**While the correlation between degrees of freedom and constants of motion is widely recognized in classical mechanics, a truly comprehensive and robust definition of integrability for quantum systems remains unclear. This has been explored recently in a study in the journal Quantum. **

*Study: Eigenstate entanglement in integrable collective spin models. Image Credit: Andrew Derr/Shutterstock.com*

The presence of an optimal solution of the model, like the Bethe ansatz based on the Yang-Baxter equation, or other properties, including a set of simple conserved quantities or Poissonian level statistics, are frequently connected with quantum integrability.

None of these characteristics, on the other hand, provides a clear characterization of integrable quantum systems that differentiates them from nonintegrable quantum systems. Since the primary purpose is to characterize quantum dynamics, a suitable metric or measure should be able to accurately divide all quantum models into two categories: integrable and nonintegrable, each with radically different dynamical behavior.

Quantum entanglement can be a defining measure for integrability in quantum theory with a tensor product structure. Various studies have found a direct link between entanglement and dynamical properties.

The qualitative characteristic of the spread of entanglement in many types of spin systems is a good illustration. There are integrable and non-integrable chains, many-body localized systems or quantum scars, as well as additional models like the Dicke model.

Another way to investigate this link is to look into entanglement in energy eigenstates and how it affects quantum dynamics.

Researchers also look at the entanglement pattern over the eigenstates, which goes beyond the average entanglement entropy (EE). Researchers discover that, based on the parameters, it exhibits a range of structures, including singular spots that correlate to singularities in the density of states. Considering these disparate EE distributions, it is worth noting that the average EE always converges to the same number.

## Discussion

*Figure 1.** Normalized average entanglement for the complete Dicke basis and the basis consisting of an equal superposition of conjugate Dicke states as a function of 1/S _{max}(≡ 1/ log(N/2 + 1)) at half bipartition. The linear fits correspond to a + b/S_{max} with intercept a fixed to 1/2. N *

*∈ [10*

^{4}, 6 × 10^{4}] for the Dicke basis, and [4 × 10^{3}, 2.8 × 10^{4}] for the superposition basis for the shown data points. Inset shows log (1 − R^{2}), where R^{2}is the coefficient of determination of the linear fit, for different fixed values of the intercept a. At a = 1/2, the values of (1−R^{2}) are 10^{−10}and 10^{−5}for the Dicke basis and the superposition basis, respectively. Image Credit: Kumari & Alhambra, 2022The EE’s values are far from optimal. Subsystems of permutation-symmetric multi-qubit systems are also permutation-symmetric, and thus their local Hilbert space dimension is permutation-symmetric.

Researchers can show that at the thermodynamic limit, the average EE over Dicke states coincides with exactly half of the maximum feasible EE.

This form of entanglement scaling is substantially slower than that seen in generic states across the entire Hilbert space, where the entanglement entropy might reach *S _{A} = N_{A}* log 2. Researchers are now looking at the numerical behavior of normalized average EE convergence over Dicke basis in the thermodynamic limit.

The possibility of recognizing integrable collective spin models utilizing average EE in vanishing bipartitions is controlled because the thermodynamic limits of the 1-qubit average EE do not correlate for two distinct bases, which are both eigenbases of the integrable LMG model for a particular choice of parameters.

Researchers now provide the major result on the LMG model’s average EE: that in the thermodynamic limit, the uniform normalized average of the half bipartition EE of all eigenstates converges to a fixed value of 1/2, regardless of parameter choice.

*Figure 2.** (a) Normalized average entanglement ** at half bipartition p = 1/2, as a function of the inverse maximal entanglement 1/S _{max} = 1/ log(N/2 + 1), given parameters (γ_{x} γ_{y} h) in the LMG model. The linear fits correspond to a + b/S_{max} with intercept a fixed to be 1/2. The inset shows a zoomed-in version of the linear fit. (b) The plot of log(1−R^{2}) for all the parameter sets in (a), when the intercept a is fixed at different values in [0.48, 0.52]. Here, R^{2} is the coefficient of determination of the linear fit for different intercept values. For a = 1/2, (1−R^{2}) varies between 10^{−3} and 10^{−6}. The number of qubits N ∈ [2×10^{3}, 10^{4}] for the data points in this figure, and the average EE are over the eigenstates from only the positive parity sector of R_{z}^{π}. Also, the eigenbasis of HLMG for (γ_{x} γ_{y} h) = (0, 0, −1) plotted here is the Dicke basis. Image Credit: Kumari & Alhambra, 2022*

The results are depicted in Figure 2(a), where the normalized average EE declines as the system size is larger, and finite-size scaling reveals that it always hits a thermodynamic limit value of 1/2.

This is supported by Figure 2(b)’s examination of the coefficient of determination, which measures the fit quality. Researchers can observe that when the intercept is fixed at 1/2, the best linear fits happen. The figures show the outcomes for a range of N qubits up to 10^{4}.

Researchers also conducted a numerical investigation for another bipartition. They confirm this with the findings in Figure 3, where they plot the average EE as a function of the subsystem size p for *N = 2 ^{13}* and discover an almost linear development reliable with a fixed ratio.

*Figure 3.** The plot of the average EE for different bipartitions p = N _{A}/N, of the eigenstates in the LMG model corresponding to (γ_{x}, γ_{y}, h) = (5, −3, 1) and for the Dicke basis. The maximum EE in the given bipartition is also plotted for comparison. N = 8192 (≡ 2^{13}) for this plot. Up to close to the half bipartition p = 1/2, all the curves are roughly straight lines. Image Credit: Kumari & Alhambra, 2022*

Researchers analyze the entanglement distribution in the spectrum, plotted in Figure 4, to better understand the applicability of the average EE. To truly comprehend the characteristics of this distribution, researchers also investigate the properties of the LMG model. Singularities (logarithmic divergences) and discontinuities can be found in the LMG model’s density of states (DOS).

*Figure 4.** Distribution of entanglement entropy at half-bipartition for all the eigenstates of the LMG model in the positive parity sector, as a function of their eigenenergy. Zones 1 to 4 corresponds to the choice of four sets of parameters (γ _{x}, γ_{y}, h) = (1/2, 1/3, 1), (2, 1/2, 1), (5, −3, 1), and (5, 3, 1) in the model. There are dips in the entanglement distribution at eigenenergies corresponding to the singularities in the density of states. For this plot, we have N = 10^{4}. Image Credit: Kumari & Alhambra, 2022*

## Conclusion

The entanglement entropy in the eigenstates of a collective spin integrable model, the LMG model, was investigated. The average EE in non-vanishing bipartitions accumulates to a value in the thermodynamic limit that correlates to the Dicke basis, as proved analytically.

The system under consideration is one of several types of solvable/integrable models described in the literature. Free fermions, 1D interacting models like XXZ and Fermi-Hubbard, and long-range interaction models like Haldane Shastry and Calogero-Sutherland models are all examples. Each of these classes is distinct from the others in terms of both physical characteristics and methods for solving them.

The findings, combined with earlier findings for 1D systems, show that integrable systems have a far from the greatest average entanglement value, which could be used to differentiate them from chaotic systems.

Nonetheless, at the thermodynamic limit, various classes of integrable models can have varied average EE values while still being far off the maximum. This also raises the idea that the average EE’s exact value defines the group of integrable models.

## Journal Reference

Kumari, M. & Alhambra, A. M. (2022) Eigenstate entanglement in integrable collective spin models. *Quantum*, 6, p. 701. Available Online: https://quantum-journal.org/papers/q-2022-04-27-701/pdf/

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