Cell discovery is the procedure in which an user equipment (UE) in a cellular network finds a suitable base station (BS) and its physical layer cell identity, in order to establish a link-layer connection. When beamforming with antenna arrays is done at both transmitter and receiver, cell discovery in mm wave systems also involves finding the correct angle of arrival (AoA) - angle of departure (AoD) alignment between the UE and the detected BS. In this paper, we consider various existing and new schemes for cell discovery, present analytical studies on their detection probability and compare them in a common framework. In the first part, we study the conventional beam sweep technique and its variations, and present their physical layer training phase in detail. While the traditional beam sweep can not directly find the identity of the detected BS, we provide modifications in its training phase to enable the cell identity detection. In the second part of the paper, exploiting the sparseness of the mm wave channels and using an equivalent compressive sensing measurement model, we develop new cell discovery schemes with lesser overheads. One such design involves mutually unbiased bases (MUB) from quantum information theory. For the MUB based training scheme, we characterize the mutual coherence parameter of the resulting sensing matrix and establish its connection to the detection probability. We also present detailed simulation studies using experimentally driven mm wave channel simulators and show that our MUB based scheme gives superior performance compared to all the other schemes.