On a generalized approach to quantum tomography via projective unitary representations of groups

Suppose that $\mathfrak {M}$ is a positive operator-valued measure on a measurable space $X$ with values in the set of all positive bounded operators $B(H)_+$ in a separable Hilbert space $H$. If $g\to U_g$ is a projective unitary representation of a group $G$ in $H$ and one can define an action of $G$ on $X$ by the rule $x\in X \to gx\in X,\ g\in G$, then $\mathfrak M$ is said to be covariant with respect to $\mathcal {U}=\{U_g,\ g\in G\}$ under the condition $U_g\mathcal {M}(B)U_g^*=\mathfrak {M}(gB)$ for all measurable subsets $B\subset X$ and $g\in G$. In quantum tomography theory we use a set of functions instead of a density operator $\rho $. Using a covariant POVM $\mathfrak M$ equipped with a projective unitary representation $\mathcal U$ we can determines two possible functions of this kind. One is $f_{\rho}(g)=Tr(\rho U_g)$ and the other is $F_{\rho }(B)=Tr(\rho \mathfrak {M}(B))$. The first function can be named a characteristic function of $\rho $, while the second one is associated with a measurement of $\rho $ by $\mathfrak {M}$. We attribute these two functions to homodyne and heterodyne measurements and discuss the connection between them.