Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems

Bei Zeng; Xie Chen; Duan-Lu Zhou; Xiao-Gang Wen

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Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems

Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen

arXiv:1508.02595v4Aug 11, 2015cond-mat.str-elquant-ph

TL;DR

The book addresses how quantum-information concepts can illuminate condensed-matter physics, especially many-body entanglement and topological order. It develops information-theoretic, Hamiltonian, error-correction, and tensor-network frameworks, with quantum conditional mutual information providing a detector of non-trivial many-body entanglement. The book’s scope remains introductory and partial, while some problems depend strongly on system dimension and realistic noise may lack nontrivial unitary subspaces.

Problem
The book asks how quantum-information ideas can supply concepts and tools for understanding condensed-matter systems, including many-body entanglement and topological order.
Source: Preface · p. 3, Structure of the Book · p. 6, §1.1 · p. 20
Method
The book develops quantum-information tools for correlation, entanglement, evolution, error correction, local Hamiltonians, and many-body systems, then applies them to condensed-matter phases.
Source: Structure of the Book · p. 6, §1.1 · p. 20, §2.1 · p. 51
Results
Quantum conditional mutual information I(A:C|B) captures irreducible tripartite correlation in gapped systems and detects non-trivial many-body entanglement, generalizing topological entanglement entropy.
Source: §5.1 · p. 126
Takeaways & Limitations
The framework connects information-theoretic quantities with the study of topological and other orders in many-body systems.
Source: Structure of the Book · p. 6, §5.1 · p. 126
The treatment is constrained by system dimension in the N-representability problem, and realistic phase-flip noise may admit no nontrivial unitary-evolution subspace.
Source: §3.1 · p. 76, §4.4.3 · p. 111

Abstract

from arXiv · show

This is the draft version of a textbook, which aims to introduce the quantum information science viewpoints on condensed matter physics to graduate students in physics (or interested researchers). We keep the writing in a self-consistent way, requiring minimum background in quantum information science. Basic knowledge in undergraduate quantum physics and condensed matter physics is assumed. We start slowly from the basic ideas in quantum information theory, but wish to eventually bring the readers to the frontiers of research in condensed matter physics, including topological phases of matter, tensor networks, and symmetry-protected topological phases.

Structure of the Book

The book develops quantum-information tools and applies them to condensed-matter problems, progressing from quantum evolution, error correction, and local Hamiltonians to entanglement, topological order, tensor networks, and SPT phases.

Structure of the Book

Quantum information concepts are introduced through correlation, entanglement, quantum evolution, circuits, and error correction before their condensed-matter applications.The early chapters also introduce the toric code and quantum-information viewpoints relevant to real quantum evolution.Source: Structure of the Book · p. 5, §2 · pp. 11–12, §3 · p. 12

4 Local Hamiltonians and Ground States

Local Hamiltonians are studied through ground-state energy, quantum marginal and N-representability problems, de Finetti extensions, and frustration-free systems.The book emphasizes locality and the hardness of determining ground-state energies, while identifying special structures that simplify practical cases.Source: Structure of the Book · pp. 5–6, §4 · p. 12

5 Gapped Quantum Systems and Entanglement Area Law

Quantum conditional mutual information generalizes topological entanglement entropy and detects non-trivial many-body entanglement in gapped systems.For sufficiently large separated regions A and C, non-zero I(A:C|B) indicates non-trivial many-body entanglement and can also detect orders beyond topological order.Source: Structure of the Book · p. 6, §5 · pp. 12–13

7 Local Transformations and Long-Range Entanglement

Topological order is introduced through its characteristic properties and then reformulated using local-unitary equivalence, gapped quantum liquids, and long-range entanglement.The framework treats topological order as stable gapped quantum liquids and symmetry-breaking order as unstable quantum liquids.Source: Structure of the Book · p. 6, §6 · pp. 13–14

8 Matrix Product State and 1D Gapped Phases

Matrix product states completely characterize 1D gapped systems, where no intrinsic topological order exists without symmetry, while tensor networks represent topological phases in 2D.The 2D discussion explains how topological order is encoded in local tensors and introduces tensor product states alongside other tensor-network forms.Source: Structure of the Book · p. 7, §7 · p. 14, §8 · pp. 14–15

10 Symmetry Protected Topological Phases

With global symmetry imposed, short-range-entangled states can occupy distinct SPT phases, with 1D bosonic SPT phases fully classified by matrix product states and higher-dimensional constructions developed systematically.The treatment includes bosonic and fermionic 1D systems and 2D SPT models using symmetry constraints and group cohomology.Source: Structure of the Book · p. 7, §10 · pp. 15–16

11 A Unification of Information and Matter.

The final chapter places quantum information and matter within a broader historical and conceptual account of physics. It argues that highly entangled qubit states may help unify light, electrons, interactions, and space.

11 A Unification of Information and Matter.

The chapter surveys four revolutions in physics before turning to the idea that physical reality may be fundamentally described by qubits rather than classical bits.It then develops an emergence approach centered on information and matter.Source: §11 · pp. 16–17
String-net liquids of qubits are presented as an example that may unify light and electrons through highly entangled quantum matter.The chapter also discusses evolving views of light, gauge theories, and where long-range-entangled quantum matter may be found.Source: §11 · p. 16

Chapter 1 Correlation and Entanglement

This chapter develops an information-theoretic framework for correlation and entanglement, progressing from classical correlations, quantum states, and reduced descriptions to quantum mutual information, Bell inequalities, and entanglement measures. It extends these concepts to composite and many-body systems, including multipartite correlations and bipartition-based analysis, while establishing basic tools for later many-body physics.

1.1 Introduction

Correlation describes dependence among parts of a composite system, while formal information-theoretic measures reveal structure beyond conventional correlation functions.The chapter motivates this broader perspective as a way to characterize phenomena such as topological phases of matter.Source: §1.1 · pp. 19–20

1.3 Quantum entanglement

The quantum discussion extends correlation through superposition and leads to a formal theory of entanglement, distinguishing it from classical correlation through Hilbert-space tensor-product structure.The chapter notes that most quantum many-body systems are entangled, with weak entanglement compatible with mean-field descriptions and strong entanglement associated with strongly correlated systems.Source: §1.1 · p. 20, §1.3 · p. 29

1.2.1 Joint probability without correlations

Classical independence is equivalent to each conditional distribution being unaffected by the other party’s outcome and to the joint distribution factorizing as pAB(ωi,λm) = pA(ωi)pB(λm).The equivalence is established in both directions and summarized as the starting point for independent probability distributions.Source: §1.2.1 · pp. 22–23

1.2.2 Correlation functions

For arbitrary random variables X and Y, the correlation function vanishes if and only if the joint probability distribution is uncorrelated.Thus, all correlation functions vanishing certifies independence, whereas one nonzero function establishes correlation; the bit example yields 0 without correlation and −1/24 with correlation.Source: §1.2.2 · pp. 25–26

1.2.3 Mutual information

Mutual information quantifies the correlation of a joint distribution with operational meaning for information transmission, without depending on the particular values assigned to X and Y.It is zero for independent variables and equals the uncertainty of either variable under perfect correlation.Source: §1.2.3 · pp. 26, 29
Shannon entropy measures the uncertainty of a probability distribution, and the binary entropy reaches its maximum when p = 1/2 while vanishing at p = 0 and p = 1.Entropy, conditional entropy, and joint entropy provide the quantities used to express mutual information.Source: §1.2.3 · pp. 26–28

1.3.1 Pure and mixed quantum states

A qubit is a state in a two-dimensional Hilbert space whose superposition differs from a classical bit, and its probability distribution depends on the chosen measurement.For |ψ⟩ = (|0⟩ + |1⟩)/√2, measuring X gives a definite value while measuring Y gives equal probabilities, showing that uncertainty is measurement-dependent.Source: §1.3.1 · pp. 29–32

1.3.2 Composite quantum systems, tensor product structure

Composite quantum systems have tensor-product Hilbert spaces, while partial tracing produces reduced states and purification represents every mixed state as part of a larger pure state.Different ensemble realizations of a mixed state correspond to different measurements on an auxiliary system but cannot be distinguished by measurements on the subsystem alone.Source: §1.3.2 · pp. 33–34

1.3.3 Pure bipartite state, Schmidt decomposition

The Schmidt decomposition expresses any pure bipartite state in orthonormal Schmidt bases with positive coefficients summing to one, and the state is uncorrelated exactly when it is a product state.The number of nonzero Schmidt coefficients satisfies n_s ≤ min{d_A,d_B}; one nonzero coefficient characterizes the product-state case.Source: §1.3.3 · pp. 35–36

1.3.4 Mixed bipartite state

For mixed bipartite states, absence of correlation is characterized by factorized joint measurement probabilities and equivalent factorized local expectations, while quantum mutual information measures total correlation.For a pure bipartite state, S_AB = 0, S_A = S_B = H({λ_i}), and I(A:B) = 2H({λ_i}).Source: §1.3.4 · p. 37

1.3.5 Bell’s inequalities

Bell’s inequalities show that quantum correlations can possess features absent from classical probability distributions, with CHSH built on a hidden-variable assumption and generalized to two-qubit Pauli measurements.The quantum version represents dichotomic variables by Pauli-operator components along Bloch-sphere directions.Source: §1.3.5 · p. 38
Quantum correlations exceed the classical CHSH bound, demonstrating that quantum correlation is stronger than its classical counterpart.Source: §1.3.5 · pp. 38–39

1.3.6 Entanglement

Entanglement is nonclassical correlation arising from superposition; for pure bipartite states, its entropy equals half the mutual information.For Schmidt coefficients {λi}, E(|ψAB⟩) = H({λi}) and I(A:B) = 2H({λi}); quantum conditional entropy may also be negative for entangled pure states.Source: §1.3.6 · pp. 39–40
The no-cloning theorem follows because linear copying of basis states would map an arbitrary superposition to an entangled state, so an unknown quantum state cannot be cloned.Source: §1.3.6 · p. 41

1.4.1 The GHZ paradox

The GHZ paradox shows that many-body quantum correlations cannot be simulated classically, because multiplying the implied local-value equations yields the contradiction −1 = 1.It is presented as a many-body analogue of Bell’s inequalities, with an even simpler form.Source: §1.4 · p. 42, §1.4.1 · p. 43

1.4.2 Many-body correlation

Irreducible k-particle correlations quantify information unavailable from lower-order reduced density matrices, yielding a nonnegative hierarchy that decomposes total correlation.For the example state ρc, CT(ρc) = 2, C2(ρc) = 2, and C3(ρc) = 0; for the GHZ state, C3(|GHZ⟩) = 1 because biparticle correlations cannot specify the relative phase between |000⟩ and |111⟩.Source: §1.4.2 · pp. 45–46

1.4.3 Many-body entanglement

Tripartite entanglement is more complex than bipartite entanglement: genuine entanglement requires nonproduct structure across every bipartition, and geometric entanglement measures distance from tripartite product states.The geometric measure is zero exactly for product states, while bipartition entropies can quantify bipartite entanglement and inform phenomena such as the entanglement area law.Source: §1.4.3 · pp. 46–47
For mixed many-body states, mutual information quantifies total correlation across bipartitions, while irreducible tripartite correlations capture correlations beyond bipartite ones.Systems with more than three particles are treated analogously, and generalized entanglement can be defined through maximal overlap with product states.Source: §1.4.3 · p. 48

1.5 Summary and further reading

The chapter develops from independence and classical correlation to entropy and mutual information, then introduces quantum entanglement as correlation beyond the classical kind.Shannon’s 1948 paper is identified as foundational for information theory, while the EPR paradox and Schrödinger’s 1935 account mark early discussions of entanglement.Source: §1.5 · p. 48
Bell’s theorem [3] establishes that hidden-variable theories cannot reproduce all quantum predictions, with the chapter presenting the CHSH inequality discussed in.The GHZ paradox extends the spirit of Bell’s theorem to more than two particles without requiring inequalities.Source: §1.5 · p. 48
The section highlights foundational developments including negative quantum conditional entropy, no-cloning, maximum entropy, irreducible correlation, separable states, and entanglement of formation.Irreducible correlation was generalized from full-rank to non-full-rank states, providing a quantum analog of the information hierarchy.Source: §1.5 · pp. 48–49
Chapters 1–3 intentionally introduce only basic quantum-information facts needed for later many-body physics, directing readers to broader references including Nielsen and Chuang, Preskill’s notes, and reviews.The chapter does not aim to present a general theory of quantum information and computation, and entanglement theory remains an active research area.Source: §1.5 · p. 49

Chapter 2 Evolution of Quantum Systems

This chapter develops quantum evolution from unitary dynamics and circuit decompositions to efficient few-body Hamiltonian simulation and open-system non-unitary dynamics. It covers universal gates, Kraus maps, Lindblad equations, and representative noise channels, connecting these topics to foundational formulations and references.

