We introduce an exact bipartite entanglement entropy between the s - and d -boson degrees of freedom in the interacting boson model (IBM), evaluated from finite- N eigenstates with good angular momentum. Unlike coherent-state approaches, the present formulation avoids mixing different angular-momentum sectors. The entanglement entropy reproduces the expected signatures of the U(5)–SU(3) [ S U ‾ (3)] and U(5)–O(6) shape transitions, while revealing a pronounced minimum at the O(6) point along the SU(3)–O(6)– S U ‾ (3) path, in sharp contrast to previous coherent-state results. We show that critical quantum fluctuations govern the local enhancement of entanglement near transition regions, whereas the entanglement hierarchy at the dynamical-symmetry limits is governed by the preserved local dynamical symmetry of the Schmidt-partitioned s - d space. The entanglement is weakest at the U(5) limit, intermediate at the O(6) limit, and strongest at the SU(3) [ S U ‾ (3)] limits, revealing an inverse correlation between preserved local dynamical symmetry and quantum entanglement in the finite- N IBM.
We introduce an exact bipartite entanglement entropy between the s - and d -boson degrees of freedom in the interacting boson model (IBM), evaluated from finite- N eigenstates with good angular momentum. Unlike coherent-state approaches, the present formulation avoids mixing different angular-momentum sectors. The entanglement entropy reproduces the expected signatures of the U(5)–SU(3) [ S U ‾ (3)] and U(5)–O(6) shape transitions, while revealing a pronounced minimum at the O(6) point along the SU(3)–O(6)– S U ‾ (3) path, in sharp contrast to previous coherent-state results. We show that critical quantum fluctuations govern the local enhancement of entanglement near transition regions, whereas the entanglement hierarchy at the dynamical-symmetry limits is governed by the preserved local dynamical symmetry of the Schmidt-partitioned s - d space. The entanglement is weakest at the U(5) limit, intermediate at the O(6) limit, and strongest at the SU(3) [ S U ‾ (3)] limits, revealing an inverse correlation between preserved local dynamical symmetry and quantum entanglement in the finite- N IBM.
Partial dynamical symmetry (PDS) is an algebraic structure in which a symmetry is neither exact nor fully broken: a subset of eigenstates keeps good quantum numbers and stays solvable while the rest mix. It is currently identified from spectroscopy. We ask whether it also leaves a purely structural mark in the eigenstates, and find, for the sd-IBM Hamiltonians studied here, that it does, though not in the magnitude of entanglement. The diagnostic is the variance of a symmetry Casimir, the label variance $\Var\,\C[G]$, which shares its zero with a block-coherence entropy and a block impurity: all three vanish when a state occupies a single Casimir sector, a single irreducible representation once conjugate representations are resolved by the cubic Casimir. Resolved state by state, the variance is numerically zero on the symmetry-selected subset and large on the mixed states, isolating the PDS states from the wavefunctions alone; the separation does not close as the boson number grows and persists at Leviatan's first- and second-order critical points, whereas the magnitude of bipartite s∣d entanglement does not separate the two families. We anchor the analysis in 168Er, reproducing Leviatan's fitted $\su3$ admixtures and his Hamiltonian-parameter-independent γ→gB(E2) ratios. The diagnostic also has an exact dynamical form, a Casimir Loschmidt echo rigid on the symmetry-pure states, and we demonstrate its readout on an IBM superconducting processor at N=3, where a single solvable/mixed exemplar reproduces the predicted pattern within finite-shot uncertainties.
