QuantumPhysics.dvi

Next, we takeρ′=ρa⊗ρb, so that

S(ρab) ≤ −kTrHaTrHb

(

ρ(lnρa⊗Ib)⊕ρ(Ia⊗lnρb)

)

≤ −kTrHa

(

ρalnρa

)

−kTrHb

(

ρblnρb

)

(16.74)

which proves 6. Properties 7 and 8 may be proven along the samelines of reasoning, and use of
the Lemma.

16.8 Examples of the use of statistical entropy

We illustrate the property ofsubadditivity, in 6, by considering a pure state of the full system, and
then viewing this pure state from the vantage point of a bi-partite subsystem. Thus we view the
full HamiltonianHab=Ha⊗Hb, and construct the density operator

ρab = |ψ〉⊗〈ψ|

|ψ〉 =

∑

α,β

Cαβ|φα;a〉⊗|φβ;b〉 (16.75)

Here,|φα;a〉is an orthonormal basis ofHaand|φβ;b〉is an orthonormal basis ofHb. Sinceρab
corresponds to a pure state, we clearly haveS(ρab) = 0. By subadditivity, however, we only have
and inequality forS(ρa) andS(ρb),

0 ≤S(ρa) +S(ρb) (16.76)

and neither quantity, in general, should be expected to haveto vanish. Physically, this result may
be interpreted as follows. Even though the full system is in apure quantum state, the fact that we
“average” over subspaceHbto getρa(and “average” over subspaceHato getρb) means that the
quantum information contained inHbhas been lost and this translates into a non-zero value of the
statistical entropy.

It is instructive to work out the corresponding density operators, and check that they do not,
in general, correspond to pure states. We have,

ρa = TrHb(ρab) =

∑

β

w(βa)|ψβ;a〉⊗〈ψβ;a|

ρb = TrHa(ρab) =

∑

α

w(αb)|ψα;b〉⊗〈ψα;b| (16.77)

Here, the weightsw(βa) andw(αb) are positive and less or equal to 1, and|ψβ;a〉, and|ψα;b〉are
normalized pure states, defined by

(

w(βa)

) (^12)
|ψβ;a〉=
∑
α
Cαβ|φα;a〉 w(βa)=
∑
α
|Cαβ|^2
(
w(αb)
)^12
|ψα;b〉=
∑
β
Cαβ|φβ;b〉 w(αb)=
∑
β
|Cαβ|^2 (16.78)