Genuine quantum correlations in quantum many-body systems: a review of recent progress

Gabriele De Chiara; Anna Sanpera

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Genuine quantum correlations in quantum many-body systems: a review of recent progress

Gabriele De Chiara, Anna Sanpera

arXiv:1711.07824v2Nov 21, 2017quant-phcond-mat.othercond-mat.str-el

TL;DR

Quantum many-body research needs correlation measures beyond bipartite entanglement to characterize states and support simulation and quantum technologies. This review synthesizes multipartite entanglement, nonlocality, discord, coherence-based correlations, and quantum metrology, highlighting universal critical scaling of global discord while noting computational and dimensionality limits.

Problem
Bipartite entanglement is important for characterizing and simulating many-body systems, but other correlations in ground, excited, and thermal states also require systematic study.
Method
The paper reviews multipartite entanglement, quantum nonlocality, quantum discord, coherence-based correlations, and quantum-metrology applications in many-body systems.
Results
Global quantum discord scales with universal critical exponents near a quantum phase transition.
Takeaways & Limitations
Quantum correlations beyond bipartite entanglement provide tools for understanding condensed matter systems and may serve as resources for quantum technologies.
Takeaways & Limitations
Characterizing or checking many-body-state nonlocality is generally NP-hard, while discord becomes NP-complete to calculate as Hilbert-space dimension increases.

Abstract

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Quantum information theory has considerably helped in the understanding of quantum many-body systems. The role of quantum correlations and in particular, bipartite entanglement, has become crucial to characterise, classify and simulate quantum many body systems. Furthermore, the scaling of entanglement has inspired modifications to numerical techniques for the simulation of many-body systems leading to the, now established, area of tensor networks. However, the notions and methods brought by quantum information do not end with bipartite entanglement. There are other forms of correlations embedded in the ground, excited and thermal states of quantum many-body systems that also need to be explored and might be utilised as potential resources for quantum technologies. The aim of this work is to review the most recent developments regarding correlations in quantum many-body systems focussing on multipartite entanglement, quantum nonlocality, quantum discord, mutual information but also other non classical measures of correlations based on quantum coherence. Moreover, we also discuss applications of quantum metrology in quantum many-body systems.

1. Introduction

Quantum information theory provides a framework for studying quantum correlations in many-body systems, supporting both physical characterization and classical simulation. This review emphasizes correlations beyond bipartite entanglement and their applications to condensed matter systems.

Quantum correlations provide a precise language for characterizing many-body systems and clarify the tensor-product structure of composite states.
Quantum discord captures non-classical correlations that can occur in some mixed states without entanglement.
A resource perspective unifies entanglement, steering, discord, nonlocality, and quantum coherence as features useful for tasks unavailable to classical states.
Quantum correlations have supported tensor-network methods, including matrix product states and extensions of DMRG to dynamics, dissipation, mixed states, and two-dimensional lattices.
The review focuses on multipartite entanglement, quantum nonlocality, quantum discord, and their applications to understanding condensed matter systems.

2. A small gallery of spin models

The review introduces spin models and their phase structures, emphasizing how solvability and correlation calculations depend on dimensionality and model parameters. It covers XY, XXZ, LMG, and spin-1 systems, including the Haldane phase.

Spin-1/2 models: The general spin-1/2 Hamiltonian uses J for nearest-neighbour coupling, γ and ∆ for interaction anisotropies, and B for a longitudinal external field.
Spin-1/2 models: In one dimension, the XY model can be diagonalised with the Jordan-Wigner transformation, enabling efficient calculation of energies, correlations, reduced states, and quantum-correlation measures.
Spin-1/2 models: The XXZ model has ferromagnetic, XY-critical, and Néel phases for ∆<−1, |∆|<1, and ∆>1, respectively.
Spin-1/2 models: At the factorising field B_f = J, the ground state has no quantum or classical correlations between spins.
Spin-1/2 models: The LMG limit becomes exactly solvable when all spins interact, while mean-field solutions become more accurate as dimensionality increases.
Spin-1 models: For spin-1 systems, the Haldane phase lies between the large-D and Néel phases and is gapped, with free edge spins and nonzero string order.

3. Bipartite entanglement in many-body systems

Bipartite entanglement is central to understanding, representing, and characterizing many-body states. Its scaling motivates tensor networks, identifies phases, and exposes topological order, while its computation remains difficult in general.

