  We first show that partial transposition for pure and mixed two-particle
states in a discrete $N$-dimensional Hilbert space is equivalent to a change in
sign of the momentum of one of the particles in the Wigner function for the
state. We then show that it is possible to formulate an uncertainty relation
for two-particle Hermitian operators constructed in terms of Schwinger
operators, and study its role in detecting entanglement in a two-particle
state: the violation of the uncertainty relation for a partially transposed
state implies that the original state is entangled. This generalizes a result
obtained for continuous-variable systems to the discrete-variable-system case.
This is significant because testing entanglement in terms of an uncertainty
relation has a physically appealing interpretation. We study the case of a
Werner state, which is a mixed state constructed as a convex combination with a
parameter $r$ of a Bell state $|\Phi^{+} \rangle$ and the completely incoherent
state, $\hat{\rho}_r = r |\Phi^{+} \rangle \langle \Phi^{+}| +
(1-r)\frac{\hat{\mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$
is obtained as a function of the dimensionality $N$, the uncertainty relation
for the partially transposed Werner state is violated and the original Werner
state is entangled.
