1
Introduction
▶
Notation
2
Matrix Product Vectors
▶
2.1
Basic definitions
2.2
Transfer maps and quantum-channel interfaces
2.3
Site-periodic tensor families
2.4
Gauge equivalence and same MPV
2.5
Injectivity and normality
2.6
Canonical form
2.7
Blocking
2.8
MPV overlap
3
The Single-Block Fundamental Theorem
▶
3.1
The multiplicative linear extension
3.2
Inner automorphism and the single-block theorem
4
Perron–Frobenius Theory for Channels and Transfer Maps
▶
4.1
Positive definiteness
4.2
Uniqueness
4.3
Existence and the Perron–Frobenius theorem
4.4
Right- and left-canonical gauges
4.5
Perron–Frobenius eigenvector existence
5
Peripheral Channel Structure and Transfer-Operator Gaps
▶
5.1
Mixed transfer operator
5.2
MPV overlap as transfer trace
5.3
Eigenvalue bound and transfer-operator gap
5.4
Transfer-operator gap — rectangular case
5.5
MPV overlap decay
5.6
Transfer-operator gap under irreducible-TP hypotheses
5.7
Overlap rigidity for normal blocks
5.8
Peripheral spectral refinements
▶
5.8.1
Cyclic decomposition of irreducible finite Kraus maps
5.9
Self-overlap and the period-one criterion
5.10
Primitive overlap convergence (complementary transfer-map gap form)
6
Wielandt Bound
▶
6.1
Cumulative span
▶
6.1.1
Paper primitivity and indices
6.2
Nonzero trace product
6.3
Eigenvector spreading
6.4
Quantum Wielandt bounds
6.5
Fixed-length matrix spanning
6.6
Primitive MPS tensors
6.7
Primitivity and normality
▶
6.7.1
From primitivity to normality
6.7.2
Primitive implies irreducible
6.7.3
From irreducibility to full spanning via Burnside’s theorem
6.7.4
From primitivity to strong irreducibility and normality
6.8
One-step padding and exact word spans
6.9
Block injectivity from TP/primitive/irreducible blocks
7
Canonical Form Reduction
▶
7.1
Invariant-subspace decomposition and irreducible blocks
▶
7.1.1
Iterated reduction to irreducible blocks
7.2
Nonzero blocks and the trace-preserving gauge
▶
7.2.1
Trace-preserving gauge for arbitrary tensors
7.3
Blocking, period removal, and primitive blocks
▶
7.3.1
Period removal by cyclic sectors
7.4
Cyclic-sector irreducibility
7.5
Translation-invariant canonical form
7.6
Normal tensors and canonical forms
▶
7.6.1
Normalization conventions
7.7
Projector criterion for canonical decomposition
7.8
Fixed points and canonical gauges of irreducible blocks
7.9
Normal canonical form
▶
7.9.1
Separated one-copy normal canonical forms
7.9.2
Canonical form from primitivity
7.10
Left-canonical and dual diagonalization of irreducible blocks
7.11
Cyclic-sector isometries and primitive reduction
▶
7.11.1
Reduction to primitive blocks
8
Basis of Normal Tensors
▶
8.1
Bases of normal tensors
8.2
Normal blocks and overlap rigidity
8.3
Block-injective consequences for BNT families
8.4
Permutation rigidity for bases of normal tensors
8.5
Newton–Girard identities and power-sum recovery
8.6
The BNT canonical form on a sector decomposition
▶
8.6.1
Sector decompositions and phase matching
8.6.2
Single-copy sector decompositions
8.6.3
Prepared-block construction of BNT canonical forms
8.7
Coefficient comparison and copy-weight recovery
▶
8.7.1
Matched-sector weight multiset equality
8.8
Strong existential matching by exact linear independence
▶
8.8.1
Bijective matching by symmetry
9
Proof of the Fundamental Theorem
▶
9.1
Matched block maps for product algebras
9.2
Coefficient identities and copy weights
9.3
The equal-MPV global gauge
9.4
The proportional and equal cases
▶
9.4.1
Unitary gauges on normalized sectors
9.4.2
The equal case
10
Symmetries, Physical String Order, and Virtual-Boundary Nondecay
▶
10.1
Virtual Local Covariance and Projective Bond Actions
10.2
Cohomology Classes of Virtual Symmetry Cocycles
▶
10.2.1
Non-triviality via the commutator phase
10.3
Physical Endpoint String Order and Virtual-Boundary Nondecay
▶
10.3.1
Periodic expectations and fixed-support limits
10.3.2
Stationary Physical Block Densities
10.3.3
Pure canonical tensors and the twisted peripheral spectrum
10.3.4
The physical endpoint selection rule at a fixed twist
10.3.5
Geometric finite-size estimates
10.3.6
Physical endpoints on dimension-bounded blocks
10.3.7
Continuous effective symmetries and identity endpoints
10.3.8
Physical endpoint string order in the AKLT and cluster states
10.3.9
A canonical tensor testing the scalar-phase convention
10.3.10
Exact periodic symmetry overlaps in the AKLT and cluster states
10.4
SPT Phase Labels and Virtual-Boundary Nondecay
▶
10.4.1
Virtual actions along symmetric paths
10.4.2
Uniform rigidity of virtual projective classes
10.4.3
Compact limits and transient periodic components
10.4.4
Reduction to the stationary support
10.4.5
Propagation on finite periodic chains
10.4.6
Virtual classes at a compact limit
10.4.7
Renormalization fixed points of SPT phases
10.4.8
Fixed points with time reversal and reflection
10.4.9
Symmetric gapped paths and bond regrouping
10.4.10
Degeneracy of the entanglement spectrum
10.4.11
Time reversal, reflection, and symmetry obstructions
11
Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
▶
11.1
The \(N\)-site Hilbert space
11.2
Local ground space \(G_L(A)\)
▶
11.2.1
The FNW boundary convention
11.3
Parent interaction and chain Hamiltonian
▶
11.3.1
The weighted FNW boundary estimate
11.3.2
The FNW lower boundary constant
11.3.3
The linear overlap estimate in FNW Lemma 6.2
11.3.4
The quadratic aggregate contribution in FNW Lemma 6.2
11.4
The geometric form of the FNW projector estimate
11.5
Intersection property and unique ground state
▶
11.5.1
Restriction maps
11.5.2
Forward direction: ground space restricts
11.5.3
Injectivity of the ground-space map
11.5.4
The intersection property
11.5.5
Unique ground state
11.6
Transport under physical blocking
11.7
Ground-space correspondence under an invertible on-site map
11.8
Periodic-boundary comparison for injective tensors
11.9
Periodic unique ground state
11.10
Injectivity-length reduction for the closure property
11.11
Periodic-boundary comparison for normal tensors
▶
11.11.1
Reverse comparison for boundary-crossing restrictions
11.11.2
Normal unique-ground-state consequences
11.12
Half-chain Schmidt spectrum and the dual fixed point
▶
11.12.1
Ordered eigenvalues and finite-size normalization
11.12.2
The unital canonical convention
11.13
Two-block uncle Hamiltonians as actual parent limits
12
Commuting Parent Hamiltonians and Spectral Gaps
▶
12.1
Commuting Local Terms for Parent Hamiltonians
12.2
Decorrelation and the idempotent-product parent-commuting condition
12.3
Spectral gap
▶
12.3.1
Martingale condition for a finite family of projections
12.3.2
Cyclic-window overlap estimates
12.3.3
The weighted virtual matrix space
12.3.4
Nested ground projections and martingale differences
12.3.5
Spectator-indexed boundary maps
12.3.6
Martingale gap consequences
12.3.7
The compatible open-chain Hamiltonian
12.4
Overlaps of distinct block ground spaces
12.5
Overlap with a sum of subspaces
12.6
Comparison of sums of orthogonal projectors
12.7
Overlap bounds with spectator sites
12.8
The normalized primitive gauge of a normal tensor
▶
12.8.1
The dimension-only prefactor at a prescribed rate
12.9
Prescribed gaps at sufficiently long interaction ranges
12.10
Continuity of the parent-Hamiltonian gap
▶
12.10.1
Additional joint boundary continuity and compactness inputs
13
Block-Injective Parent Hamiltonians and Degenerate Ground Spaces
▶
13.1
Block injectivity and degenerate ground spaces
13.2
Block separation and intersection
▶
13.2.1
Trace identities for block intersections
13.2.2
Eventual block-intersection consequences
13.3
Block-diagonal boundary conditions for periodic ground spaces
▶
13.3.1
Cyclic windows at the boundary cut
13.3.2
Periodic-boundary ground space for a block-diagonal tensor
13.4
Periodic-boundary ground space generated by a basis of normal tensors
14
Exponential Decay of Correlations
▶
14.1
Connected correlators from the transfer map
14.2
Physical correlations and exponential decay
14.3
Quantitative transfer-map gap bounds
14.4
Relation to parent Hamiltonians
14.5
Physical correlations in an arbitrary bond basis
14.6
Jordan spectral expansions
15
Concrete Examples
▶
15.1
Product states
15.2
The GHZ state
▶
15.2.1
The \(G\)-injective fixed-point tensor
15.3
The AKLT state
▶
15.3.1
Correlation length
15.4
The Majumdar–Ghosh state
15.5
The W state
15.6
Product Vacua with Boundary States
▶
15.6.1
Exact Periodic Gap in the One-Species, Zero-Phase Model
15.6.2
Critical One-Particle Dynamics and the Thermodynamic Gap
15.7
The cluster state
15.8
The Kitaev chain
▶
Scope.
