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Laser Physics Letters

Entropic uncertainty for spin-1/2 XXX chains in the presence of inhomogeneous magnetic fields and its steering via weak measurement reversals

Dong Wang, Fei Ming, Ai-Jun Huang, Wen‐Yang Sun, Ye Liu

Published Aug 22, 2017• 59 citations

Abstract

Abstract The uncertainty principle configures a low bound to the measuring precision for a pair of non-commuting observables, and hence is considerably nontrivial to quantum precision measurement in the field of quantum information theory. In this letter, we consider the entropic uncertainty relation (EUR) in the context of quantum memory in a two-qubit isotropic Heisenberg spin chain. Specifically, we explore the dynamics of EUR in a practical scenario, where two associated nodes of a one-dimensional XXX -spin chain, under an inhomogeneous magnetic field, are connected to a thermal entanglement. We show that the temperature and magnetic field effect can lead to the inflation of the measuring uncertainty, stemming from the reduction of systematic quantum correlation. Notably, we reveal that, firstly, the uncertainty is not fully dependent on the observed quantum correlation of the system; secondly, the dynamical behaviors of the measuring uncertainty are relatively distinct with respect to ferromagnetism and antiferromagnetism chains. Meanwhile, we deduce that the measuring uncertainty is dramatically correlated with the mixedness of the system, implying that smaller mixedness tends to reduce the uncertainty. Furthermore, we propose an effective strategy to control the uncertainty of interest by means of quantum weak measurement reversal. Therefore, our work may shed light on the dynamics of the measuring uncertainty in the Heisenberg spin chain, and thus be important to quantum precision measurement in various solid-state systems.

DOI: 10.1088/1612-202x/aa7b4e

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Abstract We investigate the quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a Heisenberg XYZ mixed-spin (1/2, 1) model. Coupling strength, Dzyaloshinskii–Moriya (DM) interaction and inhomogeneous magnetic field, respectively, contributing to QMA-EUR by a thermal entanglement in the hybrid-spin model are studied in detail. Furthermore, we compare the uncertainty of the bipartite hybrid model with those of qubit–qubit and qutrit–qutrit systems. Meanwhile, the effects of local PT-symmetric operation and weak measurement on the steering of entropic uncertainty are analyzed. We find that the local PT-symmetric operation can reduce the entropic uncertainty, and the entropic uncertainty can also be decreased by weak measurement reversal.

Abstract The precision of measurements for two incompatible observables in a physical system can be improved with the assistance of quantum memory. In this paper, we investigate the quantum-memory-assisted entropic uncertainty relation for a spin-1 Heisenberg model in the presence of external magnetic fields, the systemic quantum entanglement (characterized by the negativity) is analyzed as contrast. Our results show that for the XY spin chain in thermal equilibrium, the entropic uncertainty can be reduced by reinforcing the coupling between the two particles or decreasing the temperature of the environment. At zero-temperature, the strong magnetic field can result in the growth of the entropic uncertainty. Moreover, in the Ising case, the variation trends of the uncertainty are relied on the choices of anisotropic parameters. Taking the influence of intrinsic decoherence into account, we find that the strong coupling accelerates the inflation of the uncertainty over time, whereas the high magnetic field contributes to its reduction during the temporal evolution. Furthermore, we also verify that the evolution behavior of the entropic uncertainty is roughly anti-correlated with that of the entanglement in the whole dynamical process. Our results could offer new insights into quantum precision measurement for the high spin solid-state systems.

The dynamics of measurement’s uncertainty via entropy for a one-dimensional Heisenberg XYZ mode is examined in the presence of an inhomogeneous magnetic field and Dzyaloshinskii–Moriya (DM) interaction. It shows that the uncertainty of interest is intensively in connection with the filed’s temperature, the direction-oriented coupling strengths and the magnetic field. It turns out that the stronger coupling strengths and the smaller magnetic field would induce the smaller measurement’s uncertainty of interest within the current spin model. Interestingly, we reveal that the evolution of the uncertainty exhibits quite different dynamical behaviors in antiferromagnetic ( J i > 0) and ferromagnetic ( J i < 0) frames. Besides, an analytical solution related to the systematic entanglement (i.e., concurrence) is also derived in such a scenario. Furthermore, it is found that the DM-interaction is desirably working to diminish the magnitude of the measurement’s uncertainty in the region of high-temperature. Finally, we remarkably offer a resultful strategy to govern the entropy-based uncertainty through utilizing quantum weak measurements, being of fundamentally importance to quantum measurement estimation in the context of solid-state-based quantum information processing and computation.

Abstract The uncertainty principle is an inherent characteristic of quantum mechanics. This principle can be formulated in various forms. Fundamentally, this principle can be expressed in terms of the standard deviation of the measured observables. In quantum information theory, the preferred mathematical quantity to express the entropic uncertainty relation is Shannon’s entropy. In this work, we consider the generalized entropic uncertainty relation in which there is an additional particle as a quantum memory. Alice measures on her particle A and Bob, with memory particle B , predicts Alice’s measurement outcomes. It has a wide range of applications, for example, entanglement witnessing and quantum key distribution. We study the effects of the environment on the entropic uncertainty lower bound in the presence of weak measurement and measurement reversal. The dynamical model that is intended in this work is as follows: first the weak measurement is performed, second, the decoherence affects the system and finally the measurement reversal is performed on the quantum system. Here, we consider the generalized amplitude damping channel and depolarizing channel as environmental noises. We will show that in the presence of weak measurement and measurement reversal, despite the presence of environmental factors, the entropic uncertainty lower bound drops to an optimal minimum value. In fact, weak measurement and measurement reversal enhance the quantum correlation between the subsystems A and B , and thus the uncertainty of Bob regarding Alice’s measurement outcomes reduces. Based on the entropic uncertainty relation, one can also show that the weak measurement and measurement reversal can preserve the quantum secret key rate lower bound from decoherence.

Abstract In this article, we investigate the dynamics and correlations of quantum-memory-assisted entropic uncertainty, the tightness of the uncertainty, entanglement, quantum correlation and mixedness for various spin chain models with Dzyaloshinskii–Moriya (DM) interaction, including the XXZ model with DM interaction, the XY model with DM interaction and the Ising model with DM interaction. We find that the uncertainty grows to a stable value with growing temperature but reduces as the coupling coefficient, anisotropy parameter and DM values increase. It is found that the entropic uncertainty is closely correlated with the mixedness of the system. The increasing quantum correlation can result in a decrease in the uncertainty, and the robustness of quantum correlation is better than entanglement since entanglement means sudden birth and death. The tightness of the uncertainty drops to zero, apart from slight volatility as various parameters increase. Furthermore, we propose an effective approach to steering the uncertainty by weak measurement reversal.