2.1 Introduction

Quantum evolution is ideally unitary, but few-body physical Hamiltonians cannot realize arbitrary many-body unitaries directly, motivating circuit decompositions and quantum simulation.Polynomial-size circuits are called efficient, and circuit depth counts layers of parallel single- and two-qubit operations; constant-depth circuits help characterize gapped quantum phases.Source: §2.1 · pp. 51–52

2.2 Unitary evolution

Schrödinger evolution is generated by a unitary operator U_S(t,t0), reducing to exp[−iH_S(t −t0)] for a time-independent Hamiltonian.The chapter introduces this ideal evolution as the basis for quantum circuit dynamics.Source: §2.2 · p. 53

2.2.1 Single qubit unitary

Single-qubit unitaries include Pauli bit- and phase-flips, Hadamard basis transformations, and rotations, with Y and Z rotations sufficient to realize any single-qubit unitary.The Hadamard operation exchanges the Z and X bases, while the Pauli X and Z operators flip the computational state and phase, respectively.Source: §2.2.1 · pp. 53–54

2.2.2 Two-qubit unitary

Single-qubit Hamiltonians with a two-qubit Ising interaction realize controlled-Z, controlled-NOT, any controlled-U, and therefore any two-qubit unitary.Equivalently, every two-qubit unitary can be written as a product of single-qubit unitaries and controlled-NOT operations.Source: §2.2.2 · pp. 55–57

2.2.3 N-qubit unitary

Any N-qubit unitary can be generated by single-qubit terms and any non-trivial entangling two-qubit interaction, although the general construction is exponentially inefficient.The construction may require approximately 4^N two-level operations and N^2 4^N single-particle and controlled-U operations.Source: §2.2.3 · pp. 59–60

2.3 Quantum Circuits

The circuit model initializes qubits in |0⟩^⊗N, applies single- and two-qubit unitaries, and measures each qubit, which suffices to implement any N-qubit unitary in principle.Open-system dynamics are generally non-unitary because the environment is inaccessible; completely positive trace-preserving maps, Kraus operators, and Markovian master equations describe such evolution and common noise processes.Source: §2.1 · pp. 52–53, §2.3 · p. 60
Circuit size counts gates and circuit depth counts parallelizable layers; constant depth is stronger than polynomial-size efficiency and is relevant to gapped quantum phases.A constant-depth circuit has depth that does not increase with N, whereas an efficient circuit need only avoid exponential growth in size.Source: §2.3 · p. 61
Few-body Hamiltonian evolutions can be simulated efficiently with polynomially many single- and two-qubit operations, with simulation precision ε requiring poly(1/ε) operations.For commuting terms, each local evolution is directly implementable; noncommuting terms are approximated using the Lie product formula.Source: §2.3 · pp. 62–63

2.4 Open Quantum Systems

Open-system evolution is non-unitary after tracing out the environment and is represented by a Kraus operator-sum map that preserves Hermiticity, unit trace, and positivity.The Kraus representation is not unique, because changing the environment basis produces an equivalent set of Kraus operators.Source: §2.4 · pp. 63–65

2.5 Master Equation

The master equation provides a differential description of open-system dynamics, and tracing out the environment yields the system density-matrix evolution.The derivation assumes Markovian dynamics, so ρ(t + dt) is completely determined by ρ(t), an approximation that is often useful despite possible environmental memory.Source: §2.5 · p. 66, §2.5.1 · p. 66

2.5.1 The Lindblad Form

Under the Markovian approximation, the Lindblad equation separates unitary Hamiltonian evolution from environmental dissipation through Lindblad operators L_k.The Hamiltonian term generates unitary evolution, while the remaining terms describe dissipation caused by interaction with the environment.Source: §2.5.1 · pp. 66–67

2.5.2 Master equations for a single qubit

Single-qubit examples include amplitude damping, phase damping, and depolarizing dynamics, demonstrating distinct forms of non-unitary evolution.The supplied examples develop amplitude and phase damping explicitly.Source: §2.5.2 · p. 68

Amplitude Damping

Amplitude damping models spontaneous decay of a two-level atom to its ground state at rate Γ, while the off-diagonal density-matrix term decays at rate Γ.The derivation assumes a vacuum reservoir and a valid Markov approximation; the decay rate agrees with the Fermi golden rule.Source: Amplitude Damping · pp. 68, 70, 71

Phase Damping

Phase damping preserves energy while decohering the relative phase, with Γ denoting the decay rate of coherence.The diagonal density-matrix elements represent populations, whereas the off-diagonal elements encode coherence; the channel exchanges no energy with the environment.Source: Phase Damping · p. 71

Depolarizing

Depolarizing noise combines three phase-damping influences and drives a quantum state toward the completely mixed state.The channel is sufficiently simple for frequent use in theoretical studies of quantum noise.Source: Depolarizing · p. 72

2.6 Summary and further reading

Ideal quantum-system evolution is unitary and supports the circuit model, where few-body Hamiltonians enable single- and two-qubit operations.The state is represented by a density matrix, while wave-function evolution follows the Schrödinger equation.Source: §2.6 · p. 73
Arbitrary N-qubit unitaries can be constructed from arbitrary single-qubit unitaries plus one non-trivial two-qubit gate, with foundational results in and.The chapter’s treatment follows Nielsen and Chuang, drawing on, while related universality results appear in and.Source: §2.6 · p. 73
The chapter situates quantum circuits, many-body quantum simulation, and open-system dynamics within foundational and textbook references, including Deutsch, Feynman, and open-system texts.It also attributes the Lie product formula to, Kraus operators to Kraus, the Lindblad form to Lindblad, and amplitude-damping master equations to.Source: §2.6 · pp. 73–74

Chapter 3 Quantum Error-Correcting Codes

This chapter develops quantum error-correcting codes through code subspaces, noise correction, measurements, and stabilizer-based methods. It progresses from foundational codes and correction criteria to stabilizer and graph-state formalisms, the toric code, cluster states for one-way quantum computing, tensor-product-state methods, and the CWS framework.

Chapter 3 Quantum Error-Correcting Codes

Quantum error-correcting codes encode information into a subspace of the N-qubit Hilbert space so decoherence-induced errors can be corrected.Measurements can preserve coherence despite generally projecting states, motivating error-correction criteria, stabilizer formalism, and the later toric-code connection.Source: Chapter 3 Quantum Error-Correcting Codes · p. 75, §3.1 · pp. 75–76

3.1 Introduction

Although general noise is non-unitary, the entangled subspace spanned by |00⟩ and |11⟩ can evolve unitarily for an artificial phase-flip process, whereas practical phase-flip noise has no nontrivial such subspace.This contrast motivates measurement-assisted error correction rather than relying only on decoherence-free subspaces.Source: §3.1 · p. 76

3.2.1 Bit flip code

For bit-flip noise, copying basis states three times reduces error probability from order p to order p2; for p < 1/2, p′err < perr, and sufficiently large r makes it arbitrarily small.Copying basis states rather than unknown states avoids the no-cloning obstacle.Source: §3.2.1 · pp. 77–79

3.2.2 Shor’s Code

Shor’s nine-qubit code combines bit- and phase-flip correction to reduce depolarizing-noise errors from order p to order p2 and correct any single error.The code uses measurements and conditional Pauli corrections before decoding.Source: §3.2.2 · pp. 79–80, §3.3 · p. 81

3.2.3 Other noise models

The same correction strategy reduces phase-damping and amplitude-damping errors from order p to p2, and handling depolarizing noise extends to arbitrary single-qubit noise through the Pauli operator basis.Orthogonal measurements eliminate relevant cross terms, allowing superpositions of correctable Kraus operators to remain correctable.Source: §3.2.3 · p. 81

3.3 Quantum error-correcting criteria, code distance

A quantum code is a subspace whose error set is correctable when corrupted basis states remain distinguishable and each error acts proportionally to identity within the code space.For uncorrelated depolarizing noise, correcting t errors yields failure scaling of order p^(t+1); code distance characterizes the smallest operator weight that violates the correction condition.Source: §3.3 · pp. 82–84

3.4 The stabilizer formalism

Stabilizer codes replace exponentially parameterized basis states with structured states specified by a small set of commuting generators.The stabilizer code is the common +1 eigenspace of an abelian Pauli subgroup, with dimension determined by the number of independent generators.Source: §3.4 · p. 84, §3.4.2 · p. 88

3.4.1 Shor’s code

Shor’s code is the common +1 eigenspace of eight commuting Pauli generators, and its stabilizer group completely specifies the two-dimensional code space.The stabilizer projection equals |0L⟩⟨0L|+|1L⟩⟨1L| because the eight independent generators restrict the code space to at most two dimensions.Source: §3.4.1 · p. 85
Shor’s code satisfies the quantum error-correction criterion for one-qubit errors but cannot correct two errors.For two errors, a product such as Z1Z4Z7 acts as the logical X and is not proportional to the code-space projector.Source: §3.4.1 · p. 87

3.4.2 The stabilizer formalism

A stabilizer code with N qubits and N−M independent generators encodes M qubits, while its distance is the smallest weight of a Pauli operator in S⊥\S.The four-qubit example has two generators, encodes two qubits, and has distance 2.Source: §3.4.2 · p. 88

3.4.3 Stabilizer states and graph states

Stabilizer states are unique common +1 eigenstates, and graph states arise when commuting stabilizer generators are constructed from the neighborhoods of graph vertices.For the complete graph of four vertices, the four generators define a corresponding four-qubit graph state.Source: §3.4.3 · pp. 89–90

3.5 Toric code

The toric code is a local stabilizer code on a square lattice that encodes two qubits and has distance r, scaling as √N for arbitrarily large N.Its logical operators are noncontractible torus cycles, and its ground state is an equal-weight superposition of all closed loops.Source: §3.5 · pp. 90–91

3.6 Summary and further reading

The chapter situates quantum error correction historically through Shor’s code, the Knill-Laflamme condition, CSS and stabilizer-code developments, graph states, and the toric code.It notes that CSS codes are a special case of stabilizer codes and that stabilizer codes and states are locally Clifford equivalent to graph codes and graph states, respectively.Source: §3.6 · pp. 92–93
Cluster states are lattice graph states introduced as resource states for one-way quantum computing.Further discussion is available in a review article.Source: §3.6 · p. 93
Tensor-product-state methods view cluster states as valence bond solids and construct resource states that outperform cluster states in some circumstances.Reviews of these developments are cited as and.Source: §3.6 · p. 93
The codeword stabilized framework is the most general known construction framework, encompassing stabilizer codes and known good non-stabilizer codes while connecting to graph codes.Follow-up work is cited as.Source: §3.6 · p. 93

Chapter 4 Local Hamiltonians and Ground States

This chapter develops local and frustration-free Hamiltonians as a framework for understanding many-body ground states, correlations, entanglement, mean-field descriptions, and computational complexity. It introduces locality, marginal and representability problems, solvable structures, complexity results, and connections to cooling, entanglement testing, and measurement-based quantum computing.

