Quantum-memory-assisted entropic uncertainty relation in a Heisenberg XYZ chain with an inhomogeneous magnetic field

Abstract

The uncertainty principle provides a nontrivial bound to expose the precision for the outcome of the measurement on a pair of incompatible observables in a quantum system. Therefore, it is of essential importance for quantum precision measurement in the area of quantum information processing. Herein, we investigate quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a two-qubit Heisenberg spin chain. Specifically, we observe the dynamics of QMA-EUR in a realistic model there are two correlated sites linked by a thermal entanglement in the spin chain with an inhomogeneous magnetic field. It turns out that the temperature, the external inhomogeneous magnetic field and the field inhomogeneity can lift the uncertainty of the measurement due to the reduction of the thermal entanglement, and explicitly higher temperature, stronger magnetic field or larger inhomogeneity of the field can result in inflation of the uncertainty. Besides, it is found that there exists distinct dynamical behaviors of the uncertainty for ferromagnetism and antiferromagnetism chains. Moreover, we also verify that the measuring uncertainty is dramatically anti-correlated with the purity of the bipartite spin system, the greater purity can result in the reduction of the measuring uncertainty, vice versa. Therefore, our observations might provide a better understanding of the dynamics of the entropic uncertainty in the Heisenberg spin chain, and thus shed light on quantum precision measurement in the framework of versatile systems, particularly solid states.