Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

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4. Bell's Inequalities

Bell's inequalities are mathematical constraints that any local realistic theory must satisfy. Quantum mechanics violates these inequalities, demonstrating that nature cannot be both local and realistic. Experimental tests have confirmed quantum predictions, ruling out local hidden variable theories.

Local Realism

Bell's theorem addresses two fundamental assumptions:

1. Realism (Einstein):

Physical properties have definite values independent of measurement. "The moon is there even when nobody looks."

2. Locality (Einstein):

No influence can travel faster than light. Measurement on particle A cannot instantaneously affect particle B at spacelike separation.

Local Hidden Variable (LHV) Theory:

Assumes both realism and locality. Predicts correlations can be explained by shared hidden variables $\lambda$ determined at source.

$$P(a,b|A,B) = \int d\lambda \, \rho(\lambda) P_A(a|A,\lambda) P_B(b|B,\lambda)$$

Factorization reflects locality: outcome at A independent of measurement choice at B

Original Bell Inequality (1964)

Consider spin-1/2 particles in singlet state, measured along directions $\vec{a}$ and $\vec{b}$:

Define correlation function:

$$E(\vec{a}, \vec{b}) = P_{++} + P_{--} - P_{+-} - P_{-+}$$

Where $P_{ij}$ is probability of getting results $i,j \in \{+,-\}$

For three measurement directions $\vec{a}, \vec{b}, \vec{c}$, any LHV theory satisfies:

$$|E(\vec{a},\vec{b}) - E(\vec{a},\vec{c})| \leq 1 + E(\vec{b},\vec{c})$$

Quantum mechanics predicts:

$$E_{QM}(\vec{a},\vec{b}) = -\vec{a} \cdot \vec{b}$$

For appropriate choice of angles, this violates the inequality!

CHSH Inequality

Clauser-Horne-Shimony-Holt (1969) reformulated for realistic experiments:

$$S = |E(a,b) - E(a,b')| + |E(a',b) + E(a',b')| \leq 2$$

For any LHV theory, where $a, a', b, b'$ are measurement settings

Quantum violation:

For optimal measurement angles (e.g., $0°, 45°, 90°, 135°$ for polarization):

$$S_{QM}^{\max} = 2\sqrt{2} \approx 2.828$$

This is Tsirelson's bound - the maximum quantum violation

Experimental setup:

  1. Source emits entangled photon pairs
  2. Alice and Bob randomly choose measurement settings $a/a'$ and $b/b'$
  3. Measure polarization correlations
  4. Compute $S$ from statistics
  5. If $S > 2$, local realism is violated

Experimental Tests

Historical milestones:

  • 1972 - Freedman & Clauser: First test with atomic cascades, violated Bell inequality
  • 1982 - Aspect et al.: Tests with switching analyzers, ruled out local realism with high confidence
  • 2015 - Loophole-free tests: Three independent experiments (Delft, Vienna, NIST) closed all major loopholes simultaneously
  • 2022 - Nobel Prize: Awarded to Aspect, Clauser, and Zeilinger for experimental tests of Bell inequalities

Key loopholes addressed:

  • Detection loophole: Low detector efficiency could allow LHV models; requires $\eta > 82.8\%$
  • Locality loophole: Measurement settings must be spacelike separated from each other and from detection events
  • Freedom-of-choice loophole: Measurement choices must be truly random and independent

Modern experiments achieve $S \approx 2.5$ to $2.7$, strongly confirming quantum predictions

GHZ Theorem

Greenberger-Horne-Zeilinger (1989): sharper contradiction with local realism using three particles

GHZ state:

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$$

Local realism predicts:

\begin{align*} \sigma_x \otimes \sigma_y \otimes \sigma_y &: \quad +1 \\ \sigma_y \otimes \sigma_x \otimes \sigma_y &: \quad +1 \\ \sigma_y \otimes \sigma_y \otimes \sigma_x &: \quad +1 \\ \sigma_x \otimes \sigma_x \otimes \sigma_x &: \quad +1 \end{align*}

(Product of first three must equal fourth)

But quantum mechanics predicts:

$$\sigma_x \otimes \sigma_x \otimes \sigma_x : \quad -1$$

Advantage: Perfect correlation predicted - no statistical analysis needed, just single measurement set can rule out local realism!

Quantum vs. Super-Quantum Correlations

Why doesn't quantum mechanics violate CHSH maximally?

Three regimes:

  • Classical (LHV): $S \leq 2$
  • Quantum: $S \leq 2\sqrt{2}$ (Tsirelson bound)
  • No-signaling: $S \leq 4$ (Popescu-Rohrlich box)

Quantum mechanics sits between classical and maximally non-local theories!

Possible explanations:

  • Information causality principle (Pawłowski et al.)
  • No-signaling + non-trivial communication complexity
  • Macroscopic locality emerges naturally
  • Related to monogamy of entanglement

Device-Independent Protocols

Bell inequality violations enable protocols that don't require trusting devices:

1. Device-Independent Quantum Key Distribution (DIQKD):

Security guaranteed solely by Bell violation, even if devices are built by adversary

2. Device-Independent Randomness Expansion:

Generate certified random numbers from Bell tests

$$H_{\min}(A|E) \geq n \cdot [1 - h(Q)]$$

Minimum entropy guaranteed by CHSH violation

3. Quantum Certification:

Certify quantum devices and computations without detailed knowledge of their operation

Implications and Interpretations

What must we give up?

  1. Abandon local realism: Standard interpretation - nature is fundamentally non-local
  2. Abandon locality: Bohm's theory - maintains realism but allows non-local influences
  3. Abandon realism: Properties don't exist before measurement (Copenhagen-like)
  4. Superdeterminism: Measurement choices not truly free (controversial!)
  5. Many-worlds: Both outcomes occur in different branches (maintains locality and determinism)

Consensus:

Most physicists accept quantum non-locality while noting it cannot be used for superluminal signaling (no-signaling theorem). The correlation is real but cannot transmit information.

Modern Extensions

  • Network non-locality: Bell tests in quantum networks with multiple sources
  • Temporal correlations: Leggett-Garg inequalities for single system over time
  • Continuous variables: Bell tests with position/momentum rather than discrete observables
  • Macroscopic tests: Testing quantum mechanics at larger scales
  • Loopholes: Addressing measurement-independence and other subtle assumptions

Bell's Legacy: John Stewart Bell's 1964 theorem transformed quantum mechanics from philosophy to experimental science. What Einstein saw as proof of incompleteness, experiments revealed as genuine quantum non-locality. Bell's work is the foundation for quantum information science and earned a Nobel Prize for those who experimentally verified it. As Bell said: "The reasonable thing just doesn't work."

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