On the meaning and interpretation of tomography in abstract Hilbert spaces

The mechanism of describing quantum states by standard probabilities (tomograms) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of decompositions of the identity in the Hilbert space of hermitian linear operators, with trace formula as scalar product of operators. Decompositions of the identity are considered with respect to over-complete families of projectors labeled by extra parameters and containing a measure, depending on these parameters. It plays the role of a Gram-Schmidt orthonormalization kernel. When the measure is equal to one, the decomposition of identity coincides with a positive operator-valued measure (POVM) decomposition. Examples of spin tomography, photon number tomography and symplectic tomography are reconsidered in this new framework.