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Let $\mathfrak G$ and $\mathfrak {B}(\mathfrak {G})$ be a locally compact Abelian group with a Haar measure $\mu $ and the algebra of all measurable sets on $\mathfrak G$ correspondingly. Denote $B(H)_+$ the cone of positive linear operators acting in a separable Hilbert space $H$. A map $\mathfrak {M}:\mathfrak {B}(\mathfrak {G})\to B(H)_+$ is said to be a positive operator-valued measure (POVM) on $\mathfrak G$ if
$$
\mathfrak{M} (\cup_j B_j)=\sum \limits_j\mathfrak{M}(B_j), \quad B_j\cap B_k=\emptyset, \quad B_j\in \mathfrak{B}(\mathfrak {G}), j\neq k,
$$
$$
\mathfrak {M}(\emptyset )=0, \quad \mathfrak {M}(\mathfrak {G})={\rm I}.
$$
Let $\mathfrak {G}=\hat G\times G$ be the locally compact Abelian group generated by a locally compact Abelian group $(G,\nu)$ and
$\mu =\hat \nu \times \nu $ be the Haar measure on $\mathfrak {G}$. Then, a projective unitary representation $(\chi ,g)\to U_{\chi ,g}$ of $\mathfrak {G}$ in $H=L^2(G,\nu )$ defined by the formulla
$$
[U_{\chi ,g}\psi ](h)=[\chi (g)]^{1/2}\chi (h)\psi (h+g), \quad g,h\in G, \quad \chi \in \hat G, \quad \psi \in H,
$$
is irreducible and $\mathfrak M (B)=\int \limits_B U_{\chi,g} P_{0} U_{\chi, g}^*d\mu(\chi,g)$, where $P_0$ is an arbitrary one-dimensional projection on the subspace ${\mathbb{C}\xi_0}$, is a POVM on $\mathfrak{G}$ being covariant in the sense [1-2]
$$
U_{\chi,g}\mathfrak{M}(B)U_{\chi,g}^*=\mathfrak{M}(B+(\chi,g)), \quad (\chi,g)\in \mathfrak{G}, \quad B\in \mathfrak{B}(\mathfrak {G}).
$$
Given a state (a positive unit-trace operator) $\rho $ the function $F_{\rho}$ on $\mathfrak {G}$ determined as
$$
F_{\rho }(\chi ,g)=(U_{\chi ,g}\xi _0,\rho U_{\chi ,g}\xi _0)
\qquad \qquad \qquad (1)
$$
is a density of probability distribution {${ {\rm Tr}(\rho \mathfrak {M}(B)),\, B\in \mathfrak {M(G)} }$}. The POVM $\mathfrak {M}=$ {${ \mathfrak {M}(B),\ B\in \mathfrak {M(G)} }$} is said to be informationally complete if (1) can be can be converted. We show that (1) is informationally complete [3]. To prove this fact we investigate the set of contractions $T_{\chi ,g}$ generated by the POVM $\mathfrak {M}$.
References:
[1] G.G. Amosov, On quantum tomography on locally compact groups, Physics Letters A, 431 (2022), 128002, 7 pp.
[2] G.G. Amosov, On quantum channels generated by covariant positive operator-valued measures on a locally compact group, Quantum Information processing, 21 (2022), 81, 15 pp.
[3] G.G. Amosov, On constructing informationally complete covariant positive operator-valued measures, arXiv:2301.12492
