1
Deconstructing Quantum Mechanics
▶
1.1
Bipartite systems and the Schmidt decomposition
1.2
Quantum steering
1.3
Maximal weight in convex decomposition
1.4
Complete positivity from positivity
1.5
Partial trace over subsystems
1.6
Extending completely positive maps from operator systems
1.7
Extending completely positive maps from operator systems in a direct sum of matrix algebras
1.8
Composition and preparation of completely positive Kraus maps
1.9
Set spectra and spectral multiplicities
1.10
Pure states, projective rays, and Wigner’s theorem
1.11
A cyclic trace identity for rectangular powers
2
Quantum Channels and Positive Maps
▶
2.1
Positive and completely positive maps
2.2
Kraus representations and complete positivity
2.3
Quantum channels
2.4
Rectangular positive maps, trace bounds, and density matrices
2.5
Trace adjoints and rectangular Kraus maps
2.6
The Choi–Jamiolkowski representation of maps between matrix algebras
2.7
The Kraus representation theorem for rectangular matrix algebras
2.8
Tensor-factor maps and controlled partial traces
2.9
Irreducibility
2.10
Transfer maps
2.11
Peripheral spectrum and primitivity
2.12
Fixed-point projection
2.13
Peripheral eigenvalues and powering
▶
2.13.1
Periodicity removal by powering
2.13.2
Peripheral closure via adjoint fixed point
2.14
Primitivity and the complementary transfer-map gap
3
Channel Representations and Normal Forms
▶
3.1
Maximally entangled states and Choi matrices
3.2
Rectangular Choi matrices and range bounds
3.3
No information without disturbance
3.4
Representations
3.5
Further Choi-matrix identities
3.6
Representation corollaries for channel decompositions
3.7
Kraus representation theorem
3.8
Choi positivity and exact Kraus-word spans
3.9
Equivalence of ensembles by zero-padded unitary mixing
3.10
Stinespring dilation
3.11
Ordered CP maps, Radon–Nikodym, and open-system representation
3.12
POVMs and Naimark dilation
3.13
Trace-pairing expansion in transfer-matrix form
3.14
General matrix singular value decompositions
3.15
Lorentz normal form
▶
3.15.1
Canonical metric blocks for the Minkowski reduction
3.16
Determinant of a quantum channel
3.17
Determinant and Choi–Jamiołkowski operator
3.18
Self-dual channels
3.19
Transfer-matrix identities
3.20
Finite Cauchy–Schwarz equality
3.21
Trace-square and purity equality
3.22
The SIC–POVM overlap bound
3.23
Symmetric informationally complete measurements
4
Positive but Not Completely Positive Maps
▶
4.1
Trace normalization and the Lorentz cone
4.2
\(k\)-positive maps, Schmidt rank, and Choi compression
▶
4.2.1
Two-positive maps and the generalized Schwarz inequality
4.2.2
Schmidt rank and maximal overlap
4.2.3
Choi compression criteria
4.3
Trace adjoints and the positivity hierarchy
▶
4.3.1
The criterion as bounds on an infimum
4.4
Elementary positive-map examples
▶
4.4.1
The reduction map
4.4.2
Automorphisms of the positive semidefinite cone
4.5
Positivity hierarchy and two-positive maps
▶
4.5.1
The map \(T_\eta \) and strictness of the chain
4.6
The reduction criterion
4.7
The Breuer–Hall map
4.8
Choi-type maps
▶
4.8.1
Nowosad’s finite-coordinate theorem
4.9
Transposition
4.10
Decomposable positive maps
4.11
The partial transpose and the PPT property
4.12
Separable states and the PPT criterion
4.13
The Schmidt number and the full reduction criterion
4.14
Kramers degeneracy
4.15
Positive but Not Completely Positive Maps
4.16
Positive filters and trace normalization
4.17
Schmidt-rank factorization and spectral expansions
4.18
Ky Fan’s maximum principle
4.19
Right-tensor identities and Choi compressions
4.20
Closure properties of \(k\)-positive maps
4.21
Schmidt-number geometry and entanglement witnesses
4.22
Entanglement witnesses and Choi trace pairing
5
Convex Structure
6
Schwarz Inequalities and Multiplicative Domains
▶
6.1
Kadison–Schwarz inequality
6.2
