sqrt{2}\\0\end{bmatrix}$ | Odd-parity entangled structure with relative phase inversion |
Operational note: These four Bell states form a complete orthonormal basis for two-qubit Hilbert space and are central to quantum communication and error-correction workflows.