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
Operator norms in orthonormal coordinates
1.5
Quadratic forms under compression and relabelling
1.6
Scalar lower bounds for commuting products
1.7
Order of tensor products
1.8
Positive shifts of density matrices
1.9
Spectral truncation of shifted powers
1.10
Derivatives with respect to a scalar shift
1.11
Positive regulators and spectral tails
1.12
Complete positivity from positivity
1.13
Partial trace over subsystems
1.14
Extending completely positive maps from operator systems
1.15
Extending completely positive maps from operator systems in a direct sum of matrix algebras
1.16
Composition and preparation of completely positive Kraus maps
1.17
Set spectra and spectral multiplicities
1.18
Pure states, projective rays, and Wigner’s theorem
1.19
A cyclic trace identity for rectangular powers
1.20
Coherent frames and external references
1.21
A common positive compression on a fixed subspace
1.22
A norm comparison through an orthogonal projection
1.23
Isometric coordinates for an orthogonal projection
1.24
Compression of a contained projection
1.25
Commutation in isometric coordinates
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
Hermitian intertwiners of rectangular matrices
3.11
Stinespring dilation
3.12
Ordered CP maps, Radon–Nikodym, and open-system representation
▶
Prescribed pure environments and deferred partial traces.
3.13
POVMs and Naimark dilation
3.14
Trace-pairing expansion in transfer-matrix form
3.15
General matrix singular value decompositions
3.16
Lorentz normal form
▶
3.16.1
Canonical metric blocks for the Minkowski reduction
3.17
Determinant of a quantum channel
3.18
Determinant and Choi–Jamiołkowski operator
3.19
Self-dual channels
3.20
Transfer-matrix identities
3.21
Finite Cauchy–Schwarz equality
3.22
Trace-square and purity equality
3.23
The SIC–POVM overlap bound
3.24
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
7.9
Weighted geometric means and finite mean trees
▶
7.9.1
Logarithmic estimates for products and positive floors
7.9.2
Normalized derivative maps
7.9.3
Rectangular intertwiners of means and derivative maps
7.9.4
The hyperbolic-sine Fourier multipliers
7.9.5
The filtered transport path
7.9.6
Covariance of filtering and transport matrices
7.10
Weighted means on complementary projection ranges
7.11
A logarithmic lower bound on complementary ranges
7.12
Retained projection mass
7.13
Inverse powers from logarithmic expectations
7.14
The logarithm in the normalized inverse-power vector
7.15
Norm bounds from projection comparisons
7.16
Exponential bounds on invariant subspaces
7.17
A lower bound from a compressed inverse
7.18
Sublinear remainders in the lower norm comparison
7.19
Functional calculus on complementary projection ranges
7.20
Averaging matrix lower bounds
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
Equivalent growth conditions for irreducible positive maps
8.14
Time-average and ergodicity
8.15
Spectral characterization of irreducibility for positive maps
▶
8.15.1
The earlier completely positive specialization
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
Unital embeddings of full matrix algebras
10.8
Corner transport and commuting-overlap decompositions
10.9
Schwarz maps on direct sums of matrix algebras
10.10
\(*\)-subalgebras with scalar centre
10.11
Fixed-point structure and cycle decompositions
10.12
Further irreducibility and primitivity equivalences
10.13
Multi-cycle block-permutation structure
10.14
Asymptotic Structure of Quantum Channels
10.15
Preservation under the mean-ergodic projection
10.16
Trace adjoints of ergodic projections
10.17
Weighted traces and idempotent retractions
10.18
Unitary extensions for fixed-point decompositions
10.19
Rank-one positive projections
10.20
Support absorption for finite Kraus families
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
Hermitian exponential paths
12.2
Hermitian derivatives of inner actions
12.3
Dynamical semigroups and the exponential form
12.4
Perturbation theory
12.5
GKSL/Lindblad generators
▶
12.5.1
Generator decomposition and conditional complete positivity
12.5.2
The Lindblad form
12.5.3
Characterization of conditional complete positivity
12.5.4
Completely positive semigroups and conditionally completely positive generators
12.5.5
Freedom in generator representation
12.5.6
Uniqueness of the traceless Lindblad form
12.5.7
Trace-annihilation and trace preservation
12.5.8
The GKSL/Lindblad theorem
12.5.9
Kossakowski matrix form
12.6
Dissipation generated by two Pauli matrices
12.7
Primitivity and irreducibility of QDS
▶
12.7.1
Auxiliary spectral and semigroup lemmas
12.8
Kernel of the adjoint Liouvillian
12.9
Reducibility of quantum dynamical semigroups
