Measurements in quantum theory

The state of a quantum mechanical system is characterized by its density operator $\rho$. In particular, the probability of measuring value $x$ for observable $O$ in a state described by $\rho$ is known to be [57,58]

$$p(x\vert O,\rho) = {\rm Tr} \Big(P_O(x) \, \rho \Big) .$$ (13)

This defines the likelihood model of quantum theory. The observable $O$, represented by a hermitian operator, corresponds to the condition $c$ of the previous section. The projector $P_O(x)$ = $\sum_l\mbox{$\vert\, x_l\!><\! x_l \, \vert$}$ projects on the space of eigenfunctions $\mbox{$\vert\,x_l\!>$}$ of $O$ with eigenvalue $x$, i.e., for which $O\mbox{$\vert\,x_l\!>$}$ = $x\mbox{$\vert\,x_l\!>$}$. For non-degenerate eigenvalues $P_O(x)$ = $\mbox{$\vert\, x\!><\! x \, \vert$}$.

To be specific, we will consider the measurement of particle positions, i.e., the case $O$ = $\hat x$ with $\hat x$ 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 $x_i$, independent of quantum system then

$$p(\bar x_i\vert O,\rho) = \int \! dx_i \, p(\bar x_i\vert x_i) \, p(x_i\vert O,\rho) ,$$ (14)

where we denoted the `true' coordinates by $x_i$ and the corresponding noisy output by $\bar x_i$. A simple model for $p(\bar x_i\vert x_i)$ could be a Gaussian.

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 $P_O(x)$ to the space of eigenfunctions of operator $O$ with eigenvalues consistent with the measurement result. Thus, performing multiple measurements under the assumption of a constant density operator $\rho$ 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.