Alphabeta MathArticle items · 07 — Monolithic encoding and degenerate error processes
Presents 28 items from the article and its immediate prerequisites, covering dependency level 1.
1 · Prerequisites
- Finite operator and ket conventions
- Code restriction and decoded channel types
- Complementary error records in a Kraus basis
- Bosonic loss, Kerr shifts and finite-code comparison scope
- The Axiom of Countable Choice (AC_ω)
- Finite instruments and conditional states
- Instrument outcome weights normalize
- Positive-probability instrument branches are density operators
- Finite POVMs and their outcome law
- POVM outcome weights are probabilities
- Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes
- John Preskill, Quantum Information, Chapter 3: Foundations II: Measurement and Evolution (updated October 2018) · Section 3.1.2, printed pp. 9–11, Eqs. (3.25)–(3.31); Section 3.2.4, printed pp. 16–17, Eqs. (3.51)–(3.55).
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Absorption-Emission Codes for Atomic and Molecular Quantum Information Platforms · pp. 2–3, Eq. (2) and the discussion and operators immediately following it
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- The exponential series of a bounded operator
- The complex exponential by its power series
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Finite-dimensional quantum channel
- Probability measures and probability spaces
- Expectation of a nonnegative or integrable random variable
- Quantum code as an isometric encoding and its code projector
- Ordered tensor registers, their inner product, and wire permutations
- Partial trace over the first tensor factor
- Finite channel, Kraus, Choi, and Stinespring equivalence
- Tensor products of operator matrix units form a basis
- Tensor products of finitely many linear maps
- Finite-dimensional quantum erasure channel
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ
- Fubini's theorem for L¹ functions on a sigma-finite product
- A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not
- Transverse projector and isotropic angular contraction
- The Lebesgue integral is linear on L¹(μ)
- The nonnegative Lebesgue integral
- Integrable real and complex functions, and their integrals
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- The polar surface set function on the unit sphere
- The Borel sigma-algebra of a topological space
- A measurable function between measurable spaces
- Composition with a Borel measurable outer map preserves measurability
- A continuous map has Borel preimages of Borel sets
- The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- The product sigma-algebra and its finite iterates
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Finite, sigma-finite, and semifinite measures
- The product measure of two sigma-finite measure spaces
- An absolute thermal envelope controls radial correlations
- Dominated convergence
- The modulus of an integral is bounded by the integral of the modulus
- An indicator function is measurable exactly when its set is measurable
- A map into ℝⁿ is measurable exactly when its coordinates are measurable
- Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
- The exponential function is strictly increasing
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Stationary correlation spectra and ULE kernel regularity
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Extended-real-valued measurable functions
- Measure-null sets and almost-everywhere statements relative to a measure
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Nonnegative simple measurable functions
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Finite GKSL evolution and the QEC channel interface
- Density operator
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Reference-uniform trace-norm sandwich estimates
- The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||²
- A unitary error-basis change gives orthogonal error ranges
- Fundamental theorem of calculus for Banach-valued continuous curves
- The Axiom of Choice
- Real and complex inner product spaces, with the inner product linear in the first argument
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- Finite support square roots and generalized inverses
- Choi matrix with input factor first
- The product basis is orthonormal and tensor dimensions multiply
- Tensor spin composition
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- The trace tr(A) as the sum of the diagonal entries
- Finite spin matrices
- The induced length is a norm
- Linear map between vector spaces over the same field
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- The operator norm is a norm on the space of bounded linear operators
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- The orthogonal group O(3) and the rotation group SO(3)
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- The complex L² pairing on equivalence classes
- The complex L² pairing is well-defined and satisfies Cauchy–Schwarz
- The complex exponential is entire and its complex derivative is itself
- Complex differentiability at a point implies continuity there
- Complex Lp classes and Euclidean test-function conventions
- Non-negative and positive operators
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Every linear map between finite-dimensional inner product spaces has a unique adjoint
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Entanglement fidelity on a finite logical space
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Real and complex inner-product spaces and their induced length
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Finite Kraus representation
- Kraus and Choi characterizations of complete positivity
- A finite Kraus sum is completely positive
- Trace preservation is Kraus normalization
- Exact recovery implies the Knill–Laflamme condition
- Liouville matrix and Choi reshuffling conventions
- Finite-dimensional matrix exponential
- A skew-Hermitian matrix exponential is unitary
- Banach-space Volterra and Dyson series with a norm remainder
- Every finite-dimensional normed space is Banach
- Finite matrix algebras with the operator norm are C star algebras
- Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
DefinitionDependency level 1
Monolithic encoding and degenerate error processes
DefinitionDependency level 1
Order-one absorption–emission tensor error span
DefinitionDependency level 1
Calibrated diagonal lifts and transition phase differences
DefinitionDependency level 1
Inner and outer codes and resource-overhead data
DefinitionDependency level 1
Syndrome flags and logical-state-independent environmental records
LemmaDependency level 1
The selected spin-four and spin-five codes have nontrivial cubic moments
LemmaDependency level 1
A radial transverse wave-vector integral has the isotropic factor
LemmaDependency level 1
Thermal frequency cutoffs may be removed under the envelope
LemmaDependency level 1
Relative variation of a cubic spectral jump factor
LemmaDependency level 1
Energy-scale independence on the block-diagonal input algebra
LemmaDependency level 1
Coefficient variation controls a weighted-shift operator
LemmaDependency level 1
Uniform expansion of calibrated finite channels and null branches
LemmaDependency level 1
A Knill–Laflamme decoder retaining the code syndrome
LemmaDependency level 1
Finite no-jump propagator and survival identity
LemmaDependency level 1
Transverse-polarization completeness and angular average
LemmaDependency level 1
Canonical Choi eigenvectors, Kraus rank and branch weights
LemmaDependency level 1
Downward dipole coefficient normalization contract
LemmaDependency level 1
Rotational scalars on multiplicity-resolved spin blocks
LemmaDependency level 1
Conditional jump-time phases form a positive Gram matrix
LemmaDependency level 1
Invariant finite truncation for lowering generators
LemmaDependency level 1
Recovery optimization on orthogonal flagged branches
LemmaDependency level 1
QEC matrix as an error-vector Gram operator
LemmaDependency level 1
Qubit coherence channels and fidelity functionals
LemmaDependency level 1
Two orthogonal Pauli alternatives bound recovery fidelity
LemmaDependency level 1
Dimensional boundary of a rank-one KL obstruction
LemmaDependency level 1
Rotation covariance and operator-weight sectors
LemmaDependency level 1
Finite no-jump and ordered-jump expansion
PostulateDependency level 1