▶
12.9.1
Necessary conditions for relaxation
13
Quantum Entropy
▶
13.1
Trace norm
13.2
Projective pinching
13.3
Unitary polar correction
13.4
Products of rectangular channels
13.5
Birkhoff’s theorem
13.6
Von Neumann entropy
13.7
Tripartite partial traces
13.8
Strong subadditivity
13.9
Mutual information
13.10
Entropy formulations
13.11
Mutual information: monotonicity and area-law bound
13.12
Root fidelity and purification splitting
13.13
Data processing under local channels
13.14
Conditional entropy and its continuity
13.15
Regional entropy under a change of pure state
13.16
Classical information and operator-Schmidt bounds
▶
13.16.1
Operator-Schmidt rank and marginal-support compression
13.17
Support compression for entropy functionals
▶
13.17.1
Whitened Choi estimates and sandwiched Rényi bounds
13.18
Regional entropy and two ordered families
13.19
Splitting finite regions by a complement isometry
13.20
Conditional information from two ordered families
13.21
Entropy bounds for canonical spectral tails
13.22
Typical spectral restrictions
▶
13.22.1
Normalization of typical spectral weights
13.22.2
Typical spectral restrictions of density matrices
13.22.3
Typical Schmidt truncation
13.22.4
Typical states from a spectral tail
13.23
Typical widths for linear and logarithmic cut budgets
13.24
The selected Schmidt space
13.25
A Bell projection onto the selected space
13.26
Bell projections on finitely many copies
▶
13.26.1
Typical truncation on a physical region
13.27
Regularized density-simplex minima
13.28
A controlled overlap on the physical buffer
13.29
The full-system gap on two replicas
13.30
Gaussian filtering between two Hamiltonians
▶
13.30.1
The normalized Gaussian and its tail
13.30.2
The two-Hamiltonian integral
13.30.3
Spectral gaps and ground-vector estimates
13.30.4
The filter on the original physical buffer
13.30.5
Gaussian filters under finite coordinate changes
13.30.6
Uniform choices of variance and cutoff
13.30.7
A uniform localization radius
13.30.8
Uniform truncated Gaussian approximation
13.30.9
Uniform approximation on the original physical buffer
13.31
Surprisal concentration for independent copies
13.32
Exact moments of independent finite copies
13.33
Centered exponential moments of tensor powers
13.34
Signed moments of independent-copy Schur labels
13.35
Uncentered signed Schur moments
13.36
Signed singleton moments of the good-copy density
13.37
A numerical threshold for the two independent-copy tails
13.38
Reduced densities and masses of repeated pure states
13.39
Global matrix evolution and unitary comparison
13.40
Projected interaction-picture generators
13.41
Order bounds for a support inverse
13.42
Restoring operators with singular marginals
▶
13.42.1
The reduced matrix
13.43
Norm bounds for restoring operators
13.44
Column vectors and the adjoint restoration identity
13.45
Weighted coefficients for singular restoration
13.46
The physical ground component
▶
13.46.1
Commutation recovered from a regularized filter
13.47
Spectral cutoff mass
13.48
Physical excitation counts and replica energy
13.49
Exact excitation sectors of physical replicas
13.50
Compression from bounds on excitation components
13.51
Symmetry of exact excitation components
13.52
Explicit scalar bounds for excitation components
13.53
Permutation symmetry of component marginals
13.54
The actual good auxiliary marginal
13.55
Copy symmetry and physical spectral cutoffs
13.56
The low-defect projection in the original auxiliary sectors
13.57
Mass selection among Schur sectors
13.58
Ground-state factors of excitation components
13.59
Product density on the good physical copies
13.60
Regional product density on the good copies
13.61
Joint density on the good physical and auxiliary copies
13.62
A common density for the two good regions
13.63
Symmetric support of the good physical density
13.64
The actual full good-copy density
13.65
The actual good density and subgroup expectations
13.66
Partial traces of regrouped tensor products
13.67
The common marginal after tracing the good middle region
13.68
The paired merge operators in the good-copy density
13.69
The two merge moments in the full good-copy density
13.70
The physical deficit in the actual good-copy density
13.71
The two individual paired-copy moments
13.72
The separate merge moments in the actual good density
13.73
The physical product in the actual good-copy density
13.74
Regional moments of the actual good-copy density
13.75
The three original physical label moments
13.76
The original physical marginals and their entropy difference
13.77
Original marginals in regional coordinates
13.78
A common sequence of high-mass entropy labels
13.79
The initial vector with common auxiliary labels
