Quantum memory assisted entropic uncertainty relation as a signature of quantum phase transition in the spin XXZ model

Uncertainty principle establishes a remarkable lower bound to predict the measured outcome of two non-commuting observables. In this paper, based on the quantum renormalization-group method, we study the relation between quantum-memory-assisted entropic uncertainty relation (QMA-EUR) and quantum phase transition (QPT) in the spin XXZ model. The results shows that the entropic uncertainty and the lower bound have similar traits. In addition, we propose two schemes, one is based on quantum discord and classical correlation, the other Holevo quantity and mutual information, both can tighten the bound of EUR in the presence of quantum memory. The tighter the entropy uncertainty relationship is, the higher the accuracy of the predicted results will be. Moreover, we can obtain the optimal lower bound with the help of Holevo quantity and mutual information, which have the best optimization effect in this model. Additionally, we study QPT by virtue of EUR and after a certain number of iterations, finding that the value of QMA-EUR of the whole block-block state can form two saturated values, which are related to two different phases: spin-fluid phase and Néel phase. Afterwards, we discover that the QMA-EUR of the block-block state obeys the nonanalytic and scaling properties with entropic uncertainty relation exponent associated with correlation length. Our findings show that QMA-EUR deserves to be used as an effective tool in reflecting quantum criticality for more quantum many-body systems and may also shed light on many applications in quantum physics including the quantum key distribution, the detection of QPT and the evaluation of the capacity of quantum computation in critical systems.