4.3 Local Hamiltonians

Local Hamiltonians involve few-body interactions among nearby degrees of freedom, and their ground states have distinctive correlation and entanglement properties.A k-body Hamiltonian has terms acting on at most k degrees of freedom, with k independent of system size N.Source: §4.1 · p. 98, §4.3 · p. 102

4.1 Introduction

The chapter studies finite-system ground-state correlations and entanglement while tracking how properties scale with system size N toward the thermodynamic limit.It also asks whether ground energies, wave functions, and degeneracies can be determined for local Hamiltonians.Source: §4.1 · p. 98

Chapter 4 Local Hamiltonians and Ground States

Frustration-free Hamiltonians have ground states minimizing every local term, making their ground-state energy easy to determine, but deciding frustration-freeness is generally hard.For spin-1/2 systems with only two-body interactions, frustration-freeness is decidable and there is always an unentangled ground state; beyond that setting, examples such as the toric code can be highly entangled.Source: §4.1 · p. 99

4.2 Many-body Hilbert space

Many-body Hilbert spaces admit particle-basis and occupation-basis descriptions, with bosonic and fermionic statistics constraining particle states but occupation bases retaining a tensor-product structure.For distinguishable particles with m single-particle states, the particle-basis Hilbert space is m^N dimensional; bosonic and fermionic states occupy constrained subspaces.Source: §4.2 · pp. 100–101

4.3.1 Examples

Local-Hamiltonian examples include the 1D transverse-field Ising model as 2-local and the 2D cluster-state Hamiltonian as 5-local.The examples illustrate locality relative to lattice geometry, including one- and two-dimensional systems.Source: §4.3.1 · pp. 102–103

4.3.2 The effect of locality

For local stabilizer Hamiltonians, locality constrains ground-space degeneracy R and code distance d, whereas non-local k-body systems can combine exponential R with d scaling as the square root of N.The toric code realizes the local bound with d scaling as the square root of N.Source: §4.3.2 · p. 104

4.4 Ground-state energy of local Hamiltonians

Even quantum computers are very unlikely to compute the ground-state energy of a general local Hamiltonian efficiently.The chapter frames this limitation through quantum computational complexity and the local Hamiltonian problem.Source: §4.1 · p. 98, §4.4 · p. 105

4.4.1 The local Hamiltonian problem

The local Hamiltonian problem asks whether the ground energy E0 lies above b or below a for a Hamiltonian whose terms each act on at most k qubits.This formalizes the computational question underlying ground-state-energy hardness.Source: §4.4.1 · p. 105
Local Hamiltonian ground-state problems are very unlikely to be efficiently solvable, even with a quantum computer, under inverse-polynomial energy precision.The hardness persists for nearest-neighbor interactions on a 2D square lattice and, with spin-23/2 particles, on a 1D chain.Source: §4.4.1 · pp. 105–106

4.4.2 The quantum marginal problem

The quantum marginal problem asks whether specified local density matrices arise as reduced states of a global state, and it is as hard as the local Hamiltonian problem.Although symmetric two-qubit extensions admit an elegant analytical condition, such simple spectral criteria generally do not extend to other cases.Source: §4.4.2 · pp. 108, 110

4.4.3 The N-representability problem

For bosonic or fermionic systems with single-particle dimension m ≥2N, the N-representability problem is as hard as the local Hamiltonian problem.The condition Trρ2 1 ≥Trρ2 2 is sufficient for 3-representability, but it fails to suffice in higher-dimensional single-particle spaces, where m controls the hardness.Source: §4.4.3 · pp. 111–112

4.4.4 de Finetti theorem and mean-field bosonic systems

For fixed mode number and marginal order, bosonic reduced states approach mixtures of product states, with finite de Finetti error at most O(m2k/N), justifying Hartree mean-field theory in the thermodynamic limit.The infinite-size de Finetti theorem extends to separable single-particle Hilbert spaces and yields the mean-field ground-state energy per particle for a broad class of interacting bosonic systems.Source: §4.4.4 · p. 113
For generic interacting bosonic systems, Hartree mean-field theory gives the exact ground-state energy, although the ground-state wavefunction may remain genuinely entangled.The result follows from the quantum de Finetti structure of bosonic reduced density matrices in the thermodynamic limit and does not depend on specific Hamiltonian properties.Source: §4.4.4 · pp. 114–115

4.5 Frustration-free Hamiltonians

A frustration-free Hamiltonian has a ground state minimizing every local interaction term, so its total ground-state energy equals the sum of the individual ground-state energies.The chapter introduces this class because it provides tractable examples while encompassing important many-body models.Source: §4.5 · p. 116, §4.5.2 · p. 118

4.5.1 Examples of frustration-free Hamiltonians

Frustration-free examples include ferromagnetic Ising chains, the toric code, and AKLT chains, whose explicitly constructed states minimize every local term.The AKLT state is also an SPT state: breaking the protecting symmetry removes its intrinsic long-range entanglement and allows conversion to a product state by a constant-depth circuit.Source: §4.5.1 · pp. 116–117

4.5.2 The frustration-free Hamiltonians problem

Determining whether a k-local Hamiltonian is frustration-free is as hard as the local Hamiltonian problem for k ≥3.The frustration-free formulation distinguishes exactly zero ground-state energy from energies above an inverse-polynomial threshold.Source: §4.5.2 · p. 118

4.5.3 The 2-local frustration-free Hamiltonians

For 2-local qubit Hamiltonians, frustration-freeness is decidable in polynomial time, although estimating the ground-state energy remains hard when frustration is present.Every such Hamiltonian has a ground state composed of single-qubit states and singlets up to invertible local transformations; this structure also enables characterization of the entire ground space.Source: §4.5.3 · pp. 119–121
The 2-local qubit structure does not extend generally: for single-particle dimension m ≥5, frustration-freeness is as hard as the Local Hamiltonian Problem, while m = 3,4 quantum hardness remains unknown.The AKLT state illustrates that higher-dimensional systems can have unique ground states with more complicated entanglement.Source: §4.5.3 · p. 121

4.6 Summary and further reading

The local Hamiltonian problem is hard from 5-local to 3-local and 2-local Hamiltonians, even with a quantum computer.Hardness persists for nearest-neighbour interactions on 2D square lattices and even 1D chains.Source: §4.6 · p. 122
Quantum marginal and N-representability problems have longstanding quantum-chemistry roots and are computationally hard.The quantum marginal problem’s history is reviewed in, while broader quantum computational complexity reviews appear in.Source: §4.6 · p. 122
Symmetric extendibility provides an entanglement test, with the two-qubit condition conjectured in and proved in.A state lacking symmetric extension is necessarily entangled.Source: §4.6 · p. 122
Classical and quantum de Finetti theorems, including finite versions, underpin applications in quantum information, while Hartree mean-field validity is reviewed in.The original theorem concerns exchangeable Bernoulli sequences, and recent quantum de Finetti developments are surveyed in.Source: §4.6 · pp. 122–123
Frustration-free Hamiltonian complexity is easy for qubit 2-local systems but hard for 4-local systems, with 3-local hardness established later.For 2-local qubit systems, ground states have product-or-singlet structure up to transformation L [7], and the full ground-state space is characterized in.Source: §4.6 · p. 123
The m = 5 case of 2-local frustration-free Hamiltonians is quantum-hard, whereas m = 3 is classically hard and quantum hardness for m = 3, 4 remains unknown.The 2-local construction in is frustration-free; the ALKT Hamiltonian originated in, and frustration-free systems also support measurement-based quantum computing.Source: §4.6 · p. 123

Chapter 5 Gapped Quantum Systems and Entanglement Area Law

This chapter examines how locality, dimensionality, thermodynamic-limit behavior, and energy gaps constrain correlations and ground-state entanglement in gapped quantum many-body systems. It develops area laws, topological entanglement entropy, and conditional mutual information—using the toric code, transverse-field Ising model, and varied region partitions—to identify non-trivial entanglement, distinguish quantum orders and phases, and reveal symmetry-breaking and symmetry-protected topological orders.

5.2.5 Entanglement

Conditional mutual information I(A:C|B) captures irreducible tripartite correlations in gapped systems and generalizes topological entanglement entropy into a detector of topological, symmetry-breaking, and SPT orders.For sufficiently large separated A and C, nonzero I(A:C|B) indicates nontrivial many-body entanglement; its dependence on region shapes also provides information about gapless systems such as critical central charge.Source: §5.1 · p. 126, §5.2.5 · p. 133

5.2 Quantum many-body systems

Gapped quantum many-body systems are studied in the thermodynamic limit, where locality and dimensionality support universal features involving the gap, correlation length, and entanglement.Local interactions have finite range, while macroscopic properties become insensitive to microscopic details as N →∞.Source: §5.2 · p. 127, §5.2.2 · p. 128, §5.2.1 · p. 128

5.2.3 Gap

A system is gapped when finitely many lowest states remain separated from excited states by a finite ∆, whereas gapless systems lack this finite separation and can have polynomially small level spacings.The lowest states may be exactly degenerate or exponentially close in N; the transverse Ising model is gapped for J ≠ B and gapless at |J| = |B|.Source: §5.2.3 · pp. 129–130

5.2.4 Correlation

For a unique gapped ground state, correlations decay exponentially and the correlation length is finite, while degeneracy can signal symmetry breaking or topological order through distinct correlation behavior.Ising ground states exhibit constant symmetry-breaking correlations in the full ground space, whereas every toric-code ground state has correlation length 0.Source: §5.2.4 · pp. 131–132

5.3.1 Entanglement area law

In locally interacting gapped systems, a subregion’s ground-state entanglement entropy scales with its boundary rather than its volume, enabling efficient descriptions such as tensor product states.In one, two, and three dimensions, the scaling is respectively constant, proportional to L, and proportional to L2; the area-law coefficient is nonuniversal.Source: §5.3.1 · pp. 134–135

5.3.2 Topological entanglement entropy

Topologically ordered gapped systems add a subleading constant γ to the area-law entropy, and this topological entanglement entropy reflects long-range entanglement.The chapter denotes the corresponding constant correction by Stopo in its overview and calls γ the topological entanglement entropy in the two-dimensional formulation.Source: §5.1 · p. 126, §5.3.2 · pp. 135–136
The toric-code ground state is an equal-weight superposition of closed-loop configurations, enabling the entanglement calculation through boundary-conditioned Schmidt decompositions.Closed loops arise because each vertex has an even number of strings, while plaquette operators create, annihilate, or move loops.Source: §5.3.2 · pp. 136–137
For the toric code, γ=log2, and this universal value characterizes topological order independently of microscopic system details.Extracting γ generally requires cancelling leading area-law and non-universal shape contributions through multipartition entropy combinations.Source: §5.3.2 · pp. 138–139

5.4.1 Quantum conditional mutual information

Conditional mutual information I(A:C|B) is always non-negative and generalizes Stopo as a detector of non-trivial many-body entanglement in sufficiently large, gapped systems.For topologically ordered systems with ring-shaped regions, I(A:C|B)=Stopo; nonzero values indicate long-range correlation between separated A and C conditioned on B.Source: §5.4.1 · pp. 140, 142

