Tensor Network Theory: A formalization blueprint
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
Quantum Channels and Positive Maps
▶
4.1
Positive and completely positive maps
4.2
Kraus representations and complete positivity
4.3
Quantum channels
4.4
Rectangular positive maps, trace bounds, and density matrices
4.5
Maximally entangled states and Choi matrices
4.6
Irreducibility
4.7
Transfer maps
4.8
Peripheral spectrum and primitivity
4.9
Fixed-point projection
4.10
Peripheral eigenvalues and powering
▶
4.10.1
Periodicity removal by powering
4.10.2
Peripheral closure via adjoint fixed point
4.11
Primitivity and the complementary transfer-map gap
5
Schwarz Inequalities and Multiplicative Domains
▶
5.1
Kadison–Schwarz inequality
5.2
Multiplicative domain
5.3
Trace adjoints and positive retractions
5.4
Douglas-type factorization
5.5
Peripheral Schwarz equality with a faithful fixed point
5.6
Additional results on multiplicative domains and order
▶
5.6.1
Abstract Schwarz maps and their multiplicative domains
5.6.2
Kraus specialization and full algebraic structure
5.6.3
Positive maps preserve order and spectral intervals
6
Perron–Frobenius Theory for Channels and Transfer Maps
▶
6.1
Cesàro fixed points for quantum channels
6.2
Positive definiteness
6.3
Uniqueness
6.4
Existence and the Perron–Frobenius theorem
6.5
Right- and left-canonical gauges
6.6
Similarity preserves irreducibility
6.7
Perron–Frobenius eigenvector existence
6.8
Exponential positivity for irreducible CP maps
6.9
Ergodicity of irreducible channels
6.10
Spectral radius at the Perron eigenvalue
6.11
A CP spectral characterization of irreducibility
7
Peripheral Channel Structure and Transfer-Operator Gaps
▶
7.1
Peripheral eigenvalue group structure
7.2
Mixed transfer operator
7.3
MPV overlap as transfer trace
7.4
Eigenvalue bound and transfer-operator gap
7.5
Transfer-operator gap — rectangular case
7.6
MPV overlap decay
7.7
Transfer-operator gap under irreducible-TP hypotheses
7.8
Overlap rigidity for normal blocks
7.9
Peripheral spectral refinements
▶
7.9.1
Cyclic decomposition of irreducible Schwarz maps
7.10
Self-overlap and the period-one criterion
7.11
Primitive overlap convergence (complementary transfer-map gap form)
8
Wielandt Bound
▶
8.1
Cumulative span
▶
8.1.1
Paper primitivity and indices
8.2
Nonzero trace product
8.3
Eigenvector spreading
8.4
Quantum Wielandt bounds
8.5
Fixed-length matrix spanning
8.6
Primitive MPS tensors
8.7
Primitivity and normality
▶
8.7.1
From primitivity to normality
8.7.2
Primitive implies irreducible
8.7.3
From irreducibility to full spanning via Burnside’s theorem
8.7.4
From primitivity to strong irreducibility and normality
8.8
One-step padding and exact word spans
8.9
Block injectivity from TP/primitive/irreducible blocks
9
Canonical Form Reduction
▶
9.1
Invariant-subspace decomposition and irreducible blocks
▶
9.1.1
Iterated reduction to irreducible blocks
9.2
Nonzero blocks and the trace-preserving gauge
▶
9.2.1
Trace-preserving gauge for arbitrary tensors
9.3
Blocking, period removal, and primitive blocks
▶
9.3.1
Period removal by cyclic sectors
9.4
Cyclic-sector irreducibility
9.5
Translation-invariant canonical form
9.6
Normal tensors and canonical forms
▶
9.6.1
Normalization conventions
9.7
Projector criterion for canonical decomposition
9.8
Fixed points and canonical gauges of irreducible blocks
9.9
Normal canonical form
▶
9.9.1
Separated one-copy normal canonical forms
9.9.2
Canonical form from primitivity
9.10
Left-canonical and dual diagonalization of irreducible blocks
9.11
Cyclic-sector isometries and primitive reduction
▶
9.11.1
Reduction to primitive blocks
10
Basis of Normal Tensors
