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Relativistic causality forbids superluminal signaling between distant observers. By exploiting the non-signaling principle, we derive the exact relationship between the chained Clauser---Horne---Shimony---Holt sum of correlations $$text {CHSH}_K$$CHSHK and the success probability $$P_K$$PK associated with Hardyu0027s ladder test of nonlocality for two qubits and $$K+1$$K+1 observables per qubit. Then, by invoking the Tsirelson bound for $$text {CHSH}_K$$CHSHK, the derived relationship allows us to establish an upper limit on $$P_K$$PK. In addition, we draw the connection between $$text {CHSH}_K$$CHSHK and the chained version of the Clauser---Horne (CH) inequality.

Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality