QuantumPhysics.dvi

The first two terms on the right hand side simplify as follows,
∑

i,j

(pi−p′j)|〈φi|φ′j〉|^2 =

∑

i

pi−

∑

j′

pj′= 0 (16.62)

in view of the completeness relations of the both orthonormal bases. This gives us an expression
directly in terms of the functionf,

S(ρ) +kTr

(

ρlnρ′

)

=−k

∑

i,j

pif

(

p′j

pi

)

|〈φi|φ′j〉|^2 (16.63)

To derive the lemma, we use the following simple properties of the functionf(x),

• f(1) = 0;


• f(x)>0 for allx≥0 andx 6 = 1.

The first property is obvious, the second may be established from the first by noticing thatf′(x)< 0
for 0≤x <1, andf′(x)>0 for 1< x. Thus, we havef(x)≥0 for allx≥0, and as a result, we
have the inequality of the lemma, since in all terms we havepi|〈φi|φ′j〉|^2 ≥0.

To complete the proof of the lemma, we assume that the equality holds. Problems with possible
degeneracies of the population fractions force us to be moreprecise in our analysis. Thus, we shall
decompose the density matrices as follows,

ρ=

∑N

i=0

piPi

∑N

i=0

Pi=I

ρ′=

∑N′

i=0

p′iPi′

∑N′

i=0

Pi′=I (16.64)

where we shall now assume that all population fractionspiand distinct from one another, same for
p′i, so that we can order them as follows,

p 0 = 0< p 1 < p 2 <···< pN

p′ 0 = 0< p′ 1 < p′ 2 <···< p′N′ (16.65)

We shall also assume that each contribution fori 6 = 0 is non-trivial, in the sense that

dim(Pi) = tr(Pi)≥ 1 i≥ 1

dim(Pi′) = tr(Pi′)≥ 1 i≥ 1 (16.66)

In this notation, the combination of the lemma may be recast in the following form,

S(ρ) +kTr

(

ρlnρ′

)

=−k

∑

i,j

(

p′j−pi−piln

(

p′j

pi

))

tr

(

PiPj′

)

(16.67)