Alphabeta MathArticle items · 12 — Long-time atomic channel and source-resolving Kraus operators
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 15–21.
1 · Prerequisites
- Canonical angular-momentum downward jumps
- Invariant finite angular-momentum cutoff
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Finite absorbing lowering dynamics and replacement limit
- Finite GKSL evolution and the QEC channel interface
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Equal-rate rigid-rotor benchmark parameters
- Phenomenological angular-momentum degeneracy lift
- Rotation covariance and operator-weight sectors
- Angular-momentum basis and tensor labels
- Clebsch–Gordan total-rate identity
- General bounded diagonal lift for fast-checking analysis
- Rank-one transition errors in the benchmark
- Order-one absorption–emission tensor error span
- Reference-uniform trace-norm sandwich estimates
- A Knill–Laflamme decoder retaining the code syndrome
- Uniform expansion of calibrated finite channels and null branches
- Complete recovered-cycle bound with weighted trajectory branches
- Fixed-time convergence of calibrated recovery cycles
- Calibrated recovery cycles and accumulated-error contract
- Code restriction and decoded channel types
- Minimal single-manifold encoding
- Complementary error records in a Kraus basis
- Dimensional boundary of a rank-one KL obstruction
- Transpose (Petz) recovery on a code
- Entanglement fidelity on a finite logical space
- Operational checking time
- The exponential series of a bounded operator
- Finite no-jump and ordered-jump expansion
- Finite exponential and smooth quotient remainder control
- Finite matrix algebras with the operator norm are C star algebras
- C star algebra
- Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals
- Effective Kraus rank and branch weight
- Kraus rank equals Choi rank
- Lindbladian exponentiation and Choi Kraus extraction
- Exact versus thresholded Kraus rank protocol
- One-jump span in the nine-dimensional lowering model
- Two-jump and total trajectory dimensions in the lowering model
- J equals four Aydin–Barg encoding
- Transpose fidelity from the QEC matrix
- Finite-dimensional matrix exponential
- J equals two detection encoding
- J equals five absorption-emission code
- Finite instruments and conditional states
- Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ
- Finite projective measurements
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
- Spin-two downward coefficients and detection-code output populations
- Exact two-step equal-rate cascade restricted to the detection isometry
- A uniform transpose-fidelity bound for the one-decay channel
- Orthogonal decay sectors yield a strict fidelity margin
- Strict first-threshold comparison from the cascade fidelity margin
- Exact recovery implies the Knill–Laflamme condition
- Finite channel, Kraus, Choi, and Stinespring equivalence
- Kraus and Choi characterizations of complete positivity
- Finite operator and ket conventions
- The natural logarithm as the inverse of the exponential function
- The exponential function is strictly increasing
- The exponential tends to +∞ at +∞ and to 0 at -∞
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Aydin–Barg rank-one correction and lower host angular momentum
- Repeated four-level recovery and fixed-time error bound
- Inner and outer codes and resource-overhead data
- Conditional jump-time phases form a positive Gram matrix
- Calibrated diagonal lifts and transition phase differences
- A finite Kraus sum is completely positive
- Local inversion of a fidelity first crossing
- Finite no-jump propagator and survival identity
- Jump-record instruments and conditional intensity
- Two-jump probability from a bounded hazard
- Inter-event survival and density from a conditional intensity
- Expectation of a nonnegative or integrable random variable
- The expectation of an indicator is the probability of the event
- Linearity, monotonicity, and the modulus bound for expectation
- Conditional expectation given a sigma algebra
- The trace tr(A) as the sum of the diagonal entries
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||²
- Æ rank-one correction with separated error spaces
- Finite signed-code KL product test
- Density operator
- Trace Norm
- Density operators have an orthogonal spectral ensemble
- Trace Norm Rank One Formula
- Trace Norm Duality And Norm Properties
- An explicit syndrome recovery corrects a Knill–Laflamme error family
- Trace Distance Channel Contractivity
- Trace preservation is Kraus normalization
- Finite matrix trace-norm ideal inequalities
- Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition
- The complex exponential by its power series
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- Bounded phase spread and calibrated jump-time residual
- Finite-dimensional quantum channel
- Exact recovery of an encoded quantum code
- Probability measures and probability spaces
- Random elements and real random variables
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 15
Long-time atomic channel and source-resolving Kraus operators
Physics theoremDependency level 15
Weak rotational symmetry and decoupled operator sectors
Physics theoremDependency level 15
Normalized jumps give a scalar total decay operator
Physics theoremDependency level 16
Calibrated fast checks restore ideal rank-one correction
Physics theoremDependency level 16
Minimal-code decay detection without correction
Physics theoremDependency level 16
First-order fast-check channel
Thought experimentDependency level 16
Exact and thresholded Kraus ranks for J at most two
Thought experimentDependency level 16
Aydin–Barg fidelity time curves and complete-information-loss plateau
Thought experimentDependency level 16
Zero-lift checking-time fits versus physical decay rate
Thought experimentDependency level 16
Checking-time comparison under the Jm lift
AlgorithmDependency level 17
Projective code-manifold syndrome check
Physics theoremDependency level 17
Detection-code errors retain partial logical information
Physics theoremDependency level 17
Minimal-code closed fidelity and checking time
RemarkDependency level 17
Spacing interpretation and rate-normalization qualification
RemarkDependency level 17
Checking speed, lift control, and code must be chosen jointly
RemarkDependency level 17
Jump histories, distinguishability, and limits of Kraus rank
Thought experimentDependency level 17
Changing the lift to rotational-energy scaling
AlgorithmDependency level 18
Proposed augmented Jz syndrome check for an uncalibrated lift
Physics theoremDependency level 18
Detection-code short-time checking law
Physics theoremDependency level 18
Multiple decays vanish over fixed total time with fast checks
Physics theoremDependency level 18
Null-syndrome state and first-order expansion
Physics theoremDependency level 18
Matrix elements of the fast detected-decay state
Physics theoremDependency level 19
Well-defined augmented syndrome and first-order rotation correction
Physics theoremDependency level 19
Degenerate no-decay state is preserved
Physics theoremDependency level 19
Lifted null-branch matrix elements and normalization
Physics theoremDependency level 19
Fast-interval decay approaches a correctable coherent error
Open problemDependency level 20
Accumulated error of the uncalibrated augmented checks
Physics theoremDependency level 20
No-jump phase tracking without decay-rate suppression
Physics theoremDependency level 21