▶
10.1
Bases of normal tensors
10.2
Normal blocks and overlap rigidity
10.3
Block-injective consequences for BNT families
10.4
Permutation rigidity for bases of normal tensors
10.5
Newton–Girard identities and power-sum recovery
10.6
The BNT canonical form on a sector decomposition
▶
10.6.1
Sector decompositions and phase matching
10.6.2
Single-copy sector decompositions
10.6.3
Prepared-block construction of BNT canonical forms
10.7
Coefficient comparison and copy-weight recovery
▶
10.7.1
Matched-sector weight multiset equality
10.8
Strong existential matching by exact linear independence
▶
10.8.1
Bijective matching by symmetry
11
Proof of the Fundamental Theorem
▶
11.1
Coefficient identities and copy weights
11.2
The equal-MPV global gauge
11.3
The proportional and equal source theorems
▶
11.3.1
Unitary gauges on normalized sectors
11.3.2
The equal case
12
Symmetries, Physical String Order, and Virtual-Boundary Nondecay
▶
12.1
Virtual Local Covariance and Projective Bond Actions
12.2
Cohomology Classes of Virtual Symmetry Cocycles
▶
12.2.1
Non-triviality via the commutator phase
12.3
Physical Endpoint String Order and Virtual-Boundary Nondecay
12.4
SPT Phase Labels and Virtual-Boundary Nondecay
13
Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
▶
13.1
The \(N\)-site Hilbert space
13.2
Local ground space \(G_L(A)\)
13.3
Parent interaction and chain Hamiltonian
13.4
Intersection property and unique ground state
▶
13.4.1
Restriction maps
13.4.2
Forward direction: ground space restricts
13.4.3
Injectivity of the ground-space map
13.4.4
The intersection property
13.4.5
Unique ground state
13.5
Periodic-boundary comparison for injective tensors
13.6
Periodic unique ground state
13.7
Injectivity-length reduction for the closure property
13.8
Periodic-boundary comparison for normal tensors
▶
13.8.1
Reverse comparison for boundary-crossing restrictions
13.8.2
Normal unique-ground-state consequences
14
Commuting Parent Hamiltonians and Spectral Gaps
▶
14.1
Commuting Local Terms for Parent Hamiltonians
14.2
Decorrelation and the idempotent-product parent-commuting condition
14.3
Spectral gap
▶
14.3.1
Martingale condition for a finite family of projections
14.3.2
Cyclic-window overlap estimates
14.3.3
Martingale gap consequences
15
Block-Injective Parent Hamiltonians and Degenerate Ground Spaces
▶
15.1
Block injectivity and degenerate ground spaces
15.2
Block separation and intersection
▶
15.2.1
Trace identities for block intersections
15.2.2
Eventual block-intersection consequences
15.3
Block-diagonal boundary conditions for periodic ground spaces
▶
15.3.1
Cyclic windows at the boundary cut
15.3.2
Periodic-boundary ground space for a block-diagonal tensor
15.4
Periodic-boundary ground space generated by a basis of normal tensors
16
Exponential Decay of Correlations
▶
16.1
Connected correlators from the transfer map
16.2
Spectral expansion and exponential decay
16.3
Quantitative transfer-map gap bounds
16.4
Relation to parent Hamiltonians
17
Concrete Examples
▶
17.1
The GHZ state
17.2
The AKLT state
▶
17.2.1
Correlation length
17.3
The Majumdar–Ghosh state
17.4
The W state
17.5
The cluster state
17.6
Parent Hamiltonian statements for the examples
18
Channel Representations and Normal Forms
▶
18.1
Representations
18.2
Further Choi-matrix identities
18.3
Representation corollaries for channel decompositions
18.4
Kraus representation theorem
18.5
Stinespring dilation
18.6
Ordered CP maps, Radon–Nikodym, and open-system representation
18.7
POVMs and Naimark dilation
18.8
Trace-pairing expansion in transfer-matrix form
18.9
SVD normal form (existence)
18.10
Lorentz normal form
18.11
