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Physica Scripta

Learning Rényi entropy in neural network quantum states

Han-Qing Shi, Hai-Qing Zhang

Published Oct 21, 2026• 0 citations
Publisher

Abstract

Abstract We study the Rényi entropy in a one-dimensional transverse-field quantum Ising model by virtue of the machine learning methods with neural networks. The Rényi entropy is investigated both in the static ground state and in the dynamical state after a linear quench, by employing a swapping operator acting on the states prepared from the neural networks. In the ground state, the behavior of Rényi entropy can unveil the critical point from paramagnetic to ferromagnetic phase. At the critical point, the relation between the Rényi entropy and the subsystem size from the machine learning method satisfies the predictions from conformal field theory. In the quenching dynamics, coherent oscillations of the Rényi entropy are observed after the end of the quench. These oscillations have universal frequencies stem from the superpositions of the excited states. More importantly, an asymptotic scaling of the reduced Rényi entropy is observed, implying a novel length scale away from the critical point.

DOI
10.1088/1402-4896/aeb73e

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Abstract We study the Rényi entropy in a one-dimensional transverse-field quantum Ising model by virtue of the machine learning methods with neural networks. The Rényi entropy is investigated both in the static ground state and in the dynamical state after a linear quench, by employing a swapping operator acting on the states prepared from the neural networks. In the ground state, the behavior of Rényi entropy can unveil the critical point from paramagnetic to ferromagnetic phase. At the critical point, the relation between the Rényi entropy and the subsystem size from the machine learning method satisfies the predictions from conformal field theory. In the quenching dynamics, coherent oscillations of the Rényi entropy are observed after the end of the quench. These oscillations have universal frequencies stem from the superpositions of the excited states. More importantly, an asymptotic scaling of the reduced Rényi entropy is observed, implying a novel length scale away from the critical point.
We compute the Renyi entropy in a one-dimensional transverse-field quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.
Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.
Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: One is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three, and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.
Abstract Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.
Abstract We study the properties of pseudo entropy, a new generalization of entanglement entropy, in free Maxwell field theory in d = 4 dimension. We prepare excited states by the different components of the field strengths located at different Euclidean times acting on the vacuum. We compute the difference between the pseudo Rényi entropy and the Rényi entropy of the ground state and observe that the difference changes significantly near the boundary of the subsystems and vanishes far away from the boundary. Near the boundary of the subsystems, the difference between pseudo Rényi entropy and Rényi entropy of the ground state depends on the ratio of the two Euclidean times where the operators are kept. To begin with, we develop the method to evaluate pseudo entropy of conformal scalar field in d = 4 dimension. We prepare two states by two operators with fixed conformal weight acting on the vacuum and observe that the difference between pseudo Rényi entropy and ground state Rényi entropy changes only near the boundary of the subsystems. We also show that a suitable analytical continuation of pseudo Rényi entropy leads to the evaluation of real-time evolution of Rényi entropy during quenches.
We analyze the effect of decoherence, modeled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "g function" in conformal field theory, allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the conformal field theory. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.