Multiplicative domain
6.3
Trace adjoints and positive retractions
6.4
Douglas-type factorization
6.5
Peripheral Schwarz equality with a faithful fixed point
6.6
Douglas Factorization
6.7
Block Schur Complements
6.8
Additional results on multiplicative domains and order
▶
6.8.1
Abstract Schwarz maps and their multiplicative domains
6.8.2
Kraus specialization and full algebraic structure
6.8.3
Positive maps preserve order and spectral intervals
6.8.4
Weyl monotonicity and unitary comparison
6.9
Positive Schwarz maps outside complete positivity
▶
6.9.1
A positive Schwarz map that is not completely positive
6.9.2
Two-variable operator Schwarz inequality
6.10
Schwarz Inequalities and Multiplicative Domains
6.11
Trace duality and elementary order preservation
6.12
Positive functionals and rank-one retractions
6.13
Faithful weighted traces and peripheral Schwarz equality
7
Operator Convexity and Jensen Inequalities
▶
7.1
Schwarz inequalities for positive maps
▶
7.1.1
Schwarz inequality for normal operators
7.1.2
Schwarz inequality for subnormal and commuting-dominant operators
7.2
Diagonal Jensen inequality
7.3
Trace concavity and convexity of matrix powers
7.4
Operator convexity of real powers
7.5
Operator Jensen inequality for positive maps
▶
7.5.1
Special functions and the general inequalities
7.6
Source boundary for Wolf’s unitary comparison
7.7
Lieb concavity theorem
7.8
Resolvent monotonicity toward the Lieb concavity theorem
8
Perron–Frobenius Theory for Channels and Transfer Maps
▶
8.1
Cesàro fixed points for quantum channels
8.2
The phase-weighted Cesàro formula for the peripheral projection
8.3
Positive definiteness
8.4
Uniqueness
8.5
Existence and the Perron–Frobenius theorem
8.6
Right- and left-canonical gauges
8.7
Similarity preserves irreducibility
8.8
Perron–Frobenius eigenvector existence
8.9
The Collatz–Wielandt argument for irreducible positive maps
8.10
Completely positive primitive maps
8.11
General Brouwer and stationary states
8.12
Fixed-point decomposition and span
8.13
Exponential positivity for irreducible CP maps
8.14
Ergodicity of irreducible channels
8.15
A CP spectral characterization of irreducibility
8.16
Perron–Frobenius Theory for Channels and Transfer Maps
8.17
Density matrices, Brouwer’s theorem, and Cesàro limits
8.18
Canonical-gauge algebra
8.19
Similarity bookkeeping
8.20
Auxiliary Perron reductions
8.21
Exponential truncation and scalar reformulations
9
Peripheral Channel Structure and Transfer-Operator Gaps
▶
9.1
Peripheral eigenvalue group structure
9.2
Peripheral spectral refinements
▶
9.2.1
Cyclic decomposition of irreducible finite Kraus maps
9.3
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
9.4
Auxiliary trace identities
9.5
Auxiliary trace positivity
9.6
Spectral-radius decay and overlap limits
9.7
Rank-one Perron projection
9.8
Periodicity removal
▶
9.8.1
Cyclic corners and restricted primitivity
9.8.2
Group structure and divisibility
10
Asymptotic Structure of Quantum Channels
▶
10.1
Mean-ergodic theory and fixed-point structure
10.2
Fixed-point algebra
10.3
Conditional expectation from a faithful fixed point
10.4
Stationary support
▶
10.4.1
Faithful compression onto the support sector
10.5
Wedderburn decomposition of the fixed-point algebra
10.6
Wedderburn decomposition of the fixed-point algebra (continued)
10.7
Corner transport and commuting-overlap decompositions
10.8
Schwarz maps on direct sums of matrix algebras
10.9
Fixed-point structure and cycle decompositions
10.10
Further irreducibility and primitivity equivalences
10.11
Multi-cycle block-permutation structure
10.12
Asymptotic Structure of Quantum Channels
10.13
Preservation under the mean-ergodic projection
10.14
Trace adjoints of ergodic projections
10.15
Weighted traces and idempotent retractions
10.16
Unitary extensions for fixed-point decompositions
11
Wielandt Bound
