Alphabeta MathArticle items · 10 — Thermal photon bath and partition function
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 4–6.
1 · Prerequisites
- Photon modes and field normalization
- Regularized transverse photon dipole bath model
- Finite-mode bosonic Gibbs occupations and moments
- 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
- Transverse one-photon space and helicity projectors
- Transverse rotations preserve helicity and generate total angular momentum
- Jan Dereziński, Introduction to Representations of the Canonical Commutation and Anticommutation Relations (ESI 1741, 2005) · Section 9.3, printed pp. 40–42 (PDF pages 41–43), equation (73), the left Araki–Woods construction, and Theorem 40(1),(4).
- Effective Kraus rank and branch weight
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Finite instruments and conditional states
- Exact four-level Kraus operators
- Finite operator and ket conventions
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable
- The complex exponential is entire and its complex derivative is itself
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz
- A continuous f ≥ 0 on [a,b] with ∫ₐᵇ f = 0 is identically 0
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Four-level spontaneous-decay model
- Conditional decoded state and decay coherence
- Two-path coherence Gram spectrum and Kraus weights
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- The trace tr(A) as the sum of the diagonal entries
- Finite exponential and smooth quotient remainder control
- Transpose (Petz) recovery on a code
- Finite support square roots and generalized inverses
- Support-normalized transpose recovery and completion
- Transpose recovery for a flagged coherence channel
- Four-level syndrome recovery
- The principal inverse tangent arctan:ℝ→(-π/2,π/2)
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series
- Tangent, cotangent, secant, and cosecant on their exact natural domains
- Code restriction and decoded channel types
- Qubit coherence channels and fidelity functionals
- Phenomenological angular-momentum degeneracy lift
- Calibrated diagonal lifts and transition phase differences
- Bosonic loss, Kerr shifts and finite-code comparison scope
- Stationary correlation spectra and ULE kernel regularity
- ULE kernel adjoint symmetry and rate identities
- Convergent jump integrals and Hermitian Lamb operators
- Static Bohr-frequency evaluation of the jump integral
- A first-moment bound for the bath correlation convolution
- Density operator
- Universal Lindblad equation for open quantum systems · PDF pp. 4–5, Sec. IIA1; pp. 9–10, Sec. IIIC–IVA, Eqs. (24)–(29), (32)–(33); pp. 16–18, Appendix A opening and A1–A2, Eqs. (A2)–(A3), signs following (A6), and (A9); p. 20, Appendix A4, defining expression in (A25); pp. 21–22, Appendix A6, (A38)–(A41); p. 22, Appendix B, (B2), unnumbered convolution identity immediately before (B4), and norm inequality (B4).
- Erratum: Universal Lindblad equation for open quantum systems · PDF p. 1, Corrected expressions (1), (2), (4), (5), (6)
- Transpose fidelity from the QEC matrix
- QEC matrix and logical partial trace
- Entanglement fidelity on a finite logical space
- Haar-average versus entanglement fidelity
- Finite-dimensional quantum channel
- Normalized Haar probability on a compact group
- Angular-momentum basis and tensor labels
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Order-one absorption–emission tensor error span
- Finite signed-code KL product test
- J equals five absorption-emission code
- Spaced-support moment criterion for order-one AE correction
- Both selected codes have exact tensor-rank distance three
- Marked waiting law from finite no-jump evolution
- Normalized record measure and absence of explosion
- Finite jump-record ensemble equals the GKSL channel
- Simple counting-process intensity and compensator contract
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Fubini's theorem for L¹ functions on a sigma-finite product
- Progressively measurable and predictable processes
- Orthogonal decay sectors yield a strict fidelity margin
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- The exponential series of a bounded operator
- 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 -∞
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Limits at +∞ and -∞, and infinite limits at a point
- Hilbert space
- Hilbert tensor construction: inner product, completion and product bases
- Unbounded linear operators: domain, graph and extension
- Symmetric, self-adjoint and essentially self-adjoint operators
- Self-adjoint, positive, unitary and normal operators
- Trace class operator
- Trace of a trace class operator
- Trace is absolutely convergent and basis independent
- A self-adjoint operator generates a strongly continuous unitary group
- Nuclear bounded cyclicity for trace-class operators
