  In this Letter we compute an analogue of Tsirelson's bound for Hardy's test
of nonlocality, that is, the maximum violation of locality constraints allowed
by the quantum formalism, irrespective of the dimension of the system. The
value is found to be the same as the one achievable already with two-qubit
systems, and we show that only a very specific class of states can lead to such
maximal value, thus highlighting Hardy's test as a device-independent self-test
protocol for such states. By considering realistic constraints in Hardy's test,
we also compute device-independent upper bounds on this violation and show that
these bounds are saturated by two-qubit systems, thus showing that there is no
advantage in using higher-dimensional systems in experimental implementations
of such test.
