Gleason’s Theorem and Hidden Variables

1. Projection Operators

A projection operator P is a Hermitian operator satisfying:

P² = P

These operators represent measurement outcomes in quantum mechanics. If a system is in state |\psi\rangle, the probability of measuring an outcome associated with projection P is:

P(\text{outcome}) = \langle \psi | P | \psi \rangle

2. Additivity Assumption in Quantum Mechanics

Ballentine references an additivity assumption for commuting observables:

⟨A + B⟩ = ⟨A⟩ + ⟨B⟩

This assumption is valid for classical probability but has deeper implications in quantum mechanics.

3. Gleason’s Theorem (1957)

Gleason’s theorem states that in any Hilbert space of dimension ≥3, the only probability measure that satisfies the additivity assumption must follow the Born rule:

P(E) = \text{Tr}(\rho E)

This implies that hidden-variable models must reproduce quantum probabilities exactly.

4. Dispersion-Free States and Their Impossibility

A dispersion-free state is a hypothetical state in which every observable has a definite pre-determined value (0 or 1 for projection operators). Gleason’s theorem shows that:

  • If every projection operator had a unique 0 or 1 value, the additivity assumption would be violated.
  • This means no hidden-variable theory can assign definite values to all quantum observables in a way that is consistent with quantum probability rules.

5. Bell’s Response and Contextuality

John Bell (1966) argued that:

  • Hidden-variable theories could still exist if measurement outcomes depended on the whole experimental arrangement.
  • This aligns with Bohr’s (1949) view that measurement depends on context.

6. Criticism of Gleason’s Assumptions

Ballentine criticizes Gleason’s theorem for assuming that every projection operator represents an observable, which is unrealistic. Some operators (e.g., x^2 p_x z x^2) may not correspond to real measurable quantities.

7. Conclusion

  • Gleason’s theorem proves that quantum probability must follow the Born rule.
  • Dispersion-free states (deterministic hidden-variable states) violate the additivity condition.
  • Bell’s rebuttal suggests that contextual hidden-variable models might still be possible.