15.9
Correlation independence without a renormalization fixed point
15.10
Parent Hamiltonian statements for the examples
16
Matrix Product Operators and Density Operators
▶
16.1
MPO tensors
16.2
Transfer maps
16.3
MPDOs and LPDOs
16.4
Horizontal and Vertical Canonical Forms
16.5
Strong and weak symmetries
17
Pure States: Renormalization of Matrix Product States
▶
17.1
Physical blocking and transfer idempotence
17.2
Zero-correlation-length conditions
17.3
Normal tensors and fixed-point isometries
▶
17.3.1
Auxiliary block-family hypotheses
17.4
Direct sums and the joint isometry condition
17.5
Renormalization flow and physical correlations
18
Matrix Product Unitaries
▶
18.1
The matrix product unitary condition
18.2
Simple matrix product unitary tensors
18.3
Double-layer blocking and mixed physical traces
18.4
Trace-power lemma
18.5
Full-support canonical representatives
18.6
Transfer-matrix identities
18.7
Block transfer multiplicity
18.8
Normalized transfer spectrum and conditional normality
18.9
Standard form, index, and symmetry classification
▶
18.9.1
Local characterization of symmetries
18.9.2
Equivalence under symmetries
18.9.3
Symmetry examples
18.10
Finite-region observable algebras
▶
18.10.1
Bipartite support algebras
18.10.2
Products of commuting matrix factors
18.10.3
Full-matrix presentation of a factor with scalar centre
18.11
Depth estimates for a merging tree
▶
18.11.1
Boundary Gram matrices of a parity rotation
18.11.2
Interval isometries for product-projection phase gates
18.11.3
Finite coefficient alphabets at a cut
18.11.4
Unit-circle geometry and diagonal circuits
18.12
Exact circuits for matrix product unitaries
18.13
Limits of recursion inside the original Schmidt spaces
18.14
Completions and general approximation
18.15
Uniform families without exact-length full rank
18.16
Polynomial exact circuits at arbitrary fixed cut rank
19
Finite-Group MPU Representations and Scalar Gauge Data
▶
19.1
Finite-group unitary averages
19.2
Finite-group representations by matrix product unitaries
▶
19.2.1
Rank equality and the invertible first-cut comparison
19.2.2
Unitarity of the first-cut comparison
19.2.3
The compatible decomposition and its literal gates
19.2.4
Movement gates of the inverse-compatible source factors
19.3
A nondegenerate alternating-network comparison
19.4
Staircase gates and the first source network
▶
19.4.1
The second network and unitarity of the staircase gates
19.4.2
Endpoint contraction and growth of truncated symmetries
19.4.3
Periodic sewing of the two source arcs
19.4.4
Physical closures of the source factors
19.5
Scalar three-cocycles and fusion-tensor gauge freedom
▶
19.5.1
The anomaly three-cocycle from fusion tensors
19.5.2
Reductions through action tensors
19.5.3
An anomaly obstructs invariant normal states
19.5.4
Blocks permuted by the group
19.5.5
Domain walls between permuted blocks
19.5.6
Detecting the anomaly with truncated symmetries
19.5.7
Normalization of the first source factor
19.5.8
Normalization of the weighted second source factor
19.5.9
Physical normalizations of the two source X factors
19.5.10
Uniqueness of the source decompositions
19.6
The GHZ–blocked-cluster tensor
19.7
The gauge-invariant subspace of the CZX circuit tuple
▶
19.7.1
Monomial operators and their common fixed space
19.7.2
The exact CZX matrix product unitary
19.7.3
The explicit CZX adjoint gauge
19.7.4
The displayed CZX source factors
19.7.5
Normalization of the displayed CZX comparison
19.7.6
The blocked GHZ state and its CZX invariance
19.7.7
The printed CZX action pair and its empty-word boundary
19.7.8
The CZX \(g,g\) fusion caps
19.7.9
Literal fusion–action coordinate contractions
19.7.10
The displayed CZX circuit tuple
19.7.11
Orbits, holonomy, and the exact dimension
19.7.12
A second displayed circuit tuple at three and four sites
19.8
The CZX defect domains and prescribed defect maps
19.9
The displayed CZX tuple as a completion
20
Mixed States: Renormalization of Matrix Product Operators
▶
20.1
Preliminaries for physical renormalization
20.2
Renormalization fixed points
20.3
Pure-state recovery inside the MPO formalism
20.4
PEPS boundary tensors
20.5
The unnormalized strong fixed-point relation
20.6
Zero correlation length
20.7
Purification fixed points
20.8
Saturation of the area law
20.9
Support compression for entropy functionals
20.10
Rectangular Choi matrices and a range-dimension estimate
20.11
Isometric enlargements of the physical space
20.12
Simple tensors
20.13
Cyclic local-embedding coordinates
20.14
Gibbs states of nearest-neighbor commuting Hamiltonians
20.15
Two-site and positive-length physical blocking
20.16
Simple local structure from SAL and ZCL
▶
20.16.1
Simple canonical-form weights
20.16.2
Markov decomposition and inverse-map sector factorization
20.16.3
Neighboring operators and commuting bond products
20.16.4
Closed-sector contractions and coherent rephasing
20.16.5
Normalized preparations and controlled partial traces
20.16.6
Primitivity and rank-one trace matrices
20.16.7
Refinement and coarse-graining channels
20.16.8
Local simple-MPDO structure
20.17
Single-bond commuting form from the local structure
21
Mixed States: General Case and Algebraic Structure
▶
21.1
Closed-chain operators
21.2
Algebra structure
▶
21.2.1
Diagonal \(\chi \)-matrices and the trace-power formula
21.2.2
Structure coefficients and the idempotent condition
21.2.3
A physical MPDO with a single vertical BNT element
21.2.4
Direct-sum sector conjugations
21.2.5
Ambient sector retractions and normalized embeddings
21.2.6
Trace-preserving completions and physical channels
21.3
Transfer retracts and fusion coisometries
22
Periodic MPS: irreducible form and the periodic Fundamental Theorem
▶
22.1
Periodic irreducible form and physical-index rotation
▶
22.1.1
The \(Z\)-gauge degree of freedom
22.2
Periodic Fundamental Theorem
▶
22.2.1
Common-period blocking lemmas
22.2.2
Statement of the periodic Fundamental Theorem
22.2.3
Periodic overlap dichotomy
22.3
Physical symmetries on periodic MPS
22.4
Refinement and divisibility
23
The Algebraic Fundamental Theorem
▶
23.1
Site-dependent chains at fixed length
23.2
Quadratic reconstruction of the translation-invariant canonical form
23.3
Uniqueness of the translation-invariant canonical form at a fixed ring length
23.4
One-sided inverse and physical realisation of virtual insertions
▶
23.4.1
Physical realisation of virtual insertions
23.5
The virtual bond gauge and the proportionality lemma
▶
23.5.1
Two preliminary facts
23.5.2
Same state forces a virtual bond gauge
23.5.3
Aligned gauges force proportionality
23.6
The Fundamental Theorem for injective chains
▶
23.6.1
Finite-length MPV agreement
23.7
Corollaries: translation-invariant, non-uniform, and blocked chains
▶
23.7.1
Translation-invariant chains
23.7.2
Blocked chains
23.8
Product algebra construction and per-block gauge
▶
23.8.1
Block permutation algebra
23.8.2
Per-block linear extension and the product automorphism
23.8.3
Nondegeneracy of the product trace pairing
23.9
Gauge equivalence for direct sums
▶
23.9.1
Same-index direct-sum corollaries from the single-block theorem
23.9.2
MPV invariance under reindexing
23.9.3
Converse: gauge equivalence implies same MPV
23.9.4
Canonical-form tensor as a weighted direct sum
23.9.5
Equal-case block matching with differing bond dimensions
23.10
Translation-invariant reduction and global symmetry
▶
23.10.1
Global symmetry via gauge composition
24
Projected entangled pair states and the Fundamental Theorem
▶
24.1
Finite Graph Tensor Networks
▶
24.1.1
PEPS tensors and the states they generate
24.1.2
Gauge transformations on the bonds
24.1.3
Vertex injectivity and the local inverse
24.1.4
Edge-centred decomposition of a contraction
24.2
Edge-Middle Reduction
24.3
Edge Kernels and Gauge Transport
24.4
Fundamental Theorem for Injective PEPS
▶
24.4.1
From the conjugation identity to the absorbed equality
24.5
Union Closure for Normal PEPS
24.6
Square-Lattice Normal PEPS
24.7
Torus Normal PEPS
▶
Failure of the row-and-column reduction.
Staircase-window geometry for a reference edge.
24.7.1
Staircase patch geometry
24.7.2
Overlapping-window corner cancellation
24.7.3
Single-bond peeling and the window witness
24.7.4
Both edge directions and reference witnesses
24.7.5
Unscaled coefficients and the normal-window gauge
24.7.6
Transport and the vertical reference edge
24.7.7
Corner comparison at the minimal torus sizes
24.7.8
Gauge uniqueness from injective windows
24.7.9
Complements and staircase comparisons for positive windows
24.8
Normal PEPS Fundamental Theorem
24.9
Fundamental Theorem for normal matrix product states
▶
24.9.1
Matrix product states on the closed chain
24.9.2
Translation-invariant matrix product states
24.9.3
The optimal system size \(n \ge 2L + 1\)
24.9.4
Site-dependent chains
24.10
The vertex-injective case and the residual edge scalars
24.11
Balanced Edge Scalars
▶
24.11.1
Inserted-site scalar extraction
24.12
Exact contraction on the open square grid
24.13
Physical consequences of the square-grid correspondence
25
Examples of PEPS on the torus
▶
25.1
Torus PEPS of a four-leg tensor
25.2
The GHZ state
25.3
The cluster state
25.4
The CZX model
25.5
The physical boundary symmetry of a CZX rectangle
25.6
The AKLT state
25.7
The RVB state
25.8
PEPS from classical models
26
G-injective PEPS and quantum double models
▶
26.1
The toric code and quantum double models
▶
The four-site block and its boundary.
Relative coordinates separate the second bond.
The original tensor and the four surplus registers.
The normalized fixed-point state.
