Gleason’s Theorem Explained Using Single-Particle and Two-Particle Systems

Read this post on Projection Operators first.

1. What Is Gleason’s Theorem?

Gleason’s theorem states that in a Hilbert space of dimension d \geq 3, the only valid probability measure for quantum measurements must follow the Born rule:

    \[ P(E) = \text{Tr}(\rho E) \]

where:

  • P(E) is the probability of measuring outcome E.
  • E is a **projection operator** representing a measurement.
  • \rho is the **density matrix** of the quantum state.

2. Single-Particle Spin Measurement

Consider a spin-1/2 particle (like an electron) measured along the z-axis.

Spin Observable S_z

The spin operator is:

    \[ S_z = \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]

The possible measured values (eigenvalues) are:

  • +\hbar/2 (Spin up, |+\rangle)
  • -\hbar/2 (Spin down, |-\rangle)

Projection Operators

Each measurement outcome corresponds to a projection operator:

    \[ P_+ = |+\rangle \langle +| = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \]

    \[ P_- = |-\rangle \langle -| = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \]

Measurement Probabilities

If the quantum state is |\psi\rangle = \alpha |+\rangle + \beta |-\rangle, the measurement probabilities are:

    \[ P(+\hbar/2) = \langle \psi | P_+ | \psi \rangle = |\alpha|^2 \]

    \[ P(-\hbar/2) = \langle \psi | P_- | \psi \rangle = |\beta|^2 \]

Single-Particle Measurement Diagram

       Spin Measurement Device (Stern-Gerlach)
                      |
      ↑ ( +ℏ/2 )      |       ↓ ( -ℏ/2 )
  --------------------->--------------------
        |ψ⟩ = α|+⟩ + β|−⟩

3. Two-Particle Entangled State

Consider two spin-1/2 particles in the Bell state:

    \[ |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|+\rangle_A |+\rangle_B + |-\rangle_A |-\rangle_B) \]

Observable: Total Spin Along z-Axis

The total spin operator is:

    \[ S_z^{\text{total}} = S_z^A + S_z^B \]

The possible measured values are:

  • +\hbar (both particles spin up)
  • -\hbar (both particles spin down)
  • 0 (one up, one down)

Measurement Probabilities

For the Bell state |\Phi^+\rangle, Gleason’s theorem ensures that the measurement outcomes must obey:

    \[ P(+\hbar) = \langle \Phi^+ | P_{+\hbar} | \Phi^+ \rangle = \frac{1}{2} \]

    \[ P(-\hbar) = \langle \Phi^+ | P_{-\hbar} | \Phi^+ \rangle = \frac{1}{2} \]

    \[ P(0) = \langle \Phi^+ | P_0 | \Phi^+ \rangle = 0 \]

Entanglement Measurement Diagram

        Particle A                        Particle B
       -----------                      -----------
       |  +⟩   -⟩ |                      |  +⟩   -⟩ |
       ------------------                 ------------------
                 |  Bell State: |Φ+⟩ = 1/√2 (|+⟩|+⟩ + |−⟩|−⟩)
                 |  
                 |  If A is measured as +ℏ/2, then B must be +ℏ/2.
                 |  If A is measured as -ℏ/2, then B must be -ℏ/2.

4. Conclusion

  • Gleason’s theorem proves that quantum measurement probabilities must follow the Born rule.
  • Any attempt to assign classical probabilities to measurement outcomes contradicts the additivity condition.
  • This rules out non-contextual hidden-variable theories and reinforces the fundamental role of quantum uncertainty.