Determinant of a quantum channel
19
Quantum Dynamical Semigroups
▶
19.1
Dynamical semigroups and the exponential form
19.2
Perturbation theory
19.3
GKSL/Lindblad generators
▶
19.3.1
Generator decomposition and conditional complete positivity
19.3.2
The Lindblad form
19.3.3
Characterization of conditional complete positivity
19.3.4
Completely positive semigroups and conditionally completely positive generators
19.3.5
Freedom in generator representation
19.3.6
Uniqueness of the traceless Lindblad form
19.3.7
Trace-annihilation and trace preservation
19.3.8
The GKSL/Lindblad theorem
19.3.9
Kossakowski matrix form
19.4
Dissipation generated by two Pauli matrices
19.5
Primitivity and irreducibility of QDS
▶
19.5.1
Auxiliary spectral and semigroup lemmas
19.6
Kernel of the adjoint Liouvillian
19.7
Reducibility of quantum dynamical semigroups
▶
19.7.1
Sufficient conditions for non-reducibility
20
Operator Convexity and Jensen Inequalities
▶
20.1
Schwarz inequalities for positive maps
▶
20.1.1
Schwarz inequality for normal operators
20.1.2
Schwarz inequality for subnormal and commuting-dominant operators
20.2
Diagonal Jensen inequality
20.3
Trace concavity and convexity of matrix powers
20.4
Operator convexity of real powers
20.5
Operator Jensen inequality for positive maps
20.6
Lieb concavity theorem
20.7
Resolvent monotonicity toward the Lieb concavity theorem
21
Quantum Entropy
▶
21.1
Trace norm
21.2
Von Neumann entropy
21.3
Tripartite partial traces
21.4
Strong subadditivity
21.5
Mutual information
21.6
Entropy formulations
21.7
Mutual information: monotonicity and area-law bound
21.8
Trivial-factor corollaries
22
Positive but Not Completely Positive Maps
▶
22.1
Trace normalization and the Lorentz cone
22.2
\(k\)-positive maps, Schmidt rank, and Choi compression
▶
22.2.1
Two-positive maps and the generalized Schwarz inequality
22.2.2
Schmidt rank and maximal overlap
22.2.3
Choi compression criteria
22.3
Trace adjoints and the positivity hierarchy
22.4
Elementary positive-map examples
▶
22.4.1
The reduction map
22.4.2
Automorphisms of the positive semidefinite cone
22.5
Positivity hierarchy and two-positive maps
▶
22.5.1
The map \(T_\eta \) and strictness of the chain
22.6
The reduction criterion
22.7
The Breuer–Hall map
22.8
Choi-type maps
22.9
Transposition
22.10
Decomposable positive maps
22.11
The partial transpose and the PPT property
22.12
Separable states and the PPT criterion
22.13
The Schmidt number and the full reduction criterion
22.14
Positive Schwarz maps outside complete positivity
▶
22.14.1
A positive Schwarz map that is not completely positive
23
Matrix Product Operators and Density Operators
▶
23.1
MPO tensors
23.2
Transfer maps
23.3
MPDOs and LPDOs
23.4
Horizontal and Vertical Canonical Forms
24
Pure States: Renormalization of Matrix Product States
▶
24.1
Physical blocking and transfer idempotence
24.2
Zero-correlation-length conditions
24.3
Normal tensors and fixed-point isometries
▶
24.3.1
Auxiliary block-family hypotheses
24.4
Direct sums and the joint isometry condition
24.5
Renormalization flow and physical correlations
25
Asymptotic Structure of Quantum Channels
▶
25.1
Mean-ergodic theory and fixed-point structure
25.2
Fixed-point algebra
25.3
Conditional expectation from a faithful fixed point
25.4
Stationary support
▶
25.4.1
Faithful compression onto the support sector
25.5
Wedderburn decomposition of the fixed-point algebra
25.6
Wedderburn decomposition of the fixed-point algebra (continued)
25.7
Schwarz maps on direct sums of matrix algebras
25.8
Fixed-point structure and cycle decompositions
25.9
Further irreducibility and primitivity equivalences
25.10
Multi-cycle block-permutation structure