5.4.2 Toric code in a magnetic field

Ctri(ρABC) detects topological and trivial phases in the magnetic-field toric code, with nonzero irreducible tripartite correlation characterizing the topological phase.For small fields, the system is topological; for large fields, Ctri(ρABC)=0 indicates the trivial phase.Source: §5.4.2 · pp. 142–143, Figure 5.10 · p. 143
In the thermodynamic limit, Ctri(ρABC)=Stopo=2 for 0≤hx<0.328 and both vanish for hx>0.328, whereas small systems capture only one loop contribution.For small systems, the estimated transition is approximately hx=0.34, compared with the known second-order transition at approximately hx=0.328.Source: §5.4.2 · p. 143, Figure 5.11 · p. 144, Figure 5.12 · p. 144
For a toric code in a field hy, Ctri(ρABC)≤Stopo, and both quantities reveal the first-order transition near hy=1 even on a small system.As system size increases, Ctri(ρABC) and Stopo become more similar, while Stopo is easier to calculate in large systems.Source: §5.4.2 · p. 144, Figure 5.13 · p. 145, Figure 5.14 · p. 145

5.4.3 The transverse-field Ising model

The transverse-field Ising model provides a system without topological order for any B, with a two-fold degenerate ground-state space at B=0.At B=0, the ground-state space is spanned by |0⟩⊗N and |1⟩⊗N, even for finite systems.Source: §5.4.3 · p. 145
In the transverse-field Ising model, St_topo detects symmetry-breaking order, equals 1 for |GHZ⟩, signals the two quantum phases and their transitions, and equals 0 for B > 1.With periodic boundary conditions, the partition shown in Fig. 5.20(a) makes St_topo reflect only the one-dimensional area law, diverging at the critical point B = 1 in the thermodynamic limit.Source: §5.4.3 · p. 147

5.4.4 The transverse-field cluster model

For the cluster model, St_topo signals the B = 1 transition but cannot distinguish its SPT order from symmetry-breaking order, since both symmetric ground states have St_topo = 2 at B = 0.The cluster phase is SPT-protected by D2, whereas the comparison model has symmetry-breaking order; the cluster ground state is nondegenerate with periodic boundaries.Source: §5.4.4 · p. 150
For the cluster model, Sq_topo distinguishes SPT order: it equals 2 for |ψclu(0)⟩, signals the phase transition, and approaches 0 for B > 1, while remaining nearly zero for symmetry breaking.Tracing out bulk regions does not erase the boundary-carried SPT signal, unlike the symmetry-breaking signal carried by the entire exact symmetric wavefunction.Source: §5.4.4 · pp. 151–152

5.4.5 Systems with mixed orders

The mixed-order system combines Z2 symmetry breaking with D2 SPT order: St_topo probes both, whereas Sq_topo probes only the SPT component, and small B protects these orders.The thermodynamic-limit expectation is St_topo = 0 for B > 1; finite-size calculations can cap St_topo at 2 for N = 6.Source: §5.4.5 · p. 153

5.4.6 I(A:C|B) as a detector of non-trivial many-body entanglement

In 2D, analogous constructions use cylinders or disks with boundaries, extending topological entanglement-entropy diagnostics beyond 1D chains.For gapped disk ground states, symmetric local terms may be needed to avoid a gapless boundary.Source: §5.4.6 · p. 154
For disk geometries, one cutting probes both symmetry-breaking and SPT orders, whereas another probes only SPT orders.Source: §5.4.6 · p. 154
Nonzero I(A:C|B) for regions larger than the correlation length, with A and C far apart, detects non-trivial many-body entanglement.The required separation is determined by system locality.Source: §5.4.6 · p. 154
Different cuttings of A, B, and C allow I(A:C|B) to detect different quantum orders and signal distinct quantum phase transitions.Source: §5.4.6 · p. 154

A C B

This section develops topological order and long-range entanglement as frameworks beyond symmetry breaking, using entanglement measures, quantum error-correcting codes, fractional quantum Hall states, and the toric code to characterize degeneracy, quasiparticles, and fractional statistics. It also examines valid tensor-network and sequence-based descriptions of these many-body states and their topological properties.

A C B

For product states all three generalized topological entropies vanish, whereas any nonzero quantity signals non-trivial order and can probe symmetry-breaking or SPT order without known symmetry.The quantities arise from different cuttings of a disk and are based on I(A:C|B).Source: A C B · p. 155

5.5 Gapped ground states as quantum-error-correcting codes

Degenerate ground-state spaces of symmetry-breaking and SPT systems form quantum error-correcting codes with macroscopic distances for symmetry-preserving errors.The transverse Ising code has X-distance N, while the cluster-model code has X-distance N/2.Source: §5.5 · p. 157

5.6 Entanglement in gapless systems

In critical systems, I(A:C|B) remains finite rather than diverging, depends on the A,B,C geometry, and agrees with conformal-field-theory calculations.For the transverse-field Ising transition, the value near B = 1 changes with the region ratio and can encode the central charge.Source: §5.6 · p. 159

5.7 Summary and further reading

For large, separated regions, I(A:C|B) detects gapped-system orders without requiring knowledge of symmetry and probes gapless criticality through region-shape dependence.In gapless systems, varying region ratios can reveal information such as the central charge.Source: §5.7 · pp. 160–161

Chapter 6 Introduction to Topological Order

Topological order is characterized by macroscopic phenomena such as topological degeneracy, which extend phase classification beyond Landau symmetry-breaking theory.The chapter introduces topological phenomena as signatures of topologically ordered phases at macroscopic scales.Source: Chapter 6 Introduction to Topological… · p. 167

6.1.1 Phases of matter and Landau’s symmetry breaking theory

Landau’s symmetry-breaking framework explains phases through distinct organizations of particles, such as random liquid distributions versus regular crystal lattices.Liquids retain continuous translation symmetry, whereas crystals remain invariant only under selected displacements.Source: §6.1.1 · p. 167, Figure 6.1 · p. 168
Landau’s theory identifies phases with distinct organizational symmetries: spontaneous symmetry breaking lowers a system’s continuous symmetry to a discrete one, as in the liquid–crystal transition.At zero temperature, quantum phases are ground states of quantum Hamiltonians, and conventional symmetry-breaking phases can be labeled by the Hamiltonian symmetry GH and ground-state symmetry GΦ.Source: §6.1.1 · p. 168, §6.1.2 · p. 169, §6.1.4 · p. 172

6.1.2 Quantum phases of matter and transverse-field Ising model

The transverse-field Ising model has a symmetry-breaking transition at B = 2: for B > 2 its variational ground state preserves spin flip, whereas for B < 2 two degenerate ground states break it.The variational energy is symmetric under φ → π − φ, with minima at φ = π/2 for B > 2 and at φ = π/2 ± ∆φ for B < 2.Source: §6.1.2 · p. 170, Figure 6.3 · p. 170

6.1.3 Physical ways to understand symmetry breaking in quantum theory

In quantum theory, symmetry breaking is signaled by robust emergent nearly degenerate ground states in different symmetry representations, stable against symmetry-preserving perturbations.For finite systems, the symmetric true ground state can instead contain GHZ-type entanglement, which characterizes the symmetry-breaking phase.Source: §6.1.3 · p. 171

6.2.1 The discovery of topological order

Topological order extends phase classification beyond symmetry because distinct chiral spin and fractional quantum Hall states can share identical symmetries yet differ physically.Ground-state degeneracy and non-Abelian geometric phases distinguish these orders, while pure Chern-Simons theories lead to K-matrix classification of Abelian topological orders.Source: §6.2.1 · pp. 172–174

6.3 A macroscopic definition of topological order

Topological order is defined by probes unavailable to linear response: topology-dependent ground-state degeneracy and non-Abelian geometric phases of degenerate ground states [66, 65, 81, 33].The comparison with crystal and superfluid order motivates defining phases through experimentally or numerically accessible probes.Source: §6.3 · p. 175, Table 6.1 · p. 175

6.3.1 What is ‘topological ground state degeneracy’

Topological ground-state degeneracy is finite-size approximate, becomes exact in the thermodynamic limit, survives local perturbations, and depends on spatial topology.For Z2 topological order on a genus-g surface, the degeneracy is Dg = 4g [52, 69].Source: §6.3.1 · p. 176

6.3.2 What is ‘non-Abelian geometric phase of topologically degenerate states’

On a torus, shear and squeezing loops produce unitary matrices T and S that form a projective modular representation and encode topological information such as fractional statistics.S and T, together with the path-dependent total phase factor, provide a complete characterization of 2+1-dimensional topological order.Source: §6.3.2 · p. 177

6.4.1 The essence of fractional quantum Hall states

The essence of the fractional quantum Hall effect is topological order—defined by topological ground-state degeneracy and non-Abelian geometric phases—rather than symmetry breaking.FQH states are gapped two-dimensional electron ground states with rationally quantized Hall coefficient RH in units of h/e^2, and distinct states share the same symmetry.Source: §6.2.1 · p. 173, §6.4.1 · p. 178
FQH states remain topologically nontrivial without particle-number conservation or other symmetries, through topological degeneracy and non-Abelian geometric phases.Topological entanglement entropy also characterizes this symmetry-independent topological order.Source: §6.4.1 · p. 178

6.4.2 Intuitive pictures of topological order

Topological order is pictured as a globally correlated dance generated by local energy-lowering rules, producing collective quantum fluctuations and long-range entanglement.In the FQH example, local anticlockwise motion and three-step winding rules fix the Laughlin wave function ΦFQH = ∏i<j(zi −zj)3.Source: §6.4.2 · pp. 179–180

6.5 What is the significance of topological order?

Topological order has experimentally significant properties: fractional statistics and charges, protected gapless boundary modes, and emergent gauge-theory waves.These enable topological quantum memory and computation, robust conducting boundary channels, and probes such as neutron scattering.Source: §6.5 · p. 181

6.6 Quantum liquids of unoriented strings

Unoriented-string quantum liquids arise from closed strings whose amplitudes remain unchanged under local deformation and reconnection, yielding equal-weight superpositions and distinct topological orders.Changing the reconnection sign produces |ΦSem⟩, while oriented-string liquids can be constructed in a spin-1 model and yield Maxwell-equation waves.Source: §6.6 · pp. 182–183

6.7 The emergence of fractional quantum numbers and Fermi/fractional statistics

The string-liquid states |ΦZ2⟩ and |ΦSem⟩ are distinguished by topological properties including fractional statistics and topological degeneracy on compact spaces.Open-string ends behave as point-like topological excitations because their middle segments merge with ground-state strings.Source: §6.7 · p. 183, §6.7.1 · p. 184

6.7.1 Emergence of fractional angular momenta

String ends can carry fractional quantum numbers: rotation eigenstates in the |ΦZ2⟩ state include defects with spin 1/2 (mod 1) and spin 0 (mod 1).An individual string end cannot be created alone, while an open string as a whole carries non-fractionalized quantum numbers.Source: §6.7.1 · p. 184

6.7.2 Emergence of Fermi and fractional statistics

The |ΦZ2⟩ state supports emergent fermionic quasiparticles and realizes Z2 gauge theory, whereas |ΦSem⟩ has a spin-1/4 excitation and double-semion topological order.The spin-statistics theorem follows from string deformations, reconnections, exchanges, and 360° rotations.Source: §6.7.2 · pp. 185–186
Long-range entanglement in string liquids explains how purely bosonic models can produce fractional spin and fractional statistics.Fractional quantum numbers and statistics are determined by the global dancing pattern.Source: §6.7.2 · p. 186

6.8 Topological degeneracy of unoriented string liquid

On a torus, distinct global dancing patterns within four string-winding sectors produce four locally indistinguishable ground states for both Z2 and double-semion topological orders.More generally, topological degeneracy occurs when multiple global patterns satisfy the same local rules on a compact space, while the sphere has a unique pattern.Source: §6.8 · pp. 186–187

