  In 1991, Asher Peres and William Wootters wrote a seminal paper on the
nonlocal processing of quantum information [\textit{Phys. Rev. Lett.}
\textbf{66} 1119 (1991)]. We return to their classic problem and solve it in
various contexts. Specifically, for discriminating the "double trine" ensemble
with minimum error, we prove that global operations are more powerful than
local operations with classical communication (LOCC). Even stronger, there
exists a finite gap between the optimal LOCC probability and that obtainable by
separable operations (SEP). Additionally we prove that a two-way, adaptive LOCC
strategy can always beat a one-way protocol. Our results provide the first
known instance of "nonlocality without entanglement" in two qubit pure states.
