Let the underlying density matrix of the system being measured be given
by 
.
Then the probabilities for measuring the eigenstates of M and V
are given by 
and 
respectively, where the subscript indicates the time-order of the filter
application. If the filter V is applied first, followed by the filter
M, then the joint probability of seeing eigenstates of M
and V is given by 
.
It is tempting to answer the question posed above with the quantity below,
the time-ordered mutual information between the two filters
where the probability distribution 
is the reduction of
over 
.
This is indeed the mutual information between the measurements of M
and V, assuming that the filter
V is applied before the filter
M. However, in the question posed above, no actual measurement of
V is to be made! In fact, the proposed measurement of
V and
the measurement of M are conditionally independent once
is specified.
What has been ignored, and is needed, is that the measurement of M
is indirectly a measurement of ,
and it is this
that would have been measured by V. In other words, the probability
that
was the underlying distribution, given that some eigenvalue of M
was measured, and then the probability that V was measured on this
underlying ,
must appear in the information between the unmade measurement of V
and the measurement of M. In order to do this, the prior probability
of the underlying
must be specified, as is seen in the following development. Admittedly,
it is strange to be discussing a distribution over potential, but unmade,
measurements; yet it is consistent. The probability of a measurement that
could have been made can be defined!
As before, let the distributions of measured eigenstates of M
and V, given ,
be 
and 
respectively. Let the prior probability that
was the underlying state be given by 
,
and let the probability that
was the underlying state, given that the eigenvalue 
of M was measured, be given by 
.
This probability distribution is given by Bayes' theorem (see section 9.1),
once both
and
are given:
where 
.
The probability 
that the eigenvalue
of V would have been measured given that the eigenvalue
of M was measured is conceptually clearly given by
Now, 
,
and similarly 
.
The last of these indicates that the joint distribution conditioned on
of the unmeasured V eigenvalue and the measured M eigenvalue
may be treated as if both were measured on the same state .
Finally, applying these identities and Bayes' theorem, the unconditioned
joint distribution is easily seen to be
and as usual
so that all of the usual identities for events can be defined for the
unmeasured V events and the measured M events. The time-ordered
mutual information between the measurement and the unmeasurement is given
by
It is interesting to note that this mutual information is exactly zero
if
is known in advance (the prior probability of
is a delta function distribution), and so there must be some uncertainty
in the underlying physical state for the mutual information just defined
to be nonzero. This is admittedly the usual case. I propose that this is
the quantity that should be considered when making measurements (M)
of a system that is to yield information about some aspect of the system
that another measurement (V) would have been able to provide
more directly.
I leave it to another day to come to terms with what it means to have a probability distribution, like that just defined, over events that are assumed to not happen, and which, if happened, would fundamentally change the result of the measurement that is made. It is as if a step out of the usual probability theory has been taken.