26.1.1
Commuting constraints on overlapping regions
26.1.2
Physical parent interactions of regular \(G\)-isometric tensors
26.1.3
Regular intersection at one shared vertex
26.1.4
Group actions on torus bonds
26.1.5
The primal toric-code tensor in the parity basis
26.1.6
Four twisted sectors of the primal toric code
26.2
Local physical ranges of G-injective tensors
26.3
Comparison of invariant virtual spaces
26.4
G-injectivity under physical maps
26.5
Invertible physical comparison of G-injective tensors
26.6
Normalized local physical form
26.7
Local operators on the invariant boundary
26.8
Intersection of local boundary spaces along a strip
26.9
Ground spaces of heterogeneous open strips
27
Ground spaces and reduced densities of PEPS regions
▶
27.1
Ground spaces of finite PEPS regions
27.2
Parent Hamiltonians on finite PEPS graphs
27.3
The one-dimensional specialization
27.4
Reduced densities of PEPS regions
27.5
Boundary support and Gibbs operators
27.6
Physical changes of coordinates and regional ground spaces
27.7
Physical transport of the common parent ground space
27.8
Unitary physical transport of parent Hamiltonians
27.9
The full-region ground space
27.10
Nesting of regional PEPS ground spaces
27.11
Comparison of finite PEPS parent Hamiltonians
27.12
Finite-overlap gap criteria
27.13
Parent spaces determined by the physical ray
27.14
Entanglement bounds on finite graphs
27.15
A counterexample to bare inverse-adjoint parent transport
27.16
The virtual Bell-bond ground space
27.17
Nearest-neighbor parents of injective PEPS
27.18
Regional supports of ordinary injective tensors
27.19
Vertex and bond coordinates
27.20
Internal Bell constraints from actual regional contractions
27.21
The physical image of the Bell product
27.22
The common one-site image space
27.23
Regional conditions after the site inverses
27.24
Exact transport of the common virtual conditions
▶
27.24.1
Physical changes on the site images
27.24.2
Matching sectors in open weighted regions
27.25
Regional support of physical bond transformations
▶
27.25.1
Products, coordinate slices, and multiplicity maps
27.25.2
One-ended boundary transformations
27.25.3
Physical coordinates and regional operator blocks
27.25.4
Actual open contractions and their product support
27.26
Positive multiplicities and complete canonical parent spaces
▶
27.26.1
Copy weights and arbitrary open boundaries
27.26.2
Global supported inverses
27.26.3
Removing weights and changing multiplicities
27.27
Fourier coordinates for regional parent spaces
▶
27.27.1
Regional coordinate changes
27.27.2
Exterior blocks and complete kernels
27.28
Semi-regular representations and physical parent kernels
▶
27.28.1
Irreducible coordinates and the regular representation
27.28.2
The oriented incident action
27.28.3
Equivalence of the complete parent spaces
27.29
Open contractions with native edge orientations
▶
27.29.1
Averaging tensors and unrestricted regional boundaries
27.29.2
Internal support and regional blocks
27.30
Coordinate changes with native edge orientations
27.31
Oriented multiplicity and physical transport
▶
27.31.1
Copy maps on open oriented regions
27.31.2
Removing weights and comparing arbitrary physical tensors
27.32
Full native semi-regular torus parent spaces
▶
27.32.1
Native geometry and the local invariant subspace
27.32.2
Commuting closures satisfy the original plaquette constraints
27.32.3
Dimension and exhaustion of the full kernel
27.32.4
Labelled bonds and the four cuts
27.32.5
Flat group labels on the actual bonds
27.32.6
Moving the commuting seams
27.32.7
Contractions with independently sized bonds
27.32.8
Matched representations and a common local inverse
27.32.9
Coherent coefficients forced by complementary cuts
27.32.10
The four-block closure property
27.32.11
Independent four-block closure classes
27.32.12
Four cut spaces with common coordinate alphabets
27.32.13
Physical inverses and matched projector contractions
27.32.14
Bond support and common-alphabet closure reconstruction
27.32.15
Partial averaging and unrestricted exterior bonds
27.32.16
Open cut intersections from removable internal labels
27.32.17
Lifted regional contractions and physical support
27.32.18
Reconstructing full cuts from lifted vectors
27.32.19
Intersections of lifted regional ranges
27.32.20
The open intersection for three four-legged tensors
27.32.21
Microscopic parent constraints and native seam boundaries
27.32.22
Nonuniform native parent classification
27.33
Orthogonalization of nested cylinders
27.34
A uniform width bound for angular reset scales
27.35
Tag encodings with regional zero resets
27.36
Regional embeddings in uniform physical dimension
27.37
Finite-domain area laws and polynomial PEPS approximation
▶
27.37.1
Exact nearest-neighbor crossing sums
27.38
Regional states under relabelling
▶
27.38.1
Finite improvement of the box exponent
27.39
Rounded scales in a safe rectangle
▶
27.39.1
Rounded collar clearance
27.40
Minimum norm of regularized regional filters
27.41
Coordinate variation and final-state marginals
27.42
Sparse coordinate strips
27.43
Finite cell counts
27.44
Translated dyadic neighborhoods
27.45
The half-open layer partition
27.46
Distances between dyadic layers
27.47
The dummy neighborhood meets only the initial layer
27.48
Fine cells and sparse belts
27.49
Unique fine cells outside the dummy region
▶
27.49.1
Capped contained dyadic squares
27.49.2
Ambient boundaries of template dilations
27.50
Entropy cost of a partial depth row
▶
27.50.1
Finite weighted dyadic summation
27.51
Safe rectangles in template coverings
27.52
Weighted coverings of template shells
27.53
Entropy from a uniform safe-box estimate
▶
27.53.1
Physical boundaries of separated templates
27.53.2
Mixed-square counts for template shells
27.54
Uniform separation of the dyadic scales
27.55
Polynomial factors in the dyadic budget
27.56
Primary birth regions and adjacent fine meshes
27.57
The number of primary identifiers in a layer
27.58
The marked points of belt cells
27.59
Separation on the translated quarter mesh
▶
27.59.1
Clipping an incident segment
27.60
The finite family of initial marks
27.61
The fan of a belt cell
▶
27.61.1
The primary containing a non-belt fine cell
27.62
Interiors of actual fan triangles
27.63
Allowed lines and the eight open fan triangles
27.64
Restriction of concentric dyadic fans
27.65
Actual initial regions in eight sectors
27.66
Colors of adjacent initial sectors
27.67
Actual initial identifiers on sector runs
27.68
Active radial interfaces near an initial mark
27.69
Opposing corners divide a side only at its midpoint
27.70
Ordered endpoints of a whole cell side
27.71
Subdivisions determined by actual corners
27.72
Contacts between actual fine cells
27.73
Dummy-neighborhood corners on fine-cell sides
27.74
Matching elementary sides under the actual subdivision
27.75
An opposing closed region for every elementary side
27.76
Uniqueness of the opposing region
27.77
Contact and reciprocal selection of opposing regions
27.78
Opposite labels on actual belt-cell sides
27.79
Merging equal-colored triangles in a cell
27.80
Geometric contacts within one cell fan
27.81
Cyclic order of the eight midpoint fan triangles
27.82
Directed rays of an elementary fan
27.83
Parity of changes of initial sector identifiers
27.84
Cardinality of the distinct active directed rays
27.85
Radial incidence and convexity in an elementary fan
27.86
Connected pieces of the midpoint fan
27.87
Connected fan pieces in a smaller closed square
27.88
Connected runs after radial deletion
27.89
Disjoint runs after radial deletion
27.90
Connected components after radial deletion
27.91
Connected components near an initial mark
27.92
The closed fine-cell cover of a primary region
27.93
Fan contacts along cell boundaries
27.94
Interfaces of belt runs with primary regions and other belt runs
27.95
Interfaces with the initial dummy region
27.96
The initial family of regions
27.97
The translated origin and integer lattice points
27.98
Disjoint interiors of the initial regions
27.99
Initial boundaries and lattice membership
27.100
The initial partition of the integer lattice
27.101
Colors across initial interfaces
27.102
Regularity of initial birth regions
27.103
Primitive frontiers of actual fans
27.104
Layers and meshes near a cell mark
27.105
Whole sides of a dyadic square
27.106
Initial frontiers near a mark
27.107
Removal squares and repair children
▶
27.107.1
Depth and descendant count of a repair branch
27.108
Entropy cancellation for ordered labelled regions
27.109
Conditional two-family entropy estimate
27.110
Regional entropy dimension bounds
27.111
Regional entropy continuity
27.112
Entropy and dimension of a selected Schmidt spectrum
27.113
Boundary entropy from birth-target information
27.114
Collar and radius scales
▶
27.114.1
The clock time and a good realization
27.115
The final entropy comparison of the amplification
27.116
Cuts with no crossing edge
▶
27.116.1
Vertex-boundary formulations
27.117
Marginal surprisal tails at a lattice cut
27.118
Conditional movement and skew estimates on lattice regions
27.119
Conditional entropy in a buffered rectangle
27.120
Entropy cost of a partial rectangle layer
27.121
The initial safe-box estimate
27.122
Nested square patches
▶
27.122.1
Dyadic counts for rectangle shells
27.123
Weighted dyadic covering of rectangle shells
27.124
Entropy of safe rectangle shells
27.125
Initial-exponent entropy of safe rectangle shells
▶
27.125.1
Initial entropy estimate for rounded collars
27.125.2
Uniform coefficients for rounded shell estimates
27.125.3
Replica limits before large-scale entropy absorption
27.126
Geometry of the dyadic distribution protocol
27.127
Finite incidence in distributed PEPS compression
27.128
Exact operator contraction for distributed PEPS compression
27.129
Whole-group tensor bound and simultaneous truncation
27.130
Extracting one ket column from an operator network
▶
27.130.1
Error and normalization of the protocol
27.130.2
Column selection after discarding a product reference
27.131
Dyadic routing of a party network into a square-lattice PEPS
▶
27.131.1
Fixing a column before routing
27.132
Encoded frames and small-patch rewrites
27.133
Changes of ownership in encoded frames
▶
27.133.1
Splitting a sheet
27.133.2
Homogeneous birth and death
27.133.3
Exchanging assignments between two sheets
27.134
Local interactions after a partial swap
27.135
Buffer operators in paired physical coordinates
27.136
A supported Gaussian operator for the physical buffer
27.137
Small-patch rewrites as allowed monomials
27.138
Elimination of normalized pair effects
27.139
Preparing and combining pair sources
27.140
Collecting local operations by party
▶
27.140.1
Physical bases and party grouping
27.140.2
Register identifications and tensor coordinates
27.140.3
Restrictions with excluded registers
27.141
One-dimensional sources on unused pairs
27.142
Common private source spaces
27.143
Restricting the source-register operators
27.144
Density expansions of source gates
27.145
Joint contractions of the free source inputs
27.146
Grouping parties into two sides
27.147
Party restrictions after changing owner labels
27.148
Free source inputs and untouched exterior gates
▶
27.148.1
Finite gate expansions
27.148.2
Affected parties and the exterior
27.148.3
Preparing only the fixed sources
27.148.4
Operations beside untouched registers
27.149
Chronological partial source expansions
27.150
Fixed sources for the chronological partial expansion
27.151
Source substitutions and corrected density terms
▶
27.151.1
Preparing the original private input
27.152
Schmidt coordinates and one Gaussian source law
27.153
Physical contractions and Gaussian source estimates
27.154
Source-only reduction of distributed densities
27.155
Corrected source inputs and actual local outputs
27.156
Physical coordinates of the two local outputs
27.157
Sampling the actual source circuit
27.158
Approximate original gate expansions
27.159
Physical output after exchanging auxiliary registers
27.160
Exact elimination of finite-dimensional messages
27.161
Gates containing bounded messages
27.162
Original input and physical-output resources
27.163
Sampling an original construction
27.164
The exterior source memory when all parties are selected
28
Positive quasi-local constraints from a global gap
▶
28.1
Truncation near a finite set
28.2
The induced lattice instance
28.3
Finite collar histories and good-history sampling
▶
28.3.1
A common marginal moment bound for physical statuses
28.4
Entropy transfer on positive-depth prefixes
28.5
Replica metrics in augmented site coordinates
28.6
The transport root in the original site coordinates
28.7
Bad-history probability
28.8
The scanner estimate from its inputs
▶
28.8.1
Rounded weights in the marginal-moment interval
28.9
The rough upper comparison for the supplied labels
28.10
Local root channels and physical shell oscillations
▶
28.10.1
Localized square-root channels
28.10.2
Differences of successive local channels
28.11
Component support and full-channel oscillation
28.12
Finite channel oscillation bounds
▶
28.12.1
Initial oscillation from physical support
28.12.2
Coefficients from the localization errors
28.12.3
Stretched-exponential localization errors
28.12.4
The full channel on an anchor component
28.12.5
Finite families of channel events
28.12.6
Uniform weighted rows of the finite event kernel
28.12.7
Kernel bounds for actual channel families
28.12.8
Actual constraints on a finite lattice domain
28.12.9
Extended graph distance and kernel support
28.13
Chronological finite channel words
28.14
Expected spatial oscillation of channel words
28.15
Excited-state decay for the exact constraints
28.16
Truncating the retained constraint product
28.17
Omitting distant events from the constraint product
▶
28.17.1
Telescoping the omitted events
28.17.2
One omitted event
28.17.3
The coupled omission estimates
28.18
Typical projectors for the amplification
28.19
The restoring operator of the amplification
29
The Multi-Block Asymmetric Fundamental Theorem
▶
29.1
Triangular word products
29.2
The word algebra and normal tensors as simple modules
29.3
Separating simple modules
29.4
The composition-series flag
29.5
The multi-block asymmetric compression theorem
29.6
Products of matrix product operators
▶
29.6.1
Fusion and action tensors with multiplicity
29.7
Structure coefficients at every positive length
29.8
The splitting criterion
▶
29.8.1
Uniqueness of split compressions
29.9
The single-block reduction of Molnár–Ge–Schuch–Cirac
29.10
Normalized split compressions
29.11
Rectangular reductions and residual algebras
30
Matrix Product Operator Symmetries and Non-Invertible Symmetries
▶
30.1
Arbitrary boundary transport
30.2
Reconstruction from arbitrary boundaries
30.3
Comparing exact decomposition coordinates
▶
Source index convention.
