Fine-grained uncertainty relations under relativistic motion

Abstract

Among various uncertainty relations, the profound fine-grained uncertainty relation is used to distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements. In this letter, we explore this uncertainty relation in a relativistic regime. For an observer undergoing a uniform acceleration who is immersed in an Unruh thermal bath, we show that the uncertainty bound is dependent on the acceleration parameter and the choice of Unruh modes. We find that the measurements in mutually unbiased bases, sharing the same uncertainty bound in the inertial frame, could be distinguished from each other for a noninertial observer. In an alternative scenario, for the observer restricted in a single rigid cavity, we show that the uncertainty bound exhibits a periodic evolution with respect to the duration of acceleration. With properly chosen cavity parameters, the uncertainty bounds could be protected. Moreover, we find that the uncertainty bound can be degraded for specific quantum measurements to violate the bound exhibited in the nonrelativistic limit, which can be attributed to the entanglement generation between cavity modes during a particular epoch. Several implications of our results are discussed.