Gapped one-dimensional ground states obey an entanglement area law, whereas generic multipartite pure states obey a volume law.
Bipartite entanglement underlies tensor-network representations such as MPS, PEPS, and MERA for approximating many-body ground states.
Bipartite entanglement characterizes quantum phase transitions and identifies topological quantum matter through long-range and topological entanglement.
Determining many-body ground states is generally quantum NP-hard, motivating accurate tensor-network approximations and related variational constructions.
For pure bipartite states, a single Schmidt coefficient indicates a product state, whereas more than one indicates entanglement.
The Schmidt coefficients are basis-independent eigenvalues of both reduced density matrices and fully specify the pure state relative to its bipartition.

Entanglement entropy.

For pure bipartite states, operational entanglement measures lead to entanglement entropy, which is computed directly from the Schmidt coefficients. This makes the entropy a general measure across subsystem dimensions and partitions.

Entanglement of distillation and entanglement cost bound any other entanglement measure as E_D ≤ E ≤ E_C.
For pure states, entanglement cost and distillation coincide and yield the entanglement entropy.
The entropy of entanglement is E_E(|ψ_AB⟩) = −Σ_i λ_i log(λ_i), expressed using the Schmidt coefficients.
The same entropy construction applies to every bipartition of an N-spin system, including k spins versus N−k spins.

Renyi Entropies.

Rényi entropies extend entanglement characterization beyond the von Neumann entropy by weighting different sectors of the entanglement spectrum. The section places these measures alongside mixed-state entanglement measures such as concurrence and negativity.

Rényi entropies: Unlike the entanglement entropy, Rényi entropies enhance different entanglement-spectrum sectors depending on the value of n.The entanglement entropy gives greater weight to middle Schmidt eigenvalues, whereas Rényi entropies select sectors according to n.
Mixed states: For pure bipartite states, entanglement measures are functions of the reduced-density-matrix eigenvalues, whereas mixed states generally require distinct measures.The two-qubit concurrence is a notable mixed-state exception because it is equivalent to entanglement cost.
Mixed states: Negativity measures entanglement through the negative eigenvalues of a subsystem’s partially transposed density matrix.For subsystem dimensions m and n, the number of negative partial-transpose eigenvalues is bounded by (m − 1)(n − 1).
Geometric measures: The relative entropy of entanglement uses quantum relative entropy relative to the set of separable states, while robustness quantifies the minimum amount introduced in its definition.Quantum relative entropy is not a distance because it is asymmetric and violates the triangle inequality.

Entanglement localisation.

Entanglement localisation probes long-range entanglement by measuring how much entanglement can be created between distant spins through measurements on the remaining system. Related entanglement diagnostics connect spectrum scaling and pairwise measures to phase transitions, while their applicability has explicit limitations.

Entanglement localisation: Entanglement localisation defines the maximum average entanglement that local measurements and classical communication can create between two selected spins.It targets long-range correlations rather than ordinary pairwise entanglement between nearby sites.
Entanglement localisation: The entanglement localisation length ξ_E is the typical distance over which local measurements on the remaining parties can create maximally entangled states.This length has been used to detect topological phases that other bipartite measures may miss.
Entanglement scaling: Ground-state entanglement entropy follows an area law, scaling as L^(D−1), while critical gapless points add logarithmic growth with subsystem size.This scaling reflects the locality of many-body Hamiltonians.
Entanglement spectrum: The full entanglement spectrum contains ground-state information linked to Hamiltonian symmetries and edge properties that a single entanglement measure can miss.The Schmidt coefficients arise from a bipartition and provide the spectrum used in such analyses.
Quantum phase transitions: In the Ising chain, concurrence remains continuous at the second-order transition, but its first derivative becomes singular in the thermodynamic limit.Finite-size scaling of concurrence recovers the critical exponents of the Ising transition.
Quantum phase transitions: Pairwise entanglement can signal transition order in 2-local Hamiltonians, but the correspondence may fail near multicritical points, in some models, and for finite numerical lattices.The stated correspondence applies only to transitions associated with discontinuities of the ground-state energy.
Topological order: Topological entanglement appears as a size-independent correction to the area law, but detecting it numerically in frustrated higher-dimensional systems is difficult.Alternative estimates of entanglement scaling have therefore been developed for relevant ground states.
Quantum marginal problem: Minimizing over physically realizable reduced density matrices remains generally NP-hard, so local marginals do not by themselves make ground-state energy minimization easy.The quantum de Finetti theorem instead identifies conditions under which interacting ground states can be approximated by product states.

4. Multipartite entanglement

The review surveys multipartite entanglement measures and their applications to quantum many-body systems, including geometric, global, and genuine multipartite entanglement. These measures reveal scaling behavior, phase transitions, topological order, and multipartite correlations beyond bipartite entanglement.

