Alphabeta MathRadiation, spin and code interfaces
Supply the regulated photon model, thermal moments, spectral kernels and conditional ULE contracts together with angular-momentum and dipole identities. Use the same spin conventions for absorption-emission code criteria, and specify the bosonic, inner/outer encoding and continuous-monitoring interfaces needed for platform comparisons.
1 · Prerequisites
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Fourier inversion on Schwartz space
- The Axiom of Countable Choice (AC_ω)
- Schwartz space and its seminorms
- Integrable real and complex functions, and their integrals
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- A nonnegative integral over a null set vanishes
- Universal Lindblad equation for open quantum systems · p. 9, Eqs. (25)–(26) and surrounding text
- Schwartz/distribution spaces and Hilbert Fock conventions
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- Bosonic and fermionic Fock operators with exact domains
- One-mode factorization of symmetric Fock space
- Hermite recurrence
- Hermite completeness
- The p-series for a real exponent p converges exactly when p is greater than one
- Gottesman, Kitaev and Preskill, Encoding a qubit in an oscillator · Section IV, Eq. (27), position-space codewords with code dimension n=2; Section V, Eqs. (30)–(31) and the paragraph following Eq. (34).
- Finite instruments and conditional states
- Finite GKSL generators and units
- Finite-dimensional quantum channel
- Probability measures and probability spaces
- Random elements and real random variables
- Density operator
- The Borel sigma-algebra of a topological space
- Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Continuous quantum error correction via quantum feedback control · Section II A–B, PDF pp. 2–3, equations (1)–(14).
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- Normed vector space over an absolutely valued field
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Divergence to +∞ and to -∞
- Limits and Cauchy sequences of reals
- Complete ordered field (least-upper-bound property)
- Limit superior and limit inferior of a real sequence as infₙ sup_k ≥ n xₖ and supₙ inf_k ≥ n xₖ in ℝ̄
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Tensor spin composition
- The Wigner–Eckart Theorem, Physics 221A Notes 20 · Sections 6–7, equations (43)–(44) and the selection rules following equation (45)
- Finite spin matrices
- The exponential series of a bounded operator
- The induced length is a norm
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- 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)
- Finite operator and ket conventions
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- The trace tr(A) as the sum of the diagonal entries
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
Definition
Stationary correlation spectra and ULE kernel regularity
Definition
Bosonic loss, Kerr shifts and finite-code comparison scope
Definition
Continuous monitoring versus pulsed syndrome protocols
Lemma
Finite-preparation residual budget and its remote-boundary limit
Theorem
Wigner–Eckart with explicit magnetic-label contract
Lemma
Rotational scalars on multiplicity-resolved spin blocks
Lemma