Alphabeta MathArticle items · 11 — Universal Lindblad equation with correction
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 6–15.
1 · Prerequisites
- ULE system–bath coupling and short bath memory
- Finite GKSL evolution and the QEC channel interface
- ULE jump construction under independent bath hypotheses
- ULE residual bound and the modified-state distinction
- The Axiom of Countable Choice (AC_ω)
- The Axiom of Choice
- Matrix differentials obey the sum rule, product rule, and adjoint rule
- Transpose (Petz) recovery on a code
- Entanglement fidelity on a finite logical space
- Near-optimality for maximally mixed entanglement fidelity
- Equivalence of logical and code-supported physical decoding
- Bath correlation matrix and Fourier convention
- Finite support square roots and generalized inverses
- Stationary correlation spectra and ULE kernel regularity
- Four-level spontaneous-decay model
- Operational checking time
- Cubic transpose-fidelity loss
- Exact four-level transpose fidelity
- Local inversion of a fidelity first crossing
- The exponential series of a bounded operator
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- Continuity and derivatives of positive-base real powers
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- Four-level syndrome recovery
- Optimal phase-corrected four-level recovery
- Exact four-level Kraus operators
- Decay Kraus weights and photon overlap
- Four-level fidelity measures for the unrotated decoder
- Qubit coherence channels and fidelity functionals
- Transpose recovery for a flagged coherence channel
- Finite exponential and smooth quotient remainder control
- Explicit four-level transpose decoder
- Telescoping error budget for repeated finite channels
- Code restriction and decoded channel types
- Thermal photon bath and partition function
- Dimensionless dipole and ULE coupling identification
- Regularized transverse photon dipole bath model
- Finite-mode bosonic Gibbs occupations and moments
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Trace is absolutely convergent and basis independent
- Conditional decoded state and decay coherence
- Cubic short-time loss and recoverable limits
- The complex exponential is entire and its complex derivative is itself
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Continuous functions have an oriented Riemann integral
- The oriented Riemann integral is linear in continuous integrands
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- Time-local interaction-picture convention
- The state space: density operators, pure and mixed states, and purity
- Bosonic and fermionic Fock operators with exact domains
- A scalar normalization survives a two-point functional
- An absolute thermal envelope controls radial correlations
- Finite-cell sums converge on a compact frequency shell
- A radial transverse wave-vector integral has the isotropic factor
- Thermal frequency cutoffs may be removed under the envelope
- Trace class is a two sided Banach operator ideal
- Fourier inversion on Schwartz space
- Schwartz derivatives are integrable
- Fubini's theorem for L¹ functions on a sigma-finite product
- The indefinite integral of a nonnegative measurable function is a measure
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- The derivatives of sine and cosine are cosine and minus sine
- Complex integration by parts on intervals and decaying lines
- Quarter-turn values and shifts by pi/2 and pi
- Transverse-polarization completeness and angular average
- Spherical vector components and transition-frequency convention
- Photon modes and field normalization
- Dominated convergence
- Trace class operator
- Diagonal trace-class operators on ℓ²(ℕ,ℂ)
- Absolute value and singular values of a compact operator
- The total variation |nu|(E) from countable measurable partitions
- Square-summable orthogonal families have norm-convergent finite sums
- Angular-momentum emitter, photon bath, and dipole approximation
- Rank-one angular matrix elements
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Static ULE formulas and principal-value kernel
- Rotational scalars on multiplicity-resolved spin blocks
- Wigner–Eckart with explicit magnetic-label contract
- Finite GKSL generators and units
- Order-one absorption–emission tensor error span
- Angular-momentum basis and tensor labels
- Phenomenological angular-momentum degeneracy lift
- Finite diagonal lift and its relative frequency window
- Relative variation of a cubic spectral jump factor
- Coefficient variation controls a weighted-shift operator
- Energy-scale independence on the block-diagonal input algebra
- Cross-block coherences can change Choi weights in a finite cascade
- Invariant finite truncation for lowering generators
- Bosonic loss, Kerr shifts and finite-code comparison scope
- Density operator
- Quantum code as an isometric encoding and its code projector
- Finite-dimensional quantum channel
- Exact recovery of an encoded quantum code
- Kraus rank and Choi representation
- QEC matrix and logical partial trace
- Transpose fidelity from the QEC matrix
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Liouville matrix and Choi reshuffling conventions
- Rotation covariance and operator-weight sectors
- Exact versus thresholded Kraus rank protocol
- Calibrated diagonal lifts and transition phase differences
- J equals five absorption-emission code
- Spaced-support moment criterion for order-one AE correction
- Knill–Laflamme criterion
- J equals four Aydin–Barg encoding
- Finite signed-code KL product test
- Low-rank tensor families are spin polynomials
- Pair-product correction detects quadratic spin errors
- The selected spin-four and spin-five codes have nontrivial cubic moments
- The selected codes have magnetic-support gaps two and three
- Both selected codes have exact tensor-rank distance three
- Class of codes correcting absorptions and emissions · §II, PDF p. 2, Eqs. (1)–(4), and the distance paragraph continuing on PDF p. 3; §III A, PDF pp. 3–4, Theorem III.1; PDF p. 4, Construction III.1, Theorem III.3 and Examples 1–2.
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 6
Universal Lindblad equation with correction
TheoremDependency level 6
Factor-two guarantee for transpose recovery
DefinitionDependency level 7
Jump correlators and bath-memory parameters
Physics theoremDependency level 7
Four-level high-fidelity checking time
Physics theoremDependency level 7
Exact fidelity–Kraus-weight identities and short-time bound saturation
Physics theoremDependency level 7
Repeated four-level recovery and fixed-time error bound
Physics theoremDependency level 7
Photon-number expansion of the diagonal bath correlator
Physics theoremDependency level 8
Gradual which-path leakage and inverse splitting timescale
Physics theoremDependency level 8
Thermal contractions and absorption contribution
Physics theoremDependency level 8
ULE correction bound and validity condition
Physics theoremDependency level 8
Interaction-picture ULE jumps and Lamb shift
Physics theoremDependency level 9
Commutator and spontaneous-plus-stimulated emission contribution
Physics theoremDependency level 10
Isotropic thermal bath time correlator
Physics theoremDependency level 11
Thermal bath spectral density and frequency signs
Physics theoremDependency level 12
Off-diagonal correlators vanish in an isotropic photon bath
Physics theoremDependency level 12
Vacuum limit and cubic free-space spectral density
Physics theoremDependency level 13
Canonical angular-momentum downward jumps
Physics theoremDependency level 13
Isotropic Lamb shift preserves magnetic degeneracy
Physics theoremDependency level 13
Three spherical emission channels
DefinitionDependency level 14
Rank-one transition errors in the benchmark
PostulateDependency level 14
Equal-rate rigid-rotor benchmark parameters
Physics theoremDependency level 14
Small-lift amplitude approximation and energy-scale scope
Physics theoremDependency level 14
Invariant finite angular-momentum cutoff
RemarkDependency level 14
Proposed platform-independent benchmark and qualified novelty claim
AlgorithmDependency level 15
Lindbladian exponentiation and Choi Kraus extraction
PostulateDependency level 15
General bounded diagonal lift for fast-checking analysis
Physics theoremDependency level 15
Æ rank-one correction with separated error spaces
Physics theoremDependency level 15