1
Introduction
▶
Notation
2
Matrix Product Vectors
▶
2.1
Basic definitions
2.2
Site-periodic tensor families
2.3
Gauge equivalence and same MPV
2.4
Injectivity and normality
2.5
Canonical form
2.6
Blocking
2.7
MPV overlap
3
The Single-Block Fundamental Theorem
▶
3.1
The multiplicative linear extension
3.2
Inner automorphism and the single-block theorem
4
Transfer Maps and Quantum-Channel Interfaces
▶
4.1
Irreducibility
4.2
Transfer maps
5
Perron–Frobenius Theory for Channels and Transfer Maps
▶
5.1
Positive definiteness
5.2
Uniqueness
5.3
Existence and the Perron–Frobenius theorem
5.4
Right- and left-canonical gauges
5.5
Perron–Frobenius eigenvector existence
6
Peripheral Channel Structure and Transfer-Operator Gaps
▶
6.1
Mixed transfer operator
6.2
MPV overlap as transfer trace
6.3
Eigenvalue bound and transfer-operator gap
6.4
Transfer-operator gap — rectangular case
6.5
MPV overlap decay
6.6
Transfer-operator gap under irreducible-TP hypotheses
6.7
Overlap rigidity for normal blocks
6.8
Peripheral spectral refinements
▶
6.8.1
Cyclic decomposition of irreducible finite Kraus maps
6.9
Self-overlap and the period-one criterion
6.10
Primitive overlap convergence (complementary transfer-map gap form)
7
Wielandt Bound
▶
7.1
Cumulative span
▶
7.1.1
Paper primitivity and indices
7.2
Nonzero trace product
7.3
Eigenvector spreading
7.4
Quantum Wielandt bounds
7.5
Fixed-length matrix spanning
7.6
Primitive MPS tensors
7.7
Primitivity and normality
▶
7.7.1
From primitivity to normality
7.7.2
Primitive implies irreducible
7.7.3
From irreducibility to full spanning via Burnside’s theorem
7.7.4
From primitivity to strong irreducibility and normality
7.8
One-step padding and exact word spans
7.9
Block injectivity from TP/primitive/irreducible blocks
8
Canonical Form Reduction
▶
8.1
Invariant-subspace decomposition and irreducible blocks
▶
8.1.1
Iterated reduction to irreducible blocks
8.2
Nonzero blocks and the trace-preserving gauge
▶
8.2.1
Trace-preserving gauge for arbitrary tensors
8.3
Blocking, period removal, and primitive blocks
▶
8.3.1
Period removal by cyclic sectors
8.4
Cyclic-sector irreducibility
8.5
Translation-invariant canonical form
8.6
Normal tensors and canonical forms
▶
8.6.1
Normalization conventions
8.7
Projector criterion for canonical decomposition
8.8
Fixed points and canonical gauges of irreducible blocks
8.9
Normal canonical form
▶
8.9.1
Separated one-copy normal canonical forms
8.9.2
Canonical form from primitivity
8.10
Left-canonical and dual diagonalization of irreducible blocks
8.11
Cyclic-sector isometries and primitive reduction
▶
8.11.1
Reduction to primitive blocks
9
Basis of Normal Tensors
▶
9.1
Bases of normal tensors
9.2
Normal blocks and overlap rigidity
9.3
Block-injective consequences for BNT families
9.4
Permutation rigidity for bases of normal tensors
9.5
Newton–Girard identities and power-sum recovery
9.6
The BNT canonical form on a sector decomposition
▶
9.6.1
Sector decompositions and phase matching
9.6.2
Single-copy sector decompositions
9.6.3
Prepared-block construction of BNT canonical forms
9.7
Coefficient comparison and copy-weight recovery
▶
9.7.1
Matched-sector weight multiset equality
9.8
Strong existential matching by exact linear independence
▶
9.8.1
Bijective matching by symmetry
10
Proof of the Fundamental Theorem
▶
10.1
Matched block maps for product algebras
10.2
Coefficient identities and copy weights
10.3
The equal-MPV global gauge
10.4
The proportional and equal cases
▶
10.4.1
Unitary gauges on normalized sectors
10.4.2
The equal case
11
Symmetries, Physical String Order, and Virtual-Boundary Nondecay
▶
11.1
Virtual Local Covariance and Projective Bond Actions
11.2
Cohomology Classes of Virtual Symmetry Cocycles
▶
11.2.1
Non-triviality via the commutator phase
11.3
Physical Endpoint String Order and Virtual-Boundary Nondecay
11.4
SPT Phase Labels and Virtual-Boundary Nondecay
12
Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
▶
12.1
The \(N\)-site Hilbert space
12.2
Local ground space \(G_L(A)\)
12.3
Parent interaction and chain Hamiltonian
12.4
Intersection property and unique ground state
▶
12.4.1
Restriction maps
12.4.2
Forward direction: ground space restricts
12.4.3
Injectivity of the ground-space map
12.4.4
The intersection property
12.4.5
Unique ground state
12.5
Transport under physical blocking
12.6
Periodic-boundary comparison for injective tensors
12.7
