We present a method for measuring quantum states encoded in temporal modes of photons. The basis for the multilevel quantum states is defined based on modes propagating in a dispersive medium, which is a fiber in this case. The propagation and time-resolved single photon detection allows us to define a positive-operator valued measurement (POVM). The POVM depends on the amount of dispersion and the characteristics of a detector. This framework is numerically tested by performing quantum state tomography on a large number of states for a set of realistic experimental settings. Finally, the average fidelity between the expected and reconstructed states for qubits and qutrits is computed.

In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the formρ = Lρ and the macroscopic information about the system is provided by the mean values m i (t j The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system, i.e. minimal value of r and p as well as the properties of the observables {Q i } r i=1 .