Einstein-Podolsky-Rosen Steering in Two-sided Sequential Measurements with One Entangled Pair

Non-locality and quantum measurement are two fundamental topics in quantum theory and theirinterplay attracts intensive focus since the discovery of Bell theorem. Non-locality sharing amongmultiple observers is predicted and experimentally observed. However, only one-sided sequentialcase, i.e., one Alice and multiple Bobs is widely discussed and little is known about two-sided case.Here, we theoretically and experimentally explore the non-locality sharing in two-sided sequentialmeasurements case in which one entangled pair is distributed to multiple Alices and Bobs. Weexperimentally observed double EPR steering among four observers in the photonic system for thefirst time. In the case that all observers adopt the same measurement strength, it is observedthat double EPR steering can be demonstrated simultaneously while double Bell-CHSH inequalityviolations are shown to be impossible. The exact formula relating Bell quantity and sequential weakmeasurements for arbitrary many Alices and Bobs is also derived, showing that no more doubleBell-CHSH inequality violations is possible under unbiased input condition. The results not onlydeepen our understanding of relation between sequential measurements and non-locality but alsomay find important applications in quantum information tasks.