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
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Code restriction and decoded channel types
- Absorption-Emission Codes for Atomic and Molecular Quantum Information Platforms · pp. 2–3, Eq. (2) and the discussion and operators immediately following it
- Finite-dimensional quantum channel
- Quantum code as an isometric encoding and its code projector
- Ordered tensor registers, their inner product, and wire permutations
- Partial trace over the first tensor factor
- Finite channel, Kraus, Choi, and Stinespring equivalence
- Tensor products of operator matrix units form a basis
- Tensor products of finitely many linear maps
- Tensor spin composition
- Linear map between vector spaces over the same field
- Finite operator and ket conventions
- Stationary correlation spectra and ULE kernel regularity
- Integrable real and complex functions, and their integrals
- Closure properties of measurable functions used by the integral
- Complex Lp classes and Euclidean test-function conventions
- Wigner–Eckart with explicit magnetic-label contract
- 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
- Exact recovery implies the Knill–Laflamme condition
- An explicit syndrome recovery corrects a Knill–Laflamme error family
- Exact recovery of an encoded quantum code
- Finite Kraus representation
- Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T
- The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|
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
Order-one absorption–emission tensor error span
Definition
Inner and outer codes and resource-overhead data
Lemma
Tensor-family components equal coherent Clebsch–Gordan transition maps
Lemma
Real coefficient arrays for the signed spin-four isometry
Lemma
ULE kernel adjoint symmetry and rate identities
Theorem
Spaced-support moment criterion for order-one AE correction
Lemma
Finite spanning-list compression and channel qualification
Lemma
Signed mixed binomial sums cancel at order one
Lemma