6.9.1 Toric code model and string condensation

The toric-code ground state is an equal-weight superposition of closed strings and has four-fold topological degeneracy on a torus, implying nontrivial topological order.The model is exactly soluble because its vertex and plaquette operators commute; the degeneracy is protected against local perturbations by a code distance of order L.Source: §6.9.1 · p. 189, §6.9.4 · p. 192

6.9.2 Local and topological excitations

Topological excitations are localized-energy quasiparticles that cannot be created or annihilated by any finite set of local operators, unlike local quasiparticles.Topological quasiparticle types are equivalent when they differ only by local quasiparticles; only topological excitations can carry nontrivial statistics and fractional quantum numbers.Source: §6.9.1 · p. 189, §6.9.2 · p. 190

6.9.3 Three types of quasiparticles

The Z2 topological state has three nontrivial quasiparticle types: e violates the closed-string rule, m violates the equal-amplitude rule, and ε is their bound state.Together with the trivial excitation, these form four excitation types.Source: §6.9.3 · pp. 190–191

6.9.4 Three types of string operators

Open type-I, type-II, and type-III string operators create pairs of e, m, and ε excitations, respectively.Closed strings winding around a torus connect degenerate ground states only through operators of length L, making the degeneracy robust to local perturbations.Source: §6.9.4 · pp. 191–192

6.9.5 Statistics of ends of strings

The hopping algebra of string ends determines their statistics: e and m are bosons, whereas ε is a fermion.An open string operator acts as a hopping operator for its endpoints, so exchanging particles is encoded in the operator algebra.Source: §6.9.5 · p. 195

6.10 Summary and further reading

Generalized local dancing rules encoded by valid complex tensors provide a systematic theory of certain 2+1-dimensional topological orders with gapped boundaries and yield their degeneracies and fractional statistics.The chapter frames topological order through topological degeneracy, non-Abelian geometric phases, and microscopic global-dance wave functions.Source: §6.9 · p. 187, §6.10 · p. 195
Integer sequences {S_a} describe local dancing rules in fractional quantum Hall wave functions, although not every sequence defines a valid global pattern.The sequences represent patterns of zeros and require additional conditions for global validity.Source: §6.10 · p. 195
All sequences satisfying the required conditions yield a systematic theory for a class of fractional quantum Hall states.The theory is constructed by identifying every valid sequence.Source: §6.10 · p. 196
Valid sequences determine topological properties including ground-state degeneracy and fractional charges.These properties can be calculated directly from the valid sequences.Source: §6.10 · p. 196

Chapter 7 Local Transformations and Long-Range Entanglement

This chapter develops a unified framework for understanding gapped quantum phases and topological order through local transformations, local-unitary equivalence, and renormalization. It characterizes stable gapped quantum liquids, fixed-point wave functions, ground-state spaces, and transitions between universality classes.

Chapter 7 Local Transformations and Long-Range Entanglement

Topological order arises from long-range many-body entanglement, while local unitary and stochastic local transformations organize topological and symmetry-breaking phases.The framework classifies stable gapped quantum liquids and distinguishes topological-order classes from symmetry-breaking classes through their response to stochastic local transformations.Source: Chapter 7 Local Transformations and… · p. 201, §7.1 · pp. 201–202

7.2 Quantum phases and phase transitions

For zero-temperature gapped systems, two Hamiltonians are in the same phase exactly when a smooth connecting path maintains a finite energy gap.The phase concept applies to an explicitly specified H-class, whose degrees of freedom, dimension, interactions, and symmetry constraints determine which connections are allowed.Source: §7.2 · pp. 203–204

7.3.1 Quantum phases and local unitary evolutions in ground states

Two gapped ground states belong to the same phase if and only if they are related by a finite-time local unitary evolution.Finite evolution time must remain independent of system size; otherwise phase separation can disappear in finite systems, with the limit controlled by interaction-propagation speed.Source: §7.3.1 · p. 207

7.3 Quantum phases and local unitary transformations

The resulting LU equivalence classes represent universality classes of gapped quantum phases, providing a ground-state formulation of phase classification.This formulation is obtained from the equivalence between gapped Hamiltonian paths and local-unitary connections between their ground states.Source: §7.3 · p. 205, §7.3.1 · pp. 206–207

7.3.2 Local unitary evolutions and local unitary quantum circuits

Local unitary evolution is equivalent to a constant-depth circuit of piecewise local unitary layers, with depth independent of system size.The evolution-to-circuit simulation uses grouping of commuting local terms and Trotter decomposition, while circuit and evolution forms differ in whether translation symmetry is explicitly broken.Source: §7.3.2 · p. 209

7.3.3 Local unitary quantum circuits and wave function renormalization

LU transformations support wave-function renormalization by removing short-range entanglement and flowing gapped states toward phase-specific fixed-point wave functions.Generalized local unitary transformations may add or remove direct-product degrees of freedom without changing the phase or long-distance entanglement structure.Source: §7.3.3 · p. 210
Generalized local-unitary transformations reduce local support dimensions while preserving phase equivalence, so renormalization flows remain within the same phase.These transformations are unitary on each region’s support space, with U†U and UU† acting as projectors.Source: §7.3.3 · p. 211
Wave-function renormalization removes nonuniversal local entanglement and flows toward fixed-point forms defined across lattice structures of different length scales.A product state remains invariant after redundant degrees of freedom are removed, while the toric-code state remains an equal-weight closed-loop superposition after coarse-graining.Source: §7.3.3 · pp. 211–213

7.4.1 Gapped quantum systems and gapped quantum phases

A gapped quantum system is a sequence of finite-size Hamiltonians with a size-independent spectral gap and stable low-energy-state count, while LU-connected systems define the same gapped phase.Boundary conditions and the chosen sequence of system sizes matter when defining the thermodynamic limit.Source: §7.4.1 · pp. 214–215

7.4.2 Gapped quantum liquid system and gapped quantum liquid phase

Gapped quantum liquids are shapeless gapped systems that absorb locally added product-state qubits through generalized local-unitary transformations, yielding well-defined thermodynamic limits and phases.The toric-code sequence is a gapped quantum liquid on arbitrary lattices, whereas isolated-qubit degeneracy can make a gapped system non-thermodynamic.Source: §7.4.2 · pp. 216–218

7.4.3 Topological order

Topological orders are stable gapped quantum liquid phases whose ground-state spaces form quantum error-correcting codes with macroscopic distance against local operators.A gapped quantum liquid need not be stable, and a stable gapped system need not be a gapped quantum liquid.Source: §7.4.3 · p. 219
Topological orders are universality classes of stable gapped quantum liquids, separated by gapless systems in continuous transitions or unstable gapped systems in first-order transitions.Gapped quantum systems also include nonliquid systems, while unstable gapped systems include symmetry-breaking systems and first-order transitions.Source: §7.4.3 · p. 220, Figure 7.12 · p. 220

7.5 Universality classes of many-body wave functions

The notion of topological order motivates introducing universality classes of many-body wave functions, which can also be used to understand topological orders.Source: §7.5 · p. 221

7.5.1 Gapped quantum liquid

A gapped quantum state is a sequence of ground-state spaces {V_Nk} associated with a sequence of Hamiltonians {H_Nk}, rather than a single wave function.For a gapped quantum liquid, this sequence of ground-state spaces is called a gapped quantum liquid and gives the same phase notion as the Hamiltonian formulation.Source: §7.5.1 · p. 221

Box 7.16 Gapped quantum liquid phase and topologically ordered phase

This section develops a framework for distinguishing gapped quantum-liquid phases, topological order, symmetry-breaking, and symmetry-protected phases through local transformations and long-range entanglement. It also examines GHZ-form states and surveys intrinsic, symmetry-protected, symmetry-breaking, and symmetry-enriched topological phases.

Box 7.16 Gapped quantum liquid phase and topologically ordered phase

Only gLU classes of stable gapped quantum liquids correspond one-to-one with topological orders; unstable gapped liquids provide counterexamples.Gapped quantum-liquid phases are equivalence classes under gLU transformations.Source: Box 7.16 Gapped quantum liquid phase… · p. 222

7.5.2 Symmetry breaking order

For the transverse Ising model, B > 1 is trivial, whereas 0 < B < 1 is nontrivial because ground-state degeneracy differs and is gLU-invariant.The ground states are non-degenerate for B > 1 and two-fold degenerate for 0 ≤B < 1, with degeneracy unstable to symmetry-breaking perturbations.Source: §7.5.2 · pp. 222–223
Symmetry-breaking GHZ states and topologically ordered states occupy nontrivial gLU classes, while GHZ states and product states can share trivial topological order.Thus gLU classes capture both topological order and symmetry-breaking order, not topological order alone.Source: §7.5.2 · pp. 223–224

7.5.3 Stochastic local transformations and long-range entanglement

Under gSL transformations, GHZ-form states convert to product states with finite success probability, whereas topologically ordered states do not and remain stable under small stochastic local transformations.This motivates defining only topologically ordered states as long-range entangled.Source: §7.5.3 · p. 225
gSL convertibility is a partial order rather than an equivalence relation: GHZ states convert to product states, but product states do not convert back.For |GHZ+_Nk(a)⟩, the gSL transformation can approach |0⟩⊗Nk while retaining success probability greater than |a|2.Source: §7.5.3 · p. 228
States with GHZ-form entanglement are treated as short-range entangled under gSL because their topological entanglement entropy is unstable under small stochastic local transformations.They are more entangled than product states under convertibility, yet sufficiently close to product states to lack long-range entanglement.Source: §7.5.3 · pp. 227, 229
Topologically ordered states are defined as long-range-entangled gapped quantum liquids that cannot be converted to product states by any gSL transformation.Topologically ordered phases are equivalence classes of long-range-entangled gapped quantum liquids not connected by gSL transformations.Source: §7.5.3 · p. 229
Topological entanglement entropy is invariant under gLU transformations and stable under small gSL transformations, distinguishing topological orders from symmetry-breaking orders.For symmetry-breaking orders, generalized topological entanglement entropy is stable only under symmetry-preserving small gSL transformations and is not invariant across all gSL classes.Source: §7.5.3 · pp. 229–230
Topologically ordered states cannot be converted to product states with finite size-independent probability, whereas GHZ-form states can, reflecting their different geometric entanglement.For topological states, geometric entanglement grows proportionally with system size, making projection probabilities exponentially small; GHZ-form states have constant geometric entanglement.Source: §7.5.3 · pp. 230–231

7.6 Symmetry-protected topological order

Symmetric LU equivalence produces distinct symmetry-protected topological phases among symmetry-preserving short-range-entangled states, including Haldane and topological-insulator examples.These phases lack ground-state degeneracy, fractional charge and statistics, and nonzero topological entanglement entropy, but may have symmetry-protected gapless edge excitations.Source: §7.6 · p. 232
With symmetries, the phase diagram includes symmetry-breaking, symmetry-protected, symmetry-breaking long-range-entangled, and symmetry-enriched topological phases.Without symmetry, short-range-entangled states form one phase, while long-range entanglement yields multiple intrinsic topological phases; symmetries further subdivide both categories.Source: §7.6 · p. 233

7.7 A new chapter in physics

Long-range entanglement can generate gauge interactions and Fermi statistics from qubits, motivating entanglement-based mathematical frameworks alongside geometric and field-theoretic descriptions.Tensor categories and group cohomology are identified as possible components of a framework for long-range entanglement and topological quantum matter.Source: §7.7 · p. 234

7.8 Summary and further reading

The chapter’s framework treats topological orders as stable gapped quantum liquids and uses gLU and small gSL transformations to distinguish them from symmetry-breaking orders.This establishes long-range entanglement as the defining feature of topological order within a broader quantum phase diagram.Source: §7.8 · pp. 234–235
GHZ-form states associated with symmetry-breaking phases are convertible to product states with some finite probability p.The geometrical entanglement case for the symmetric-breaking phase is discussed in.Source: §7.8 · p. 237
Examples span intrinsic topological order, symmetry-protected phases, topologically ordered phases with symmetry breaking, and symmetry-enriched topological phases.The examples include quantum Hall systems, chiral and Z2 spin liquids, quantum double and string-net models, the Haldane phase, topological insulators, topological superconductors, and fractionalized topological insulators.[55, 15, 56, 41, 53, 30, 19, 26, 13, 10, 37, 16, 2, 17, 32, 7, 38, 40, 20, 54, 22, 23, 39, 60, 36, 57, 29, 44]Source: §7.8 · p. 237

Part IV Gapped Topological Phases and Tensor

This part develops matrix product states and related tensor-network structures as efficient descriptions of one-dimensional gapped ground states, covering canonical forms, injectivity, parent Hamiltonians, observables, correlations, entanglement area laws, gauge freedom, and renormalization. It uses these tools to characterize one-dimensional gapped bosonic systems as short-range entangled and without intrinsic topological order, while connecting them to AKLT, DMRG, and extensions toward higher-dimensional tensor networks.