▶
11.1
Cumulative span
▶
11.1.1
Paper primitivity and indices
11.2
Nonzero trace product
11.3
Eigenvector spreading
11.4
Quantum Wielandt bounds
11.5
Wielandt Bound
11.6
Exact-word and block-injectivity support
11.7
Cumulative-span and spectral linear algebra
11.8
One-step augmentation
11.9
Blocking and fixed-length spanning
11.10
Complementary-gap consequences
11.11
Identity in the one-step span
12
Quantum Dynamical Semigroups
▶
12.1
Dynamical semigroups and the exponential form
12.2
Perturbation theory
12.3
GKSL/Lindblad generators
▶
12.3.1
Generator decomposition and conditional complete positivity
12.3.2
The Lindblad form
12.3.3
Characterization of conditional complete positivity
12.3.4
Completely positive semigroups and conditionally completely positive generators
12.3.5
Freedom in generator representation
12.3.6
Uniqueness of the traceless Lindblad form
12.3.7
Trace-annihilation and trace preservation
12.3.8
The GKSL/Lindblad theorem
12.3.9
Kossakowski matrix form
12.4
Dissipation generated by two Pauli matrices
12.5
Primitivity and irreducibility of QDS
▶
12.5.1
Auxiliary spectral and semigroup lemmas
12.6
Kernel of the adjoint Liouvillian
12.7
Reducibility of quantum dynamical semigroups
▶
12.7.1
Sufficient conditions for non-reducibility
13
Quantum Entropy
▶
13.1
Trace norm
13.2
Projective pinching
13.3
Birkhoff’s theorem
13.4
Von Neumann entropy
13.5
Tripartite partial traces
13.6
Strong subadditivity
13.7
Mutual information
13.8
Entropy formulations
13.9
Mutual information: monotonicity and area-law bound
13.10
Data processing under local channels
13.11
Classical information and operator-Schmidt bounds
▶
13.11.1
Operator-Schmidt rank and marginal-support compression
13.12
Support compression for entropy functionals
▶
13.12.1
Whitened Choi estimates and sandwiched Rényi bounds
13.13
Trivial-factor corollaries
14
Bibliography
Dependency graph
Dependency graphs
Quantum Information and Channels: A formalization blueprint
Sirui Lu, Erickson Tjoa, and J. Ignacio Cirac
Last update: August 26, 2026
1
Deconstructing Quantum Mechanics
1.1
Bipartite systems and the Schmidt decomposition
1.2
Quantum steering
1.3
Maximal weight in convex decomposition
1.4
Complete positivity from positivity
1.5
Partial trace over subsystems
1.6
Extending completely positive maps from operator systems
1.7
Extending completely positive maps from operator systems in a direct sum of matrix algebras
1.8
Composition and preparation of completely positive Kraus maps
1.9
Set spectra and spectral multiplicities
1.10
Pure states, projective rays, and Wigner’s theorem
1.11
A cyclic trace identity for rectangular powers
2
Quantum Channels and Positive Maps
2.1
Positive and completely positive maps
2.2
Kraus representations and complete positivity
2.3
Quantum channels
2.4
Rectangular positive maps, trace bounds, and density matrices
2.5
Trace adjoints and rectangular Kraus maps
2.6
The Choi–Jamiolkowski representation of maps between matrix algebras
2.7
The Kraus representation theorem for rectangular matrix algebras
2.8
Tensor-factor maps and controlled partial traces
2.9
Irreducibility
2.10
Transfer maps
2.11
Peripheral spectrum and primitivity
2.12
Fixed-point projection
2.13
Peripheral eigenvalues and powering
2.13.1
Periodicity removal by powering
2.13.2
Peripheral closure via adjoint fixed point
2.14
Primitivity and the complementary transfer-map gap
3
Channel Representations and Normal Forms
3.1
Maximally entangled states and Choi matrices
3.2
Rectangular Choi matrices and range bounds
3.3
No information without disturbance
3.4
Representations
3.5
Further Choi-matrix identities
3.6
Representation corollaries for channel decompositions
3.7
Kraus representation theorem
3.8
Choi positivity and exact Kraus-word spans
3.9
Equivalence of ensembles by zero-padded unitary mixing
3.10
Stinespring dilation