13.80
Schmidt truncation and the common-label Bell projection
13.81
Entropy of the selected Schmidt auxiliary state
13.82
Trivial-factor corollaries
14
Random Sources and Sampling
▶
14.1
Independent source slots
14.2
Rectangular trace-norm comparisons
14.3
Rectangular matrix second moments
14.4
Correction sums and deterministic realizations
15
Expected Trace Error of Corrected Sources
▶
15.1
An unbiased product-operator source
15.2
Finite branch sums and physical matrix entries
15.3
Covariance and weighted block errors
16
Partial Traces in Orthonormal Coordinates
17
Ambient Sources and Density Reduction
▶
17.1
Ambient endpoint coordinates
17.2
Subnormalized vectors and changing memories
17.3
Actual readout densities and marked error budgets
17.4
A product reference with private norm loss
18
Source Frames, Global Sampling, and Idle Registers
▶
18.1
Actual Schmidt source frames
18.2
Common private source sectors
18.3
Multilinear contraction and correction positions
18.4
A single probability space for all branch labels
18.5
Chronological fresh registers
19
Scalar Estimates for Buffered Filters
20
A Compactly Supported Spectral Filter
▶
20.1
The profile and its Cauchy estimate
20.2
The cutoff
20.3
The time kernel
20.4
Spectral filtering of a finite matrix
20.5
Positive replacement of a gapped Hamiltonian
20.6
Square-root channels
20.7
Locality of the positive constraints
20.8
Stability of a global gap
21
Marginal Surprisal Moments Under a Global Gap
▶
21.1
Scalar estimates
21.2
Modular conjugation
21.3
The moment and tail bounds
21.4
Designated supports
21.5
Normalized filters
▶
21.5.1
Sharp one-crossing estimate for polynomial patches
22
Schur Labels of Tensor Powers
▶
22.1
Labels of a finite group
22.2
Copies of a tensor power
22.3
The Schur-label estimates
22.4
Isotypic multiplicities and the surprisal bound
22.5
Merging subsystems and grouping copies
22.6
Merge moments on paired copy spaces
22.7
The two merge deficits on a common copy space
22.8
Nonnegative merge deficits and spectral projections
22.9
The compatible physical-label projection
22.10
The dimension formulas
22.11
Schur–Weyl duality and highest weights
22.12
The labels of the symmetric group are partitions
22.13
Exponential expansion of merge deficits
22.14
Weighted traces of commuting exponentials
22.15
Merge moments of excitation components
22.16
Two merge moments of an excitation component
▶
22.16.1
Physical label moments of an excitation component
22.17
Weighted exponential trace inequalities
22.18
Five-factor Hölder for the actual merge deficits
22.19
Centered moments of the Schur label
22.20
Quantitative bounds for the centered label moments
22.21
Branching and the dimension formula
22.22
The star operator and branch probabilities
22.23
The Weyl character formula
22.24
Counting excitation subsets
22.25
Labels on a uniform auxiliary pair
22.26
Observables after splitting the copies
22.27
Labels on a symmetric subsystem
22.28
Nonnegative label support on symmetric copies
22.29
Exponential errors and inverse polynomial mass
22.30
Finitely many exponential losses
22.31
Functional calculus of the finite-spectrum surprisal
22.32
Exponential upper tails of finite-copy surprisal
22.33
A common upper cutoff for the Schur labels
22.34
Physical regions and the prescribed compressed-state thresholds
22.35
Compressed physical and auxiliary sites
22.36
Copy coordinates of the actual auxiliary factor
22.37
Common cutoffs for the actual compressed state
22.38
The same auxiliary-label mass after regrouping
22.39
Regional marginals after spectral compression
22.40
The actual compressed state’s regional cutoff losses
22.41
The compressed vector as a family of physical sites
22.42
Permutation representations in product coordinates
22.43
Exponential upper tails of the actual copy labels
22.44
Spectral cutoffs of the label observable
22.45
Regional copy coordinates and cutoff mass
22.46
Transferring cutoffs between comparable regions
22.47
An auxiliary label sequence for the selected state
22.48
One Schmidt label sequence and the common-cutoff overlap
22.49
The selected Bell projection in exterior-site coordinates
22.50
Overlap with the supplied-label physical prevector
22.51
A physical test vector with the supplied auxiliary label
22.52
Label observables under restriction to two groups of copies
22.53
Dimensions of labels on the bad copies
22.54
A whole-space bound for the good auxiliary labels
22.55
The good auxiliary observable on a whole label
22.56
The good auxiliary label on an excitation component
22.57
The joint good auxiliary exponential
22.58
Inverse metrics on actual excitation components
22.59
The rough inverse estimate for excitation components
22.60