Chapter 8 Matrix Product State and 1D Gapped Phases

MPS fixed-point renormalization shows that one-dimensional boson and spin systems have no long-range-entangled phase or intrinsic topological order.All short-range-entangled states belong to the same phase when no special symmetry constraint is imposed.Source: Chapter 8 Matrix Product State and 1D… · p. 243, §8.1 · p. 244

8.1 Introduction

MPS efficiently represent one-dimensional gapped ground states, while injective MPS are gapped ground states of local one-dimensional Hamiltonians, enabling phase classification.The representation captures finite correlation length and entanglement area-law structure.Source: §8.1 · pp. 243–244

8.2.2 Double tensor

The double tensor uniquely determines MPS matrices up to a unitary transformation on the physical index, which changes local basis without changing entanglement structure.This equivalence supports renormalization transformations of the state.Source: §8.2.2 · pp. 247–248

8.2.3 Calculation of norm and physical observables

MPS norms and local observables are computable efficiently, with matrix multiplication scaling as ∼D3 and total observable evaluation linear in system size N.The double tensor provides the graphical and algebraic construction for these calculations.Source: §8.2.3 · p. 248

8.2.4 Correlation length

An MPS has finite correlation length if and only if the largest eigenvalue of its double tensor is non-degenerate.When the second-largest eigenvalue satisfies λ2 < 1, the correlation length is determined by λ2 and correlations decay exponentially.Source: §8.2.4 · pp. 249–250

8.2.5 Entanglement area law

MPS obey an exact one-dimensional entanglement area law, with entropy between any segment and the rest bounded by 2lnD.All one-dimensional gapped local-Hamiltonian ground states satisfy an area law and therefore admit efficient MPS descriptions, whereas generic two-dimensional representations require exponentially large matrices.Source: §8.2.5 · pp. 250–251

8.2.6 Gauge degree of freedom

The non-uniqueness of MPS representations is a gauge degree of freedom that becomes important for understanding one-dimensional symmetry-protected topological orders.This chapter itself restricts attention to systems without special symmetry constraints.Source: §8.1 · p. 244, §8.2.6 · p. 251

8.2.7 Projected entangled pair picture

MPS can be constructed as projected entangled pair states by projecting neighboring maximally entangled virtual spins of dimension D onto physical degrees of freedom of dimension d.The construction generalizes to higher-dimensional tensor product states.Source: §8.2.7 · p. 252
The projected entangled pair construction represents virtual maximally entangled spins projected into physical spins, giving an equivalent PEPS/MPS tensor description.The PEPS, MPS, and more general tensor product state formalisms are equivalent, though one representation may be more convenient.Source: §8.2.7 · p. 252

8.2.8 Canonical form

Canonical form decomposes an MPS into a finite, minimal set of short-range-correlated components, with possible block translation symmetry when transfer-matrix eigenvalues are degenerate in magnitude.A unique leading eigenvalue yields finite correlation length; equal-magnitude eigenvalues can produce translated components with enlarged unit cells.Source: §8.2.8 · p. 253

8.2.9 Injectivity

Injectivity identifies MPS representations with one canonical block and a unique dominant transfer-matrix eigenvalue, guaranteeing finite correlation length and enabling parent-Hamiltonian constructions.GHZ is noninjective because its dominant eigenvalue is twofold degenerate, whereas AKLT is injective; injectivity is also generic for random MPS.Source: §8.2.9 · pp. 253–255

8.2.10 Parent Hamiltonian

For injective MPS, projectors onto sufficiently large-block reduced-density-matrix supports define a frustration-free parent Hamiltonian with a unique ground state and finite energy gap.For noninjective MPS, the parent Hamiltonian remains gapped but has a degenerate ground space spanned by the canonical components and their superpositions.Source: §8.2.10 · pp. 255–257

8.3 Renormalization group transformation on MPS

MPS renormalization groups neighboring sites by diagonalizing recombined positive double tensors, retaining nonzero eigenvalues to remove local entanglement and flow toward a fixed point.Iterating the transformation reduces physical dimensions to those needed for entanglement with the rest of the system, from which universal properties can be determined.Source: §8.3 · pp. 257–258

8.4 No intrinsic topological order in 1D bosonic systems

All gapped one-dimensional bosonic systems without symmetry lie in one phase: their injective MPS flow to valence-bond fixed points and can be disentangled into product states by local unitaries.Subleading transfer-matrix contributions decay exponentially, leaving a fixed point whose physical degrees of freedom form neighboring entangled pairs; one local-unitary layer removes these pairs.Source: §8.4 · pp. 259–261
Local-unitary transformations disentangle fixed-point wave functions from products of entangled pairs into direct-product states.This illustrates the absence of intrinsic topological order in one-dimensional bosonic systems.Source: Figure 8.8 · p. 261

8.5 Summary and further reading

Injective matrix product states have finite correlation length and unique gapped local-parent-Hamiltonian ground states, enabling classification of one-dimensional gapped phases.A local-unitary renormalization group flows injective states to simple fixed points, establishing that all one-dimensional gapped bosonic phases are short-range entangled and lack intrinsic topological order.Source: §8.5 · p. 262
Matrix product states connect the AKLT model[1], finitely correlated states[4, 5], canonical forms[11], and DMRG as a variational matrix-product ansatz[10, 7], while area laws[6, 2] ensure efficient representations.The required inner dimension scales polynomially with system size[13], and a polynomial-time algorithm can find matrix product representations of one-dimensional gapped states[8].Source: §8.5 · p. 262

Chapter 9 Tensor Product States and 2D Gapped Phases

This chapter develops tensor-network representations for two-dimensional gapped phases, covering tensor product states, their area-law and PEPS structure, and extensions to MERA and tree tensor networks. It examines how local tensor structure encodes symmetry breaking, topological order, and symmetry-protected topological order, while supporting approximate contraction and efficient calculations.

9.1 Introduction

Tensor product states generalize matrix product states to higher dimensions, but their contraction is computationally hard and their efficiency for gapped ground states is unproven.Approximate contraction sacrifices accuracy, and it remains unclear which tensor product states describe gapped rather than gapless systems.Source: §9.1 · pp. 265–266

9.2 Tensor product states

Tensor product states are generally entangled, satisfy an entanglement area law, and can represent both short-range-entangled and topologically ordered states.The toric-code construction uses local tensors whose constraints enforce closed loops, giving equal amplitudes to all closed-loop configurations.Source: §9.1 · p. 266, §9.2.1 · pp. 268–269, §9.2.2.1 · p. 270

9.2.1 Definition and examples

A tensor product state assigns higher-rank local tensors to lattice sites, contracts shared inner indices, and differs from a product state because nontrivial tensors encode many-body entanglement.On a hexagonal lattice, each tensor has one physical index and inner indices along the lattice directions; the inner dimension is D.Source: §9.2.1 · pp. 267–268

9.2.2.1 Properties similar to matrix product state

For inner dimension D, the reduced-density-matrix rank is bounded by D^n, so tensor product states obey an entanglement area law proportional to the boundary length.They also admit double-tensor norm and observable calculations, gauge freedom, and a PEPS construction from virtual maximally entangled pairs.Source: §9.2.2.1 · pp. 269–270

9.2.2.2 Properties different from matrix product states

Injectivity no longer guarantees finite correlation length in higher dimensions: some injective tensor product states have polynomially decaying correlations, obscuring the gapped-state subset.Two-dimensional tensor-network contraction is generally inefficient, and approximate renormalization methods do not always have well-bounded error.Source: §9.2.2.2 · p. 271

9.2.2.3 Approximate calculation of local observables

Tensor renormalization uses singular-value decompositions and truncation at Dcut to reduce large two-dimensional networks to efficiently contractible ones, although approximately.On the hexagonal lattice, N rounds reduce a network with 3^(N+1) tensors to three tensors; on the square lattice, each round doubles the lattice constant and halves the tensor count.Source: §9.2.2.3 · pp. 271–273

9.3.1 Ising model

The transverse-field Ising model illustrates a Z2 symmetry-breaking phase with two degenerate polarized ground states, contrasted with a unique symmetric ground state when the field dominates.At J = 1 and B = 0, the two product states break Z2 and exchange under the symmetry; at J = 0 and B = 1, the unique state is |→→...→⟩.Source: §9.3.1 · p. 274, §9.3 · p. 274

9.3.2 Structural properties

The symmetric Ising tensor in the symmetry-breaking phase is non-injective, supported only on the |00...0⟩ and |11...1⟩ inner-index sectors, which form two orthogonal blocks without cross terms.This block structure encodes the two symmetry-breaking components in the local tensor representation.Source: §9.3.2 · p. 275
A Z2 symmetry acting on physical and inner indices can leave the full tensor-network state invariant because transformations on contracted inner indices cancel pairwise.The two tensor blocks are exchanged by the symmetry while the complete tensor remains invariant under the product of physical X operators.Source: §9.3.2 · p. 275

9.3.3 Symmetry breaking and the block structure of tensors

Tensor networks make quantum symmetry breaking visible through block decomposition, resolving how finite-size symmetric ground states can encode thermodynamic-limit symmetry breaking.Different blocks represent orthogonal short-range-correlated states, and the equal-weight superposition remains symmetric even though the individual states are not.Source: §9.3.3 · pp. 275–276
A tensor with one block represents a symmetric short-range-correlated phase, while multiple equal-amplitude blocks signal spontaneous symmetry breaking.For the Ising example, the J = 1,B = 0 ground states have two blocks and break symmetry, whereas the J = 0,B = 1 ground state has one block and does not.Source: §9.3.3 · p. 276

9.4.1 Toric code model

Tensor-product representations of the toric-code ground state place vertex tensors and link tensors on the square lattice, with the vertex tensor selecting even-parity inner-index configurations.The network uses T tensors at vertices and t tensors on links, connected according to the underlying square lattice.Source: §9.4.1 · p. 277, Figure 9.7 · p. 278

9.4.2 Structural properties

The toric-code tensors possess an inner Z2 symmetry, span the full Z2-invariant space, and therefore are non-injective rather than injective.The symmetry follows from even-parity inner configurations, and on a region the transformations cancel on contracted indices while acting on open boundary indices.Source: §9.4.2 · pp. 278–279

