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 no-jump propagator and survival identity
- Marked waiting law from finite no-jump evolution
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- The Axiom of Choice
- The Borel sigma-algebra of a topological space
- The product sigma-algebra and its finite iterates
- A measurable function between measurable spaces
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Counting measure is a measure
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
- The indefinite integral of a nonnegative measurable function is a measure
- Probability measures and probability spaces
- Stochastic processes and their finite-dimensional distributions
- Conditional law given a random element
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- Finite GKSL evolution and the QEC channel interface
- The exponential series of a bounded operator
- Fundamental theorem of calculus for Banach-valued continuous curves
- Fubini's theorem for L¹ functions on a sigma-finite product
- Hermitian Trace Norm Positive Negative Parts
- Trace Norm Unitary Invariance And Positive Formula
- Trace Norm Duality And Norm Properties
- Every finite-dimensional normed space is Banach
- A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space
- A uniform limit of continuous functions is continuous, so C(X,Y) is closed in Y^X under the uniform metric
- The modulus of an integral is bounded by the integral of the modulus
- Finite instruments and conditional states
- Simple counting-process intensity and compensator contract
- Progressively measurable and predictable processes
- Inter-event survival and density from a conditional intensity
- Temporal point-process models and conditional intensity
- The finite union bound
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
Normalized record measure and absence of explosion
Lemma
Finite jump-record ensemble equals the GKSL channel
Lemma
Jump-record instruments and conditional intensity
Lemma