The sharp inverse estimate for excitation components
22.61
The sharp inverse bound on the low-defect subspace
22.62
The rough inverse bound on the low-defect subspace
22.63
Compression of a contained projection
22.64
Symmetry projections under equivariant coordinates
22.65
The rough lower bound in symmetric coordinates
22.66
The sharp lower bound in symmetric coordinates
22.67
A uniform regional lower bound
22.68
A sharp regional lower bound
22.69
The sharp lower norm comparison in physical coordinates
22.70
The rough lower norm comparison in physical coordinates
22.71
A numerical lower comparison
22.72
The lower comparison at fixed total band weight
23
Replica Entropy Metrics
▶
23.1
Ratios of Gamma values
23.2
The common label function
23.3
The marked ratio
23.4
Functional calculus on an orthogonal resolution
23.5
The metrics
23.6
The marked ratio of the metrics
23.7
Similarity transforms of copy means
23.8
Unitary twirls
23.9
The gamma integral
23.10
The integral representation
23.11
The relative coherent pin
23.12
Subgroup labels and simultaneous symmetry
23.13
Removal of the bad-copy exponential
23.14
Replica metrics of nested regional bands
23.15
Bounds for the original replica metrics
23.16
Uniform quadratic-form bounds through a mean tree
23.17
Commutation of the compressed replica bands
23.18
The same-tree product in symmetric coordinates
23.19
Original labels of the normalized band-product vector
23.20
Regional coordinates in one global replica space
23.21
Inserting a tensor power at an additional site
23.22
Bell insertion in the original physical coordinates
23.23
Changes of coordinates in the inserted register
23.24
Bell restriction of replica metrics and their means
23.25
Ordered band products on the Bell range
23.26
Real powers in fixed symmetric coordinates
23.27
Inverse powers in the supplied symmetric coordinates
23.28
The rough upper bound for the original prevector
23.29
The supplied Bell witness in physical coordinates
23.30
Enumerating the selected auxiliary register
23.31
Enumeration in fixed compressed coordinates
23.32
The rough upper bound in physical coordinates
23.33
Replica metrics under regional grouping
23.34
The original regions of the grouped metric
23.35
Original auxiliary labels in physical-copy coordinates
23.36
Symmetry projections under equivariant coordinates
23.37
Physical coordinate covariance of the low-defect projection
23.38
One low-defect projection for every regional cut
23.39
The original gap in the common replica coordinates
23.40
Defect mass of the normalized band-product vector
23.41
The inverse metric after splitting the copies
23.42
A common floor on the symmetric space
23.43
Spectral compression of a commuting exponential
23.44
Matrix lower comparisons
23.45
A polynomial floor on complete symmetric copies
24
The Conditional Movement Estimate
25
Comparing Marginal Phases
▶
25.1
The resolvent defect of a compression
25.2
The faithful case
25.3
The general case
26
The Conditional Skew Estimate
▶
26.1
The scalar function
26.2
The imaginary axis
26.3
The estimate
27
Entropy and Energy Transport on Symmetric Replicas
▶
27.1
Coherent measures and Fourier transport
27.2
The logarithmic passage
27.3
The assembled estimate
27.4
The transport root in symmetric coordinates
27.5
Uniform floors for band metrics and mean trees
27.6
Uniform floors for commuting band products
27.7
Filtered-norm bounds
27.8
Floors for transport roots
27.9
Poissonized finite words
▶
27.9.1
Concatenation and finite fibers
27.10
Time composition of Poissonized words
▶
27.10.1
Words with a marked letter
27.11
The insertion-time weight identity
27.12
Insertion into Poissonized words
▶
27.12.1
Letter counts and multiplicities
27.13
Independent Poisson letter counts
27.14
Uniform conditioning on a finite event
27.15
Independent counts and uniform word order
27.16
Tagged interleavings
27.17
Independent thinning of ordered words
27.18
Relabeling Poissonized words
27.19
Thinning by a subset of the alphabet
27.20
Occupation of retained prefixes
27.21
Integrable real prefix occupation
27.22
The bounded real Poisson word jump formula
27.23
Weighted growth of Poisson word observables
27.24
Exponential profiles of extended distance
27.25
Spatial growth under the Poisson word law
27.26
Polynomial moments of stretched exponentials
27.27
Incidence kernels of metric balls
27.28
Stretched-exponential sums of ball kernels
27.29
Poisson word growth from quadratic shell counts
27.30
Finite words of square-root factors
27.31
Chronological products under concatenation
27.32
Contraction-word decay with spectators
27.33
Poissonized excited-state decay
27.34
The logarithmic consequence of a Bell overlap
28
Bibliography
Dependency graph
Dependency graphs
