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Entropic uncertainty relations in the spin-1 Heisenberg model

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Abstract

The uncertainty relations build an intrinsic lower bound to the measurement precision for arbitrary two incompatible observables and hence deemed as a backbone in quantum theory, strikingly distinguishing from classical physics. In this work, we examine the quantum-memory-assisted entropic uncertainty relations (QMA-EUR) in two-qutrit spin-1 Heisenberg XYZ and XXX chains under homogeneous magnetic fields, respectively. We specifically derive the dynamical evolution of QMA-EUR for various incompatible measurements on Pauli operators (mutually unbiased bases) and SU(3) generators in the Heisenberg XXX and XYZ models when spin A is the object to be measured and B is served as quantum memory during information processing. Notably, it has been found in the case of mutual unbiased-base measurements that, firstly, the larger coupling strength J between A and B can induce the degradation of the measurement’s uncertainty; secondly, the entropic uncertainty is extremely dependent on the coupling strength and the external magnetic field, which would bring the uncertainty on the inflation. In addition, we unveil the dynamical behaviors of the uncertainty measured on SU(3) generators, and it proves that the uncertainty’s dynamics are subtly different from those in the former. Moreover, we explore the relationship between the entropic uncertainty and the system’s entanglement (negativity) and declare that the dynamics of the entropic uncertainty of interest are approximately anti-correlated with that of negativity. At last, the entanglement’s dynamics of the probed system in the XXX and XYZ models are revealed and analyzed by means of negativity, respectively. Hence, our observations might pave the way to understand the dynamical traits of the entropic uncertainty during measurement-based information processing.

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Acknowledgements

This work was supported by the National Science Foundation of China under Grant Nos. 61601002 and 11575001, Anhui Provincial Natural Science Foundation (Grant No. 1508085QF139) and the fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701).

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  1. School of Physics and Material Science, Anhui University, Hefei, 230601, China

    Wei-Nan Shi, Fei Ming, Dong Wang & Liu Ye

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  1. Wei-Nan Shi
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Correspondence to Dong Wang.

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Shi, WN., Ming, F., Wang, D. et al. Entropic uncertainty relations in the spin-1 Heisenberg model. Quantum Inf Process 18, 70 (2019). https://doi.org/10.1007/s11128-019-2196-7

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