Alphabeta MathNoise dynamics and calibrated cycles
Connect finite GKSL dynamics and jump-time Gram matrices to phase averaging, asymptotic checking times, repeated-cycle error budgets, invariant truncations and numerical Choi calculations. Place calibrated-cycle data beside the dynamical and channel-composition inputs it uses.
1 · Prerequisites
- Finite GKSL evolution and the QEC channel interface
- Real and complex inner product spaces, with the inner product linear in the first argument
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- The exponential series of a bounded operator
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Every linear map between finite-dimensional inner product spaces has a unique adjoint
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Liouville matrix and Choi reshuffling conventions
- Finite spin matrices
- Finite-dimensional matrix exponential
- A skew-Hermitian matrix exponential is unitary
- Finite operator and ket conventions
- The complex exponential by its power series
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Finite-dimensional quantum channel
- Probability measures and probability spaces
- Expectation of a nonnegative or integrable random variable
- Fundamental theorem of calculus for Banach-valued continuous curves
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Finite support square roots and generalized inverses
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Conditional jump-time phases form a positive Gram matrix
- Choi matrix with input factor first
- Finite Kraus representation
- Hilbert–Schmidt operator and Hilbert–Schmidt norm
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Finite instruments and conditional states
- Completely positive maps
- Support-normalized transpose recovery and completion
- Qubit coherence channels and fidelity functionals
- Code restriction and decoded channel types
- The direct sum of an indexed family of modules
- Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius
- The exponential function is smooth and (exp)'=exp
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- The exponential dominates every fixed nonnegative integer power at +∞
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
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
Lemma
Invariant finite truncation for lowering generators
Lemma
Rotation covariance and operator-weight sectors
Definition
Calibrated diagonal lifts and transition phase differences
Lemma
Finite no-jump propagator and survival identity
Lemma
Two-path coherence Gram spectrum and Kraus weights
Lemma
Transpose recovery for a flagged coherence channel
Lemma