30.4
Canonical parents for adjoint-closed boundary ranges
30.5
Normal blocks and arbitrary-boundary adjoints
▶
30.5.1
From periodic duality to exact similarities
30.5.2
Literal block sums and canonical parents
30.5.3
A derived algebraic counterexample
30.6
Adjoint boundaries from star-compatible coalgebras
30.7
Triangular physical padding
30.8
Mixed orientation of the fusion pentagon
30.9
A common fusion family for operators and states
30.10
Identifying the fusion and action comparisons
30.11
The multiplicity mixed pentagon from the exact source maps
▶
Source index correction and scope.
30.12
Invertible symmetries
30.13
Fusion algebras of matrix product operators
30.14
Non-invertible symmetries
30.15
Symmetric matrix product states
30.16
The zipper condition
30.17
Mixed interpolation and endpoint supports
▶
30.17.1
Local sectors at the first endpoint
30.17.2
Exact action on the mixed interpolation
30.17.3
Fusion of the mixed operator family
30.17.4
Three-site injectivity of the mixed operator
30.17.5
Common symbols along the mixed interpolation
30.17.6
Comparison at the first endpoint
30.17.7
Extended boundary spaces on open intervals
30.17.8
Joint extended boundary spaces
30.17.9
The mixed family with shared endpoint alphabets
30.17.10
Joint rectangular columns at a singular endpoint
30.17.11
Joint compression at the two boundary sites
30.17.12
The actual support in the joint boundary frame
30.17.13
The common core for every ordered pair of blocks
30.17.14
Dependent finite spectator coordinates
30.17.15
The degenerate mixed endpoint on a ring
30.17.16
Exchanging the two periodic endpoints
30.17.17
Open gaps from changes at the two boundary sites
30.17.18
A uniform gap along the closed mixed interpolation
30.18
Physical isometries and periodic gaps
▶
30.18.1
Commuting support penalties
30.18.2
Orthogonal decomposition along an isometry
30.18.3
Periodic ground spaces and gap bounds
30.19
Mixed paths in arbitrary physical alphabets
▶
30.19.1
Polar support and full endpoint embeddings
30.19.2
The continuous positive interaction
30.19.3
Ground lines and a volume-uniform gap
30.20
Attaching the canonical parents
31
Physics Examples of the Multi-Block Compression Theorem
▶
31.1
The GHZ state as a compression target
31.2
A product state from a repeated block
▶
31.2.1
Compression and the absence of a gauge
31.2.2
A Jordan extension detected by a boundary
31.3
The CZX symmetry
▶
31.3.1
The bond-two tensor and the periodic operator
31.3.2
The tensor printed in the review
31.3.3
The decorated representative
31.3.4
Asymmetric compression of the square
31.3.5
A non-simple family with an oscillating coefficient
31.3.6
Scalar three-cocycles on the group of order two
31.4
The anomalous \(\mathbb {Z}_2\) symmetry of the Levin–Gu edge
31.5
Kramers–Wannier duality
▶
31.5.1
The duality kernel and its square
31.5.2
Action on ground states
31.5.3
The duality as a bimodule over the spin-flip algebra
31.6
Fibonacci fusion operators
▶
31.6.1
Admissible configurations and the vacuum operator
31.6.2
The golden chain
31.6.3
Two periodic states and the Fibonacci action
31.6.4
Asymmetric compression of the stacked square
31.6.5
The categorical anomaly
31.7
The Ising anyon chain
▶
31.7.1
The periodic operators of the three tensors
31.7.2
One-site injectivity by exact arithmetic
31.7.3
The weighted direct sum and its local product
31.7.4
The gauge and the block decomposition
31.7.5
Compression onto the \(\sigma \) tensor
31.7.6
The length-dependent product coefficient
31.7.7
Coefficients from the Ising fusion rules
31.7.8
Action on a fusion-tree product state
31.7.9
Contraction of a matrix-unit factor
31.7.10
The three-object weighted direct sum
31.8
Group matrix product operators from a three-cocycle
▶
31.8.1
Fusion tensors and the inverse cocycle
31.8.2
The group of order two and the decorated CZX operator
31.8.3
The cyclic groups
31.8.4
The group \(\mathbb {Z}_2\times \mathbb {Z}_2\)
31.9
The anomalous \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry and its condensation defect
▶
31.9.1
The condensation defect and its periodic fusion table
31.9.2
The sixteen pair blocks of the stacked defect
31.9.3
The anomaly class of the full group
31.9.4
The eight scalar representatives
31.10
The anomalous \(\mathbb {Z}_3\) symmetry and its condensation defect
▶
31.10.1
Fusion of the anomalous representation
31.10.2
The group family
31.10.3
The condensation defect
31.11
The \(\mathbb {Z}_3\) clock symmetry
31.12
Mixed Bell bonds and renormalization fixed points
▶
31.12.1
Independent mixed Bell bonds
31.12.2
Mixed Bell bonds with an even-parity flag
31.12.3
Trace-preserving fixed-point maps
31.12.4
Vertical components and their multiplication
31.12.5
Representative rescaling
31.12.6
The one-label fixed point
31.12.7
The graded quantum-dimer twist
32
Matrix Product Operator Algebras
▶
32.1
Fusion tensors and action tensors
32.2
Gauge freedom of fusion and action tensors
32.3
The associator of a group-like algebra
32.4
Cohomological obstruction to symmetric states
32.5
Weak Hopf algebras, dualities, and density operators
33
Local Quantum Circuits
▶
33.1
Operators on sets of sites and local circuits
33.2
Two-site gates suffice
33.3
Circuits with measurements
33.4
Adaptive channels on arbitrary inputs
33.5
Teleportation and long-range gates
33.6
Local channel circuits and conversions
33.7
Approximate local channel conversions
33.8
Fixed physical ports and bounded local registers
▶
33.8.1
Compiling bounded native protocols
33.8.2
Original physical-port input and output
33.8.3
Unitary circuits with one final environment discard
33.9
Lieb–Robinson propagation on finite graphs
▶
Finite-range interactions on a graph.
Normalized partial traces.
33.10
Supported projected relative evolution
33.11
Physical-site oscillation with a spectator
▶
33.11.1
The on-site unitary supremum
33.11.2
Averaging physical sites
34
Sequential Generation of Matrix Product States
▶
34.1
Sequential schemes
34.2
Isometric decompositions
34.3
Resources of the successive decompositions
34.4
Cut ranks
34.5
Sequential generation with an ancilla
34.6
The deterministic transition scheme
34.7
Minimal resources
34.8
Sequential generation without an ancilla
34.9
Preparation by a staircase of local gates
35
Preparation of Matrix Product States by Log-Depth Circuits
▶
35.1
Polar decomposition of a blocked tensor
▶
35.1.1
Polar decomposition of a matrix
35.1.2
A tensor as a map on the virtual pair space
35.1.3
Polar factors of a tensor
35.2
The fixed-point pair state of a normal tensor
35.3
The approximating state
35.4
Exact factorizations of the isometry
35.5
Approximation error
▶
35.5.1
Quadratic overlap estimates for ordered blocks
35.6
Depth of the preparation
▶
35.6.1
Exact preparation with vanishing subleading spectrum
35.7
Conversion through compatible coherent seeds
▶
35.7.1
Selected vectors with different sector probabilities
35.8
Canonical sector encoders
▶
35.8.1
An exact whole-encoder construction
35.8.2
Uniform coherent conversion of the canonical sector encodings
35.8.3
Polynomial accuracy at every sufficiently large length
35.9
The tree circuit as a multiscale network
35.10
Inhomogeneous short-range correlated states
35.11
Uniform contraction of contiguous blocks
35.12
Normalized preparation from ordered mixing
35.13
Explicit blocking scales and physical depth
35.14
Varying density-matrix references
▶
Discussion and outlook.
35.15
Quantitative preparation with varying bond dimensions
35.16
Preparation from local window mixing
▶
Scope and boundary example.