26
Mixed States: Renormalization of Matrix Product Operators
▶
26.1
Preliminaries for physical renormalization
26.2
Renormalization fixed points
26.3
Pure-state recovery inside the MPO formalism
26.4
Zero correlation length
26.5
Purification fixed points
26.6
Saturation of the area law
26.7
Simple tensors
26.8
Gibbs states of nearest-neighbor commuting Hamiltonians
26.9
Two-site and positive-length physical blocking
26.10
Simple local structure from SAL and ZCL
▶
26.10.1
Simple canonical-form weights
26.10.2
Markov decomposition and inverse-map sector factorization
26.10.3
Neighboring operators and commuting bond products
26.10.4
Closed-sector contractions and coherent rephasing
26.10.5
Normalized preparations and controlled partial traces
26.10.6
Primitivity and rank-one trace matrices
26.10.7
Refinement and coarse-graining channels
26.10.8
Local simple-MPDO structure
26.11
Single-bond commuting form from the local structure
27
Mixed States: General Case and Algebraic Structure
▶
27.1
Recall of the vertical canonical form and closed-chain operators
27.2
Algebra structure
▶
27.2.1
Diagonal \(\chi \)-matrices and the trace-power formula
27.2.2
The BNT-label algebra law and idempotent trace vector
27.3
Fusion isometries
1
Appendices
▼
A
Schwarz Inequalities and Multiplicative Domains: Supporting Results
▶
A.1
Trace duality and elementary order preservation
A.2
Positive functionals and rank-one retractions
A.3
Faithful weighted traces and peripheral Schwarz equality
B
Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
▶
B.1
Density matrices, Brouwer’s theorem, and Cesàro limits
B.2
Canonical-gauge algebra
B.3
Similarity bookkeeping
B.4
Auxiliary Perron reductions
B.5
Exponential truncation and scalar reformulations
C
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
▶
C.1
Mixed-transfer powers and overlap traces
C.2
Frobenius estimates and eigenvalue bounds
C.3
Peripheral intertwiners and gauge rigidity
C.4
Spectral-radius decay and overlap limits
C.5
Rank-one Perron projection
C.6
Periodicity removal
D
Wielandt Bound: Supporting Results
▶
D.1
Exact-word and block-injectivity support
D.2
Cumulative-span and spectral linear algebra
D.3
One-step augmentation
D.4
Blocking and fixed-length spanning
D.5
Complementary-gap consequences
D.6
Identity in the one-step span
E
Canonical Form Reduction: Supporting Results
▶
E.1
Similarities, strict splitting, and zero blocks
E.2
Projector closure and isometric corner decompositions
E.3
Perron gauges and preservation under conjugation
E.4
Blocking identities and primitive stability
E.5
Support compression and cyclic-sector families
E.6
Prepared normal-form consequences
F
Basis of Normal Tensors: Supporting Results
▶
F.1
Elementary normal-tensor and Gram-matrix support
F.2
Change of basis, power sums, and sector bookkeeping
F.3
Direct-sum separation and prepared-family bookkeeping
G
Proof of the Fundamental Theorem: Supporting Results
▶
G.1
Matched phases and coefficient extraction
G.2
Coefficient identities from gauge-phase equalities
G.3
The scalar-threaded proportional identity
G.4
Direct-sum conjugation
G.5
Additional proportional consequences
G.6
Matched flattened coordinates
G.7
Permutation gauges and literal coordinates
G.8
Unitary witness refinements
H
Symmetries and Virtual-Boundary Nondecay: Supporting Results
▶
H.1
Permutation reindexing and identity bookkeeping
H.2
Gauge ratios and scalar uniqueness
H.3
Cocycle equivalence and coboundaries
H.4
Stationary-boundary twist support
H.5
Kraus mixing, transfer expansions, and a common TP gauge
H.6
Boundary and limit support
H.7
Routine universality corollary
I