3.11
Ordered CP maps, Radon–Nikodym, and open-system representation
3.12
POVMs and Naimark dilation
3.13
Trace-pairing expansion in transfer-matrix form
3.14
General matrix singular value decompositions
3.15
Lorentz normal form
3.15.1
Canonical metric blocks for the Minkowski reduction
3.16
Determinant of a quantum channel
3.17
Determinant and Choi–Jamiołkowski operator
3.18
Self-dual channels
3.19
Transfer-matrix identities
3.20
Finite Cauchy–Schwarz equality
3.21
Trace-square and purity equality
3.22
The SIC–POVM overlap bound
3.23
Symmetric informationally complete measurements
4
Positive but Not Completely Positive Maps
4.1
Trace normalization and the Lorentz cone
4.2
\(k\)-positive maps, Schmidt rank, and Choi compression
4.2.1
Two-positive maps and the generalized Schwarz inequality
4.2.2
Schmidt rank and maximal overlap
4.2.3
Choi compression criteria
4.3
Trace adjoints and the positivity hierarchy
4.3.1
The criterion as bounds on an infimum
4.4
Elementary positive-map examples
4.4.1
The reduction map
4.4.2
Automorphisms of the positive semidefinite cone
4.5
Positivity hierarchy and two-positive maps
4.5.1
The map \(T_\eta \) and strictness of the chain
4.6
The reduction criterion
4.7
The Breuer–Hall map
4.8
Choi-type maps
4.8.1
Nowosad’s finite-coordinate theorem
4.9
Transposition
4.10
Decomposable positive maps
4.11
The partial transpose and the PPT property
4.12
Separable states and the PPT criterion
4.13
The Schmidt number and the full reduction criterion
4.14
Kramers degeneracy
4.15
Positive but Not Completely Positive Maps
4.16
Positive filters and trace normalization
4.17
Schmidt-rank factorization and spectral expansions
4.18
Ky Fan’s maximum principle
4.19
Right-tensor identities and Choi compressions
4.20
Closure properties of \(k\)-positive maps
4.21
Schmidt-number geometry and entanglement witnesses
4.22
Entanglement witnesses and Choi trace pairing
5
Convex Structure
6
Schwarz Inequalities and Multiplicative Domains
6.1
Kadison–Schwarz inequality
6.2
Multiplicative domain
6.3
Trace adjoints and positive retractions
6.4
Douglas-type factorization
6.5
Peripheral Schwarz equality with a faithful fixed point
6.6
Douglas Factorization
6.7
Block Schur Complements
6.8
Additional results on multiplicative domains and order
6.8.1
Abstract Schwarz maps and their multiplicative domains
6.8.2
Kraus specialization and full algebraic structure
6.8.3
Positive maps preserve order and spectral intervals
6.8.4
Weyl monotonicity and unitary comparison
6.9
Positive Schwarz maps outside complete positivity
6.9.1
A positive Schwarz map that is not completely positive
6.9.2
Two-variable operator Schwarz inequality
6.10
Schwarz Inequalities and Multiplicative Domains
6.11
Trace duality and elementary order preservation
6.12
Positive functionals and rank-one retractions
6.13
Faithful weighted traces and peripheral Schwarz equality
7
Operator Convexity and Jensen Inequalities
7.1
Schwarz inequalities for positive maps
7.1.1
Schwarz inequality for normal operators
7.1.2
Schwarz inequality for subnormal and commuting-dominant operators
7.2
Diagonal Jensen inequality
7.3
Trace concavity and convexity of matrix powers
7.4
Operator convexity of real powers
7.5
Operator Jensen inequality for positive maps
7.5.1
Special functions and the general inequalities
7.6
Source boundary for Wolf’s unitary comparison
7.7
Lieb concavity theorem
7.8
Resolvent monotonicity toward the Lieb concavity theorem
8
Perron–Frobenius Theory for Channels and Transfer Maps
8.1
Cesàro fixed points for quantum channels
8.2
The phase-weighted Cesàro formula for the peripheral projection
8.3
Positive definiteness
8.4
Uniqueness
8.5
Existence and the Perron–Frobenius theorem
8.6
Right- and left-canonical gauges
8.7
Similarity preserves irreducibility
8.8
Perron–Frobenius eigenvector existence
8.9