35.16.1
Local windows on varying bond spaces
35.17
Tensors that are not normal
▶
35.17.1
Blocks with multiplicities
35.17.2
Blocks whose states overlap
35.17.3
Repeated blocks whose states overlap
35.17.4
Weights with a phase
35.17.5
Overlapping blocks at the second-order rate
35.17.6
The actual canonical block pairs
35.18
Preparation with measurements
35.19
The tree circuit with measurements
35.20
Registers of several sites and the depth \(O(\log \log (N/\varepsilon ))\)
35.21
Trees on blocks of any length
▶
35.21.1
Tensors that are not normal, with measurements
35.21.2
Overlapping blocks with measurements, for every chain length
35.22
Bounded physical leaves and arbitrary chain lengths
35.23
Non-normal canonical states in double-logarithmic depth
36
Interaction counting in lattice domains
▶
36.1
Walks through overlapping supports
36.2
Ambient lattice counting
36.3
Original-support budgets
36.4
Exact lattice constants
36.5
Chains reaching a target
36.6
Exponential generating series
36.7
Graph-distance propagation in lattice domains
36.8
Localization from an arbitrary region
36.9
A supported Gaussian integral
36.10
Exact finite-size PEPS representations
36.11
The finite-size approximation conclusion
36.12
Removing finitely many exceptional sizes
▶
36.12.1
Assembly from a compressed dyadic operator network
1
Appendices
▶
A
Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
▶
A.1
Canonical-gauge algebra
B
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
▶
B.1
Mixed-transfer powers and overlap traces
B.2
Auxiliary trace positivity
B.3
Rectangular spectral-radius formulation
B.4
Peripheral intertwiners and gauge rigidity
B.5
Spectral-radius decay and overlap limits
B.6
Rank-one Perron projection
B.7
Periodicity removal
C
Wielandt Bound: Supporting Results
▶
C.1
Exact-word and block-injectivity support
C.2
Cumulative-span and spectral linear algebra
C.3
One-step augmentation
C.4
Blocking and fixed-length spanning
C.5
Complementary-gap consequences
C.6
Identity in the one-step span
D
Canonical Form Reduction: Supporting Results
▶
D.1
Similarities, strict splitting, and zero blocks
D.2
Projector closure and isometric corner decompositions
D.3
Perron gauges and preservation under conjugation
D.4
Blocking identities and primitive stability
D.5
Support compression and cyclic-sector families
D.6
Prepared normal-form consequences
E
Basis of Normal Tensors: Supporting Results
▶
E.1
Elementary normal-tensor and Gram-matrix support
E.2
Change of basis, power sums, and sector bookkeeping
E.3
Direct-sum separation and prepared-family bookkeeping
F
Proof of the Fundamental Theorem: Supporting Results
▶
F.1
Matched phases and coefficient extraction
F.2
Coefficient identities from gauge-phase equalities
F.3
The scalar-threaded proportional identity
F.4
Direct-sum conjugation
F.5
Additional proportional consequences
F.6
Matched flattened coordinates
F.7
Permutation gauges and literal coordinates
F.8
Unitary witness refinements
G
Symmetries and Virtual-Boundary Nondecay: Supporting Results
▶
G.1
Permutation reindexing and identity bookkeeping
G.2
Gauge ratios and scalar uniqueness
G.3
Cocycle equivalence and coboundaries
G.4
Stationary-boundary twist support
G.5
Kraus mixing, transfer expansions, and a common TP gauge
G.6
Boundary and limit support
G.7
Universality of virtual-boundary nondecay
H
Relative Commutants of Full Matrix Algebras: Supporting Results
I
Physics Examples of the Multi-Block Compression Theorem: Supporting Results
▶
I.1
Explicit gauges and exact arithmetic
I.2
Adapted bases and scaled matrix units
I.3
A sign-weighted pair with a non-split extension
J
Bibliography
Dependency graph
Dependency graphs
Tensor Network Theory: A formalization blueprint
Sirui Lu, Erickson Tjoa, and J. Ignacio Cirac
Last update: October 14, 2026
1
Introduction
Notation
2
Matrix Product Vectors
2.1
Basic definitions
2.2
Transfer maps and quantum-channel interfaces
2.3
Site-periodic tensor families
2.4
Gauge equivalence and same MPV
2.5
Injectivity and normality
2.6
Canonical form
2.7
Blocking
2.8
MPV overlap
3
The Single-Block Fundamental Theorem
3.1
The multiplicative linear extension
3.2
Inner automorphism and the single-block theorem
4
Perron–Frobenius Theory for Channels and Transfer Maps
4.1
Positive definiteness
4.2
Uniqueness
4.3
Existence and the Perron–Frobenius theorem
4.4
Right- and left-canonical gauges
4.5
Perron–Frobenius eigenvector existence
5
Peripheral Channel Structure and Transfer-Operator Gaps
5.1
Mixed transfer operator
5.2
MPV overlap as transfer trace
5.3
Eigenvalue bound and transfer-operator gap
5.4
Transfer-operator gap — rectangular case
5.5
MPV overlap decay
5.6
Transfer-operator gap under irreducible-TP hypotheses
5.7
Overlap rigidity for normal blocks
5.8
Peripheral spectral refinements
5.8.1
Cyclic decomposition of irreducible finite Kraus maps
5.9
Self-overlap and the period-one criterion
5.10
Primitive overlap convergence (complementary transfer-map gap form)
6
Wielandt Bound
6.1
Cumulative span
6.1.1
Paper primitivity and indices
6.2
Nonzero trace product
6.3
Eigenvector spreading
6.4
Quantum Wielandt bounds
6.5
Fixed-length matrix spanning
6.6
Primitive MPS tensors
6.7
Primitivity and normality
6.7.1
From primitivity to normality
6.7.2
Primitive implies irreducible
6.7.3
From irreducibility to full spanning via Burnside’s theorem
6.7.4
From primitivity to strong irreducibility and normality
6.8
One-step padding and exact word spans
6.9
Block injectivity from TP/primitive/irreducible blocks
7
Canonical Form Reduction
7.1
Invariant-subspace decomposition and irreducible blocks
7.1.1
Iterated reduction to irreducible blocks
7.2
Nonzero blocks and the trace-preserving gauge
7.2.1
Trace-preserving gauge for arbitrary tensors
7.3
Blocking, period removal, and primitive blocks
7.3.1
Period removal by cyclic sectors
7.4
Cyclic-sector irreducibility
7.5
Translation-invariant canonical form
7.6
Normal tensors and canonical forms
7.6.1
Normalization conventions
7.7
Projector criterion for canonical decomposition
7.8
Fixed points and canonical gauges of irreducible blocks
7.9
Normal canonical form
7.9.1
Separated one-copy normal canonical forms
7.9.2
Canonical form from primitivity
7.10
Left-canonical and dual diagonalization of irreducible blocks
7.11
Cyclic-sector isometries and primitive reduction
7.11.1
Reduction to primitive blocks
8
Basis of Normal Tensors
8.1
Bases of normal tensors
8.2
Normal blocks and overlap rigidity
8.3
Block-injective consequences for BNT families
8.4
Permutation rigidity for bases of normal tensors
8.5
Newton–Girard identities and power-sum recovery
8.6
The BNT canonical form on a sector decomposition
8.6.1
Sector decompositions and phase matching
8.6.2
Single-copy sector decompositions
8.6.3
Prepared-block construction of BNT canonical forms
8.7
Coefficient comparison and copy-weight recovery
8.7.1
Matched-sector weight multiset equality
8.8
Strong existential matching by exact linear independence
8.8.1
Bijective matching by symmetry
9
Proof of the Fundamental Theorem
9.1
Matched block maps for product algebras
9.2
Coefficient identities and copy weights
9.3
The equal-MPV global gauge
9.4
The proportional and equal cases
9.4.1
Unitary gauges on normalized sectors
9.4.2
The equal case
10
Symmetries, Physical String Order, and Virtual-Boundary Nondecay
10.1
Virtual Local Covariance and Projective Bond Actions
10.2
Cohomology Classes of Virtual Symmetry Cocycles
10.2.1
Non-triviality via the commutator phase
10.3
Physical Endpoint String Order and Virtual-Boundary Nondecay
10.3.1
Periodic expectations and fixed-support limits
10.3.2
Stationary Physical Block Densities
10.3.3
Pure canonical tensors and the twisted peripheral spectrum
10.3.4
The physical endpoint selection rule at a fixed twist
10.3.5
Geometric finite-size estimates
10.3.6
Physical endpoints on dimension-bounded blocks
10.3.7
Continuous effective symmetries and identity endpoints
10.3.8
Physical endpoint string order in the AKLT and cluster states
10.3.9
A canonical tensor testing the scalar-phase convention
10.3.10
Exact periodic symmetry overlaps in the AKLT and cluster states
10.4
SPT Phase Labels and Virtual-Boundary Nondecay
10.4.1
Virtual actions along symmetric paths
10.4.2
Uniform rigidity of virtual projective classes
10.4.3
Compact limits and transient periodic components
10.4.4
Reduction to the stationary support
10.4.5
Propagation on finite periodic chains
10.4.6
Virtual classes at a compact limit
10.4.7
Renormalization fixed points of SPT phases
10.4.8
Fixed points with time reversal and reflection
10.4.9
Symmetric gapped paths and bond regrouping
10.4.10
Degeneracy of the entanglement spectrum
10.4.11
Time reversal, reflection, and symmetry obstructions
11
Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
11.1
The \(N\)-site Hilbert space
11.2
Local ground space \(G_L(A)\)
11.2.1
The FNW boundary convention
11.3
Parent interaction and chain Hamiltonian
11.3.1
The weighted FNW boundary estimate
11.3.2
The FNW lower boundary constant
11.3.3
The linear overlap estimate in FNW Lemma 6.2
11.3.4
The quadratic aggregate contribution in FNW Lemma 6.2
11.4
The geometric form of the FNW projector estimate
11.5
Intersection property and unique ground state
11.5.1
Restriction maps
11.5.2
Forward direction: ground space restricts
11.5.3
Injectivity of the ground-space map
11.5.4
The intersection property
11.5.5
Unique ground state
11.6
Transport under physical blocking
11.7
Ground-space correspondence under an invertible on-site map
11.8
Periodic-boundary comparison for injective tensors
11.9
Periodic unique ground state
11.10
Injectivity-length reduction for the closure property
11.11
Periodic-boundary comparison for normal tensors
11.11.1
Reverse comparison for boundary-crossing restrictions
11.11.2
Normal unique-ground-state consequences
11.12
Half-chain Schmidt spectrum and the dual fixed point
11.12.1
Ordered eigenvalues and finite-size normalization
11.12.2
The unital canonical convention
11.13
Two-block uncle Hamiltonians as actual parent limits
12
Commuting Parent Hamiltonians and Spectral Gaps
12.1
Commuting Local Terms for Parent Hamiltonians
12.2
Decorrelation and the idempotent-product parent-commuting condition
12.3
Spectral gap
12.3.1
Martingale condition for a finite family of projections
12.3.2
Cyclic-window overlap estimates
12.3.3
The weighted virtual matrix space
12.3.4
Nested ground projections and martingale differences
12.3.5
Spectator-indexed boundary maps
12.3.6
Martingale gap consequences
12.3.7
The compatible open-chain Hamiltonian
12.4
Overlaps of distinct block ground spaces
12.5
Overlap with a sum of subspaces
12.6
Comparison of sums of orthogonal projectors
12.7
Overlap bounds with spectator sites
12.8
The normalized primitive gauge of a normal tensor
12.8.1
The dimension-only prefactor at a prescribed rate
12.9
Prescribed gaps at sufficiently long interaction ranges
12.10
Continuity of the parent-Hamiltonian gap
12.10.1
Additional joint boundary continuity and compactness inputs
13
Block-Injective Parent Hamiltonians and Degenerate Ground Spaces
13.1
Block injectivity and degenerate ground spaces
13.2
Block separation and intersection
13.2.1
Trace identities for block intersections
13.2.2
Eventual block-intersection consequences
13.3
Block-diagonal boundary conditions for periodic ground spaces
13.3.1
Cyclic windows at the boundary cut
13.3.2
Periodic-boundary ground space for a block-diagonal tensor
13.4
Periodic-boundary ground space generated by a basis of normal tensors
14
Exponential Decay of Correlations
14.1
Connected correlators from the transfer map
14.2
Physical correlations and exponential decay
14.3
Quantitative transfer-map gap bounds
14.4
Relation to parent Hamiltonians
14.5
Physical correlations in an arbitrary bond basis
14.6
Jordan spectral expansions
15
Concrete Examples
15.1
Product states
15.2
The GHZ state
15.2.1
The \(G\)-injective fixed-point tensor
15.3
The AKLT state
15.3.1
Correlation length
15.4
The Majumdar–Ghosh state
15.5
The W state
15.6
Product Vacua with Boundary States
15.6.1
Exact Periodic Gap in the One-Species, Zero-Phase Model
15.6.2
Critical One-Particle Dynamics and the Thermodynamic Gap
15.7
The cluster state
15.8
The Kitaev chain
Scope.