Periodic unique ground state
12.8
Injectivity-length reduction for the closure property
12.9
Periodic-boundary comparison for normal tensors
▶
12.9.1
Reverse comparison for boundary-crossing restrictions
12.9.2
Normal unique-ground-state consequences
12.10
Half-chain Schmidt spectrum and the dual fixed point
13
Commuting Parent Hamiltonians and Spectral Gaps
▶
13.1
Commuting Local Terms for Parent Hamiltonians
13.2
Decorrelation and the idempotent-product parent-commuting condition
13.3
Spectral gap
▶
13.3.1
Martingale condition for a finite family of projections
13.3.2
Cyclic-window overlap estimates
13.3.3
Spectator-indexed boundary maps
13.3.4
Martingale gap consequences
13.3.5
The compatible open-chain Hamiltonian
14
Block-Injective Parent Hamiltonians and Degenerate Ground Spaces
▶
14.1
Block injectivity and degenerate ground spaces
14.2
Block separation and intersection
▶
14.2.1
Trace identities for block intersections
14.2.2
Eventual block-intersection consequences
14.3
Block-diagonal boundary conditions for periodic ground spaces
▶
14.3.1
Cyclic windows at the boundary cut
14.3.2
Periodic-boundary ground space for a block-diagonal tensor
14.4
Periodic-boundary ground space generated by a basis of normal tensors
15
Exponential Decay of Correlations
▶
15.1
Connected correlators from the transfer map
15.2
Spectral expansion and exponential decay
15.3
Quantitative transfer-map gap bounds
15.4
Relation to parent Hamiltonians
16
Concrete Examples
▶
16.1
The GHZ state
16.2
The AKLT state
▶
16.2.1
Correlation length
16.3
The Majumdar–Ghosh state
16.4
The W state
16.5
The cluster state
16.6
Correlation independence without a renormalization fixed point
16.7
Parent Hamiltonian statements for the examples
17
Matrix Product Operators and Density Operators
▶
17.1
MPO tensors
17.2
Transfer maps
17.3
MPDOs and LPDOs
17.4
Horizontal and Vertical Canonical Forms
18
Pure States: Renormalization of Matrix Product States
▶
18.1
Physical blocking and transfer idempotence
18.2
Zero-correlation-length conditions
18.3
Normal tensors and fixed-point isometries
▶
18.3.1
Auxiliary block-family hypotheses
18.4
Direct sums and the joint isometry condition
18.5
Renormalization flow and physical correlations
19
Matrix Product Unitaries
▶
19.1
The matrix product unitary condition
19.2
Simple matrix product unitary tensors
19.3
Double-layer blocking and mixed physical traces
19.4
Trace-power lemma
19.5
Full-support canonical representatives
19.6
Transfer-matrix identities
19.7
Block transfer multiplicity
19.8
Normalized transfer spectrum and conditional normality
19.9
Standard form, index, and symmetry classification
▶
19.9.1
Local characterization of symmetries
19.9.2
Equivalence under symmetries
19.9.3
Symmetry examples
19.10
Finite-region observable algebras
▶
19.10.1
Bipartite support algebras
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
The unnormalized strong fixed-point relation
20.5
Zero correlation length
20.6
Purification fixed points
20.7
Saturation of the area law
20.8
Support compression for entropy functionals
20.9
Rectangular Choi matrices and a range-dimension estimate
20.10
Isometric enlargements of the physical space
20.11
Simple tensors
20.12
Cyclic local-embedding coordinates
20.13
Gibbs states of nearest-neighbor commuting Hamiltonians
20.14
Two-site and positive-length physical blocking
20.15
Simple local structure from SAL and ZCL
▶
20.15.1
Simple canonical-form weights
20.15.2
Markov decomposition and inverse-map sector factorization
20.15.3
Neighboring operators and commuting bond products
20.15.4
Closed-sector contractions and coherent rephasing
20.15.5
Normalized preparations and controlled partial traces
20.15.6
Primitivity and rank-one trace matrices
20.15.7
Refinement and coarse-graining channels
20.15.8
Local simple-MPDO structure
20.16
Single-bond commuting form from the local structure
21
Mixed States: General Case and Algebraic Structure
▶
21.1
Recall of the vertical canonical form and closed-chain operators
21.2
Algebra structure
▶
21.2.1
Diagonal \(\chi \)-matrices and the trace-power formula
21.2.2
The BNT-label algebra law and idempotent trace vector
21.2.3
An all-length bond-two singleton base model
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
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
Routine universality corollary
H
Bibliography
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Dependency graphs
