© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2410.00515 · Physical review. B./Physical review. B

Exploring entanglement in finite-size quantum systems with degenerate ground state

V. S. Okatev, O. M. Sotnikov, Vladimir V. Mazurenko

Published Feb 28, 2025• 2 citations

Abstract

We develop an approach for characterizing nonlocal quantum correlations in spin systems with exactly or nearly degenerate ground states. Starting with linearly independent degenerate eigenfunctions calculated with exact diagonalization we generate a finite set of their random linear combinations with Haar measure, which guarantees that these combinations are uniformly distributed in the space spanned by the initial eigenstates. Estimating the von Neumann entropy of the random wave functions helps to reveal previously unknown features of the quantum correlations in the phases with degeneracy of the ground state. For instance, spin spiral phase of the quantum magnet with Dzyaloshinskii-Moriya interaction is characterized by the enhancement of the entanglement entropy, which can be qualitatively explained by the changes in behavior of two- and three-spin correlation functions. To establish the connection between our theoretical findings and real experiments we elaborate on the problem of estimating observables on the basis of the single-shot measurements of numerous degenerate eigenstates. By the example of solving a simple Ising model it is shown that digital quantum simulations performed with quantum computers can provide accurate description of the entanglement of the degenerate systems even in the presence of noise. We also discuss the ground-state properties of degenerate fermionic and bosonic models, which can be useful for theoretical analysis of the experiments with ultracold atoms.

DOI: 10.1103/physrevb.111.054443

Related papers

EXPLORING ENTANGLEMENT IN QUANTUM SYSTEMS WITH DEGENERATE GROUND STATE

V. S. Okatev, O. M. Sotnikov, В. В. Мазуренко

We develop an approach to characterize correlations in quantum systems with degenerate ground states. It includes generation of the random linear combinations of the eigenfunctions obtained with exact diagonalization and calculation of their entanglement entropy. For tests the Ising model was used.

•Jan 1, 2025•0 citations

Entanglement Properties Of Quantum Many-Body Wave Functions

J. W. Clark, Aikaterini Mandilara, MANFRED L. RISTIG

Entanglement properties of the trial many-body wave functions in variational treatments of the transverse Ising model in two, three, and four dimensions are investigated. Based on data for magnetizations and correlation functions generated by for symmetry-breaking Hartree-Jastrow trial ground states, results are presented for several measures of bipartite entanglement, including the von Neumann entropy of a single spin in the lattice, the concurrence between two spins, and a lower bound on the localizable entanglement, as functions of coupling strength. A simple relation is established between the Hartree-Jastrow states and the nilpotent polynomial representation of multipartite entanglement. In addition, some general comments are made on the entanglement properties of classes of many-body wave functions commonly adopted in theories of strongly correlated quantum systems.

arXiv:cond-mat/0512558•Dec 21, 2005•1 citation

Shannon and entanglement entropies of one- and two-dimensional critical wave functions

Jean-Marie Stéphan, Shunsuke Furukawa, Grégoire Misguich +1 more

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant ${S}_{0}$. In $c=1$ critical systems (Tomonaga-Luttinger liquids), we find that ${S}_{0}$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 $XXZ$ chain. In a massive (crystal) phase, ${S}_{0}$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a $c=1/2$ critical point.

arXiv:0906.1153•Physical Review B•Nov 20, 2009•187 citations

Topological phases and topological entropy of two-dimensional systems with finite correlation length

Stefanos Papanikolaou, Kumar S. Raman, Eduardo Fradkin

We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.

arXiv:0709.0729•Physical Review B•Dec 19, 2007•49 citations

Entanglement of degenerate ground state for spin 1 and 1/2 pair

You‐Quan Li, Guo‐Qiang Zhu, Xuean Zhao

We study the entanglement feature of the ground state of a system composed of spin 1 and 1/2 parts. The concurrence vector is shown to be consistent with the measurement of von Neumann entropy for such system. In the light of the ground state degeneracy, we suggest a average concurrence to measure the entanglement of Hilbert subspace. The entanglement property of both a general superposition as well as the mixture of the degenerate ground states are discussed by means of average concurrence and the negativity respectively.

arXiv:quant-ph/0408155•Aug 25, 2004•0 citations

Excited-eigenstate entanglement properties of XX spin chains with random long-range interactions

