QuantumPhysics.dvi

(Wang) #1

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)
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