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arXiv:2605.20661

Entangling Power: A Probe of Symmetry and Integrability in Quantum Many-Body Systems

Ian Low, Pallab Goswami

quant-phcond-mat.str-elhep-phhep-thnucl-thSubmitted May 20, 2026• Announced May 21, 2026• 0 citations
v1 · May 20, 2026v2 · Aug 24, 2026
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Abstract

The entangling power of a unitary operator quantifies its ability to generate entanglement from product states and provides a natural probe of quantum many-body dynamics. Entanglement extremization at points of enhanced symmetry has previously been observed in high-energy scattering. In this work we compute the time-averaged entangling power of anisotropic Heisenberg spin chains across two-site models and finite-size systems, as well as the entangling power of the two-magnon $S$-matrix in the thermodynamic limit. For two-site models we establish a monotonic hierarchy: the entangling power decreases as the symmetry group grows, reaching its minimum at the $SU(2)$ XXX point. Finite-size XXZ chains exhibit sharp dips at the $SU(2)$ points $Δ=\pm 1$ and the free-fermion point $Δ=0$, with the free-fermion dip decaying much more slowly with system size. In the thermodynamic limit, we decompose the two-magnon $S$-matrix into quantum logic gates -- Identity, SWAP, and $σ_z\otimesσ_z$ -- and show that the entangling power vanishes for all scattering energies at the $SU(2)$ points, where the $S$-matrix reduces to the Identity gate, while the free-fermion point achieves the maximum -- the opposite of the finite-size many-body behavior. The entangling power can serve as an {\em operator} diagnostic for symmetry and selected aspects of integrability in quantum simulations of spin-chain dynamics.

Comments
A vibe-coding project; 27 pages, 9 figures, and 4 appendices

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The entangling power of a unitary operator quantifies its ability to generate entanglement from product states and provides a natural probe of quantum many-body dynamics. Entanglement extremization at points of enhanced symmetry has previously been observed in high-energy scattering. In this work we compute the time-averaged entangling power of anisotropic Heisenberg spin chains across two-site models and finite-size systems, as well as the entangling power of the two-magnon SSS-matrix in the thermodynamic limit. For two-site models we establish a monotonic hierarchy: the entangling power decreases as the symmetry group grows, reaching its minimum at the SU(2)SU(2)SU(2) XXX point. Finite-size XXZ chains exhibit sharp dips at the SU(2)SU(2)SU(2) points Δ=±1Δ=\pm 1Δ=±1 and the free-fermion point Δ=0Δ=0Δ=0, with the free-fermion dip decaying much more slowly with system size. In the thermodynamic limit, we decompose the two-magnon SSS-matrix into quantum logic gates -- Identity, SWAP, and σz⊗σzσ_z\otimesσ_zσz​⊗σz​ -- and show that the entangling power vanishes for all scattering energies at the SU(2)SU(2)SU(2) points, where the SSS-matrix reduces to the Identity gate, while the free-fermion point achieves the maximum -- the opposite of the finite-size many-body behavior. The entangling power can serve as an {\em operator} diagnostic for symmetry and selected aspects of integrability in quantum simulations of spin-chain dynamics.
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The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=∑a=1neiθaPaU=\sum_{a=1}^{n}e^{iθ_a}P_aU=∑a=1n​eiθa​Pa​, where eiθae^{iθ_a}eiθa​ are the eigenvalues, PaP_aPa​ the corresponding eigen-projectors, and nnn is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n−1)(n-1)(n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2n−12^{n-1}2n−1 points on the torus where every relative phase is 000 or πππ, which we define as \textit{corners}. Up to an overall phase, UUU at each corner is a generalized reflection R=I−2QR=\mathbb{I}-2QR=I−2Q satisfying R2=IR^2=\mathbb{I}R2=I, where QQQ is the sum of spectral projectors whose relative phase is πππ. At the corner the entangling power is expressed in terms of seven local-unitary invariants of QQQ. A unitary gate UUU can be realized as a corner of some projector family if and only if U2∝IU^2\propto\mathbb{I}U2∝I, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N)SU(N)SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.
Abstract The entanglement evolution after a quantum quench became one of the tools to distinguish integrable versus chaotic (non-integrable) quantum many-body dynamics. Following this line of thoughts, here we propose that the revivals in the entanglement entropy provide a finite-size diagnostic benchmark for the purpose. Indeed, integrable models display periodic revivals manifested in a dip in the block entanglement entropy in a finite system. On the other hand, in chaotic systems, initial correlations get dispersed in the global degrees of freedom (information scrambling) and such a dip is suppressed. We show that while for integrable systems the height of the dip of the entanglement of an interval of fixed length decays as a power law with the total system size, upon breaking integrability a much faster decay is observed, signalling strong scrambling. Our results are checked by exact numerical techniques in free-fermion and free-boson theories, and by time-dependent density matrix renormalisation group in interacting integrable and chaotic models.
The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis {|j,m⟩}m=−jj in any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to 1/2 of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).