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Universal Lindblad equation for open quantum systems · PDF p. 1, abstract; pp. 4–6, Secs. II and IIA
- Pauli-complementary qubit recovery fidelity inequality
- Recovery optimization on orthogonal flagged branches
- Kraus rank and Choi representation
- Kraus rank equals Choi rank
- Exact state evolution from a four-level codeword
- Density operators have an orthogonal spectral ensemble
- Densely defined, closed and closable operators, and cores
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Hilbert tensor construction: inner product, completion and product bases
- Bounded tensor maps: extension, norm estimate, adjoint and commutation
- Finite-preparation residual budget and its remote-boundary limit
- Mass and first moment of the ULE scalar envelope
- Bounded quadrant modification and generator norms
- Derivative of the quadrant modification
- Modified-operator distance and residual composition
- Universal Lindblad equation for open quantum systems · PDF pp. 18–22, Appendix A, Eqs. (A21), (A25), (A31), (A35)–(A41), Appendix B, Eq. (B2); pp. 23–26, Appendix C, Eqs. (C1)–(C22), (C27), (C29)–(C31)
- Erratum: Universal Lindblad equation for open quantum systems · PDF p.1, opening paragraph and corrected expressions (4)–(6)
- Monolithic encoding and degenerate error processes
- Complementary error records in a Kraus basis
- Dimensional boundary of a rank-one KL obstruction
- Passive subspace and subsystem protection interface
- Syndrome flags and logical-state-independent environmental records
- Exact recovery implies the Knill–Laflamme condition
- Equivalence of logical and code-supported physical decoding
- Finite Kraus representation
- Finite channel, Kraus, Choi, and Stinespring equivalence
- The Near-optimal Performance of Quantum Error Correction Codes · PDF p. 2, Eqs. (2)-(5) and Theorem 1; PDF p. 8 (supplement p. 2), Eq. (S7), Lemma 1 and Eq. (S8)
- Time-local interaction-picture convention
- Integrable real and complex functions, and their integrals
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Finite GKSL evolution and the QEC channel interface
- Inter-event survival and density from a conditional intensity
- Temporal point-process models and conditional intensity
- The finite union bound
- Schwartz/distribution spaces and Hilbert Fock conventions
- Bosonic and fermionic Fock operators with exact domains
- Adjoint of a densely defined operator
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- Quarter-turn values and shifts by pi/2 and pi
- The zero sets of sine and cosine and the least positive common period 2 pi
- Signs, monotonicity intervals, and ranges of sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
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
PostulateDependency level 4
Thermal photon bath and partition function
PropositionDependency level 4
Canonical weight as maximally mixed branch probability
Physics theoremDependency level 4
Degenerate four-level amplitude-damping form
Physics theoremDependency level 4
Decay Kraus weights and photon overlap
Physics theoremDependency level 4
Explicit four-level transpose decoder
Physics theoremDependency level 4
Exact four-level transpose fidelity
Physics theoremDependency level 4
Branch-averaged decoded state fidelity
Physics theoremDependency level 4
Source-dependent versus uniform Zeeman shifts
Physics theoremDependency level 4
ULE jump construction under independent bath hypotheses
TheoremDependency level 4
Haar-average fidelity versus entanglement fidelity
DefinitionDependency level 5
J equals four Aydin–Barg encoding
LemmaDependency level 5
Jump-record instruments and conditional intensity
LemmaDependency level 5
Strict first-threshold comparison from the cascade fidelity margin
PostulateDependency level 5
ULE system–bath coupling and short bath memory
Physics theoremDependency level 5
Optimal phase-corrected four-level recovery
Physics theoremDependency level 5
Onset of the third four-level Kraus operator
Physics theoremDependency level 5
Four-level fidelity measures for the unrotated decoder
Physics theoremDependency level 5
Long-time four-level transpose plateau
Physics theoremDependency level 5
Cubic short-time loss and recoverable limits
Physics theoremDependency level 5
Static ULE formulas and principal-value kernel
Physics theoremDependency level 5
ULE residual bound and the modified-state distinction
RemarkDependency level 5
Frequency shifts, which-path records, and passive protection
TheoremDependency level 5
Near-optimality for maximally mixed entanglement fidelity
DefinitionDependency level 6
Bath correlation matrix and Fourier convention
LemmaDependency level 6
Two-jump probability from a bounded hazard
PostulateDependency level 6
Dimensionless dipole and ULE coupling identification
Physics theoremDependency level 6
Transient later-time advantage of the unrotated decoder
Physics theoremDependency level 6