Device-Independent Bounds for Hardy’s Experiment

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 a test.

Figure 1

Schematic diagram for the Hardy’s test scenario.

Figure 2

Upper and lower bounds on maximum Hardy’s probability, $p_{\mathrm{Hardy}}$, in terms of the bound $ϵ$ on the constraint probabilities. The solid (blue) line is the upper bound, computed from the set ${\mathcal{Q}}_3$; the dotted (red) line is the lower bound, computed from two-qubit systems; the dashed (black) line is the local bound.