QCQI - Chapter 3: Multiple Qubits, Entanglement & Measurement

Learning Objectives¶

• Build and recognize entangled vs. product states.

• Partial trace, reduced states, and local vs. global properties.

• Bell states, correlations, and CHSH tests.

Composite States¶

$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$; product if $ad-bc=0$.

Eureka! Rank$(M)=1$ $\Leftrightarrow$ separable (pure two-qubit).

Reduced States¶

$\rho_A=\text{Tr}_B[\rho_{AB}]$, $\rho_B=\text{Tr}_A[\rho_{AB}]$.

For $|\Phi^+\rangle$, $\rho_A=I/2$.

Eureka! Maximally entangled $\Rightarrow$ locally maximally mixed.

Bell Basis & CHSH¶

$|\Phi^\pm\rangle,|\Psi^\pm\rangle$ show strong correlations.

$S=E(A_0,B_0)+E(A_0,B_1)+E(A_1,B_0)-E(A_1,B_1)$.

Classical $|S|\le2$, quantum up to $2\sqrt2$.

Eureka! Entanglement enables $|S|>2$.

Schematic: Bell Preparation¶

Hands-on Notebook¶

Companion notebook (multi-backend): L03_QCQI_Ch03_Multiple_Qubits_Entanglement_and_Measurement

Set backend in the first cell to cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab¶

• Prepare all four Bell states; compute $\rho_A$ and $S(\rho_A)$.

• Estimate CHSH $S$ at different angles; compare with $2\sqrt2$ bound.