The Collatz–Wielandt argument for irreducible positive maps
8.10
Completely positive primitive maps
8.11
General Brouwer and stationary states
8.12
Fixed-point decomposition and span
8.13
Exponential positivity for irreducible CP maps
8.14
Ergodicity of irreducible channels
8.15
A CP spectral characterization of irreducibility
8.16
Perron–Frobenius Theory for Channels and Transfer Maps
8.17
Density matrices, Brouwer’s theorem, and Cesàro limits
8.18
Canonical-gauge algebra
8.19
Similarity bookkeeping
8.20
Auxiliary Perron reductions
8.21
Exponential truncation and scalar reformulations
9
Peripheral Channel Structure and Transfer-Operator Gaps
9.1
Peripheral eigenvalue group structure
9.2
Peripheral spectral refinements
9.2.1
Cyclic decomposition of irreducible finite Kraus maps
9.3
Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
9.4
Auxiliary trace identities
9.5
Auxiliary trace positivity
9.6
Spectral-radius decay and overlap limits
9.7
Rank-one Perron projection
9.8
Periodicity removal
9.8.1
Cyclic corners and restricted primitivity
9.8.2
Group structure and divisibility
10
Asymptotic Structure of Quantum Channels
10.1
Mean-ergodic theory and fixed-point structure
10.2
Fixed-point algebra
10.3
Conditional expectation from a faithful fixed point
10.4
Stationary support
10.4.1
Faithful compression onto the support sector
10.5
Wedderburn decomposition of the fixed-point algebra
10.6
Wedderburn decomposition of the fixed-point algebra (continued)
10.7
Corner transport and commuting-overlap decompositions
10.8
Schwarz maps on direct sums of matrix algebras
10.9
Fixed-point structure and cycle decompositions
10.10
Further irreducibility and primitivity equivalences
10.11
Multi-cycle block-permutation structure
10.12
Asymptotic Structure of Quantum Channels
10.13
Preservation under the mean-ergodic projection
10.14
Trace adjoints of ergodic projections
10.15
Weighted traces and idempotent retractions
10.16
Unitary extensions for fixed-point decompositions
11
Wielandt Bound
11.1
Cumulative span
11.1.1
Paper primitivity and indices
11.2
Nonzero trace product
11.3
Eigenvector spreading
11.4
Quantum Wielandt bounds
11.5
Wielandt Bound
11.6
Exact-word and block-injectivity support
11.7
Cumulative-span and spectral linear algebra
11.8
One-step augmentation
11.9
Blocking and fixed-length spanning
11.10
Complementary-gap consequences
11.11
Identity in the one-step span
12
Quantum Dynamical Semigroups
12.1
Dynamical semigroups and the exponential form
12.2
Perturbation theory
12.3
GKSL/Lindblad generators
12.3.1
Generator decomposition and conditional complete positivity
12.3.2
The Lindblad form
12.3.3
Characterization of conditional complete positivity
12.3.4
Completely positive semigroups and conditionally completely positive generators
12.3.5
Freedom in generator representation
12.3.6
Uniqueness of the traceless Lindblad form
12.3.7
Trace-annihilation and trace preservation
12.3.8
The GKSL/Lindblad theorem
12.3.9
Kossakowski matrix form
12.4
Dissipation generated by two Pauli matrices
12.5
Primitivity and irreducibility of QDS
12.5.1
Auxiliary spectral and semigroup lemmas
12.6
Kernel of the adjoint Liouvillian
12.7
Reducibility of quantum dynamical semigroups
12.7.1
Sufficient conditions for non-reducibility
13
Quantum Entropy
13.1
Trace norm
13.2
Projective pinching
13.3
Birkhoff’s theorem
13.4
Von Neumann entropy
13.5
Tripartite partial traces
13.6
Strong subadditivity
13.7
Mutual information
13.8
Entropy formulations
13.9
Mutual information: monotonicity and area-law bound
13.10
Data processing under local channels
13.11
Classical information and operator-Schmidt bounds
13.11.1
Operator-Schmidt rank and marginal-support compression
13.12
Support compression for entropy functionals
13.12.1
Whitened Choi estimates and sandwiched Rényi bounds
13.13
Trivial-factor corollaries
14
Bibliography