so that the probability for measuring the eigenvalues± ̄h/2 are actually equal to one another. This
is not really surprising, since both eigenstates±of the electronaentered into the state|Φ〉. In
fact, the expectation value ofna·Sain the state|Φ〉vanishes for anyna,
〈Φ|(
na·Sa)
|Φ〉= 0 (17.10)This is an immediate result of the rotation invariance of thestate|Φ〉.
Therefore, the probabilities for measuring the eigenvalues ± ̄h/2 forna·Saare equal to one
another, for any orientationna. This behavior of the observableSza, from the viewpoint of the
subsystema, is unlike that of any pure state ofa, but in fact precisely coincides with the behavior
of an unpolarized state ofa. We see that by summing, or “averaging”, over the states in the
subsystemb, the full system reduces to subsystema, and the state|Φ〉, which is a pure state of
the full systemHab, reduces to a mixed state of the subsystema. Mathematically, this may be
expressed as follow,
〈Φ|Saz|Φ〉 = trHab(
|Φ〉〈Φ|Sza)= trHatrHb(
|Φ〉〈Φ|Sza)= trHa(
ρaSaz)
(17.11)where we have defined
ρa≡trHb(
|Φ〉〈Φ|)
(17.12)This quantity may be computed explicitly, using the form of|Φ〉, and we get
ρa =1
2
trHb(
|z+;z−〉−|z−;z+〉)(
<z+;z−|−〈z−;z+|)=
1
2
(
|z+〉〈z+|+|z−〉〈z−|)
=1
2
Ia (17.13)The valueρa=Ia/2 indeed confirms that subsystemaappears in an unpolarized state. From the
point of view of subsystema, the state|Φ〉isentangledwith subsystemb
17.2 Entangled states from non-entangled states
Entangled states naturally appear in the time evolution of asystem that was originally in a non-
entangled state. This may be illustrated concretely in the above two spin 1/2 system as well. Let
the time evolution be determined by the simplest non-trivial Hamiltonian,
H=
2 ω
̄h
Sa·Sb=
ω
̄h(
S^2 −3
2
̄h^2)
(17.14)