  The hypothetical nonlocal box (\textsf{NLB}) proposed by Popescu and Rohrlich
allows two spatially separated parties, Alice and Bob, to exhibit stronger than
quantum correlations. If the generated correlations are weak, they can
sometimes be distilled into a stronger correlation by repeated applications of
the \textsf{NLB}. Motivated by the limited distillability of \textsf{NLB}s, we
initiate here a study of the distillation of correlations for nonlocal boxes
that output quantum states rather than classical bits (\textsf{qNLB}s). We
propose a new protocol for distillation and show that it asymptotically
distills a class of correlated quantum nonlocal boxes to the value $1/2
(3\sqrt{3}+1) \approx 3.098076$, whereas in contrast, the optimal non-adaptive
parity protocol for classical nonlocal boxes asymptotically distills only to
the value 3.0. We show that our protocol is an optimal non-adaptive protocol
for 1, 2 and 3 \textsf{qNLB} copies by constructing a matching dual solution
for the associated primal semidefinite program (SDP). We conclude that
\textsf{qNLB}s are a stronger resource for nonlocality than \textsf{NLB}s. The
main premise that develops from this conclusion is that the \textsf{NLB} model
is not the strongest resource to investigate the fundamental principles that
limit quantum nonlocality. As such, our work provides strong motivation to
reconsider the status quo of the principles that are known to limit nonlocal
correlations under the framework of \textsf{qNLB}s rather than \textsf{NLB}s.

Quantum Physics