Optimality of semiquantum nonlocality in the presence of high inconclusive rates

Quantum nonlocality is a counterintuitive phenomenon that lies beyond the purview of causal influences. Recently, Bell inequalities have been generalized to the case of quantum inputs, leading to a powerful family of semiquantum Bell inequalities that are capable of detecting any entangled state. Here, we focus on a different problem and investigate how the local indistinguishability of quantum inputs and postselection may affect the requirements to detect semiquantum nonlocality. To this end, we consider a semiquantum nonlocal game based on locally indistinguishable qubit inputs, and derive its postselected local and quantum bounds by using a connection to the local distinguishability of quantum states. Interestingly, we find that the postselected local bound is independent of the measurement efficiency, and the achievable postselected Bell violation increases with decreasing measurement efficiency.

Figure 1

Operational interpretation. Alice's and Bob's measurement choices are encoded into trusted qubit systems and then sent to their respective untrusted PQM devices. The PQM devices share a bipartite state (denoted by ${\varphi }_{{\mathsf{A}}^{\prime }{\mathsf{B}}^{\prime }}$) which may or may not be entangled. To test for entanglement, Alice and Bob compute Eq. (2): if the inequality is violated, they conclude ${\varphi }_{{\mathsf{A}}^{\prime }{\mathsf{B}}^{\prime }}$ is entangled, otherwise, the experiment is not conclusive. It is useful to mention that like standard Bell experiments, the PQM devices and the source device are all part of the test.

Figure 2

Quantum violation vs efficiency. The vertical axis is the postselected Bell value $S\left(\gamma \right)$ and the horizontal axis is the measurement efficiency $\gamma \in (0,1]$. The (black) dashed line is given by the maximum quantum bound, Eq. (5), which is obtained using general two-qubit measurements. The (red) solid line is the postselected local bound, Eq. (6). The (blue) shaded area is due to the pretty good quantum strategy.