Geometric entanglement: Geometric entanglement quantifies a state's distance from the closest product state and is defined as an extensive quantity.A larger maximum product-state overlap indicates less entanglement.
Geometric entanglement: In critical one-dimensional systems, geometric entanglement grows logarithmically with block size until saturation at the correlation length.Numerical studies relate its scaling coefficient to the conformal field theory central charge, while analytical work derives a central-charge-dependent lower bound.
Geometric entanglement: Geometric entanglement detects the BKT transition in the XXZ chain through a cusp caused by an abrupt change in the closest product state.Two-body-correlation measures such as concurrence remain unable to detect this transition because relevant correlations are analytic at the critical point.
Geometric entanglement: For the finite 2D Ising system, geometric entanglement, entanglement entropy, and single-site entropy show sharp cusps near the critical field.Their derivatives become discontinuous near h ≈3.25, close to the quantum Monte Carlo estimate h ≈3.04.
Geometric entanglement: Geometric entanglement also characterizes topological order through an area-law term plus a constant contribution that is nonzero only in topologically ordered systems.For the toric code under string tension, geometric entanglement and bipartite Rényi-based measures are compared across the transition between polarized and topological phases.
Genuine multipartite entanglement: Global entanglement averages the single-qubit linear entropy, measuring how each qubit is entangled with the rest using single-particle measurements.Genuine multipartite entanglement is additionally studied through k-separability, energy-based witnesses, the generalized geometric measure, and rotationally invariant-state methods.

5. Quantum nonlocality in many-body systems

Quantum nonlocality in many-body systems remains difficult to characterise, but symmetry-based and energy-based Bell-inequality constructions extend detection to larger systems. These approaches reveal nonlocality in selected many-body states and models while exposing computational limits.

Foundations: Nonlocality is defined by correlations that cannot be reproduced by a local hidden variable model and is tested through Bell inequalities.Bell inequalities bound correlations between independent local observables and define facets of the local polytope.
Foundations: The relevant probability description scales exponentially: mN(dN −1) independent conditions arise for N parties, m measurements, and d outputs.This device-independent formulation uses conditional probabilities for measurement outcomes across all parties.
Many-body challenges: Although pure entangled many-body states are nonlocal, characterising or checking ground- and excited-state nonlocality is generally NP-hard.The difficulty follows from constructing Bell inequalities using all conditional probabilities.
Many-body Bell inequalities: Symmetry constraints simplify the local polytope and permit genuine many-body Bell inequalities expressed through one- and two-body correlators.This simplification enabled experimental certification of genuine multipartite Bell-inequality violations in squeezed Bose–Einstein condensates.
Many-body Bell inequalities: Two-body Bell inequalities can detect entangled states that are positive under partial transpose with respect to one partition, including Dicke-diagonal symmetric states.The construction also applies to Dicke ground states of the Lipkin–Meshkov–Glick model and has been proposed for neutral atoms in optical lattices.
Energy-based detection: Ground-state energies can test nonlocality when dynamical programming supplies the classical local-hidden-variable bound for translationally invariant spin Hamiltonians.A ground-state energy below the corresponding classical bound signals nonlocal correlations, including in models not solvable by Jordan–Wigner transformation.

6. Other quantum correlations

Quantum discord, mutual information, and coherence-based measures extend the study of correlations beyond entanglement, especially in strongly correlated lattice systems. Discord can remain informative when entanglement vanishes and can behave differently across thermal, spatial, dynamical, and dimensional settings.

Quantum discord: Quantum discord captures genuinely quantum correlations present in some separable, non-entangled states.It is defined as the difference between mutual information and measurement-based one-way classical information.
Definitions: Mutual information quantifies information in a bipartite state that is inaccessible from its reduced states, while one-way classical information depends on measurements on one subsystem.The measurement-based conditional entropy averages the information retrieved from outcomes that condition the other subsystem.
Higher-dimensional systems: Quantum discord detects phase transitions in the triangular-lattice XXZ model, whereas it is identically zero for every qubit pair in the deformed toric model.The latter model instead requires a global quantum-correlation measure to identify its critical point.
Thermal and spatial correlations: Discord can increase with temperature, remain nonzero for distant spins, and detect quantum phase transitions after two-qubit entanglement vanishes.These behaviours were reported for thermal states of Heisenberg chains and related spin systems.
Disordered systems: In random spin chains, discord and concurrence exhibit different distance-dependent correlation decays, while the connection between discord length and conventional correlation length remains unresolved.The review identifies this relationship as an open problem.
Dynamics: Discord dynamics avoids the collapses characteristic of entanglement and can signal entanglement revival when its amount is sufficiently large.Discord and entanglement may also differ qualitatively in their long-time behaviour, including ergodic versus nonergodic dynamics.
Experiments: Discord has been measured in NMR systems and solid-state spin chains, where it persists above the Curie temperature after entanglement disappears.Experiments have also observed discord dynamics between an ion’s spin and motional degrees of freedom.

