-------------------------
S1e. Open quantum systems
-------------------------

Open quantum systems are usually modelled in a stochastic way
to account for the unpredictability of the measurement process.
(Note that a measurement is any non-negligible interaction with the
environment, whether or not it is observed by something deserving
the name 'detector' or 'observer').

In the simplest setting in which states can be assumed to
be pure and measurements occur at definite, a priori known times
and have a negligible duration, an open quantum system is a discrete 
stochastic process with values psi(t) in the Hilbert space of state 
vectors, normalized to norm 1. Between two consecutive measurements,
the system is assumed to be closed.

Thus between two consecutive measurements at times t' and t''>t', 
the normalized state psi(t) evolves according to the Schroedinger 
equation 
   i hbar psidot = H psi,
so that 
   psi(t''-0)= P psi(t'+0),   P = exp (i/hbar (t'-t'')H).      (1)
(In the interaction picture, H=0 and psi remains constant between 
measurements.)

A measurement at time t is assumed to happen in infinitesimal time 
and replaces psi(t-0) independent of other measurements with 
probability p_s by
   psi(t+0)= P_s psi(t-0)/p_s   if p_s>0,                      (2)
where the P_s are linear operators determined by the experimental 
arrangement, satisfying the relation
   sum_s P_s^*P_s = 1,                                         (3)
and 
   p_s=|P_s\psi(t-0)|^2                                        (4)
guarantees that psi(t+0) remains normalized. Clearly the p_s are
nonnegative and by (3), they sum up to 1 (since psi(t-0) is normalized).
(For measurements with more than countably many possible outcomes,
one must replace the probabilities by probability densities and the
sums by integrals.)
Thus this is a well-defined stochastic process.

A von-Neumann measurement of a self-adjoint linear operator A
corresponds to the special case where P_s is an orthogonal projector 
to the eigenspace corresponding to the eigenvalue a_s of A
(respective to the set of eigenvalues corresponding to the s-th
interval in a partition of the continuous spectrum of A.)

   
If the measurement at different times has the same (or different) 
nature, the P_s at these times are the same (or different).
It is possible to introduce 'empty measurements' at arbitrary 
intermediate times with a trivial sum over a singleton s, where P_s=1.

For continuous measurements (where the open system cannot be considered
closed at all but a discrete number of times), one needs to take
a continuum limit of the above description. Depending how one takes
the limit, one gets quantum diffusion processes or quantum jump 
processes. In this case, the density matrix for the associated 
deterministic expectation evolves according to a Lindblad dynamics.

Realistic measurements (i.e. those taking into account the unavoidable 
uncertainty) are not modelled by von-Neumann measurements, but rather
by positive operator valued measures, short POVMs. These are well 
explained in 
    http://en.wikipedia.org/wiki/POVM 

For more on real measurement processes (as opposed to the 
von-Neumann measurement caricature treated in typical textbooks
of quantum mechanics), see, e.g.,
    V.B. Braginsky and F.Ya. Khalili,
    Quantum measurement, 
    Cambridge Univ. Press, Cambridge 1992