Experimental investigation of entropic uncertainty relations and coherence uncertainty relations

The uncertainty relation usually is one of the most important features in quantum mechanics and is the backbone of quantum theory, which distinguishes it from the rule in its classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states with high fidelity, i.e., Bell-like states and Bell-like diagonal states, we carry out local projective measurements over a complete set of mutually unbiased bases on the measured subsystem. In terms of quantum tomography, the density matrices of the initial states and the postmeasurement states are reconstructed. It shows that our experimental results coincide with the theoretical predictions very well. Additionally, we also verify that the lower bounds of both the entropy-based and coherence-based uncertainty can be tightened by imposing the Holevo quantity and mutual information, and the entropic uncertainty is inversely correlated with the coherence. Our demonstrations might offer an insight into their uncertainty relations and their connection to quantum coherence in quantum information science, which might be applicable to the security analysis of quantum key distributions.