Quantum Information and Channels: A formalization blueprint
Sirui Lu, Erickson Tjoa, and J. Ignacio Cirac
Last update: October 14, 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
Operator norms in orthonormal coordinates
1.5
Quadratic forms under compression and relabelling
1.6
Scalar lower bounds for commuting products
1.7
Order of tensor products
1.8
Positive shifts of density matrices
1.9
Spectral truncation of shifted powers
1.10
Derivatives with respect to a scalar shift
1.11
Positive regulators and spectral tails
1.12
Complete positivity from positivity
1.13
Partial trace over subsystems
1.14
Extending completely positive maps from operator systems
1.15
Extending completely positive maps from operator systems in a direct sum of matrix algebras
1.16
Composition and preparation of completely positive Kraus maps
1.17
Set spectra and spectral multiplicities
1.18
Pure states, projective rays, and Wigner’s theorem
1.19
A cyclic trace identity for rectangular powers
1.20
Coherent frames and external references
1.21
A common positive compression on a fixed subspace
1.22
A norm comparison through an orthogonal projection
1.23
Isometric coordinates for an orthogonal projection
1.24
Compression of a contained projection
1.25
Commutation in isometric coordinates
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
Hermitian intertwiners of rectangular matrices
3.11
Stinespring dilation
3.12
Ordered CP maps, Radon–Nikodym, and open-system representation
Prescribed pure environments and deferred partial traces.
3.13
POVMs and Naimark dilation
3.14
Trace-pairing expansion in transfer-matrix form
3.15
General matrix singular value decompositions
3.16
Lorentz normal form
3.16.1
Canonical metric blocks for the Minkowski reduction
3.17
Determinant of a quantum channel
3.18
Determinant and Choi–Jamiołkowski operator
3.19
Self-dual channels
3.20
Transfer-matrix identities
3.21
Finite Cauchy–Schwarz equality
3.22
Trace-square and purity equality
3.23
The SIC–POVM overlap bound
3.24
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
7.9
Weighted geometric means and finite mean trees
7.9.1
Logarithmic estimates for products and positive floors
7.9.2
Normalized derivative maps
7.9.3
Rectangular intertwiners of means and derivative maps
7.9.4
The hyperbolic-sine Fourier multipliers
7.9.5
The filtered transport path
7.9.6
Covariance of filtering and transport matrices
7.10
Weighted means on complementary projection ranges
7.11
A logarithmic lower bound on complementary ranges
7.12
Retained projection mass
7.13
Inverse powers from logarithmic expectations
7.14
The logarithm in the normalized inverse-power vector
7.15
Norm bounds from projection comparisons
7.16
Exponential bounds on invariant subspaces
7.17
A lower bound from a compressed inverse
7.18
Sublinear remainders in the lower norm comparison
7.19
Functional calculus on complementary projection ranges
7.20
Averaging matrix lower bounds
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
Equivalent growth conditions for irreducible positive maps
8.14
Time-average and ergodicity
8.15
Spectral characterization of irreducibility for positive maps
8.15.1
The earlier completely positive specialization
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
Unital embeddings of full matrix algebras
10.8
Corner transport and commuting-overlap decompositions
10.9
Schwarz maps on direct sums of matrix algebras
10.10
\(*\)-subalgebras with scalar centre
10.11
Fixed-point structure and cycle decompositions
10.12
Further irreducibility and primitivity equivalences
10.13
Multi-cycle block-permutation structure
10.14
Asymptotic Structure of Quantum Channels
10.15
Preservation under the mean-ergodic projection
10.16
Trace adjoints of ergodic projections
10.17
Weighted traces and idempotent retractions
10.18
Unitary extensions for fixed-point decompositions
10.19
Rank-one positive projections
10.20
Support absorption for finite Kraus families
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
Hermitian exponential paths
12.2
Hermitian derivatives of inner actions
12.3
Dynamical semigroups and the exponential form
12.4
Perturbation theory
12.5
GKSL/Lindblad generators
12.5.1
Generator decomposition and conditional complete positivity
12.5.2
The Lindblad form
12.5.3
Characterization of conditional complete positivity
12.5.4
Completely positive semigroups and conditionally completely positive generators
12.5.5
Freedom in generator representation
12.5.6
Uniqueness of the traceless Lindblad form
12.5.7
Trace-annihilation and trace preservation
12.5.8
The GKSL/Lindblad theorem
12.5.9
Kossakowski matrix form
12.6
Dissipation generated by two Pauli matrices
12.7
Primitivity and irreducibility of QDS
12.7.1
Auxiliary spectral and semigroup lemmas