15.9
Correlation independence without a renormalization fixed point
15.10
Parent Hamiltonian statements for the examples
16
Matrix Product Operators and Density Operators
16.1
MPO tensors
16.2
Transfer maps
16.3
MPDOs and LPDOs
16.4
Horizontal and Vertical Canonical Forms
16.5
Strong and weak symmetries
17
Pure States: Renormalization of Matrix Product States
17.1
Physical blocking and transfer idempotence
17.2
Zero-correlation-length conditions
17.3
Normal tensors and fixed-point isometries
17.3.1
Auxiliary block-family hypotheses
17.4
Direct sums and the joint isometry condition
17.5
Renormalization flow and physical correlations
18
Matrix Product Unitaries
18.1
The matrix product unitary condition
18.2
Simple matrix product unitary tensors
18.3
Double-layer blocking and mixed physical traces
18.4
Trace-power lemma
18.5
Full-support canonical representatives
18.6
Transfer-matrix identities
18.7
Block transfer multiplicity
18.8
Normalized transfer spectrum and conditional normality
18.9
Standard form, index, and symmetry classification
18.9.1
Local characterization of symmetries
18.9.2
Equivalence under symmetries
18.9.3
Symmetry examples
18.10
Finite-region observable algebras
18.10.1
Bipartite support algebras
18.10.2
Products of commuting matrix factors
18.10.3
Full-matrix presentation of a factor with scalar centre
18.11
Depth estimates for a merging tree
18.11.1
Boundary Gram matrices of a parity rotation
18.11.2
Interval isometries for product-projection phase gates
18.11.3
Finite coefficient alphabets at a cut
18.11.4
Unit-circle geometry and diagonal circuits
18.12
Exact circuits for matrix product unitaries
18.13
Limits of recursion inside the original Schmidt spaces
18.14
Completions and general approximation
18.15
Uniform families without exact-length full rank
18.16
Polynomial exact circuits at arbitrary fixed cut rank
19
Finite-Group MPU Representations and Scalar Gauge Data
19.1
Finite-group unitary averages
19.2
Finite-group representations by matrix product unitaries
19.2.1
Rank equality and the invertible first-cut comparison
19.2.2
Unitarity of the first-cut comparison
19.2.3
The compatible decomposition and its literal gates
19.2.4
Movement gates of the inverse-compatible source factors
19.3
A nondegenerate alternating-network comparison
19.4
Staircase gates and the first source network
19.4.1
The second network and unitarity of the staircase gates
19.4.2
Endpoint contraction and growth of truncated symmetries
19.4.3
Periodic sewing of the two source arcs
19.4.4
Physical closures of the source factors
19.5
Scalar three-cocycles and fusion-tensor gauge freedom
19.5.1
The anomaly three-cocycle from fusion tensors
19.5.2
Reductions through action tensors
19.5.3
An anomaly obstructs invariant normal states
19.5.4
Blocks permuted by the group
19.5.5
Domain walls between permuted blocks
19.5.6
Detecting the anomaly with truncated symmetries
19.5.7
Normalization of the first source factor
19.5.8
Normalization of the weighted second source factor
19.5.9
Physical normalizations of the two source X factors
19.5.10
Uniqueness of the source decompositions
19.6
The GHZ–blocked-cluster tensor
19.7
The gauge-invariant subspace of the CZX circuit tuple
19.7.1
Monomial operators and their common fixed space
19.7.2
The exact CZX matrix product unitary
19.7.3
The explicit CZX adjoint gauge
19.7.4
The displayed CZX source factors
19.7.5
Normalization of the displayed CZX comparison
19.7.6
The blocked GHZ state and its CZX invariance
19.7.7
The printed CZX action pair and its empty-word boundary
19.7.8
The CZX \(g,g\) fusion caps
19.7.9
Literal fusion–action coordinate contractions
19.7.10
The displayed CZX circuit tuple
19.7.11
Orbits, holonomy, and the exact dimension
19.7.12
A second displayed circuit tuple at three and four sites
19.8
The CZX defect domains and prescribed defect maps
19.9
The displayed CZX tuple as a completion
20
Mixed States: Renormalization of Matrix Product Operators
20.1
Preliminaries for physical renormalization
20.2
Renormalization fixed points
20.3
Pure-state recovery inside the MPO formalism
20.4
PEPS boundary tensors
20.5
The unnormalized strong fixed-point relation
20.6
Zero correlation length
20.7
Purification fixed points
20.8
Saturation of the area law
20.9
Support compression for entropy functionals
20.10
Rectangular Choi matrices and a range-dimension estimate
20.11
Isometric enlargements of the physical space
20.12
Simple tensors
20.13
Cyclic local-embedding coordinates
20.14
Gibbs states of nearest-neighbor commuting Hamiltonians
20.15
Two-site and positive-length physical blocking
20.16
Simple local structure from SAL and ZCL
20.16.1
Simple canonical-form weights
20.16.2
Markov decomposition and inverse-map sector factorization
20.16.3
Neighboring operators and commuting bond products
20.16.4
Closed-sector contractions and coherent rephasing
20.16.5
Normalized preparations and controlled partial traces
20.16.6
Primitivity and rank-one trace matrices
20.16.7
Refinement and coarse-graining channels
20.16.8
Local simple-MPDO structure
20.17
Single-bond commuting form from the local structure
21
Mixed States: General Case and Algebraic Structure
21.1
Closed-chain operators
21.2
Algebra structure
21.2.1
Diagonal \(\chi \)-matrices and the trace-power formula
21.2.2
Structure coefficients and the idempotent condition
21.2.3
A physical MPDO with a single vertical BNT element
21.2.4
Direct-sum sector conjugations
21.2.5
Ambient sector retractions and normalized embeddings
21.2.6
Trace-preserving completions and physical channels
21.3
Transfer retracts and fusion coisometries
22
Periodic MPS: irreducible form and the periodic Fundamental Theorem
22.1
Periodic irreducible form and physical-index rotation
22.1.1
The \(Z\)-gauge degree of freedom
22.2
Periodic Fundamental Theorem
22.2.1
Common-period blocking lemmas
22.2.2
Statement of the periodic Fundamental Theorem
22.2.3
Periodic overlap dichotomy
22.3
Physical symmetries on periodic MPS
22.4
Refinement and divisibility
23
The Algebraic Fundamental Theorem
23.1
Site-dependent chains at fixed length
23.2
Quadratic reconstruction of the translation-invariant canonical form
23.3
Uniqueness of the translation-invariant canonical form at a fixed ring length
23.4
One-sided inverse and physical realisation of virtual insertions
23.4.1
Physical realisation of virtual insertions
23.5
The virtual bond gauge and the proportionality lemma
23.5.1
Two preliminary facts
23.5.2
Same state forces a virtual bond gauge
23.5.3
Aligned gauges force proportionality
23.6
The Fundamental Theorem for injective chains
23.6.1
Finite-length MPV agreement
23.7
Corollaries: translation-invariant, non-uniform, and blocked chains
23.7.1
Translation-invariant chains
23.7.2
Blocked chains
23.8
Product algebra construction and per-block gauge
23.8.1
Block permutation algebra
23.8.2
Per-block linear extension and the product automorphism
23.8.3
Nondegeneracy of the product trace pairing
23.9
Gauge equivalence for direct sums
23.9.1
Same-index direct-sum corollaries from the single-block theorem
23.9.2
MPV invariance under reindexing
23.9.3
Converse: gauge equivalence implies same MPV
23.9.4
Canonical-form tensor as a weighted direct sum
23.9.5
Equal-case block matching with differing bond dimensions
23.10
Translation-invariant reduction and global symmetry
23.10.1
Global symmetry via gauge composition
24
Projected entangled pair states and the Fundamental Theorem
24.1
Finite Graph Tensor Networks
24.1.1
PEPS tensors and the states they generate
24.1.2
Gauge transformations on the bonds
24.1.3
Vertex injectivity and the local inverse
24.1.4
Edge-centred decomposition of a contraction
24.2
Edge-Middle Reduction
24.3
Edge Kernels and Gauge Transport
24.4
Fundamental Theorem for Injective PEPS
24.4.1
From the conjugation identity to the absorbed equality
24.5
Union Closure for Normal PEPS
24.6
Square-Lattice Normal PEPS
24.7
Torus Normal PEPS
Failure of the row-and-column reduction.
Staircase-window geometry for a reference edge.
24.7.1
Staircase patch geometry
24.7.2
Overlapping-window corner cancellation
24.7.3
Single-bond peeling and the window witness
24.7.4
Both edge directions and reference witnesses
24.7.5
Unscaled coefficients and the normal-window gauge
24.7.6
Transport and the vertical reference edge
24.7.7
Corner comparison at the minimal torus sizes
24.7.8
Gauge uniqueness from injective windows
24.7.9
Complements and staircase comparisons for positive windows
24.8
Normal PEPS Fundamental Theorem
24.9
Fundamental Theorem for normal matrix product states
24.9.1
Matrix product states on the closed chain
24.9.2
Translation-invariant matrix product states
24.9.3
The optimal system size \(n \ge 2L + 1\)
24.9.4
Site-dependent chains
24.10
The vertex-injective case and the residual edge scalars
24.11
Balanced Edge Scalars
24.11.1
Inserted-site scalar extraction
24.12
Exact contraction on the open square grid
24.13
Physical consequences of the square-grid correspondence
25
Examples of PEPS on the torus
25.1
Torus PEPS of a four-leg tensor
25.2
The GHZ state
25.3
The cluster state
25.4
The CZX model
25.5
The physical boundary symmetry of a CZX rectangle
25.6
The AKLT state
25.7
The RVB state
25.8
PEPS from classical models
26
G-injective PEPS and quantum double models
26.1
The toric code and quantum double models
The four-site block and its boundary.
Relative coordinates separate the second bond.
The original tensor and the four surplus registers.
The normalized fixed-point state.
