The general density matrixρabof the combined system may be parametrized as follows,
ρab=ρa⊗ρb+
∑^3
i,j=1
Cijσia⊗σbj (16.55)
whereCijis an arbitrary 3×3 real matrix. For any choice ofCij, we have
trHa(ρab) = ρb
trHb(ρab) = ρa (16.56)
so that the reduced density matricesρa andρbare independent ofCij. Only forCij= 0 is the
density matrixρabthe tensor product ofρaandρb.
16.7.3 Lemma
For any two density operatorρandρ′in the same Hilbert spaceH, we have the inequality,
S(ρ)≤−kTr
(
ρlnρ′
)
(16.57)
with equality being attained iffρ′=ρ.
To prove this, we write both density matrices in diagonal form, in orthonormal bases,
ρ =
∑
i
|φi〉pi〈φi|
ρ′ =
∑
i
|φ′i〉p′i〈φ′i| (16.58)
The bases|φi〉and|φ′i〉are in general different. We now consider
S(ρ) +kTr
(
ρlnρ′
)
= −k
∑
i
pilnpi+k
∑
i,j
pilnp′j|〈φi|φ′j〉|^2
= k
∑
i,j
pi(lnp′j−lnpi)|〈φi|φ′j〉|^2 (16.59)
where we have used the completeness relation
∑
j|φ
′
j〉〈φ
′
j|=Iin recasting the simple sum on the
first line in the form of a double sum on the last line. Next, we introduce the function
f(x)≡x− 1 −lnx (16.60)
We setx=p′j/piand recast the above double sum in the following way,
S(ρ) +kTr
(
ρlnρ′
)
=k
∑
i,j
pi
(
1 −
p′j
pi
−f
(
p′j
pi
))
|〈φi|φ′j〉|^2 (16.61)