12.8
Kernel of the adjoint Liouvillian
12.9
Reducibility of quantum dynamical semigroups
12.9.1
Necessary conditions for relaxation
13
Quantum Entropy
13.1
Trace norm
13.2
Projective pinching
13.3
Unitary polar correction
13.4
Products of rectangular channels
13.5
Birkhoff’s theorem
13.6
Von Neumann entropy
13.7
Tripartite partial traces
13.8
Strong subadditivity
13.9
Mutual information
13.10
Entropy formulations
13.11
Mutual information: monotonicity and area-law bound
13.12
Root fidelity and purification splitting
13.13
Data processing under local channels
13.14
Conditional entropy and its continuity
13.15
Regional entropy under a change of pure state
13.16
Classical information and operator-Schmidt bounds
13.16.1
Operator-Schmidt rank and marginal-support compression
13.17
Support compression for entropy functionals
13.17.1
Whitened Choi estimates and sandwiched Rényi bounds
13.18
Regional entropy and two ordered families
13.19
Splitting finite regions by a complement isometry
13.20
Conditional information from two ordered families
13.21
Entropy bounds for canonical spectral tails
13.22
Typical spectral restrictions
13.22.1
Normalization of typical spectral weights
13.22.2
Typical spectral restrictions of density matrices
13.22.3
Typical Schmidt truncation
13.22.4
Typical states from a spectral tail
13.23
Typical widths for linear and logarithmic cut budgets
13.24
The selected Schmidt space
13.25
A Bell projection onto the selected space
13.26
Bell projections on finitely many copies
13.26.1
Typical truncation on a physical region
13.27
Regularized density-simplex minima
13.28
A controlled overlap on the physical buffer
13.29
The full-system gap on two replicas
13.30
Gaussian filtering between two Hamiltonians
13.30.1
The normalized Gaussian and its tail
13.30.2
The two-Hamiltonian integral
13.30.3
Spectral gaps and ground-vector estimates
13.30.4
The filter on the original physical buffer
13.30.5
Gaussian filters under finite coordinate changes
13.30.6
Uniform choices of variance and cutoff
13.30.7
A uniform localization radius
13.30.8
Uniform truncated Gaussian approximation
13.30.9
Uniform approximation on the original physical buffer
13.31
Surprisal concentration for independent copies
13.32
Exact moments of independent finite copies
13.33
Centered exponential moments of tensor powers
13.34
Signed moments of independent-copy Schur labels
13.35
Uncentered signed Schur moments
13.36
Signed singleton moments of the good-copy density
13.37
A numerical threshold for the two independent-copy tails
13.38
Reduced densities and masses of repeated pure states
13.39
Global matrix evolution and unitary comparison
13.40
Projected interaction-picture generators
13.41
Order bounds for a support inverse
13.42
Restoring operators with singular marginals
13.42.1
The reduced matrix
13.43
Norm bounds for restoring operators
13.44
Column vectors and the adjoint restoration identity
13.45
Weighted coefficients for singular restoration
13.46
The physical ground component
13.46.1
Commutation recovered from a regularized filter
13.47
Spectral cutoff mass
13.48
Physical excitation counts and replica energy
13.49
Exact excitation sectors of physical replicas
13.50
Compression from bounds on excitation components
13.51
Symmetry of exact excitation components
13.52
Explicit scalar bounds for excitation components
13.53
Permutation symmetry of component marginals
13.54
The actual good auxiliary marginal
13.55
Copy symmetry and physical spectral cutoffs
13.56
The low-defect projection in the original auxiliary sectors
13.57
Mass selection among Schur sectors
13.58
Ground-state factors of excitation components
13.59
Product density on the good physical copies
13.60
Regional product density on the good copies
13.61
Joint density on the good physical and auxiliary copies
13.62
A common density for the two good regions
13.63
Symmetric support of the good physical density
13.64
The actual full good-copy density
13.65
The actual good density and subgroup expectations
13.66
Partial traces of regrouped tensor products
13.67
The common marginal after tracing the good middle region
13.68
The paired merge operators in the good-copy density
13.69
The two merge moments in the full good-copy density
13.70
The physical deficit in the actual good-copy density
13.71
The two individual paired-copy moments
13.72
The separate merge moments in the actual good density
13.73
The physical product in the actual good-copy density
13.74
Regional moments of the actual good-copy density
13.75
The three original physical label moments
13.76
The original physical marginals and their entropy difference
13.77
Original marginals in regional coordinates
13.78