9.4.3 Topological property from local tensors

Defect-line Z strings create two endpoint charge excitations whose locations, but not connecting paths, matter; braiding these charges has trivial statistics.The tensor network changes only near the two endpoints, and the defect line can be freely deformed while keeping those endpoints fixed.Source: §9.4.3 · p. 280, Figure 9.10 · p. 280
A nontrivial torus loop operator maps between degenerate ground states that no local operator can distinguish, realizing a logical operation on the fourfold-degenerate ground space.The loop can be moved without changing its effect because two loops in the same direction enclose a region whose tensor network is invariant.Source: §9.4.3 · p. 281

9.4.4 Stability under symmetry constraint

Breaking the inner Z2 symmetry makes the toric-code topological entanglement entropy vanish in the thermodynamic limit, whereas sufficiently small symmetry-preserving variations retain γ = 1.The symmetry-breaking perturbation allows odd vertex configurations and open strings, while symmetry-preserving variations are generated by local physical operations.Source: |ψεtoric code⟩= ∑ · pp. 282, 284
Breaking the tensor-product tensor’s Z2 symmetry can destroy the toric code’s topological order, demonstrating the necessity of the symmetry constraint.The chapter identifies an internal tensor symmetry as essential for topological order and discusses its stability under tensor variation.Source: |ψεtoric code⟩= ∑ · pp. 284–285, §9.6 · p. 288

9.5 Other forms of tensor network representation

Beyond matrix and tensor product states, MERA and tree tensor networks provide alternative representations for many-body systems with different forms of entanglement.The chapter presents these two examples as extensions of tensor-network methods for many-body states.Source: §9.5 · p. 285

9.5.1 Multiscale entanglement renormalization ansatz

MERA uses multilayer disentanglers and isometries to represent one-dimensional gapless states whose entanglement logarithmically violates the area law.Its layers mimic renormalization-group transformations, and the network can be efficiently contracted for norms and local observables.Source: §9.5.1 · pp. 285–286

9.5.2 Tree tensor network state

A tree tensor network has a loop-free structure, and any bipartition obtained by cutting one link has Schmidt rank bounded by the inner-index dimension χ.Reduced-density-matrix calculations and time evolution scale at most linearly with the number of spins and polynomially with χ.Source: §9.5.2 · pp. 286–287

9.6 Summary and further reading

The chapter concludes that tensor product states describe two-dimensional symmetry-breaking and topological phases, while block structure tracks symmetry breaking and internal tensor symmetry supports topological order.It also surveys prior work on projected entangled pair states, approximate contraction, tensor-network toric-code and string-net states, gauge symmetry, and stability conditions.Source: §9.6 · p. 288

Chapter 10 Symmetry Protected Topological Phases

This chapter develops symmetry-based classifications of short-range-entangled phases, focusing on protected edge states and projective representations in one-dimensional bosonic systems. It also extends the framework to fermions, broader symmetries and dimensions, and exactly solvable SPT constructions.

Chapter 10 Symmetry Protected Topological Phases

SPT phases are gapped and short-range entangled, but symmetry protects boundary states that remain degenerate or gapless unless symmetry is broken.Without enforcing symmetry, their ground states can be smoothly deformed into a trivial product state.Source: Chapter 10 Symmetry Protected… · p. 291, §10.1 · p. 291

10.1 Introduction

Matrix product states classify 1D bosonic SPT phases by relating symmetry-preserving fixed-point states to projective representations of G.The classification treats spin and boson systems alike because both consist of commuting local degrees of freedom.Source: §10.1 · p. 292, §10.2.1 · p. 294
Fermionic SPT phases are classified by mapping 1D fermion systems to spins through the Jordan-Wigner transformation, with fermion parity providing an unbreakable intrinsic Z2 symmetry.A bosonic Z2 symmetry-breaking phase maps to a fermionic topological phase with nontrivial edge states.Source: §10.1 · p. 292

10.2.1 Examples

The AKLT chain has a symmetric, unique gapped periodic ground state but fourfold-degenerate open-chain ground states from uncoupled spin 1/2 edge modes protected by SO(3).A 2π rotation gives spin 1/2 edges a −1 phase, preventing their symmetric deformation to trivial spin-0 edges without closing the bulk gap.Source: §10.2.1 · pp. 293–294
The cluster-state chain has twofold-degenerate edge states and a fourfold-degenerate open-chain ground space protected by Z2 × Z2 symmetry.The two Z2 generators act anticommutatively on each edge, so the degeneracy cannot be removed without symmetry breaking or a bulk phase transition.Source: §10.2.1 · pp. 293–294

Box 10.3 Classification of bosonic symmetry protected topological phases

1D bosonic gapped phases preserving an on-site symmetry group G are labeled by projective representations of G, with additional U(1) distinctions from one-dimensional representations.The classification applies when the one-dimensional representations α(G) form a finite group; for G = U(1), further classification according to α(U(1)) exists.Source: Box 10.3 Classification of bosonic… · p. 295

10.2.2.1 Symmetric RG transformation and fixed point

Symmetry-preserving renormalization keeps track of the on-site representation and maps gapped states to fixed points whose two site degrees of freedom carry projective representations.Although each full site transforms linearly and the total state remains invariant, individual degrees of freedom form neighboring singlets projectively.Source: §10.2.2.1 · pp. 295, 296, 298

10.2.2.2 Example

In the MPS fixed point, the matrices M(g) multiply up to a phase, establishing a projective representation that distinguishes nontrivial SPT order from a trivial product state.For the trivial Ising symmetric state, the projective factor is ω(g,h) = 1; for AKLT, the fixed-point action is generated by an effective spin 1/2 and neighboring virtual spins form singlets.Source: §10.2.2.1 · pp. 297–298, §10.2.2.2 · pp. 299–300
For SO(3), spin-1/2 edge modes form projective rather than linear representations, producing fourfold open-chain ground-state degeneracy protected by spin rotation symmetry.A symmetry-breaking magnetic field removes this degeneracy.Source: §10.2.2.2 · p. 300

10.2.2.3 Projective representation and edge state

Nontrivial projective representations, classified by H2(G,U(1)), require multidimensional edge spaces and produce symmetry-protected degeneracy around a gapped, nondegenerate bulk.Equivalent factor systems define the same projective class, while tensor products add their classes.Source: §10.2.2.3 · pp. 301–302

10.2.2.4 Equivalence between symmetric fixed point states

One-dimensional bosonic SPT phases with on-site symmetry G correspond one-to-one with projective-representation classes: equal classes connect by symmetric LU transformations, while different classes cannot.The distinction persists under symmetric LU transformations as long as the system remains gapped and short-range correlated.Source: §10.2.2.4 · pp. 303–304

10.2.3 Time reversal symmetry

Time-reversal-invariant one-dimensional bosonic systems have two symmetry-preserving phases, distinguished by m = ±1, with m = −1 yielding nontrivial projective edge degeneracy.Time reversal is anti-unitary, and the two phases are equivalent precisely when they share the same m value.Source: §10.2.3 · pp. 305–306

10.2.4.1 Translation invariance only

Translation-invariant one-dimensional gapped ground states form a single phase when no other symmetries are imposed.A symmetry-preserving smooth LU or adiabatic path connects translation-invariant short-range-correlated states to a disentangled product-state fixed point without closing the gap.Source: §10.2.4.1 · p. 307

10.2.4.2 Translation invariance and on-site symmetry

With translation and a nontrivial on-site projective symmetry, a one-dimensional symmetric ground state cannot be short-range correlated.If both symmetries remain unbroken, the system is gapless; if either breaks, the ground state is degenerate.Source: §10.2.4.2 · p. 308

Box 10.8 Nonexistence of short range correlated half integer spin chain

This section develops the classification and construction of one-dimensional fermionic and two-dimensional bosonic symmetry-protected topological phases using symmetry constraints, tensor networks, and group cohomology. It connects bulk tensor-network symmetry actions to anomalous boundary physics and summarizes the resulting scope across interacting bosonic, spin, and fermionic systems.

Box 10.8 Nonexistence of short range correlated half integer spin chain

1D half-integer-spin systems with translation and SO(3) symmetry are always gapless or have degenerate ground states.If translation symmetry is explicitly broken by dimerization, such chains can instead be gapped and short-range correlated.Source: Box 10.8 Nonexistence of short range… · pp. 308–309
For translation-invariant one-dimensional bosonic systems with on-site linear symmetry G, gapped symmetric phases are classified by (ω,α), combining a projective-representation class and a one-dimensional representation.For G = Z_n there are n phases, while integer-spin SO(3) chains have two, including the AKLT and trivial product-state phases.Source: Box 10.8 Nonexistence of short range… · pp. 309–310
A one-dimensional conserved-boson system at fractional filling must be gapless if it preserves both U(1) and translation symmetry.In contrast, two-dimensional fractional quantum Hall states of bosons can be gapped while preserving both symmetries.Source: Box 10.8 Nonexistence of short range… · p. 310

10.2.4.3 Translation invariance and parity symmetry

Translation-invariant one-dimensional bosonic systems with parity symmetry have four distinct gapped symmetric phases labeled by α(P) = ±1 and β(P) = ±1.The four classes cannot be connected by parity-preserving local-unitary transformations; the AKLT/Haldane phase is protected by parity symmetry.Source: §10.2.4.3 · pp. 311–312

10.3 Topological phases in 1D fermion systems

Fermionic one-dimensional phases can be classified by Jordan–Wigner mapping to spin systems, followed by spin-phase classification and mapping back to fermions.The approach applies directly to open chains; with periodic boundaries, the mapping is effectively local for gapped systems with fixed total fermion parity.Source: §10.3 · pp. 312–313, §10.3.1 · p. 314

10.3.1 Jordan Wigner transformation

The Jordan–Wigner transformation maps local bulk fermion terms to local spin terms, but boundary-crossing terms become nonlocal unless total fermion parity is fixed.Larger local Hilbert spaces can be embedded into a fermionic mode tensor a bosonic space before applying the transformation.Source: §10.3.1 · p. 314

10.3.2 Fermion parity symmetry only

With only fermion parity symmetry, there are two fermionic phases: a topologically trivial phase and a topological phase with Majorana edge modes.The topological phase corresponds under Jordan–Wigner duality to the spin-chain symmetry-breaking phase; an open Majorana chain has two boundary zero modes and a two-fold ground-state degeneracy, whereas the periodic chain has a unique ground state.Source: §10.3.2 · pp. 315–316
With fermion parity symmetry alone, one-dimensional fermion systems have two gapped phases: one with a Majorana edge mode and one without.Their distinct edge states cannot be connected by physical fermionic perturbations without closing the gap.Source: §10.3.2 · p. 317

10.3.3 Fermion parity and T 2 = 1 time reversal

With fermion parity and T^2 = 1 time reversal, one-dimensional fermion systems have eight different gapped phases, including four symmetric phases and phases with Majorana boundary modes.The classification combines symmetry-preserving, time-reversal-breaking, and Majorana-boundary cases.Source: §10.3.3 · pp. 317–318

10.3.4 Fermion parity and T 2̸ = 1 time reversal

For T^2 = Pf time reversal, one-dimensional fermion systems have two gapped phases, including a time-reversal symmetry-protected topological phase.The two phases comprise one symmetric phase and one time-reversal-breaking phase with a Majorana boundary mode.Source: §10.3.4 · pp. 318–319

10.3.5 Fermion number conservation

With U(1) charge conservation, one-dimensional fermion systems have infinitely many phases labeled by different average fermion numbers per site.U(1) cannot break and has no nontrivial projective representation, while its infinite family of one-dimensional representations labels the phases.Source: §10.3.5 · p. 319

