QuantumPhysics.dvi

Thus, the logarithm of the number of micro-states Ω is an extensive quantity, proportional to the
lengthNof the message. The statistical entropy is the corresponding intensive quantity, up to a
multiplicative factor.

Some basic properties of the statistical entropy are as follows.

1. Positivity,S(p 1 ,···,pn)≥0;


2. The minimum isS= 0, attained when all probability assignments are 0, exceptfor a single
   entrypj= 1;


3. The maximum is attained when all probabilities are equal,pi= 1/n, givingSmax= lnn.

16.7 Quantum statistical entropy

A quantum mechanical macro-state is characterized by the density operatorρ. The state may be
obtained as an incoherent mixture oforthogonalpure states|φi〉, with population fractionspi, so
that

ρ=

∑

i

|φi〉pi〈φi|

∑

i

pi= 1 (16.39)

The population fractionspimay be measuredsimultaneouslyby the commuting (actually orthog-
onal) observablesPi=|φi〉〈φi|. Thus, thepimay be viewed essentially as classical probabilities,
and the statistical entropy may be generalized immediately,^14

S(ρ) =−kTr

(

ρlnρ

)

(16.40)

Here again,kis a positive constant, left arbitrary in information theory, but equal to Boltzmann’s
constantkBfor statistical mechanics. This is the statistical entropyfor a macro-state specified by
the density operatorρ, and is sometimes referred to as the von Neumann entropy.

Some fundamental properties of the statistical entropy aregiven as follows,

1. Positivity,S(ρ)≥0;


2. MinimumisS(ρ) = 0 attained if and only ifρis a projection operator and thus corresponds
   to a pure state;


3. Maximumis attained as follows. If the possible probabilities are non-zero for a subspace
   Hn⊂Hof finite dimensionn, then the maximum entropy isSmax=klnn;


4. Invarianceunder conjugation of the density operator by a unitary transformation. In partic-
   ular, the entropy is invariant under time evolution, under the assumption that the population
   fractions remain unchanged in time;

(^14) Note that when the density matrix is written as a weighted sum involving non-orthogonal states as in
(16.6), the weightswicannot be measured simultaneously, as the corresponding statesare not orthogonal.
Thus, the entropy isnot equalto the replacementpi→wi.