An overview on single apparatus quantum measurements

Given the state of a quantum system, one can calculate the expectation value of any observable
of the system. However, the inverse problem of determining the state by performing different
measurements is not a trivial task. In various experimental setups it is reasonably straightforward
to reconstruct the state of a quantum system employing linear tomographic technique. In this way
the elements of the density matrix can be linearly related to a set of measured quantities. But
since different observables of a quantum system are not commuting with each other, one often has
to perform series of successive measurements of observables which cannot be done simultaneously.
Simultaneous measurement of observables cost less time and energy and is more beneficial. In
this paper we review the strategy of quantum state tomography with simultaneous measurement of
commuting observables. This can be done by introducing an assistant system of which the state
is known. We show that the interaction between the assistant and the system of interest within
different frame works allows the reconstruction of the state of the system. Specifically, we consider
a two-level system and reconstruct its initial state by introducing an assistant which can be either
another two-level system or a single cavity mode of the electromagnetic field.