The above postulates are more or less "natural". They are in agreement
with the existing ideas of non-unitary
evolution. So, for
instance, in [20] the authors considered the ionization
model. They wrote: According to the usual procedure the
ionization probability 
should be given by
.
Even if our postulates are
natural, it is worthwhile to notice that EEQT allows
us to interpret them, to understand them and to derive them, in terms of
classical Markov processes.
First of all let us see that the above formula for 
can be understood
in terms of an inhomogeneous Poisson decision process as follows. Assume
the evolution starts with some quantum state 
, of norm one, as
above. Define the positive function 
as
where
Then 
above happens to be nothing but the first-jump probability
of the inhomogeneous Poisson process with intensity 
.
It is instructive to see that this is indeed the case. To this
end let us divide the interval 
into n subintervals
of length 
. Denote 
The inhomogeneous Poisson process of intensity 
consists then of
taking independent decisions jump--or--not--jump\
in each time subinterval.
The probability for jumping in the k-th subinterval is assumed to be
(that is why 
is called the
intensity\
of the process).
Thus the probability 
of not jumping up to time t is
Let us show that 
can be identified with 
given by Eq.
(7).
To this end notice that
Thus 
, given by Eq. (1) satisfies the same differential
equation as
given by Eq. (7). Because 
, it
follows that 
, and so 
indeed is the first
jump
probability of the inhomogeneous Poisson process with intensity 
.
This observation is useful but rather trivial. It can not yet stand
for a justification of the formula (1) - this for the simple
reason that the jump process above, based upon a continuous observation
of the variable 
and registering the time instant of its jump, is not
a Markovian process. It would become
Markovian if we know 
, but to know 
we must
know 
. This leads us to consider pairs 
,
where
is the Hilbert space vector describing quantum state, and 
is the yes-no state of the counter. Then 
evolves deterministically
according to the formula (2), the intensity function 
is computed on the spot, and the Poisson decision process described above
is responsible for the jump of value of 
- in our case
it corresponds to a "click" of the counter. The time of the click is a
random variable 
, well defined and computable by the above prescription.
This prescription sheds some light onto the meaning of the quantum state vector 
.
We see that 
codes in itself information that is being used by
a decision mechanism working in an entirely classical way - the
only randomness we need is that of a biased (by 
)
classical roulette. Until we ask why the bias is determined
by this particular functional of the quantum state, until then we
do not have to invoke more esoteric concepts of quantum probability
-- whatever they mean.
But, in fact, it is possible to understand somewhat more, still in pure
classical terms. We will not need this extra knowledge in the rest
of this paper, but we think it is worthwhile to sketch here at least
the idea.
In the reasoning above we were interested only in what governs the
time of the first jump, when the counter clicks. But in reality
nothing ends with this click. A photon, for instance, when detected,
is transformed into another form of energy. So, if we want to
continue our process in time, after 
, we must feed it
with an extra information: how is the quantum state transformed
as the result of the jump. So, in general, we have a classical
variable 
that can take finitely many, denumerably many,
a continuum, or more, possible values, and to each ordered pair
there corresponds an operator
. The transition 
is called an event, and to each event there corresponds
a transformation of the quantum state
.
In the case of a counter there is only one 
. In general, when
there are several 
-s, we need to tell not only when to jump,
but also where to jump. One obtains in this way a piecewise deterministic
Markov process on pure states of the total system: (quantum object,
classical monitoring device). It can be then shown [6,22]
that this process, including
the jump rate formula (5) follows uniquely from the simplest
possible Liouville equation that couples the two systems.
