The state of a quantum mechanical system
is characterized by its density operator .
In particular,
the probability of measuring value for observable
in a state described by
is known to be
[57,58]
(13) |
To be specific, we will consider the measurement of particle positions, i.e., the case = with being the multiplication operator in coordinate space. However, the formalism we will develop does not depend on the particular kind of measured observable. It would be possible, for example, to mix measurements of position and momentum (see, for example, Section 3.2.5).
For the sake of simplicity, we will assume that no classical noise
is added by the measurement process.
It is straightforward, however, to include
a classical noise factor in the likelihood function.
If, for example, the classical noise
is, conditioned on ,
independent of quantum system
then
(14) |
In contrast to the (ideal) measurement of a classical system, the measurement of a quantum system changes the state of the system. In particular, the measurement process acts as projection to the space of eigenfunctions of operator with eigenvalues consistent with the measurement result. Thus, performing multiple measurements under the assumption of a constant density operator requires special care to ensure the correct preparation of the quantum system before each measurement. In particular, considering a quantum statistical system at finite temperature, as we will do in the the next section, the time between two consecutive measurements should be large enough to allow thermalization of the system.