QuantumPhysics.dvi

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)