Alphabeta MathArticle items · 09 — Exact state evolution from a four-level codeword
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 2–4.
1 · Prerequisites
- Four-level spontaneous-decay model
- Finite operator and ket conventions
- Finite GKSL evolution and the QEC channel interface
- Finite no-jump and ordered-jump expansion
- Density operator
- Pure state and ray
- A unit-vector outer product is a density operator
- Monolithic encoding and degenerate error processes
- Code restriction and decoded channel types
- Inner and outer codes and resource-overhead data
- The Axiom of Countable Choice (AC_ω)
- Order-one absorption–emission tensor error span
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Wigner–Eckart with explicit magnetic-label contract
- Tensor spin composition
- Absorption-Emission Codes for Atomic and Molecular Quantum Information Platforms · PDF p. 3 (journal p. 260601-3), Eq. (3), its immediately following coefficient identification, and Eqs. (4)–(5) with their surrounding paragraphs
- Entanglement fidelity on a finite logical space
- Finite support square roots and generalized inverses
- QEC matrix as an error-vector Gram operator
- Support-normalized transpose recovery and completion
- Finite Kraus representation
- Partial trace over the first tensor factor
- Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison
- The Near-optimal Performance of Quantum Error Correction Codes · Accepted manuscript, Theorem 1, Eq. (4), and discussion following it
- Angular-momentum basis and tensor labels
- Transpose (Petz) recovery on a code
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- Kraus rank and Choi representation
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Exact versus thresholded Kraus rank protocol
- Low-rank tensor families are spin polynomials
- Nonvanishing downward coefficients for spin one and two
- Distinct complex exponentials have full interval Gram rank
- The induced length is a norm
- 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
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- The complex exponential by its power series
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums
- Clebsch–Gordan total-rate identity
- Complete recovered-cycle bound with weighted trajectory branches
- Telescoping error budget for repeated finite channels
- Finite instruments and conditional states
- Instrument outcome weights normalize
- The finite union bound
- Schwarz and trace-norm channel contraction
- Fidelity
- Fuchs Van De Graaf
- Finite channel, Kraus, Choi, and Stinespring equivalence
- Finite matrix trace cyclicity and nondegeneracy
- Spin-two downward coefficients and detection-code output populations
- Downward dipole coefficient normalization contract
- Fundamental theorem of calculus for Banach-valued continuous curves
- Fubini's theorem for L¹ functions on a sigma-finite product
- The Axiom of Choice
- Exact recovery implies the Knill–Laflamme condition
- An explicit syndrome recovery corrects a Knill–Laflamme error family
- Tensor-family components equal coherent Clebsch–Gordan transition maps
- Finite spanning-list compression and channel qualification
- Real coefficient arrays for the signed spin-four isometry
- Signed mixed binomial sums cancel at order one
- Signed diagonal binomial moments agree at order one
- Class of codes correcting absorptions and emissions · §II, PDF p. 2, Eqs. (1)–(3); §III A, Theorem III.1, conditions (C1)–(C4), PDF pp. 3–4
- An affine family of lifts and couplings
- Calibrated diagonal lifts and transition phase differences
- Complementary error records in a Kraus basis
- Angular-momentum emitter, photon bath, and dipole approximation
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Regularized transverse photon dipole bath model
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- Four-level syndrome recovery
- Two-path coherence Gram spectrum and Kraus weights
- Bosonic loss, Kerr shifts and finite-code comparison scope
- Syndrome flags and logical-state-independent environmental records
- Environment-state independence under KL
- Knill–Laflamme criterion
- QEC matrix and logical partial trace
- Spherical vector components and transition-frequency convention
- The selected spin-four and spin-five codes have nontrivial cubic moments
- Feasibility and range of transpose entanglement fidelity
- Logical Kraus operators of the completed transpose decoder
- Support of an integrated finite trajectory family
- Choi matrix with input factor first
- A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections
- Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections
- Borel Darboux integrands in finite dimension
- Continuous monitoring versus pulsed syndrome protocols
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
Physics theoremDependency level 2
Exact state evolution from a four-level codeword
RemarkDependency level 2
Fault-tolerant overhead and monolithic inner codes
TheoremDependency level 2
Spaced-support moment criterion for order-one AE correction
TheoremDependency level 2
Independent transpose Gram fidelity formula
DefinitionDependency level 3
J equals five absorption-emission code
DefinitionDependency level 3
Operational checking time
DefinitionDependency level 3
Effective Kraus rank and branch weight
LemmaDependency level 3
Pair-product correction detects quadratic spin errors
LemmaDependency level 3
One-jump span in the nine-dimensional lowering model
LemmaDependency level 3
Cross-block coherences can change Choi weights in a finite cascade
LemmaDependency level 3
Fixed-time convergence of calibrated recovery cycles
LemmaDependency level 3
A uniform transpose-fidelity bound for the one-decay channel
LemmaDependency level 3
Exact two-step equal-rate cascade restricted to the detection isometry
LemmaDependency level 3
Finite signed-code KL product test
Open problemDependency level 3
Degeneracy lifting that also modifies jump operators
PostulateDependency level 3
Phenomenological angular-momentum degeneracy lift
PostulateDependency level 3
Photon modes and field normalization
Physics theoremDependency level 3
Conditional decoded state and decay coherence
Physics theoremDependency level 3
Exact four-level Kraus operators
RemarkDependency level 3
Bosonic anharmonicity and finite-energy GKP tradeoff
RemarkDependency level 3
Logical information leakage versus erasure flags
TheoremDependency level 3
Kraus rank equals Choi rank
TheoremDependency level 3
Transpose fidelity from the QEC matrix
TheoremDependency level 3
Rank-one angular matrix elements
LemmaDependency level 4
Both selected codes have exact tensor-rank distance three
LemmaDependency level 4
Equivalence of logical and code-supported physical decoding
LemmaDependency level 4
Two-jump and total trajectory dimensions in the lowering model
LemmaDependency level 4
Orthogonal decay sectors yield a strict fidelity margin
Open problemDependency level 4