A common sequence of high-mass entropy labels
13.79
The initial vector with common auxiliary labels
13.80
Schmidt truncation and the common-label Bell projection
13.81
Entropy of the selected Schmidt auxiliary state
13.82
Trivial-factor corollaries
14
Random Sources and Sampling
14.1
Independent source slots
14.2
Rectangular trace-norm comparisons
14.3
Rectangular matrix second moments
14.4
Correction sums and deterministic realizations
15
Expected Trace Error of Corrected Sources
15.1
An unbiased product-operator source
15.2
Finite branch sums and physical matrix entries
15.3
Covariance and weighted block errors
16
Partial Traces in Orthonormal Coordinates
17
Ambient Sources and Density Reduction
17.1
Ambient endpoint coordinates
17.2
Subnormalized vectors and changing memories
17.3
Actual readout densities and marked error budgets
17.4
A product reference with private norm loss
18
Source Frames, Global Sampling, and Idle Registers
18.1
Actual Schmidt source frames
18.2
Common private source sectors
18.3
Multilinear contraction and correction positions
18.4
A single probability space for all branch labels
18.5
Chronological fresh registers
19
Scalar Estimates for Buffered Filters
20
A Compactly Supported Spectral Filter
20.1
The profile and its Cauchy estimate
20.2
The cutoff
20.3
The time kernel
20.4
Spectral filtering of a finite matrix
20.5
Positive replacement of a gapped Hamiltonian
20.6
Square-root channels
20.7
Locality of the positive constraints
20.8
Stability of a global gap
21
Marginal Surprisal Moments Under a Global Gap
21.1
Scalar estimates
21.2
Modular conjugation
21.3
The moment and tail bounds
21.4
Designated supports
21.5
Normalized filters
21.5.1
Sharp one-crossing estimate for polynomial patches
22
Schur Labels of Tensor Powers
22.1
Labels of a finite group
22.2
Copies of a tensor power
22.3
The Schur-label estimates
22.4
Isotypic multiplicities and the surprisal bound
22.5
Merging subsystems and grouping copies
22.6
Merge moments on paired copy spaces
22.7
The two merge deficits on a common copy space
22.8
Nonnegative merge deficits and spectral projections
22.9
The compatible physical-label projection
22.10
The dimension formulas
22.11
Schur–Weyl duality and highest weights
22.12
The labels of the symmetric group are partitions
22.13
Exponential expansion of merge deficits
22.14
Weighted traces of commuting exponentials
22.15
Merge moments of excitation components
22.16
Two merge moments of an excitation component
22.16.1
Physical label moments of an excitation component
22.17
Weighted exponential trace inequalities
22.18
Five-factor Hölder for the actual merge deficits
22.19
Centered moments of the Schur label
22.20
Quantitative bounds for the centered label moments
22.21
Branching and the dimension formula
22.22
The star operator and branch probabilities
22.23
The Weyl character formula
22.24
Counting excitation subsets
22.25
Labels on a uniform auxiliary pair
22.26
Observables after splitting the copies
22.27
Labels on a symmetric subsystem
22.28
Nonnegative label support on symmetric copies
22.29
Exponential errors and inverse polynomial mass
22.30
Finitely many exponential losses
22.31
Functional calculus of the finite-spectrum surprisal
22.32
Exponential upper tails of finite-copy surprisal
22.33
A common upper cutoff for the Schur labels
22.34
Physical regions and the prescribed compressed-state thresholds
22.35
Compressed physical and auxiliary sites
22.36
Copy coordinates of the actual auxiliary factor
22.37
Common cutoffs for the actual compressed state
22.38
The same auxiliary-label mass after regrouping
22.39
Regional marginals after spectral compression
22.40
The actual compressed state’s regional cutoff losses
22.41
The compressed vector as a family of physical sites
22.42
Permutation representations in product coordinates
22.43
Exponential upper tails of the actual copy labels
22.44
Spectral cutoffs of the label observable
22.45
Regional copy coordinates and cutoff mass
22.46
Transferring cutoffs between comparable regions
22.47
An auxiliary label sequence for the selected state
22.48
One Schmidt label sequence and the common-cutoff overlap
22.49
The selected Bell projection in exterior-site coordinates
22.50
Overlap with the supplied-label physical prevector
22.51
A physical test vector with the supplied auxiliary label
22.52
Label observables under restriction to two groups of copies
22.53
Dimensions of labels on the bad copies
22.54
A whole-space bound for the good auxiliary labels
22.55
The good auxiliary observable on a whole label
22.56
The good auxiliary label on an excitation component
22.57
The joint good auxiliary exponential
22.58
Inverse metrics on actual excitation components
22.59
