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 generators and units
- Finite completely positive trace-preserving maps
- Finite GKSL generates CPTP semigroups
- Every finite-dimensional normed space is Banach
- The exponential series of a bounded operator
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- The real exponential function and the number e by a power series
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- The induced length is a norm
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- Real powers for positive bases, with the zero-base positive-exponent convention
- 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
- 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
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- Finite-dimensional quantum channel
- Schwarz and trace-norm channel contraction
- Ordered tensor registers, their inner product, and wire permutations
- Finite matrix trace-norm ideal inequalities
- Finite registers and their ordered computational bases
- Dirac Notation Conversion
- Completely positive maps
- A finite Kraus sum is completely positive
- Non-negative and positive operators
- Choi matrix with input factor first
- Tensor products of finitely many linear maps
- Linear map between vector spaces over the same field
- A finite list v : n → V is an ordered basis if and only if every x ∈ V equals ∑_i<n λᵢ vᵢ for exactly one λ : n → F; those scalars are the coordinates of x in that ordered basis
- Banach space
- A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- Banach-space Volterra and Dyson series with a norm remainder
- Finite matrix algebras with the operator norm are C star algebras
- Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals
- Finite operator and ket conventions
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- Density operator
- Probability measures and probability spaces
- The complex L² pairing on equivalence classes
- The complex L² pairing is well-defined and satisfies Cauchy–Schwarz
- The Borel sigma-algebra of a topological space
- A measurable function between measurable spaces
- The complex exponential is entire and its complex derivative is itself
- Complex differentiability at a point implies continuity there
- A continuous map has Borel preimages of Borel sets
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Complex Lp classes and Euclidean test-function conventions
- The Lebesgue integral is linear on L¹(μ)
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
Finite GKSL evolution and the QEC channel interface
Lemma
Finite exponential and smooth quotient remainder control
Lemma
Local inversion of a fidelity first crossing
Lemma
Telescoping error budget for repeated finite channels
Definition
Liouville matrix and Choi reshuffling conventions
Lemma
Finite no-jump and ordered-jump expansion
Lemma