Y. Mohdeb, Javad Vahedi, Stefan Kettemann

Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\ensuremath{\alpha}$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interactions. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy-level spacings, as obtained by ED for up to $N\ensuremath{\sim}18$ spins, we find indications of a delocalization transition at ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$ EE displays an algebraic growth with the subsystem size $l, {S}_{l}\ensuremath{\sim}{l}^{\ensuremath{\beta}}$, with $0<\ensuremath{\beta}<1$. We find that ${\ensuremath{\alpha}}^{*}\ensuremath{\approx}1$ coincides with the delocalization transition ${\ensuremath{\alpha}}_{c}$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to rainbow proliferation for very long-range interactions $\ensuremath{\alpha}\ensuremath{\ll}1$. We also investigate the effective temperature dependence of the EE, allowing us to study the half-chain EE of eigenstates at different energy densities, where we find that the crossover in EE occurs at ${\ensuremath{\alpha}}^{*}<1$.

arXiv:2201.10607•Physical review. B./Physical review. B•Sep 14, 2022•7 citations

We develop an approach for characterizing nonlocal quantum correlations in spin systems with exactly or nearly degenerate ground states. Starting with linearly independent degenerate eigenfunctions calculated with exact diagonalization we generate a finite set of their random linear combinations with Haar measure, which guarantees that these combinations are uniformly distributed in the space spanned by the initial eigenstates. Estimating the von Neumann entropy of the random wave functions helps to reveal previously unknown features of the quantum correlations in the phases with degeneracy of the ground state. For instance, spin spiral phase of the quantum magnet with Dzyaloshinskii-Moriya interaction is characterized by the enhancement of the entanglement entropy, which can be qualitatively explained by the changes in behavior of two- and three-spin correlation functions. To establish the connection between our theoretical findings and real experiments we elaborate on the problem of estimating observables on the basis of the single-shot measurements of numerous degenerate eigenstates. By the example of solving a simple Ising model it is shown that digital quantum simulations performed with quantum computers can provide accurate description of the entanglement of the degenerate systems even in the presence of noise. We also discuss the ground-state properties of degenerate fermionic and bosonic models, which can be useful for theoretical analysis of the experiments with ultracold atoms.

Entanglement properties of the trial many-body wave functions in variational treatments of the transverse Ising model in two, three, and four dimensions are investigated. Based on data for magnetizations and correlation functions generated by for symmetry-breaking Hartree-Jastrow trial ground states, results are presented for several measures of bipartite entanglement, including the von Neumann entropy of a single spin in the lattice, the concurrence between two spins, and a lower bound on the localizable entanglement, as functions of coupling strength. A simple relation is established between the Hartree-Jastrow states and the nilpotent polynomial representation of multipartite entanglement. In addition, some general comments are made on the entanglement properties of classes of many-body wave functions commonly adopted in theories of strongly correlated quantum systems.

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant

{S}_{0}

. In

c=1

critical systems (Tomonaga-Luttinger liquids), we find that

{S}_{0}

is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2

XXZ

chain. In a massive (crystal) phase,

{S}_{0}

is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a

c=1/2

critical point.

We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.

We study the entanglement feature of the ground state of a system composed of spin 1 and 1/2 parts. The concurrence vector is shown to be consistent with the measurement of von Neumann entropy for such system. In the light of the ground state degeneracy, we suggest a average concurrence to measure the entanglement of Hilbert subspace. The entanglement property of both a general superposition as well as the mixture of the degenerate ground states are discussed by means of average concurrence and the negativity respectively.

Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\ensuremath{\alpha}$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interactions. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy-level spacings, as obtained by ED for up to $N\ensuremath{\sim}18$ spins, we find indications of a delocalization transition at ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$ EE displays an algebraic growth with the subsystem size $l, {S}_{l}\ensuremath{\sim}{l}^{\ensuremath{\beta}}$, with $0<\ensuremath{\beta}<1$. We find that ${\ensuremath{\alpha}}^{*}\ensuremath{\approx}1$ coincides with the delocalization transition ${\ensuremath{\alpha}}_{c}$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to rainbow proliferation for very long-range interactions $\ensuremath{\alpha}\ensuremath{\ll}1$. We also investigate the effective temperature dependence of the EE, allowing us to study the half-chain EE of eigenstates at different energy densities, where we find that the crossover in EE occurs at ${\ensuremath{\alpha}}^{*}<1$.