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

Cheng‐Cheng Liu, Ze-Wei Sun, Xiao‐Gang Fan, Ding Zhi-yong, Juan He, Tao Wu, Liu Ye

Published Apr 3, 2025• 6 citations

Abstract

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.

DOI: 10.1038/s41598-025-95765-6

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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.

Abstract Based on the quantum-memory-assisted entropic uncertainty relation (QMA-EUR), we study the quantum phase transition (QPT) of a spin-1/2 frustrated Heisenberg chain and show that QMA entropy can be a useful tool to detect QPT. For the six-site case, we choose the reduced two-site as the detecting system and obtain the analytical result of QMA entropy and its bound. We find that the QMA entropy of the ground state is discontinuous at the first-order QPT point. Meanwhile for the excited state, the catastrophe point of QMA entropy approaches the infinite-order QPT point, which is confirmed by the numerical results in the 8-, 10- and 12-site cases. Moreover, we study the behavior of QMA entropy undergoing the dephasing process. Interestingly, two-site QMA entropy can still indicate the QPT point when the state degenerates into a mixed state. In this sense, QMA entropy is more effective than quantum correlation in reflecting the quantum criticality.

The uncertainty principle determines the distinction between the classical and the quantum world. This principle states that it is not possible to measure two incompatible observables with the desired accuracy simultaneously. In quantum information theory, the Shannon entropy is used as an appropriate measure to express the uncertainty relation. Improving the bound of the entropic uncertainty relation is of great importance. The bound can be varied by considering an extra quantum system as the quantum memory which is correlated with the measured quantum system. One can extend the bipartite quantum-memory-assisted entropic uncertainty relation to the tripartite one in which the memory is split into two parts. Here, a lower bound is obtained for the tripartite quantum-memory-assisted entropic uncertainty relation. This lower bound has two extra terms compared with the lower bound in Renes and Boileau [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 110, 020402 (2013)] which depends on the conditional von Neumann entropy, the Holevo quantity, and the mutual information. It is shown that the bound is tighter than other bounds derived earlier. It also leads to a lower bound for the quantum secret key rate. In addition, it is applied to obtain the states for which both the strong subadditivity and the Koashi-Winter inequalities turn into equalities.

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 X Y Z 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 ( J < 0 ) and antiferromagnetism ( J > 0 ) 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.

The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables cannot be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. M. Berta et al. [Nat. Phys. 6, 659 (2010)] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the presence of a quantum memory by adding an additional term depending on the Holevo quantity and mutual information. We conclude that our lower bound will be tightened with respect to that of Berta et al. when the accessible information about measurements outcomes is less than the mutual information about the joint state. Some examples have been investigated for which our lower bound is tighter than Berta et al.'s lower bound. Using our lower bound, a lower bound for the entanglement of formation of bipartite quantum states has been obtained, as well as an upper bound for the regularized distillable common randomness.

Entropic uncertainty relations demonstrate the intrinsic uncertainty of nature from an information-theory perspective. Recently, a quantum-memory-assisted entropic uncertainty relation for multiple measurements was proposed by Wu et al. [Phys. Rev. A 106, 062219 (2022)]. Interestingly, the quantum-memory-assisted entropic uncertainty relation for multiple measurement settings can be further generalized. In this work, we propose two complementary multipartite quantum-memory-assisted entropic uncertainty relations and our lower bounds depend on values of complementarity of the observables, (conditional) von Neumann entropies, Holevo quantities, and mutual information. As an illustration, we provide several typical cases to exhibit that our bounds are tighter and outperform the previous bounds.

The uncertainty principle is a striking and fundamental feature in quantum mechanics, distinguishing it from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and the Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound of Renes and Boileau [Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit

X

-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes and Boileau's bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound one can attain the tighter bound of the quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.