26.1.1
Commuting constraints on overlapping regions
26.1.2
Physical parent interactions of regular \(G\)-isometric tensors
26.1.3
Regular intersection at one shared vertex
26.1.4
Group actions on torus bonds
26.1.5
The primal toric-code tensor in the parity basis
26.1.6
Four twisted sectors of the primal toric code
26.2
Local physical ranges of G-injective tensors
26.3
Comparison of invariant virtual spaces
26.4
G-injectivity under physical maps
26.5
Invertible physical comparison of G-injective tensors
26.6
Normalized local physical form
26.7
Local operators on the invariant boundary
26.8
Intersection of local boundary spaces along a strip
26.9
Ground spaces of heterogeneous open strips
27
Ground spaces and reduced densities of PEPS regions
27.1
Ground spaces of finite PEPS regions
27.2
Parent Hamiltonians on finite PEPS graphs
27.3
The one-dimensional specialization
27.4
Reduced densities of PEPS regions
27.5
Boundary support and Gibbs operators
27.6
Physical changes of coordinates and regional ground spaces
27.7
Physical transport of the common parent ground space
27.8
Unitary physical transport of parent Hamiltonians
27.9
The full-region ground space
27.10
Nesting of regional PEPS ground spaces
27.11
Comparison of finite PEPS parent Hamiltonians
27.12
Finite-overlap gap criteria
27.13
Parent spaces determined by the physical ray
27.14
Entanglement bounds on finite graphs
27.15
A counterexample to bare inverse-adjoint parent transport
27.16
The virtual Bell-bond ground space
27.17
Nearest-neighbor parents of injective PEPS
27.18
Regional supports of ordinary injective tensors
27.19
Vertex and bond coordinates
27.20
Internal Bell constraints from actual regional contractions
27.21
The physical image of the Bell product
27.22
The common one-site image space
27.23
Regional conditions after the site inverses
27.24
Exact transport of the common virtual conditions
27.24.1
Physical changes on the site images
27.24.2
Matching sectors in open weighted regions
27.25
Regional support of physical bond transformations
27.25.1
Products, coordinate slices, and multiplicity maps
27.25.2
One-ended boundary transformations
27.25.3
Physical coordinates and regional operator blocks
27.25.4
Actual open contractions and their product support
27.26
Positive multiplicities and complete canonical parent spaces
27.26.1
Copy weights and arbitrary open boundaries
27.26.2
Global supported inverses
27.26.3
Removing weights and changing multiplicities
27.27
Fourier coordinates for regional parent spaces
27.27.1
Regional coordinate changes
27.27.2
Exterior blocks and complete kernels
27.28
Semi-regular representations and physical parent kernels
27.28.1
Irreducible coordinates and the regular representation
27.28.2
The oriented incident action
27.28.3
Equivalence of the complete parent spaces
27.29
Open contractions with native edge orientations
27.29.1
Averaging tensors and unrestricted regional boundaries
27.29.2
Internal support and regional blocks
27.30
Coordinate changes with native edge orientations
27.31
Oriented multiplicity and physical transport
27.31.1
Copy maps on open oriented regions
27.31.2
Removing weights and comparing arbitrary physical tensors
27.32
Full native semi-regular torus parent spaces
27.32.1
Native geometry and the local invariant subspace
27.32.2
Commuting closures satisfy the original plaquette constraints
27.32.3
Dimension and exhaustion of the full kernel
27.32.4
Labelled bonds and the four cuts
27.32.5
Flat group labels on the actual bonds
27.32.6
Moving the commuting seams
27.32.7
Contractions with independently sized bonds
27.32.8
Matched representations and a common local inverse
27.32.9
Coherent coefficients forced by complementary cuts
27.32.10
The four-block closure property
27.32.11
Independent four-block closure classes
27.32.12
Four cut spaces with common coordinate alphabets
27.32.13
Physical inverses and matched projector contractions
27.32.14
Bond support and common-alphabet closure reconstruction
27.32.15
Partial averaging and unrestricted exterior bonds
27.32.16
Open cut intersections from removable internal labels
27.32.17
Lifted regional contractions and physical support
27.32.18
Reconstructing full cuts from lifted vectors
27.32.19
Intersections of lifted regional ranges
27.32.20
The open intersection for three four-legged tensors
27.32.21
Microscopic parent constraints and native seam boundaries
27.32.22
Nonuniform native parent classification
27.33
Orthogonalization of nested cylinders
27.34
A uniform width bound for angular reset scales
27.35
Tag encodings with regional zero resets
27.36
Regional embeddings in uniform physical dimension
27.37
Finite-domain area laws and polynomial PEPS approximation
27.37.1
Exact nearest-neighbor crossing sums
27.38
Regional states under relabelling
27.38.1
Finite improvement of the box exponent
27.39
Rounded scales in a safe rectangle
27.39.1
Rounded collar clearance
27.40
Minimum norm of regularized regional filters
27.41
Coordinate variation and final-state marginals
27.42
Sparse coordinate strips
27.43
Finite cell counts
27.44
Translated dyadic neighborhoods
27.45
The half-open layer partition
27.46
Distances between dyadic layers
27.47
The dummy neighborhood meets only the initial layer
27.48
Fine cells and sparse belts
27.49
Unique fine cells outside the dummy region
27.49.1
Capped contained dyadic squares
27.49.2
Ambient boundaries of template dilations
27.50
Entropy cost of a partial depth row
27.50.1
Finite weighted dyadic summation
27.51
Safe rectangles in template coverings
27.52
Weighted coverings of template shells
27.53
Entropy from a uniform safe-box estimate
27.53.1
Physical boundaries of separated templates
27.53.2
Mixed-square counts for template shells
27.54
Uniform separation of the dyadic scales
27.55
Polynomial factors in the dyadic budget
27.56
Primary birth regions and adjacent fine meshes
27.57
The number of primary identifiers in a layer
27.58
The marked points of belt cells
27.59
Separation on the translated quarter mesh
27.59.1
Clipping an incident segment
27.60
The finite family of initial marks
27.61
The fan of a belt cell
27.61.1
The primary containing a non-belt fine cell
27.62
Interiors of actual fan triangles
27.63
Allowed lines and the eight open fan triangles
27.64
Restriction of concentric dyadic fans
27.65
Actual initial regions in eight sectors
27.66
Colors of adjacent initial sectors
27.67
Actual initial identifiers on sector runs
27.68
Active radial interfaces near an initial mark
27.69
Opposing corners divide a side only at its midpoint
27.70
Ordered endpoints of a whole cell side
27.71
Subdivisions determined by actual corners
27.72
Contacts between actual fine cells
27.73
Dummy-neighborhood corners on fine-cell sides
27.74
Matching elementary sides under the actual subdivision
27.75
An opposing closed region for every elementary side
27.76
Uniqueness of the opposing region
27.77
Contact and reciprocal selection of opposing regions
27.78
Opposite labels on actual belt-cell sides
27.79
Merging equal-colored triangles in a cell
27.80
Geometric contacts within one cell fan
27.81
Cyclic order of the eight midpoint fan triangles
27.82
Directed rays of an elementary fan
27.83
Parity of changes of initial sector identifiers
27.84
Cardinality of the distinct active directed rays
27.85
Radial incidence and convexity in an elementary fan
27.86
Connected pieces of the midpoint fan
27.87
Connected fan pieces in a smaller closed square
27.88
Connected runs after radial deletion
27.89
Disjoint runs after radial deletion
27.90
Connected components after radial deletion
27.91
Connected components near an initial mark
27.92
The closed fine-cell cover of a primary region
27.93
Fan contacts along cell boundaries
27.94
Interfaces of belt runs with primary regions and other belt runs
27.95
Interfaces with the initial dummy region
27.96
The initial family of regions
27.97
The translated origin and integer lattice points
27.98
Disjoint interiors of the initial regions
27.99
Initial boundaries and lattice membership
27.100
The initial partition of the integer lattice
27.101
Colors across initial interfaces
27.102
Regularity of initial birth regions
27.103
Primitive frontiers of actual fans
27.104
Layers and meshes near a cell mark
27.105
Whole sides of a dyadic square
27.106
Initial frontiers near a mark
27.107
Removal squares and repair children
27.107.1
Depth and descendant count of a repair branch
27.108
Entropy cancellation for ordered labelled regions
27.109
Conditional two-family entropy estimate
27.110
Regional entropy dimension bounds
27.111
Regional entropy continuity
27.112
Entropy and dimension of a selected Schmidt spectrum
27.113
Boundary entropy from birth-target information
27.114
Collar and radius scales
27.114.1
The clock time and a good realization
27.115
The final entropy comparison of the amplification
27.116
Cuts with no crossing edge
27.116.1
Vertex-boundary formulations
27.117
Marginal surprisal tails at a lattice cut
27.118
Conditional movement and skew estimates on lattice regions
27.119
Conditional entropy in a buffered rectangle
27.120
Entropy cost of a partial rectangle layer
27.121
The initial safe-box estimate
27.122
Nested square patches
27.122.1
Dyadic counts for rectangle shells
27.123
Weighted dyadic covering of rectangle shells
27.124
Entropy of safe rectangle shells
27.125
Initial-exponent entropy of safe rectangle shells
27.125.1
Initial entropy estimate for rounded collars
27.125.2
Uniform coefficients for rounded shell estimates
27.125.3
Replica limits before large-scale entropy absorption
27.126
Geometry of the dyadic distribution protocol
27.127
Finite incidence in distributed PEPS compression
27.128
Exact operator contraction for distributed PEPS compression
27.129
Whole-group tensor bound and simultaneous truncation
27.130
Extracting one ket column from an operator network
27.130.1
Error and normalization of the protocol
27.130.2
Column selection after discarding a product reference
27.131
Dyadic routing of a party network into a square-lattice PEPS
27.131.1
Fixing a column before routing
27.132
Encoded frames and small-patch rewrites
27.133
Changes of ownership in encoded frames
27.133.1
Splitting a sheet
27.133.2
Homogeneous birth and death
27.133.3
Exchanging assignments between two sheets
27.134
Local interactions after a partial swap
27.135
Buffer operators in paired physical coordinates
27.136
A supported Gaussian operator for the physical buffer
27.137
Small-patch rewrites as allowed monomials
27.138
Elimination of normalized pair effects
27.139
Preparing and combining pair sources
27.140
Collecting local operations by party
27.140.1
Physical bases and party grouping
27.140.2
Register identifications and tensor coordinates
27.140.3
Restrictions with excluded registers
27.141
One-dimensional sources on unused pairs
27.142
Common private source spaces
27.143
Restricting the source-register operators
27.144
Density expansions of source gates
27.145
Joint contractions of the free source inputs
27.146
Grouping parties into two sides
27.147
Party restrictions after changing owner labels
27.148
Free source inputs and untouched exterior gates
27.148.1
Finite gate expansions
27.148.2
Affected parties and the exterior
27.148.3
Preparing only the fixed sources
27.148.4
Operations beside untouched registers
27.149
Chronological partial source expansions
27.150
Fixed sources for the chronological partial expansion
27.151
Source substitutions and corrected density terms
27.151.1
Preparing the original private input
27.152
Schmidt coordinates and one Gaussian source law
27.153
Physical contractions and Gaussian source estimates
27.154
Source-only reduction of distributed densities
27.155
Corrected source inputs and actual local outputs
27.156
Physical coordinates of the two local outputs
27.157
Sampling the actual source circuit
27.158
Approximate original gate expansions
27.159
Physical output after exchanging auxiliary registers
27.160
Exact elimination of finite-dimensional messages
27.161
Gates containing bounded messages
27.162
Original input and physical-output resources
27.163
Sampling an original construction
27.164
The exterior source memory when all parties are selected
28
Positive quasi-local constraints from a global gap
28.1
Truncation near a finite set
28.2
The induced lattice instance
28.3
Finite collar histories and good-history sampling
28.3.1
A common marginal moment bound for physical statuses
28.4
Entropy transfer on positive-depth prefixes
28.5
Replica metrics in augmented site coordinates
28.6
The transport root in the original site coordinates
28.7
Bad-history probability
28.8
The scanner estimate from its inputs
28.8.1
Rounded weights in the marginal-moment interval
28.9
The rough upper comparison for the supplied labels
28.10
Local root channels and physical shell oscillations
28.10.1
Localized square-root channels
28.10.2
Differences of successive local channels
28.11
Component support and full-channel oscillation
28.12
Finite channel oscillation bounds
28.12.1
Initial oscillation from physical support
28.12.2
Coefficients from the localization errors
28.12.3
Stretched-exponential localization errors
28.12.4
The full channel on an anchor component
28.12.5
Finite families of channel events
28.12.6
Uniform weighted rows of the finite event kernel
28.12.7
Kernel bounds for actual channel families
28.12.8
Actual constraints on a finite lattice domain
28.12.9
Extended graph distance and kernel support
28.13
Chronological finite channel words
28.14
Expected spatial oscillation of channel words
28.15
Excited-state decay for the exact constraints
28.16
Truncating the retained constraint product
28.17
Omitting distant events from the constraint product
28.17.1
Telescoping the omitted events
28.17.2
One omitted event
28.17.3
The coupled omission estimates
28.18
Typical projectors for the amplification
28.19
The restoring operator of the amplification
29
The Multi-Block Asymmetric Fundamental Theorem
29.1
Triangular word products
29.2
The word algebra and normal tensors as simple modules
29.3
Separating simple modules
29.4
The composition-series flag
29.5
The multi-block asymmetric compression theorem
29.6
Products of matrix product operators
29.6.1
Fusion and action tensors with multiplicity
29.7
Structure coefficients at every positive length
29.8
The splitting criterion
29.8.1
Uniqueness of split compressions
29.9
The single-block reduction of Molnár–Ge–Schuch–Cirac
29.10
Normalized split compressions
29.11
Rectangular reductions and residual algebras
30
Matrix Product Operator Symmetries and Non-Invertible Symmetries
30.1
Arbitrary boundary transport
30.2
Reconstruction from arbitrary boundaries
30.3
Comparing exact decomposition coordinates
Source index convention.
