Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0807.4741

Entanglement and ground states of gapped Hamiltonians

Spyridon Michalakis

Published Jul 29, 2008• 0 citations
arXivHTMLPDF

Abstract

We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for $V$ a finite subset of $Z^d$, we let $H_V$ denote a Hamiltonian on $V$ with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix $ρ_A$ that describes the finite-volume restriction to a region $A$ of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of $ρ_A$ is bounded by a uniform constant $C$. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten $p$-norm of quantum channels, for $p > 1$. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.

DOI
10.48550/arxiv.0807.4741

Related papers

Entanglement in Finitely Correlated Spin States

Spyridon Michalakis, Bruno Nachtergaele

We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.

arXiv:math-ph/0606018•Physical Review Letters•Oct 4, 2006•11 citations

Ground state entanglement in one-dimensional translationally invariant quantum systems

Sandy Irani

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {Hn} for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any state in the ground space of Hn and any m, there are regions of size m with entanglement entropy Ω(min{m,n}). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Δ, where Δ is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Δ. Previously, the best known such bound was logarithmic in 1/Δ.

arXiv:0901.1107•Journal of Mathematical Physics•Feb 1, 2010•43 citations

Pure gapped ground states of spin chains are short-range entangled

Wojciech De Roeck, Martin Fraas, Bruno de O. Carvalho

We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.

math-phCross-listedarXiv:2511.14699•Nov 19, 2025•0 citations

Approximating the ground state of gapped quantum spin systems

Eman Hamza, Spyridon Michalakis, Bruno Nachtergaele +1 more

Abstract. We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z d. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset X ⊂ V the ground state projector can be approximated by the product of two projections, one supported on X and one supported on X c, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue. 1.

•Apr 29, 2009•18 citations

Quantum spin chains and random matrix theory

H. J. Wells

The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.

arXiv:1410.1666•Oct 7, 2014•4 citations

Entanglement vs. gap for one-dimensional spin systems

Matthew B. Hastings, Dorit Aharonov, Daniel Gottesman

We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap {Delta} is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small {Delta}. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in 1/{Delta}, whereas previously studied systems had the entropy of all intervals bounded by a constant times log(1/{Delta}).

Journal of Mathematical Physics•Jan 1, 2008•1 citation
We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for VVV a finite subset of ZdZ^dZd, we let HVH_VHV​ denote a Hamiltonian on VVV with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix ρAρ_AρA​ that describes the finite-volume restriction to a region AAA of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of ρAρ_AρA​ is bounded by a uniform constant CCC. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten ppp-norm of quantum channels, for p>1p > 1p>1. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.
We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.
We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {Hn} for the infinite chain. The spectral gap of Hn is Ω(1/poly(n)). Moreover, for any state in the ground space of Hn and any m, there are regions of size m with entanglement entropy Ω(min{m,n}). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Δ, where Δ is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Δ. Previously, the best known such bound was logarithmic in 1/Δ.
We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.
Abstract. We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z d. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset X ⊂ V the ground state projector can be approximated by the product of two projections, one supported on X and one supported on X c, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue. 1.
The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.
We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap {Delta} is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small {Delta}. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in 1/{Delta}, whereas previously studied systems had the entropy of all intervals bounded by a constant times log(1/{Delta}).