The rough inverse estimate for excitation components
22.60
The sharp inverse estimate for excitation components
22.61
The sharp inverse bound on the low-defect subspace
22.62
The rough inverse bound on the low-defect subspace
22.63
Compression of a contained projection
22.64
Symmetry projections under equivariant coordinates
22.65
The rough lower bound in symmetric coordinates
22.66
The sharp lower bound in symmetric coordinates
22.67
A uniform regional lower bound
22.68
A sharp regional lower bound
22.69
The sharp lower norm comparison in physical coordinates
22.70
The rough lower norm comparison in physical coordinates
22.71
A numerical lower comparison
22.72
The lower comparison at fixed total band weight
23
Replica Entropy Metrics
23.1
Ratios of Gamma values
23.2
The common label function
23.3
The marked ratio
23.4
Functional calculus on an orthogonal resolution
23.5
The metrics
23.6
The marked ratio of the metrics
23.7
Similarity transforms of copy means
23.8
Unitary twirls
23.9
The gamma integral
23.10
The integral representation
23.11
The relative coherent pin
23.12
Subgroup labels and simultaneous symmetry
23.13
Removal of the bad-copy exponential
23.14
Replica metrics of nested regional bands
23.15
Bounds for the original replica metrics
23.16
Uniform quadratic-form bounds through a mean tree
23.17
Commutation of the compressed replica bands
23.18
The same-tree product in symmetric coordinates
23.19
Original labels of the normalized band-product vector
23.20
Regional coordinates in one global replica space
23.21
Inserting a tensor power at an additional site
23.22
Bell insertion in the original physical coordinates
23.23
Changes of coordinates in the inserted register
23.24
Bell restriction of replica metrics and their means
23.25
Ordered band products on the Bell range
23.26
Real powers in fixed symmetric coordinates
23.27
Inverse powers in the supplied symmetric coordinates
23.28
The rough upper bound for the original prevector
23.29
The supplied Bell witness in physical coordinates
23.30
Enumerating the selected auxiliary register
23.31
Enumeration in fixed compressed coordinates
23.32
The rough upper bound in physical coordinates
23.33
Replica metrics under regional grouping
23.34
The original regions of the grouped metric
23.35
Original auxiliary labels in physical-copy coordinates
23.36
Symmetry projections under equivariant coordinates
23.37
Physical coordinate covariance of the low-defect projection
23.38
One low-defect projection for every regional cut
23.39
The original gap in the common replica coordinates
23.40
Defect mass of the normalized band-product vector
23.41
The inverse metric after splitting the copies
23.42
A common floor on the symmetric space
23.43
Spectral compression of a commuting exponential
23.44
Matrix lower comparisons
23.45
A polynomial floor on complete symmetric copies
24
The Conditional Movement Estimate
25
Comparing Marginal Phases
25.1
The resolvent defect of a compression
25.2
The faithful case
25.3
The general case
26
The Conditional Skew Estimate
26.1
The scalar function
26.2
The imaginary axis
26.3
The estimate
27
Entropy and Energy Transport on Symmetric Replicas
27.1
Coherent measures and Fourier transport
27.2
The logarithmic passage
27.3
The assembled estimate
27.4
The transport root in symmetric coordinates
27.5
Uniform floors for band metrics and mean trees
27.6
Uniform floors for commuting band products
27.7
Filtered-norm bounds
27.8
Floors for transport roots
27.9
Poissonized finite words
27.9.1
Concatenation and finite fibers
27.10
Time composition of Poissonized words
27.10.1
Words with a marked letter
27.11
The insertion-time weight identity
27.12
Insertion into Poissonized words
27.12.1
Letter counts and multiplicities
27.13
Independent Poisson letter counts
27.14
Uniform conditioning on a finite event
27.15
Independent counts and uniform word order
27.16
Tagged interleavings
27.17
Independent thinning of ordered words
27.18
Relabeling Poissonized words
27.19
Thinning by a subset of the alphabet
27.20
Occupation of retained prefixes
27.21
Integrable real prefix occupation
27.22
The bounded real Poisson word jump formula
27.23
Weighted growth of Poisson word observables
27.24
Exponential profiles of extended distance
27.25
Spatial growth under the Poisson word law
27.26
Polynomial moments of stretched exponentials
27.27
Incidence kernels of metric balls
27.28
Stretched-exponential sums of ball kernels
27.29
Poisson word growth from quadratic shell counts
27.30
Finite words of square-root factors
27.31
Chronological products under concatenation
27.32
Contraction-word decay with spectators
27.33
Poissonized excited-state decay
27.34
The logarithmic consequence of a Bell overlap
28
Bibliography