30.4
Canonical parents for adjoint-closed boundary ranges
30.5
Normal blocks and arbitrary-boundary adjoints
30.5.1
From periodic duality to exact similarities
30.5.2
Literal block sums and canonical parents
30.5.3
A derived algebraic counterexample
30.6
Adjoint boundaries from star-compatible coalgebras
30.7
Triangular physical padding
30.8
Mixed orientation of the fusion pentagon
30.9
A common fusion family for operators and states
30.10
Identifying the fusion and action comparisons
30.11
The multiplicity mixed pentagon from the exact source maps
Source index correction and scope.
30.12
Invertible symmetries
30.13
Fusion algebras of matrix product operators
30.14
Non-invertible symmetries
30.15
Symmetric matrix product states
30.16
The zipper condition
30.17
Mixed interpolation and endpoint supports
30.17.1
Local sectors at the first endpoint
30.17.2
Exact action on the mixed interpolation
30.17.3
Fusion of the mixed operator family
30.17.4
Three-site injectivity of the mixed operator
30.17.5
Common symbols along the mixed interpolation
30.17.6
Comparison at the first endpoint
30.17.7
Extended boundary spaces on open intervals
30.17.8
Joint extended boundary spaces
30.17.9
The mixed family with shared endpoint alphabets
30.17.10
Joint rectangular columns at a singular endpoint
30.17.11
Joint compression at the two boundary sites
30.17.12
The actual support in the joint boundary frame
30.17.13
The common core for every ordered pair of blocks
30.17.14
Dependent finite spectator coordinates
30.17.15
The degenerate mixed endpoint on a ring
30.17.16
Exchanging the two periodic endpoints
30.17.17
Open gaps from changes at the two boundary sites
30.17.18
A uniform gap along the closed mixed interpolation
30.18
Physical isometries and periodic gaps
30.18.1
Commuting support penalties
30.18.2
Orthogonal decomposition along an isometry
30.18.3
Periodic ground spaces and gap bounds
30.19
Mixed paths in arbitrary physical alphabets
30.19.1
Polar support and full endpoint embeddings
30.19.2
The continuous positive interaction
30.19.3
Ground lines and a volume-uniform gap
30.20
Attaching the canonical parents
31
Physics Examples of the Multi-Block Compression Theorem
31.1
The GHZ state as a compression target
31.2
A product state from a repeated block
31.2.1
Compression and the absence of a gauge
31.2.2
A Jordan extension detected by a boundary
31.3
The CZX symmetry
31.3.1
The bond-two tensor and the periodic operator
31.3.2
The tensor printed in the review
31.3.3
The decorated representative
31.3.4
Asymmetric compression of the square
31.3.5
A non-simple family with an oscillating coefficient
31.3.6
Scalar three-cocycles on the group of order two
31.4
The anomalous \(\mathbb {Z}_2\) symmetry of the Levin–Gu edge
31.5
Kramers–Wannier duality
31.5.1
The duality kernel and its square
31.5.2
Action on ground states
31.5.3
The duality as a bimodule over the spin-flip algebra
31.6
Fibonacci fusion operators
31.6.1
Admissible configurations and the vacuum operator
31.6.2
The golden chain
31.6.3
Two periodic states and the Fibonacci action
31.6.4
Asymmetric compression of the stacked square
31.6.5
The categorical anomaly
31.7
The Ising anyon chain
31.7.1
The periodic operators of the three tensors
31.7.2
One-site injectivity by exact arithmetic
31.7.3
The weighted direct sum and its local product
31.7.4
The gauge and the block decomposition
31.7.5
Compression onto the \(\sigma \) tensor
31.7.6
The length-dependent product coefficient
31.7.7
Coefficients from the Ising fusion rules
31.7.8
Action on a fusion-tree product state
31.7.9
Contraction of a matrix-unit factor
31.7.10
The three-object weighted direct sum
31.8
Group matrix product operators from a three-cocycle
31.8.1
Fusion tensors and the inverse cocycle
31.8.2
The group of order two and the decorated CZX operator
31.8.3
The cyclic groups
31.8.4
The group \(\mathbb {Z}_2\times \mathbb {Z}_2\)
31.9
The anomalous \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry and its condensation defect
31.9.1
The condensation defect and its periodic fusion table
31.9.2
The sixteen pair blocks of the stacked defect
31.9.3
The anomaly class of the full group
31.9.4
The eight scalar representatives
31.10
The anomalous \(\mathbb {Z}_3\) symmetry and its condensation defect
31.10.1
Fusion of the anomalous representation
31.10.2
The group family
31.10.3
The condensation defect
31.11
The \(\mathbb {Z}_3\) clock symmetry
31.12
Mixed Bell bonds and renormalization fixed points
31.12.1
Independent mixed Bell bonds
31.12.2
Mixed Bell bonds with an even-parity flag
31.12.3
Trace-preserving fixed-point maps
31.12.4
Vertical components and their multiplication
31.12.5
Representative rescaling
31.12.6
The one-label fixed point
31.12.7
The graded quantum-dimer twist
32
Matrix Product Operator Algebras
32.1
Fusion tensors and action tensors
32.2
Gauge freedom of fusion and action tensors
32.3
The associator of a group-like algebra
32.4
Cohomological obstruction to symmetric states
32.5
Weak Hopf algebras, dualities, and density operators
33
Local Quantum Circuits
33.1
Operators on sets of sites and local circuits
33.2
Two-site gates suffice
33.3
Circuits with measurements
33.4
Adaptive channels on arbitrary inputs
33.5
Teleportation and long-range gates
33.6
Local channel circuits and conversions
33.7
Approximate local channel conversions
33.8
Fixed physical ports and bounded local registers
33.8.1
Compiling bounded native protocols
33.8.2
Original physical-port input and output
33.8.3
Unitary circuits with one final environment discard
33.9
Lieb–Robinson propagation on finite graphs
Finite-range interactions on a graph.
Normalized partial traces.
33.10
Supported projected relative evolution
33.11
Physical-site oscillation with a spectator
33.11.1
The on-site unitary supremum
33.11.2
Averaging physical sites
34
Sequential Generation of Matrix Product States
34.1
Sequential schemes
34.2
Isometric decompositions
34.3
Resources of the successive decompositions
34.4
Cut ranks
34.5
Sequential generation with an ancilla
34.6
The deterministic transition scheme
34.7
Minimal resources
34.8
Sequential generation without an ancilla
34.9
Preparation by a staircase of local gates
35
Preparation of Matrix Product States by Log-Depth Circuits
35.1
Polar decomposition of a blocked tensor
35.1.1
Polar decomposition of a matrix
35.1.2
A tensor as a map on the virtual pair space
35.1.3
Polar factors of a tensor
35.2
The fixed-point pair state of a normal tensor
35.3
The approximating state
35.4
Exact factorizations of the isometry
35.5
Approximation error
35.5.1
Quadratic overlap estimates for ordered blocks
35.6
Depth of the preparation
35.6.1
Exact preparation with vanishing subleading spectrum
35.7
Conversion through compatible coherent seeds
35.7.1
Selected vectors with different sector probabilities
35.8
Canonical sector encoders
35.8.1
An exact whole-encoder construction
35.8.2
Uniform coherent conversion of the canonical sector encodings
35.8.3
Polynomial accuracy at every sufficiently large length
35.9
The tree circuit as a multiscale network
35.10
Inhomogeneous short-range correlated states
35.11
Uniform contraction of contiguous blocks
35.12
Normalized preparation from ordered mixing
35.13
Explicit blocking scales and physical depth
35.14
Varying density-matrix references
Discussion and outlook.
35.15
Quantitative preparation with varying bond dimensions
35.16
Preparation from local window mixing
Scope and boundary example.
35.16.1
Local windows on varying bond spaces
35.17
Tensors that are not normal
35.17.1
Blocks with multiplicities
35.17.2
Blocks whose states overlap
35.17.3
Repeated blocks whose states overlap
35.17.4
Weights with a phase
35.17.5
Overlapping blocks at the second-order rate
35.17.6
The actual canonical block pairs
35.18
Preparation with measurements
35.19
The tree circuit with measurements
35.20
Registers of several sites and the depth \(O(\log \log (N/\varepsilon ))\)
35.21
Trees on blocks of any length
35.21.1
Tensors that are not normal, with measurements
35.21.2
Overlapping blocks with measurements, for every chain length
35.22
Bounded physical leaves and arbitrary chain lengths
35.23
Non-normal canonical states in double-logarithmic depth
36
Interaction counting in lattice domains
36.1
Walks through overlapping supports
36.2
Ambient lattice counting
36.3
Original-support budgets
36.4
Exact lattice constants
36.5
Chains reaching a target
36.6
Exponential generating series
36.7
Graph-distance propagation in lattice domains
36.8
Localization from an arbitrary region
36.9
A supported Gaussian integral
36.10
Exact finite-size PEPS representations
36.11
The finite-size approximation conclusion
36.12
Removing finitely many exceptional sizes
36.12.1
Assembly from a compressed dyadic operator network
1
Appendices
A
Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
A.1
Canonical-gauge algebra
B
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
B.1
Mixed-transfer powers and overlap traces
B.2
Auxiliary trace positivity
B.3
Rectangular spectral-radius formulation
B.4
Peripheral intertwiners and gauge rigidity
B.5
Spectral-radius decay and overlap limits
B.6
Rank-one Perron projection
B.7
Periodicity removal
C
Wielandt Bound: Supporting Results
C.1
Exact-word and block-injectivity support
C.2
Cumulative-span and spectral linear algebra
C.3
One-step augmentation
C.4
Blocking and fixed-length spanning
C.5
Complementary-gap consequences
C.6
Identity in the one-step span
D
Canonical Form Reduction: Supporting Results
D.1
Similarities, strict splitting, and zero blocks
D.2
Projector closure and isometric corner decompositions
D.3
Perron gauges and preservation under conjugation
D.4
Blocking identities and primitive stability
D.5
Support compression and cyclic-sector families
D.6
Prepared normal-form consequences
E
Basis of Normal Tensors: Supporting Results
E.1
Elementary normal-tensor and Gram-matrix support
E.2
Change of basis, power sums, and sector bookkeeping
E.3
Direct-sum separation and prepared-family bookkeeping
F
Proof of the Fundamental Theorem: Supporting Results
F.1
Matched phases and coefficient extraction
F.2
Coefficient identities from gauge-phase equalities
F.3
The scalar-threaded proportional identity
F.4
Direct-sum conjugation
F.5
Additional proportional consequences
F.6
Matched flattened coordinates
F.7
Permutation gauges and literal coordinates
F.8
Unitary witness refinements
G
Symmetries and Virtual-Boundary Nondecay: Supporting Results
G.1
Permutation reindexing and identity bookkeeping
G.2
Gauge ratios and scalar uniqueness
G.3
Cocycle equivalence and coboundaries
G.4
Stationary-boundary twist support
G.5
Kraus mixing, transfer expansions, and a common TP gauge
G.6
Boundary and limit support
G.7
Universality of virtual-boundary nondecay
H
Relative Commutants of Full Matrix Algebras: Supporting Results
I
Physics Examples of the Multi-Block Compression Theorem: Supporting Results
I.1
Explicit gauges and exact arithmetic
I.2
Adapted bases and scaled matrix units
I.3
A sign-weighted pair with a non-split extension
J
Bibliography