Positive but Not Completely Positive Maps: Supporting Results
▶
I.1
Positive filters and trace normalization
I.2
Schmidt-rank factorization and spectral expansions
I.3
Ky Fan’s maximum principle
I.4
Right-tensor identities and Choi compressions
I.5
Closure properties of \(k\)-positive maps
J
Asymptotic Structure of Quantum Channels: Supporting Results
▶
J.1
Preservation under the mean-ergodic projection
J.2
Trace adjoints of ergodic projections
J.3
Weighted traces and idempotent retractions
J.4
Unitary extensions for fixed-point decompositions
K
Bibliography
Dependency graph
1 Appendices
A
Schwarz Inequalities and Multiplicative Domains: Supporting Results
A.1
Trace duality and elementary order preservation
A.2
Positive functionals and rank-one retractions
A.3
Faithful weighted traces and peripheral Schwarz equality
B
Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
B.1
Density matrices, Brouwer’s theorem, and Cesàro limits
B.2
Canonical-gauge algebra
B.3
Similarity bookkeeping
B.4
Auxiliary Perron reductions
B.5
Exponential truncation and scalar reformulations
C
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
C.1
Mixed-transfer powers and overlap traces
C.2
Frobenius estimates and eigenvalue bounds
C.3
Peripheral intertwiners and gauge rigidity
C.4
Spectral-radius decay and overlap limits
C.5
Rank-one Perron projection
C.6
Periodicity removal
C.6.1
Cyclic corners and restricted primitivity
C.6.2
Group structure and divisibility
D
Wielandt Bound: Supporting Results
D.1
Exact-word and block-injectivity support
D.2
Cumulative-span and spectral linear algebra
D.3
One-step augmentation
D.4
Blocking and fixed-length spanning
D.5
Complementary-gap consequences
D.6
Identity in the one-step span
E
Canonical Form Reduction: Supporting Results
E.1
Similarities, strict splitting, and zero blocks
E.2
Projector closure and isometric corner decompositions
E.3
Perron gauges and preservation under conjugation
E.4
Blocking identities and primitive stability
E.5
Support compression and cyclic-sector families
E.6
Prepared normal-form consequences
F
Basis of Normal Tensors: Supporting Results
F.1
Elementary normal-tensor and Gram-matrix support
F.2
Change of basis, power sums, and sector bookkeeping
F.2.1
Proportional matching and finite bijection extraction
F.3
Direct-sum separation and prepared-family bookkeeping
G
Proof of the Fundamental Theorem: Supporting Results
G.1
Matched phases and coefficient extraction
G.2
Coefficient identities from gauge-phase equalities
G.3
The scalar-threaded proportional identity
G.4
Direct-sum conjugation
G.5
Additional proportional consequences
G.6
Matched flattened coordinates
G.7
Permutation gauges and literal coordinates
G.8
Unitary witness refinements
H
Symmetries and Virtual-Boundary Nondecay: Supporting Results
H.1
Permutation reindexing and identity bookkeeping
H.1.1
Linear physical-index rotations
H.2
Gauge ratios and scalar uniqueness
H.3
Cocycle equivalence and coboundaries
H.4
Stationary-boundary twist support
H.5
Kraus mixing, transfer expansions, and a common TP gauge
H.6
Boundary and limit support
H.7
Routine universality corollary
I
Positive but Not Completely Positive Maps: Supporting Results
I.1
Positive filters and trace normalization
I.2
Schmidt-rank factorization and spectral expansions
I.3
Ky Fan’s maximum principle
I.4
Right-tensor identities and Choi compressions
I.5
Closure properties of \(k\)-positive maps
J
Asymptotic Structure of Quantum Channels: Supporting Results
J.1
Preservation under the mean-ergodic projection
J.2
Trace adjoints of ergodic projections
J.3
Weighted traces and idempotent retractions
J.4
Unitary extensions for fixed-point decompositions
K
Bibliography