Group theoretical quantum tomography (Q2731808)

The problem of determining the state in quantum optics (quantum tomography) is considered. For solving this problem various techniques were elaborated (Vogel, Risken (1989), Smithey, Beck, Raymer, Faridani (1993), D'Ariano, Macchiavello, Paris (1994), Breitenbach, Schiller, Mlynek (1997)). To refer to these techniques the name quantum tomography is used, since they are based on the Radon transform, as in medical tomography imaging. The common feature (D'Ariano 1977) of these methods is the use of a set of observables \(\{Q_n:n\in X\}\) called quorum, parametrized by a space \(X\). It is pointed out, that the quantum state described by positive trace-class of trace one operators \(T\) on Hilbert space \({\mathcal H}\), as well as the physical quantities defined as expectation values \(T_r(AT)\) for arbitrary self-adjoint Hilbert-Schmidt operators \(T\) on \({\mathcal H}\) are completely determined in terms of \(T_r \{Q_nT\}\). The fundamental property of the quorum is that any observable \(T_r (AT)\) can be expressed as an integral transform on the space \(X\) by averaging the function \(\sigma(A)\), called the estimator of \(A\). NEWLINENEWLINENEWLINEIn the paper, the recently proposed general method to realize a quorum and define the estimators in terms of suitable representations of Lie groups (D'Ariano (2000), Paini (1999, 2000)) is developed. As a result, the mathematical foundation of this method in quantum tomography is given by using the theory of square-integrable representations of unimodular Lie groups. The presented mathematical theory is applied to two examples directly connected with the experimental situation: the homodyne tomography related to the Weyl-Heisenberg group and the angular momentum tomography associated with the rotation group \(SU(2)\).