having probability 
as already mentioned. The density matrix is therefore given by
and the 
values are dropped here in the representation of the 
eigenvectors. Based on this probability distribution we have the measurement
entropy,
(almost always) even when k=N, the number of spins in the
system. However, the two entropies are related, and their difference is
a physically meaningful quantity. The fact that they are not equal is shown
next, and the generality of the argument to follow indicates that care
must be taken when defining the entropy of a system from measurements of
the states of that system.
Consider the distribution of measured
states given in equation 8.18.
We can make the following expansion of this as
where
and where 
,
and the 
and 
are also 
eigenstates, etc.
Now from equation 8.27
note that for k=N we have 
,
where the subscript on 
has been dropped. Because of the orthonormality of the various vectors
involved 
.
Note also that 
.
Making the appropriate substitutions in the log sum inequality we find
. Measurement increases entropy. Given a density matrix 
describing a quantum mechanical system of the form 
,
where the
are orthonormal, and an orthonormal measurement basis 
,
define the intrinsic entropy 
and the measurement entropy 
,
where 
.
Then 
,
with equality iff 
is independent of 
.
Proof: By the log sum inequality of chapter 2
we have
Consider the quantity on the left, cancel the 
's
and sum over ,
then consider the quantity on the right, sum over
where these sums appear, finally change signs on both sides, and using
equation 8.23
find that
Note that the entropy of the reduced measurements (k<N) may be less than the intrinsic entropy simply because of the reduction in dimensionality involved.
It is important to note that quantities like 
need to be carefully considered. For instance, the notion of making simultaneous
measurements of
and
is impossible, because the operators for these eigenvalues do not generally
commute. It is possible to time-order the "measurements" (now called filters),
and then quantities like 
appear, where the M filter is applied first, followed by the V
filter. This quantity indicates the probability that the quantum object
follows the route through channel
of M, then through channel
of V. This allows us to perform strange inferences of retrodiction,
like 
,
the probability that the object passed through channel
of M having been found in channel
of V, addressed briefly in the next section. We can go much further
though, and do in the next section: we may ask the question what is
the information in one measurement that would have been available in another
measurement that could have been, but wasn't, made?.
In summary, we have the relationship that 
.
In figure 8.1
an example of the relationship
is demonstrated. (Note that the logarithms are base e, and that 
appears as b in the axis label.)
Figure 8.1: Typical plot of the entropy of the density matrix 
(narrow dashes) and the entropy of the measured values of the z-components
of the spins 
(wide dashes) for the two-spin case and order two entropies. Note .