  Extended non-local games are a generalization of monogamy-of-entanglement
games, played by two quantum parties and a quantum referee that performs a
measurement on their local quantum system. Along the lines of the NPA
hierarchy, the optimal winning probability of those games can be upper bounded
by a hierarchy of semidefinite programs (SDPs) converging to the optimal value.
Here, we show that if one extends such games by considering constraints and
loss, motivated by experimental errors and loss through quantum communication,
the convergence of the SDPs to the optimal value still holds. We give
applications of this result, and we compute SDPs that show tighter security of
protocols for relativistic bit commitment, quantum key distribution, and
quantum position verification.
