QCQI – Chapter 3 Detailed Summary: Multiple Qubits, Entanglement & Measurement - Quantum Computing & Quantum Information

Learning Objectives¶

Build multi-qubit states using tensor products and identify product vs. entangled states.
Compute reduced states via partial trace and reason about local vs. global properties.
Prepare the Bell basis, evaluate correlations, and test the CHSH inequality.
Understand projective measurements on multi-qubit systems and readout correlations.

Tensor Products & Composite Systems¶

For systems $A$ and $B$ , the joint space is $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ . A general pure two-qubit state is $|\psi\rangle = a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle, \quad |a|^2+|b|^2+|c|^2+|d|^2=1.$

Product vs. entangled. $|\psi\rangle$ is a product state iff the $2\times2$ matrix $M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ has rank 1 (i.e., $ad-bc=0$ ).

Eureka! Entanglement is basis-independent: if $ad-bc\neq 0$ (pure two-qubit case), no single-qubit basis change can factor the state.

Reduced States & Partial Trace¶

For $\rho_{AB}=|\psi\rangle\!\langle \psi|$ (or mixed), the reduced states are $\rho_A=\operatorname{Tr}_B[\rho_{AB}],\qquad \rho_B=\operatorname{Tr}_A[\rho_{AB}].$ For the Bell state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ , one finds $\rho_A=\rho_B=I/2$ .

Eureka! “Locally maximally mixed, globally pure” is a hallmark of bipartite entanglement.

Bell Basis & Correlations¶

The Bell states are $|\Phi^\pm\rangle=\tfrac{1}{\sqrt2}(|00\rangle\pm|11\rangle),\quad |\Psi^\pm\rangle=\tfrac{1}{\sqrt2}(|01\rangle\pm|10\rangle).$ They exhibit perfect (anti-)correlations along matching axes.

CHSH Inequality (Sketch)¶

Choose two measurement settings per party: $A_0,A_1$ for Alice and $B_0,B_1$ for Bob with outcomes $\pm1$ . Define correlators $E(A_i,B_j)=\langle A_i\otimes B_j\rangle$ . The CHSH quantity is $S=E(A_0,B_0)+E(A_0,B_1)+E(A_1,B_0)-E(A_1,B_1).$ Local hidden variable (LHV) theories obey $|S|\le 2$ . Quantum mechanics attains $|S|\le 2\sqrt2$ (Tsirelson).

Eureka! Bell violations are operational signatures of nonclassicality; entanglement enables $|S|>2$ with suitable settings.

Projective Measurements on Two Qubits¶

A computational-basis measurement applies $P_{ab}=|ab\rangle\!\langle ab|$ on $AB$ . Joint probabilities are $p(ab)=\langle \psi|P_{ab}|\psi\rangle$ ; classical correlations emerge from the joint distribution.

Worked Examples¶

Bell preparation. Apply $H$ on qubit $A$ and CNOT $_{A\to B}$ to $|00\rangle$ to get $|\Phi^+\rangle$ .

Concurrence (pure two-qubit). $C=2|ad-bc|$ . $C=1$ for Bell states; $C=0$ for product states.

Entanglement entropy (pure). $S(\rho_A)=S(\rho_B)$ ; for Bell states $S=1$ (bits).

Pseudo-code Aids¶

Pseudocode — Partial trace of a 2-qubit state (trace out $B$ )

Input: $4\times 4$ density matrix $\rho_{AB}$ (basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ ) Output: $2\times 2$ reduced state $\rho_A$ Partition $\rho_{AB}$ into four $2\times2$ blocks $\begin{pmatrix}R_{00}&R_{01}\\ R_{10}&R_{11}\end{pmatrix}$ Return $\rho_A = R_{00}+R_{11}$

Pseudocode — CHSH correlator from sampled outcomes

Input: settings $A_i,B_j\in\{\pm1\}$ implemented by basis-rotations; shots $S$ For each shot: sample outcomes $a,b\in\{\pm1\}$ ; accumulate $ab$ Return $E(A_i,B_j)=\frac{1}{S}\sum_{s=1}^S a_sb_s$

Pseudocode — Bell-state preparation (H+CNOT)

Input: start in $|00\rangle$ Apply $H$ on qubit $A$ ; apply CNOT with control $A$ and target $B$ Output: $|\Phi^+\rangle$

Schematics¶

Bell Preparation¶

CHSH Layout (conceptual)¶

Hands-on Notebook (Multi-Backend)¶

Run: L03_QCQI_Ch03_Multiple_Qubits_Entanglement_and_Measurement

In the first cell set backend to one of: cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab Ideas¶

Prepare each Bell state and measure $\rho_A$ to verify maximal mixedness.
Estimate CHSH $S$ for several angle pairs; approach $2\sqrt2$ for optimal settings.
Add depolarizing noise and observe degradation of $S$ and entanglement entropy.