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
- Canonical Choi eigenvectors, Kraus rank and branch weights
- Liouville matrix and Choi reshuffling conventions
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Completely positive maps
- Finite Kraus representation
- Calibrated diagonal lifts and transition phase differences
- Conditional jump-time phases form a positive Gram matrix
- Probability measures and probability spaces
- Expectation of a nonnegative or integrable random variable
- Linearity, monotonicity, and the modulus bound for expectation
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- The complex exponential is entire and its complex derivative is itself
- Mean value inequality for a differentiable Banach-valued curve
- Fundamental theorem of calculus for Banach-valued continuous curves
- Bochner integral norm inequality
- The Axiom of Countable Choice (AC_ω)
- Random elements and real random variables
- Measure-null sets and almost-everywhere statements relative to a measure
- A measurable function between measurable spaces
- A continuous map has Borel preimages of Borel sets
- Complex Lp classes and Euclidean test-function conventions
- Integrable real and complex functions, and their integrals
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Every finite-dimensional normed space is Banach
- Code restriction and decoded channel types
- Telescoping error budget for repeated finite channels
- Finite instruments and conditional states
- Filtration and filtered probability space
- Sigma-algebras
- Measures are monotone
- The finite union bound
- Finite matrix trace-norm ideal inequalities
- Finite no-jump propagator and survival identity
- The Axiom of Choice
- The Borel sigma-algebra of a topological space
- The product sigma-algebra and its finite iterates
- Measure kernel and probability kernel
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- The measure with density f relative to μ
- Monotone convergence for the integral
- Measurability of integration against a kernel
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
Exact versus thresholded Kraus rank protocol
Lemma
Bounded phase spread and calibrated jump-time residual
Definition
Calibrated recovery cycles and accumulated-error contract
Lemma