each unitary transformation one could perform a unitary operation $U$ that diagonalizes $A$ with respect to the computational basis and measure with respect to this basis.

However, the description of measurements by self-adjoint operators is not general enough. Most general measurements are described by positive operator valued measures (POVMs). A POVM is defined as follows [@Davies2]. Let $\Omega$ be the set of possible outcomes and $\Sigma$ be a sigma-algebra of measurable subsets of $\Omega$. Let ${{\cal P}}$ be the set of positive operators acting on the Hilbert space ${{\cal H}}$. Then a POVM $A$ is a map $A: \Sigma \rightarrow
{{\cal P}}, m \mapsto A_m,$ with the following properties:

1.  For all countable families $(m_j)$ of mutually disjoint sets $m_j$ one has $$A_{\cup_j m_j} = \sum_j A_{m_j}\,,$$ where the infinite sum converges in the weak operator topology.

2.  $A_{\Omega}={\bf 1}$.

The probability for obtaining an outcome in the set $m$ is given by ${\rm tr}(\rho A_m)$. When the set $\Omega$ of possible outcomes is finite or countably infinite a POVM is uniquely given by a family $(A_j)$ of positive operators such that $p_j={\rm tr}(\rho A_j)$ is the probability for obtaining the outcome $j$. We only consider POVMs with a finite 