10.4 2D symmetry protected topological order

Two-dimensional bosonic SPT phases can have gapped short-range-entangled bulks and symmetry-protected gapless boundaries, as illustrated by the AKLT and CZX models.The AKLT edge requires translation and spin-rotation symmetry, whereas the CZX edge is robust under disorder with an internal symmetry preserved.Source: §10.4 · pp. 319–320

10.4.1.1 Bulk definition and boundary state

The 2D AKLT state is nontrivial only when translation and spin-rotation symmetries are both preserved; spin rotation alone permits a gapped dimerized boundary.Its bulk is short-range entangled and symmetric, while boundary spin-1/2 degrees of freedom form a gapless chain when translation remains unbroken.Source: §10.4.1.1 · pp. 320–321

10.4.1.2 Tensor network representation

In the AKLT tensor network, physical spin rotations induce gauge transformations on virtual indices that form projective SO(3) representations, encoding the SPT boundary structure.The displayed tensor is a zero-correlation-length representative; tensors in the same phase may have finite correlation length and larger bond dimension while transforming similarly.Source: §10.4.1.2 · pp. 322–323

10.4.2 2D CZX model

The CZX model realizes nontrivial two-dimensional SPT order protected only by on-site Z2 symmetry, whose non-onsite boundary symmetry forbids a gapped symmetric edge.Its bulk uses plaquette entanglement and a nonfactorizable on-site symmetry, while local Hamiltonian terms preserve the Z2 symmetry.Source: §10.4.2 · pp. 323–324, §10.4.2.1 · pp. 324–325

10.4.2.1 Bulk definition

The closed-surface CZX ground state is unique, gapped, symmetric, and short-range entangled, although symmetry-preserving local unitaries cannot disentangle its plaquettes.Small symmetry-preserving perturbations retain the gap, symmetry, and short-range entanglement; the obstruction appears only with a boundary.Source: §10.4.2.1 · pp. 325–326

10.4.2.2 Boundary description

For two boundaries together, entangled effective-spin pairs form a symmetric short-range-entangled state, unlike the isolated-boundary case.Projecting each paired pair to the |˜0˜0⟩, |˜1˜1⟩ subspace makes the symmetry onsite, consistent with recovering the closed-surface state after gluing.Source: §10.4.2.2 · p. 328

10.4.2.4 Matrix product unitary operators and its relation to 3 cocycle

For CZX, φ(CZX,CZX,CZX) = −1; this minus one cannot be removed by redefining phases, so φ is a nontrivial 3-cocycle for the Z2 group.Source: §10.4.2.4 · p. 331

10.4.2.5 Nontrivial 3-cocycle of MPUO and nonexistence of SRE symmetric state

A one-dimensional MPUO symmetry can support a symmetric short-range-entangled state only for a trivial 3-cocycle, proving the CZX model has nontrivial 2D SPT order.The proof uses matrix-product-state gauge reductions and shows that the nontrivial cocycle equation cannot be satisfied by any phase choices.Source: §10.4.2.5 · pp. 331, 333, 334
The isolated CZX boundary cannot have a gapped symmetric ground state: it must either be gapless or break the Z2 symmetry.The effective boundary symmetry is non-onsite, and the transverse-field cluster Hamiltonian provides an explicit gapless example.Source: §10.4.2.2 · p. 327, §10.4.2.5 · p. 333

10.4.2.6 Tensor network representation

The CZX ground state has a simple tensor-network representation: each tensor part is nonzero exactly when its physical and two inner indices share the same value.This network represents the tensor product of plaquettes in the state |0000⟩+|1111⟩.Source: §10.4.2.6 · p. 334
The CZX tensor’s Z2 gauge transformation is not even projective, yet it corresponds to the effective boundary symmetry action for states in the same SPT phase.Applying the symmetry twice is trivial on the full tensor, while the boundary retains the associated nontrivial action.Source: §10.4.2.6 · pp. 334–335

10.5.1 Group cohomology

Group cohomology defines cocycles and coboundaries whose quotient H^n(G,U(1)) organizes symmetry data, with H^1 giving one-dimensional representations and H^2 classifying projective representations.The construction extends to anti-unitary symmetries through their nontrivial action on U(1) phases.Source: §10.5.1 · pp. 335–337

10.5.2 SPT model from group cohomology

Using an H^{n+1} group cocycle, the construction produces n-dimensional SPT models whose boundaries carry the corresponding projective or matrix-product-operator symmetry action.For one-dimensional models, paired projective representations yield a linear on-site representation and boundary factors ω and ω−1; in two dimensions, 3-cocycles produce nontrivial SPT phases precisely when the cocycle is nontrivial.Source: §10.5.2 · pp. 337–339

10.6 Summary and further reading

The chapter establishes complete 1D SPT classification, nontrivial projective edge representations, and systematically constructed higher-dimensional phases whose 2D edges remain gapless unless symmetry breaks.One-dimensional fermionic classifications follow through the Jordan–Wigner transformation.Source: §10.6 · p. 339
The discussion situates these results within the Haldane, AKLT, CZX, and fermionic topological-phase literature, while noting that interacting fermionic SPT phases remain incompletely understood.Key prior developments include the Haldane gap and edge spin-1/2 states [18, 14, 31], AKLT’s solvable point [2], and the CZX and group-cocycle constructions,.Source: §10.6 · pp. 339–340

Chapter 11 A Unification of Information and Matter

This chapter presents long-range entanglement as a unifying foundation for matter, space, gauge fields, emergent particles, and topological quantum matter. It develops this viewpoint through emergence and string-net liquids, while surveying realizations of topological order, material challenges, and prospects for artificial vacua.

Chapter 11 A Unification of Information and Matter

Long-range-entangled qubits may unify matter by producing gauge fields as entanglement fluctuations and fermions as entanglement defects.The chapter suggests that this space of long-range-entangled qubits could underlie nearly all elementary particles and serve as an origin of matter.Source: Chapter 11 A Unification of… · p. 345

11.1 Four revolutions in physics

Physics progresses through successive unifications: particles under Newtonian mechanics, fields under Maxwell theory, spacetime and gravity under relativity, and particle-wave matter under quantum theory.The chapter identifies a possible fourth stage in which matter and space arise from information or qubits.Source: §11.1 · pp. 345–346, §11.1.1 · p. 347, §11.1.2 · p. 348

11.1.1 Mechanical revolution

Newtonian mechanics unified terrestrial and planetary motion, while Maxwell theory unified electricity, magnetism, and light as electromagnetic waves.These revolutions replaced apparently separate phenomena with common particle- and field-based descriptions.Source: §11.1.1 · pp. 346–347, §11.1.2 · pp. 348–349

11.1.3 Relativity revolution

Relativity unified space with time and gravity with spacetime geometry, while connecting gauge fields to the relativity of local directions and quantum phases.The chapter also describes gravitational waves as propagating distortions of space and electromagnetism as a gauge-field phenomenon.Source: §11.1.3 · pp. 350–352

11.1.4 Quantum revolution

Quantum theory unifies particle-like and wave-like matter while introducing entangled states with no classical analogue.Experiments show that electrons behave as waves and light as photons, requiring quantum theory beyond classical particle and field descriptions.Source: §11.1.4 · pp. 352–353
Geometry may be emergent rather than fundamental because quantum gravity excludes physical separations below the Planck length and classical field theories do not explain what waves.The chapter therefore treats Maxwell and Einstein theories as effective long-distance descriptions rather than microscopic foundations.Source: §11.1.4 · pp. 353–354

11.2 It from qubit, not bit

The proposed “matter = information” view follows from E = hf and E = mc^2, linking frequency, energy, mass, and information.This extends Wheeler’s “it from bit” idea while raising the problem of deriving gauge fields, fermionic statistics, and elementary-particle equations from qubits.Source: §11.2 · pp. 354–355
A “qubit ether” models space as an ocean of qubits whose waves, bubbles, and defects could represent elementary particles and matter.A simple qubit medium can produce scalar density waves but fails to generate Maxwell or Yang–Mills waves, so a richer microscopic structure is required.Source: §11.2 · pp. 355–356
The “it from qubit” proposal asserts that elementary particles emerge from a long-range-entangled qubit ether, unlike Wheeler’s “it from bit” picture.The resulting collective excitations can satisfy Maxwell, Yang-Mills, or Dirac/Weyl equations.Source: §11.2 · p. 357

11.3.1 Two approaches

The emergence approach treats vacuum as the ground state of a qubit ether and matter as its excitations, replacing the search for smaller constituents with collective organization.Phonons motivate this perspective because they are particle-like collective motions rather than composites of smaller parts.Source: §11.3.1 · p. 358, Figure 11.15 · p. 358

11.3.2 Principle of emergence

Material properties arise from particle organization: ordered crystals resist compression and shear, whereas disordered liquids resist only compression and therefore flow freely.The resulting wave content and mechanical equations are emergent consequences of the underlying organization.Source: §11.3.2 · pp. 360–361

11.3.3 String-net liquid of qubits unifies light and electrons

String-net liquids of long-range-entangled qubits unify light and electrons: string fluctuations yield Maxwell waves, while open-string ends yield electrons with Fermi statistics.The construction produces gauge and Grassmann fields from local qubit models containing only complex scalar fields at the cutoff scale.Source: §11.3.3 · pp. 363–365

11.3.4 Evolving views for light and gauge theories

Gauge fields need not be fundamental or protected by exact gauge symmetry: local qubit models can generate gauge interactions, making gauge fields fluctuations of long-range entanglement.This reframes string-net theory as an entanglement-based account of gauge fields, with strings describing patterns of long-range entanglement.Source: §11.3.4 · p. 367
Gauge theory can be viewed as a theory of long-range entanglement, tracing evolving descriptions of light and gauge interactions from particle beams to waves in long-range entanglement.This perspective treats the entanglement picture as capturing gauge theory’s essence rather than identifying it primarily with curved fiber bundles.Source: §11.3.4 · p. 367
Long-range entanglement can unify gauge bosons and fermions as emergent excitations of local qubit models in any dimension and for any gauge group.Gauge bosons appear as string-density waves, while fermions appear as string ends; the vacuum’s entanglement pattern may determine observed particle content and structure.Source: §11.3.4 · p. 368

11.3.5 Where to find long-range entangled quantum matter?

Topological order is a new quantum phase characterized by long-range entanglement and described systematically through global patterns, local dancing rules, string-nets, unitary fusion categories, and patterns of zeros.The local-rule approach produces explicit Hamiltonians for string-net and fractional quantum Hall states, but these often require difficult three-body or more complicated interactions.Source: §11.3.5 · p. 368
The exactly soluble 1997 model established a realization of Z2 topological order, following earlier proposals involving chiral and time-reversal- and parity-symmetric spin liquids.The Z2 spin liquid has spin-charge separation and emergent fermions, with a low-energy Z2 gauge theory.Source: §11.3.5 · p. 369
Numerical calculations indicate that the J1-J2-J3 Heisenberg model on the Kagome lattice has a gapped chiral spin-liquid ground state.This connects a concrete spin model to a chiral topological phase.Source: §11.3.5 · p. 369
Herbertsmithite ZnCu3(OH)6Cl2 shows no spin ordering or finite-temperature phase transition down to 50mK despite J1≈150K, suggesting a two-dimensional spin liquid.Its gapless or gapped character remains unresolved because both U(1) and Z2 spin liquids have been proposed.Source: §11.3.5 · p. 369
Topological order can produce emergent gauge theories, fractional and non-Abelian statistics, fractional charge, and conducting boundaries, motivating artificial photons, electrons, and vacua from oriented string-net liquids.Realizing such phases in simple Hamiltonians and real materials remains an open challenge; proposed routes include flat-band materials for high-temperature fractional quantum Hall states.Source: §11.3.5 · pp. 369–370