6.2. Quantum correlations based on response functions

Response-function-based quantum correlation functions provide a statistical-mechanical measure that compares ordinary correlations with a perturbative response. Applied to hard-core bosons and quantum rotors, the measure captures quantum structure in imaginary time and real space and displays signatures near thermal transitions.

Definition: The quantum correlation function compares two-point fluctuations with the response of one region to a perturbation applied in another.It vanishes automatically in classical systems through the fluctuation-dissipation relation, but need not vanish quantum mechanically.
Applications: The measure was applied to hard-core bosons and quantum rotors on a two-dimensional square lattice across normal and superfluid phases.Both models undergo a BKT transition at T < TBKT, where the correlation length diverges.
Applications: Quantum Monte Carlo calculations compare QCF gQ(r), standard correlations g(r), and site-to-site discord D(r) as functions of separation and temperature.The comparison is shown for both hard-core bosons and quantum rotors.
Results: Discord and standard correlations are larger and decay more slowly than the QCF, while hard-core-boson discord decays algebraically in the superfluid phase.Discord is also singular at the BKT transition despite that transition being driven by thermal fluctuations.
Results: The QCF contains the structure of quantum correlations in imaginary time and real space and therefore carries more information than standard correlation functions in this sense.Discord instead depends on the reduced density matrix of two sites.

6.3. Global discord in spin chains

Global quantum discord extends symmetric bipartite discord to multipartite systems and can diagnose quantum criticality through universal finite-size scaling. Studies of spin chains find discord signatures at phase transitions, including temperature-dependent behavior and efficient tensor-network evaluation.

Definition and interpretation: Global quantum discord generalizes symmetric bipartite discord to multipartite states using local measurements and relative-entropy disturbances.The definition compares the global disturbance of a state with the sum of local disturbances, typically minimizing over local projective measurements.
Critical-point detection: The peak of global quantum discord detects the ferro-paramagnetic phase transition in an Ising chain with periodic boundary conditions.
Finite-size scaling: Universal scaling of the first derivative of global quantum discord was observed near criticality in transverse-field Ising, cluster-Ising, and XY chains.For the Ising chain, the scaling uses the finite-size critical field Bm and critical exponents associated with the Ising universality class.
Finite-size scaling: For the Ising transition, the correlation-length exponent is ν = 1 and the global-discord scaling exponent is approximately ω ≈−1.5.The scaled curves collapse near the critical point, providing evidence that global discord follows universal critical behavior.
Spin-1 chains: In a spin-1 chain, the first derivative of nearest-neighbour symmetric discord diverges with system size at the second-order Néel-Haldane transition but shows an inflection near the third-order Haldane-Large-D transition.The second derivative near the latter transition yields UC ≃0.9667 and ν = 1.6 ± 0.1, compatible with previous calculations.
Thermal states and computation: Temperature generally suppresses global discord, although some cases exhibit the opposite behavior, while matrix-product-state methods enable more efficient approximate block-discord calculations.

6.4. Quantum coherence-based correlations and Wigner-Yanase skew information

Quantum coherence provides correlation measures based on noncommutativity with local observables. In spin chains, derivatives and susceptibilities of coherence-based quantities can show critical behavior and finite-size scaling.

Wigner-Yanase skew information: The Wigner-Yanase skew information measures noncommutativity between a state and an observable, representing measurement uncertainty caused by coherence in the observable’s eigenbasis.A modified quantity, IL(ρ, K), was also proposed as a coherence measure.
Coherence-based correlations: A bipartite coherence-based correlation measure formed from local Wigner-Yanase skew information quantifies global quantum coherence and vanishes for classically correlated states.
Critical behavior: The first derivative of local quantum coherence diverges near the quantum phase transition in the one-dimensional anisotropic XY chain.For the same model, the divergence at criticality is logarithmic in system size.
Critical behavior: Finite-size scaling of local quantum coherence has been extended to other interacting and topological models, while finite-temperature Wigner-Yanase skew information can collapse onto a common scaling curve.
Global coherence: Coherence susceptibility diverges in several one- and two-dimensional models, including the Ising, XX, and Kitaev honeycomb models.The susceptibility is defined as the derivative of coherence with respect to a continuous state parameter λ.

