Majorana signatures in the tripartite uncertainty relations with quantum memory

Quantumness imposes a fundamental limit on measurement accuracy. The paradigmatic cases are Heisenberg's uncertainty relation in the original formulation, Robertson's formulation, and improved uncertainty relations. However, the more universal measures are given in terms of quantum entropies. Uncertainties of measurements done on one quantum system correlated with another quantum system constitute a more intriguing question. Quantum correlations can influence the lower bound of uncertainties, and the reason for this is the quantum memory. In this article, we study uncertainties of measurements performed on one quantum dot correlated with the second one through the superconductor, hosting the Majorana boundary modes. We prove that the Majorana quasiparticles allow the uncertainties to reach the minimal possible lower bound. By rigorous theoretical considerations, we obtain the result of experimental relevance expressed in terms of only two parameters: the overlap between Majorana modes and their coupling strength with the quantum dots. We show that the overlap between Majorana modes reduces quantum uncertainties, which is a general result of fundamental importance. We also propose the protocol to measure spins in both quantum dots, consecutively, and demonstrate that the result of the second measurement would depend on the presence of Majorana quasiparticles. This could serve as an indirect tool for their empirical observation, which is of importance for the ongoing discussions concerning unambiguous detection of the Majorana quasiparticles in nanoscopic hybrid structures.