Within the framework of quantum mechanics over a quadratic extension of the
non-Archimedean field of p-adic numbers, we provide a general definition of a
quantum state relying on a general algebraic approach and on a p-adic model of
probability theory. As in the standard complex case, a distinguished set of
physical states are related to a notion of trace for a certain class of bounded
operators and, in fact, we show that one can define a suitable space of trace
class operators in the non-Archimedean setting, as well. The analogies, but
also the several (highly non-trivial) differences, with respect to the case of
standard quantum mechanics in a complex Hilbert space are analyzed.
