Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: entangled states and the Bell basis, the EPR argument, the singlet correlation, Bell’s inequality, no signalling and no cloning, and the measures of entanglement. No worked example sits above the answers.

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∣ Φ^{+} ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2}

ρ^_{A} = tr_{B} ρ^_{A B}

S (ρ) = - tr (ρ lo g ρ)

∣ S_{CHSH} ∣ \leq 2

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Intuition

Bring the chapter together: entangled states and the Bell basis, the EPR argument, the singlet correlation, Bell’s inequality, no signalling and no cloning, and the measures of entanglement. No worked example sits above the answers.

A judge weighs two accounts of the same event and finds that no single story fits both. Bell’s inequality is that judgement, made with numbers.

Before you practise

$E(\mathbf{a},\mathbf{b})=-\cos\theta$ for the singlet; local answers keep $|S|\le2$ , quantum mechanics reaches $2\sqrt2$ ; $B$ ’s reduced state ignores everything done to $A$ .

S=E(\mathbf{a},\mathbf{b})+E(\mathbf{a},\mathbf{b}')+E(\mathbf{a}',\mathbf{b})-E(\mathbf{a}',\mathbf{b}')

Reminders

Entangled pure states have mixed parts; the Bell states have completely mixed parts.
Only equal or orthogonal states can be copied by one device.
Pure-state entanglement is the entropy of a part; for two qubits, $C=2|c_{00}c_{11}-c_{01}c_{10}|$ .

Applications

Practice

Try it

The four measured correlations are $-0.6$ , $-0.6$ , $-0.6$ and $+0.6$ , the last for $\mathbf{a}',\mathbf{b}'$ . What is $|S|$ ?