whereS=Sa+Sbis the total spin. The Hamiltonian is diagonal on the states|Φ〉, and|T^0 ,±〉, and
has eigenvalues−3 ̄hω/2 and ̄hω/2 respectively. Consider now a state|Ψ(t)〉for which|Ψ(0)〉=
|z+;z−〉, then time evolution under HamiltonianHproduces
|Ψ(t)〉=eiωt/^2 (cos(ωt)|z+;z−〉−isin(ωt)|z−;z+〉) (17.15)The state|Ψ(0)〉was chosen to benon-entangled, but we see that time evolution entangles and
then un-entangles the state. In particular, we may compute the density matrix
ρa(t) = trHb|Ψ(t)〉〈Ψ(t)|
= cos^2 (ωt)|z+〉〈z+|+ sin^2 (ωt)|z−〉〈z−| (17.16)This state is non-entangled whent=kπ/(2ω) for any integerk, and maximally entangled (unpo-
larized) whent=π/(4ω) +kπ/(2ω) for any integerk.
17.3 The Schmidt purification theorem
Consider now, more generally, a full quantum system with Hilbert spaceHab, built out of two
subsystemsaandbwith respective Hilbert spacesHaandHb, so thatHab=Ha⊗Hb. In this
general system, let|Ψ〉be apure stateinHab. The division of the system into the two subsystems
aandballows us to define the two density operators,
ρa ≡ trHb(|Ψ〉〈Ψ|)
ρb ≡ trHa(|Ψ〉〈Ψ|) (17.17)By construction,ρais a density operator for subsystema, whileρbis a density matrix for subsystem
b. The density operatorsρaandρbare not independent. In particular, we always have rank(ρa) =
rank(ρb). In fact, even more precise relations hold betweenρaandρb, which we now exhibit.
The Schmidt purification theorem states that, for any|Ψ〉 ∈ Hab, there exists an orthonormal
set|i,a〉inHaand an orthonormal set|i,b〉inHb, such that
|Ψ〉=∑
i√
pi|i,a〉⊗|i,b〉 (17.18)Notice that the sum is over a common indexi, even though the Hilbert spacesHaandHbneed not
have the same dimension. The numberspiare real and satisfy 0≤pi≤1. Finally, note that the
sets{|i,a〉}iand{|i,b〉}idepend on the state|Ψ〉.
To prove this theorem, we begin by decomposing the pure statein an orthonromal basis|i,a〉
ofHaand|m,b′〉ofHb,
|Ψ〉=∑i,mCim|i,a〉⊗|m,b′〉 (17.19)We choose the basis|i,a〉to be such thatρais diagonal in this basis, and given by the sum,
∑
ipi|i,a〉〈i,a| (17.20)