Self-calibrating tomography for multidimensional systems

Quantum state tomography relies on the ability to perform well-characterized unitary operations, which are not always available. This places restrictions on which physical systems can be characterized using such techniques. We develop a formalism that circumvents the requirement for well-characterized unitary operations by treating unknown parameters in the state and unitary process on the same footing, thereby simultaneously performing state and partial process tomography. Our formalism is generalized to d-level systems, and we provide a specific example for a V-type three-level atomic system whose transition dipole moments are not known. We show that it is always possible to retrieve not only the unknown state but also the process parameters, except for a set of zero measure in the state-parameter space.