QuantumPhysics.dvi

17.11Three particle entangled states

Even more drastic violations of hidden variable type theories may be obtained by considering the
decay of an entangled state into three particles. We imaginea system of three spin 1/2 particles,
with associated 8-dimensional Hilbert spaceHabc=Ha⊗Hb⊗Hc. Instead of working with spin
components, we shall work directly with components of the Pauli matricesσai,σbiandσciwhere
i=x,y,z.

A basis for all observables on this Hilbert space is given by 64 matricesσIa⊗σJb⊗σcK, where
Jcan take the valuesx,y,zor 0, andσ^0 =Iis the identity matrix. Of special interest are sets
of observables that mutually commute with one another. The maximal number is 3, excluding the
identity operator itself. For example, one could pick the set

Ia⊗Ib⊗σcz

Ia⊗σzb⊗Ic

σza⊗Ib⊗Ic (17.58)

Any operator commuting with all three must be functionally dependent on these three, as may be
easily verified.

But it is possible to choose a more interesting set of mutually commuting observables that
shows some degree of entanglement. One may also choose,

Σa = σxa⊗σby⊗σcy

Σb = σya⊗σxb⊗σcy

Σc = σya⊗σyb⊗σxc (17.59)

Using the commutation relations for theσiand the fact that the matrix product commutes with
the tensor product, such as for example, (σia⊗σjb)(σi
′
a⊗σ

j′

b) = (σ

i

aσ

i′

a)⊗(σ

j

bσ

j′
b). we easily verify
that these operators mutually commute,

[Σa,Σb] = [Σb,Σc] = [Σc,Σa] = 0 (17.60)

Furthermore, we have

(Σa)^2 = (Σb)^2 = (Σc)^2 =I (17.61)

so that each of these operators has eigenvalues±1. We now define a state|Φ〉which satisfies,

Σa|Φ〉= Σb|Φ〉= Σc|Φ〉= +|Φ〉 (17.62)

The condition becomes easier to solve if translated to the equivalent conditions

ΣaΣb|Φ〉= ΣbΣc|Φ〉= ΣcΣa|Φ〉= +|Φ〉 (17.63)