Tensor Network Theory: A formalization blueprint
Sirui Lu, Erickson Tjoa, and J. Ignacio Cirac
Last update: August 26, 2026
1
Introduction
Notation
2
Matrix Product Vectors
2.1
Basic definitions
2.2
Site-periodic tensor families
2.3
Gauge equivalence and same MPV
2.4
Injectivity and normality
2.5
Canonical form
2.6
Blocking
2.7
MPV overlap
3
The Single-Block Fundamental Theorem
3.1
The multiplicative linear extension
3.2
Inner automorphism and the single-block theorem
4
Transfer Maps and Quantum-Channel Interfaces
4.1
Irreducibility
4.2
Transfer maps
5
Perron–Frobenius Theory for Channels and Transfer Maps
5.1
Positive definiteness
5.2
Uniqueness
5.3
Existence and the Perron–Frobenius theorem
5.4
Right- and left-canonical gauges
5.5
Perron–Frobenius eigenvector existence
6
Peripheral Channel Structure and Transfer-Operator Gaps
6.1
Mixed transfer operator
6.2
MPV overlap as transfer trace
6.3
Eigenvalue bound and transfer-operator gap
6.4
Transfer-operator gap — rectangular case
6.5
MPV overlap decay
6.6
Transfer-operator gap under irreducible-TP hypotheses
6.7
Overlap rigidity for normal blocks
6.8
Peripheral spectral refinements
6.8.1
Cyclic decomposition of irreducible finite Kraus maps
6.9
Self-overlap and the period-one criterion
6.10
Primitive overlap convergence (complementary transfer-map gap form)
7
Wielandt Bound
7.1
Cumulative span
7.1.1
Paper primitivity and indices
7.2
Nonzero trace product
7.3
Eigenvector spreading
7.4
Quantum Wielandt bounds
7.5
Fixed-length matrix spanning
7.6
Primitive MPS tensors
7.7
Primitivity and normality
7.7.1
From primitivity to normality
7.7.2
Primitive implies irreducible
7.7.3
From irreducibility to full spanning via Burnside’s theorem
7.7.4
From primitivity to strong irreducibility and normality
7.8
One-step padding and exact word spans
7.9
Block injectivity from TP/primitive/irreducible blocks
8
Canonical Form Reduction
8.1
Invariant-subspace decomposition and irreducible blocks
8.1.1
Iterated reduction to irreducible blocks
8.2
Nonzero blocks and the trace-preserving gauge
8.2.1
Trace-preserving gauge for arbitrary tensors
8.3
Blocking, period removal, and primitive blocks
8.3.1
Period removal by cyclic sectors
8.4
Cyclic-sector irreducibility
8.5
Translation-invariant canonical form
8.6
Normal tensors and canonical forms
8.6.1
Normalization conventions
8.7
Projector criterion for canonical decomposition
8.8
Fixed points and canonical gauges of irreducible blocks
8.9
Normal canonical form
8.9.1
Separated one-copy normal canonical forms
8.9.2
Canonical form from primitivity
8.10
Left-canonical and dual diagonalization of irreducible blocks
8.11
Cyclic-sector isometries and primitive reduction
8.11.1
Reduction to primitive blocks
9
Basis of Normal Tensors
9.1
Bases of normal tensors
9.2
Normal blocks and overlap rigidity
9.3
Block-injective consequences for BNT families
9.4
Permutation rigidity for bases of normal tensors
9.5
Newton–Girard identities and power-sum recovery
9.6
The BNT canonical form on a sector decomposition
9.6.1
Sector decompositions and phase matching
9.6.2
Single-copy sector decompositions
9.6.3
Prepared-block construction of BNT canonical forms
9.7
Coefficient comparison and copy-weight recovery
9.7.1
Matched-sector weight multiset equality
9.8
Strong existential matching by exact linear independence
9.8.1
Bijective matching by symmetry
10
Proof of the Fundamental Theorem
10.1
Matched block maps for product algebras
10.2
Coefficient identities and copy weights
10.3
The equal-MPV global gauge
10.4
The proportional and equal cases
10.4.1
Unitary gauges on normalized sectors
10.4.2
The equal case
11
Symmetries, Physical String Order, and Virtual-Boundary Nondecay
11.1
Virtual Local Covariance and Projective Bond Actions
11.2
Cohomology Classes of Virtual Symmetry Cocycles
11.2.1
Non-triviality via the commutator phase
11.3
Physical Endpoint String Order and Virtual-Boundary Nondecay
11.4
SPT Phase Labels and Virtual-Boundary Nondecay
12
Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
12.1
The \(N\)-site Hilbert space
12.2
Local ground space \(G_L(A)\)
12.3
Parent interaction and chain Hamiltonian
12.4
Intersection property and unique ground state
12.4.1
Restriction maps
12.4.2
Forward direction: ground space restricts
12.4.3
Injectivity of the ground-space map
12.4.4
The intersection property
12.4.5
Unique ground state
12.5
