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arXiv:2610.17038

Entanglement Sudden Death in Gapped Ground States

Robert Ferydouni, Thomas Jackson, Samuel O. Scalet

quant-phcond-mat.stat-mechcond-mat.str-elmath-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

Spatial sudden death of entanglement refers to the disappearance of quantum correlations between subsystems after a certain distance called the entanglement length. Specifically, given a density matrix $ρ_{ABC}$ on a tripartite system $ABC$ with $B$ spatially separating $A$ and $C$, the entanglement length is the distance $\ell(A,C)$ after which the reduced density matrix $ρ_{AC}$, given by tracing out the $B$ subsystem, is separable. We show in the case of gapped ground states in any dimension that the entanglement length is finite on finite subsystems. Restricting to one-dimensional systems, we prove that normal matrix product states have a finite entanglement length independent of $A$ and $C$, and that a finite pure state with such uniform entanglement length has a matrix product state representation with bond dimension independent of $A$ and $C$. Consequently, in one dimension, entanglement length distinguishes matrix product states from all other pure states.

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Spatial sudden death of entanglement refers to the disappearance of quantum correlations between subsystems after a certain distance called the entanglement length. Specifically, given a density matrix ρABCρ_{ABC}ρABC​ on a tripartite system ABCABCABC with BBB spatially separating AAA and CCC, the entanglement length is the distance ℓ(A,C)\ell(A,C)ℓ(A,C) after which the reduced density matrix ρACρ_{AC}ρAC​, given by tracing out the BBB subsystem, is separable. We show in the case of gapped ground states in any dimension that the entanglement length is finite on finite subsystems. Restricting to one-dimensional systems, we prove that normal matrix product states have a finite entanglement length independent of AAA and CCC, and that a finite pure state with such uniform entanglement length has a matrix product state representation with bond dimension independent of AAA and CCC. Consequently, in one dimension, entanglement length distinguishes matrix product states from all other pure states.
We consider the relationship between correlations and entanglement in gapped quantum systems, with application to matrix product state representations. We prove that there exist gapped one-dimensional local Hamiltonians such that the entropy is exponentially large in the correlation length, and we present strong evidence supporting a conjecture that there exist such systems with arbitrarily large entropy. However, we then show, under an assumption on the density of states which is believed to be satisfied by many physical systems such as the fractional quantum Hall effect, that an efficient matrix product state representation of the ground state exists in any dimension. Finally, we comment on the implications for numerical simulation.
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.
For a given Hamiltonian HHH on a multipartite quantum system, one is interested in finding the energy E0{E}_{0}E0​ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}$ of HHH among all product states. For several concrete model Hamiltonians, we investigate the difference ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremath{-}{E}_{0}$, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.
Entanglement sudden death (ESD), the complete loss of entanglement in finite time, is demonstrated to occur in a class of bipartite states of qudit pairs of any finite dimension d>2d>2d>2, when prepared in so-called ``isotropic states'' and subject to multilocal dephasing noise alone. This extends previous results for qubit pairs [T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006)] to all qudit pairs with d>2d>2d>2.
The occurrence of entanglement sudden death in the evolution of a bipartite system depends on both the initial state and the channel responsible for the evolution. An extreme case is that of entanglement-breaking channels, which are channels that, acting on only one of the subsystems, drives them to full disentanglement, regardless of the initial state. In general, one can find certain combinations of initial states and channels acting on one or both subsystems that can result in entanglement sudden death or not. Neither the channel nor the initial state are responsible for this effect but their combination. In this paper we show that, for two entangled qubits, when entanglement sudden death occurs, the evolution can be mapped to that of an effective entanglement-breaking channel on a modified initial state. Our results allow to anticipate which states will suffer entanglement sudden death or not for a given evolution. An experiment with polarization-entangled photons demonstrates the utility of this result in a variety of cases.