Analysis of the U(5)↔SU(3) transitional region from an entanglement-based perspective
The degree of entanglement between components of a bipartite system can serve as a useful criterion for studying Quantum Phase Transitions (QPT). The von Neumann entropy and generalized concurrence serve as measures, while Schmidt number and Schmidt gap are quantifiers; together, they are key tools for identifying QPT. Using these tools, we have examined the QPT in the [Formula: see text] region within the first version of the semiclassical approximation of the Interacting Boson Model (IBM-1). Additionally, the Kullback-Leibler divergence (KLD) was employed as a criterion to compare two probability distribution functions in order to identify the transition region and the critical point. The theoretical results indicated that the Schmidt number and Schmidt gap, the von Neumann entropy, generalized concurrence, and the KLD can characterize the transition behavior and critical point (X(5)) in this region. To verify these results in real nuclei, the Dysprosium isotopic chain was selected. Using the suggested entanglement measures, the symmetry limits and the critical point were identified. The results obtained are fully consistent with experimental evidence.
The degree of chaos in the simplest interacting boson model (IBM-1) is compared with what we call the ``dynamical symmetry content'' of the system. The latter is represented by the information entropy of the eigenfunctions with respect to bases associated with dynamical symmetries of the IBM-1, and expresses thus the localization of actual eigenfunctions in these symmetry bases. The wave-function entropy is shown to be a sensitive tool for monitoring the processes of a single dynamical symmetry breaking or transitions between two and more symmetries. For the IBM-1 Hamiltonians studied here, the known features related to chaos, namely, the dependence of chaotic measures on the Hamiltonian parameters (position in the Casten triangle) and on the angular momentum, turn out to be correlated with the behavior of the wave-function entropy.
Symmetry-Reduced Variational Quantum Simulation of the U(5)→SU(3) Quantum Phase Transition in the Interacting Boson Model
The spherical-to-deformed quantum phase transition of the Interacting Boson Model (IBM) is investigated using the variational quantum eigensolver (VQE). The U(5)--SU(3) transitional Hamiltonian is studied with χ=−7/2. We develop a symmetry-preserving, minimum-qubit VQE framework for collective nuclear models, achieving a substantial reduction in qubit requirements without compromising the finite-size quantum-phase-transition physics. The transition is characterized through the normalized d-boson occupation and ground-state energy derivatives. Finite-size results are found to approach the analytic critical point ξc=8/17≃0.470588, with an independent order-parameter extrapolation yielding ξ∞=0.46986(61). The VQE reproduces ground-state energies and structural observables to numerical precision. These results demonstrate the potential of symmetry-reduced VQE for efficient quantum simulations of collective nuclear dynamics and quantum phase transitions.
Background: One of the fundamental problems of quantum information science is quantum entanglement. Recently, quantum phase transition in nuclear systems is studied in connection with quantum entanglement.Purpose: In this paper, the use of entanglement entropy as a suitable signal for the study of quantum phase transition in the even-even and odd-A nuclei is investigated. The effect of the coupling of a single fermion to a boson core on entanglement entropy is studied.Method: By use of the affine SU(1,1) Lie algebra and through the Schmidt decomposition in the framework of the interacting boson model (IBM) and interacting boson-fermion model (IBFM), entanglement entropy in the even-even and odd-A are obtained. The entanglement entropy is used for tracking and studying the shape phase transition in these nuclei.Results: The entanglement entropy in the IBM and the IBFM is calculated. The entanglement entropy values of the low-lying states of 122−−134Xe,102−−110Pd, and 123−−133Xe were calculated and analyzed. It is found that entanglement entropy is a suitable order parameter to detect shape phase transition in nuclear systems.Conclusions: The obtained results indicate that the entropy of entanglement is sensitive to the shape-phase transition between spherical and $\ensuremath{\gamma}$-unstable regions in nuclei. The results show that entanglement entropy is a powerful tool for identifying shape phase transitions in nuclei. It is found that the coupling of the single fermion with angular momentum j to the even-even system does not change the geometry imposed by the boson core performing the transition and only the entanglement entropy values have been shifted by the addition of the odd particle with respect to the even case.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2) symmetry, constrained to fixed total spin J and magnetization Jz=0. Focusing on spin-21 lattices and subsystem fractions f<21, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume V. In addition to the expected leading volume law term, we prove the existence of a 21logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2) symmetry shapes average entanglement.