Transport under physical blocking
12.6
Periodic-boundary comparison for injective tensors
12.7
Periodic unique ground state
12.8
Injectivity-length reduction for the closure property
12.9
Periodic-boundary comparison for normal tensors
12.9.1
Reverse comparison for boundary-crossing restrictions
12.9.2
Normal unique-ground-state consequences
12.10
Half-chain Schmidt spectrum and the dual fixed point
13
Commuting Parent Hamiltonians and Spectral Gaps
13.1
Commuting Local Terms for Parent Hamiltonians
13.2
Decorrelation and the idempotent-product parent-commuting condition
13.3
Spectral gap
13.3.1
Martingale condition for a finite family of projections
13.3.2
Cyclic-window overlap estimates
13.3.3
Spectator-indexed boundary maps
13.3.4
Martingale gap consequences
13.3.5
The compatible open-chain Hamiltonian
14
Block-Injective Parent Hamiltonians and Degenerate Ground Spaces
14.1
Block injectivity and degenerate ground spaces
14.2
Block separation and intersection
14.2.1
Trace identities for block intersections
14.2.2
Eventual block-intersection consequences
14.3
Block-diagonal boundary conditions for periodic ground spaces
14.3.1
Cyclic windows at the boundary cut
14.3.2
Periodic-boundary ground space for a block-diagonal tensor
14.4
Periodic-boundary ground space generated by a basis of normal tensors
15
Exponential Decay of Correlations
15.1
Connected correlators from the transfer map
15.2
Spectral expansion and exponential decay
15.3
Quantitative transfer-map gap bounds
15.4
Relation to parent Hamiltonians
16
Concrete Examples
16.1
The GHZ state
16.2
The AKLT state
16.2.1
Correlation length
16.3
The Majumdar–Ghosh state
16.4
The W state
16.5
The cluster state
16.6
Correlation independence without a renormalization fixed point
16.7
Parent Hamiltonian statements for the examples
17
Matrix Product Operators and Density Operators
17.1
MPO tensors
17.2
Transfer maps
17.3
MPDOs and LPDOs
17.4
Horizontal and Vertical Canonical Forms
18
Pure States: Renormalization of Matrix Product States
18.1
Physical blocking and transfer idempotence
18.2
Zero-correlation-length conditions
18.3
Normal tensors and fixed-point isometries
18.3.1
Auxiliary block-family hypotheses
18.4
Direct sums and the joint isometry condition
18.5
Renormalization flow and physical correlations
19
Matrix Product Unitaries
19.1
The matrix product unitary condition
19.2
Simple matrix product unitary tensors
19.3
Double-layer blocking and mixed physical traces
19.4
Trace-power lemma
19.5
Full-support canonical representatives
19.6
Transfer-matrix identities
19.7
Block transfer multiplicity
19.8
Normalized transfer spectrum and conditional normality
19.9
Standard form, index, and symmetry classification
19.9.1
Local characterization of symmetries
19.9.2
Equivalence under symmetries
19.9.3
Symmetry examples
19.10
Finite-region observable algebras
19.10.1
Bipartite support algebras
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
The unnormalized strong fixed-point relation
20.5
Zero correlation length
20.6
Purification fixed points
20.7
Saturation of the area law
20.8
Support compression for entropy functionals
20.9
Rectangular Choi matrices and a range-dimension estimate
20.10
Isometric enlargements of the physical space
20.11
Simple tensors
20.12
Cyclic local-embedding coordinates
20.13
Gibbs states of nearest-neighbor commuting Hamiltonians
20.14
Two-site and positive-length physical blocking
20.15
Simple local structure from SAL and ZCL
20.15.1
Simple canonical-form weights
20.15.2
Markov decomposition and inverse-map sector factorization
20.15.3
Neighboring operators and commuting bond products
20.15.4
Closed-sector contractions and coherent rephasing
20.15.5
Normalized preparations and controlled partial traces
20.15.6
Primitivity and rank-one trace matrices
20.15.7
Refinement and coarse-graining channels
20.15.8
Local simple-MPDO structure
20.16
Single-bond commuting form from the local structure
21
Mixed States: General Case and Algebraic Structure
21.1
Recall of the vertical canonical form and closed-chain operators
21.2
Algebra structure
21.2.1
Diagonal \(\chi \)-matrices and the trace-power formula
21.2.2
The BNT-label algebra law and idempotent trace vector
21.2.3
An all-length bond-two singleton base model
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
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
Routine universality corollary
H
Bibliography