Explain CHSH inequality in quantum mechanics.

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1. Background

The CHSH inequality is a specific type of Bell inequality, formulated by John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH) in 1969. Bell inequalities test whether correlations between measurements on two systems can be explained by local hidden variable (LHV) theories, i.e., classical physics assumptions, or if they require quantum mechanics.

  • Locality: Measurement outcomes on one particle do not depend on actions performed on a distant particle.

  • Realism: Measurement outcomes are predetermined by hidden variables.

Quantum mechanics can violate these inequalities, showing that nature is non-local in the quantum sense.

2. Setup

Consider two parties, traditionally called Alice and Bob, who share an entangled quantum state. Each of them can choose two possible measurement settings:

  • Alice chooses A0A_0 or A1A_1

  • Bob chooses B0B_0 or B1B_1

Each measurement outcome is ±1.

The CHSH combination is defined as:

S=⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩S = \langle A_0 B_0 \rangle + \langle A_0 B_1 \rangle + \langle A_1 B_0 \rangle - \langle A_1 B_1 \rangle

Where ⟨AiBj⟩\langle A_i B_j \rangle is the expectation value of the product of Alice's and Bob's outcomes.

3. CHSH Inequality

Under local hidden variable theories, the absolute value of SS is bounded:

∣S∣≤2|S| \leq 2

This is the CHSH inequality. If a theory obeys locality and realism, the correlations cannot exceed 2.

4. Quantum Violation

Quantum mechanics allows certain entangled states (like the Bell state) to violate this bound. For example:

  • Consider the singlet state:

∣ψ⟩=12(∣01⟩−∣10⟩)|\psi\rangle = \frac{1}{\sqrt{2}} \left( |01\rangle - |10\rangle \right)

  • If Alice and Bob choose their measurement axes cleverly (at certain angles), quantum mechanics predicts:

∣S∣≤22≈2.828|S| \leq 2\sqrt{2} \approx 2.828

This is the Tsirelson bound, the maximum quantum violation.

5. Implications

  1. Classical vs Quantum: Any value of ∣S∣>2|S| > 2 cannot be explained by classical physics or local hidden variables.

  2. Entanglement Evidence: Experiments measuring CHSH inequality violations confirm quantum entanglement.

  3. Quantum Information: CHSH inequality is foundational for quantum cryptography, device-independent protocols, and randomness generation.

✅ Summary Table

FeatureClassical (LHV)Quantum Mechanics
MaximumS
Theory typeLocal realismNon-local entanglement
Measurement outcomes±1±1
ImplicationSatisfies CHSHCan violate CHSH

If you want, I can draw a diagram showing Alice, Bob, measurement axes, and CHSH correlation, which makes this concept much easier to visualize.

Read More  :

What is Bell’s theorem?

What is quantum annealing vs simulated annealing?

What is the role of entanglement entropy?

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