6.5. Mutual information

Mutual information captures total classical and quantum correlations when bipartite entanglement entropy is insufficient, including mixed, thermal, and nonequilibrium many-body states. Its variants and multipartite extensions reveal correlation scaling, information propagation, and many-body localization.

Quantum mutual information: Mutual information quantifies total classical and quantum correlations between subsystems, making it applicable when block states are mixed or the system is at non-zero temperature.
Quantum mutual information: Quantum mutual information is non-negative, vanishes only for product states, and is monotone under completely positive trace-preserving maps.
Scaling laws: Unlike ground-state entanglement entropy for gapped short-range one-dimensional Hamiltonians, mutual information obeys an area law regardless of the energy gap or lattice dimensionality.
Nonequilibrium behavior: Nonequilibrium steady states formed by joining regions at different temperatures can violate the area law, with mutual information growing logarithmically with block size.Similar results were reported for the spin-1/2 XY chain.
Experimental measurement: An optical-lattice experiment measured block Rényi entropies and mutual information after a quench, finding volume-law behavior for both quantities.
Information variants: Rényi-Shannon mutual-information scaling is less coherent across systems than von Neumann mutual-information scaling, although critical spin chains show logarithmic Rényi-Shannon entropy growth with a central-charge-dependent prefactor.Rényi entropy with n > 1 can be negative and lacks the same operational meaning as ordinary quantum mutual information.
Multipartite total correlations: Total mutual information generalizes quantum mutual information to multipartite systems by including both quantum and classical correlations.In random Heisenberg chains, it scales approximately proportionally to N in the ergodic phase but remains constant with system size in the many-body-localized phase.
Multipartite total correlations: Total correlations are more robust to the initial state than nearest-neighbour concurrence and distinguish ergodic from many-body-localized dynamics more clearly.The review focuses on total-correlation results rather than surveying many-body localization comprehensively.

6.6. Metrology of strongly correlated systems

Quantum metrology links parameter-estimation precision to quantum Fisher information and multipartite entanglement in many-body systems. The review covers entanglement detection, critical scaling, thermometry, and sensing near dynamical phase transitions.

Quantum Fisher information: The quantum Fisher information sets a lower bound on phase-estimation uncertainty and connects metrological sensitivity with multipartite entanglement.For pure states, it equals the variance of the generator O.
Entanglement detection: For N-qubit separable states, violating the quantum Fisher-information inequality certifies some entanglement, although entangled states can also satisfy the inequality.
Criticality and metrology: Quantum Fisher information of ground and thermal states develops non-analyticities and critical scaling near quantum phase transitions.
Experimental access: Dynamic susceptibility can be related to quantum Fisher information, so experimentally accessible susceptibility measurements may reveal multipartite entanglement through inequality violations.The discussed example uses an operator such as the total magnetization of a spin chain.
Topological systems: Scaling quantum Fisher information detects multipartite entanglement in the Kitaev chain and relates it to the model’s topological properties.
Quench dynamics: After a transverse-field quench in the quantum Ising chain, the long-time asymptotic state contains genuine multipartite entanglement beyond two-partite entanglement.
Quantum thermometry: For temperature estimation, energy-based quantum Fisher information is related to Hamiltonian variance and specific heat, but sensitivity decreases at high temperature and individual-probe thermometry encounters low-temperature difficulties.Alternative schemes estimate spin-chain temperature using collective operators or local thermal susceptibilities.
Open quantum systems: Parameter estimation in open quantum systems can be enhanced near dynamical phase transitions in the steady state.

7. Discussion

The review presents quantum correlations as central to understanding, classifying, and simulating quantum many-body systems, while highlighting experimental progress and unresolved challenges, especially in two dimensions.

Quantum correlations provide a unified perspective on quantum many-body systems beyond band theory, BCS theory, and Ginzburg-Landau theory.
Entanglement theory supports efficient classical simulation techniques, while machine-learning methods are being explored for diagnosing states and their quantum correlations.
Experimental platforms including ion traps, optical lattices, atomic gases, superconducting materials, and quantum Hall systems are investigated as quantum simulators for new phases.
Characterization of phases and transitions is well understood for one-dimensional systems in terms of entanglement and other quantum correlations.
Two-dimensional systems remain substantially less clear, while quantum steering, non-signalling theory, and generalized resource theories may provide further diagnostic tools.