FOUNDATIONS OF QUANTUM MECHANICS

VII. 6. AN ALGEBRAIC PROOF WITHOUT INEQUALITIES 159
Consider a composite system of three spin 1/2 fermions with pure states in the direct product
Hilbert space C 2 ⊗ C 2 ⊗ C 2 = C 8 . We look at 10 physical quantities which correspond to the
spin operators represented in the Mermin pentagon, figure VII. 8. In this diagram σy
1 is shorthand
for σ y (1) ⊗ 11 (2) ⊗ 11 (3), and σy 1 σy 2 σx 3 is likewise for σ y (1) ⊗ σ y (2) ⊗ σ x (3), etc. On every
straight line through the Mermin pentagon we find four commuting operators. These operators are
products of commuting operators with eigenvalues ±1 and therefore have eigenvalues ±1 also.
σ 1 y
σ 1 x σ 2 x σ 3 x σ 1 y σ 2 y σ 3 x σ 1 y σ 2 x σ 3 y σ 1 x σ 2 y σ 3 y
σ 3 x
σ 3 y
σ 1 x
σ 2 y
σ 2 x
Figure VII. 8: The Mermin pentagon
Using the properties of the Pauli matrices (III. 122), p. 66, it can be shown that
(
σx (1) ⊗ σ y (2) ⊗ σ y (3) ) ( σ y (1) ⊗ σ x (2) ⊗ σ y (3) ) ( σ y (1) ⊗ σ y (2) ⊗ σ x (3) )
= − σ x (1) ⊗ σ x (2) ⊗ σ x (3), (VII. 77)
where we note that the four operators acting in C 8 commute. Consequently, they have a simultaneous
eigenstate in C 8 , having eigenvalue +1 for the three operators on the left - hand side of the equation,
and eigenvalue −1 for the operator on the right - hand side. The entangled state in C 8 ,
|GHZ⟩ := 1 2
√
2
(
|z ↑⟩ ⊗ |z ↑⟩ ⊗ |z ↑⟩ − |z ↓⟩ ⊗ |z ↓⟩ ⊗ |z ↓⟩
)
, (VII. 78)
is such a state.
We assume that the three particles are already far away from each other and are moving still further
apart, and the composite system is, as far as spin is concerned, in the state |GHZ⟩. A measurement of
two particles, of which we assume that it does not influence the third particle in any way, determines
the value of the third particle because, according to quantum mechanics, the product of the outcomes
of measurement is determined.