Alphabeta MathSupporting prerequisites · 01 — Simple counting-process intensity and compensator contract
Presents 18 supporting prerequisites, covering published library prerequisites and dependency levels 0–2.
1 · Prerequisites
- Continuous-time filtrations and all-pairs martingales
- Continuous-time adapted processes and martingales
- Progressively measurable and predictable processes
- Conditional expectation as an ae class
- The Axiom of Choice
- Finite-dimensional quantum channel
- Every finite-dimensional channel has a finite isometric dilation
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Finite-dimensional von Neumann entropy
- Ensemble and purification
- Purifications exist and are unitarily equivalent after padding
- Partial trace over a finite tensor factor
- Partial trace over the first tensor factor
- Finite wire permutations are unitary basis permutations
- An isometry followed by partial trace is a channel
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Finite-dimensional channels compose and tensor
- Density operator
- If V = ⨁_i<n Uᵢ with every Uᵢ finite-dimensional, then V is finite-dimensional and dim_F V = ∑_i<n dim_F Uᵢ; in particular dim_F(U ⊕ W) = dim_F U + dim_F W
- For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥
- Finite registers and their ordered computational bases
- Real and complex inner-product spaces and their induced length
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Linear map between vector spaces over the same field
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- Over a commutative ring, M⊗_RN is an R-module with r(m⊗ n)=(rm)⊗ n=m⊗(rn)
- The elementary tensors of two bases form the product basis of the tensor product
- 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
- Rᵐ⊗_RRⁿ≅ Rᵐⁿ with the product basis, and dim_F(V⊗_FW)=dim_FV dim_FW
- Finite iterated tensor products represent multilinear maps independently of parenthesization
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition
- The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector
- The rank of a linear map is the number of its nonzero singular values
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Orthogonal and unitary operators form groups, and their determinants have modulus one
- Total momentum multipliers on transverse symmetric Fock sectors
- Radiation extension of the framework: transverse mode and photon assignments
- State and composition postulate P1: Hilbert spaces, preparations and composite systems
- Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes
- Identical-particle and spin assignments P6: permutation sectors and finite-spin representations
- Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators
- Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions
- The state space of the framework: preparations, pure rays and density operators
- Assigned sharp observables and the outcome law of the framework
- Dimensions, SI unit conventions and quantity types of the framework
- Declared species sectors and their preparations: the framework's reading of P6
- Hilbert direct-sum construction
- Fock sector projections
- Number operator domain
- Transverse continuum free Fock Hamiltonian
- Restricted momentum and energy multipliers of the transverse one-photon space
- The fixed-mode isometry of Fock space and the transfer of number and ladder operators
- Orthonormal families, complete orthonormal systems and Hilbert bases
- Hilbert space
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The Axiom of Choice
- The polar surface set function on the unit sphere
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ
- Borel change of variables from the compact-support formula and Radon uniqueness
- The Euclidean inverse function theorem
- Every affine hyperplane of ℝⁿ, and hence every proper linear subspace, is Lebesgue null
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Fubini's theorem for L¹ functions on a sigma-finite product
- Monotone convergence for the integral
- Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity
- The derivatives of sine and cosine are cosine and minus sine
- The addition formulas for sine and cosine
- Signs, monotonicity intervals, and ranges of sine and cosine
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- Parity and the Pythagorean identity for sine and cosine
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- The Borel sigma-algebra of a topological space
- A continuous map has Borel preimages of Borel sets
- Measures on sigma-algebras
- Schwartz/distribution spaces and Hilbert Fock conventions
- Orthogonality and the orthogonal complement
- Orthogonal complements are closed
- Hilbert tensor construction: inner product, completion and product bases
- Permutation sector projections on the n-fold tensor power
- C star algebra
- Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- The natural numbers ℕ (von Neumann)
- Closed adjoint sector domains
- Bounded fixed-N operator
- Creation contraction bounds
- Euclidean L² product unitary: L²(Rᵃ)⊗̂L²(Rᵇ) ≅ L²(Rᵃ⁺ᵇ)
- Cauchy-Schwarz inequality for L²
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Kato-Rellich theorem
- Relative boundedness with respect to an operator
- Symmetric, self-adjoint and essentially self-adjoint operators
- Adjoint of a densely defined operator
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Finite completely positive trace-preserving maps
- Finite GKSL generators and units
- Every finite-dimensional normed space is Banach
- Finite matrix trace cyclicity and nondegeneracy
- The exponential series of a bounded operator
- Complete finite Choi-Kraus equivalence
- Channel duals and norm closure
- Finite matrix spectral and state calculus
- Trace-class cross-log entropy inequality
- Trace-class Gibbs normalization for a semibounded self-adjoint operator
- Summable canonical differentiation
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Bounded borel pvm integral
- The exponential series of a bounded operator
- Finite operator and ket conventions
- The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1}
- Linear independence: a finite list v : n → V is independent when ∑_i<n λᵢ vᵢ = 0_V forces every λᵢ = 0_F, and a subset S ⊆ V is independent when every injective finite list into S is independent
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- span(S) is exactly the set of linear combinations of finite lists of elements of S, and span(∅) = {0_V}
- If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V
- The induced length is a norm
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- The product basis is orthonormal and tensor dimensions multiply
- Rank and nullity of a linear map with finite-dimensional domain
- In finite dimension, W^⊥⊥=W and dim W+dim W^⊥=dim V
- The Axiom of Countable Choice (AC_ω)
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Euclidean balls have positive finite Lebesgue measure
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- Integrable real and complex functions, and their integrals
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Lebesgue integral is linear on L¹(μ)
- Finite support square roots and generalized inverses
- Choi matrix with input factor first
- Continuity of a map of topological spaces at a point and globally
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Restriction of a measure to a measurable set
- The restriction of a measure to a measurable set is a measure
- Finite spin blocks, spherical tensors and Clebsch–Gordan phases
- Downward dipole coefficient normalization contract
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
Simple counting-process intensity and compensator contract
Definition
Complementary channels and coherent information
Definition
Finite-dimensional quantum erasure channel
Definition
The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
Definition
Ordered tensor registers, their inner product, and wire permutations
Theorem
Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison
Definition
The radiation sector of the framework: modes and the number, energy and momentum assignments
Lemma
Sphere integrals in angle coordinates
Lemma
One-mode factorization of symmetric Fock space
Lemma
Banach-space Volterra and Dyson series with a norm remainder
Proposition
Finite spin matrices
Theorem
Cutoff dipole relative bound
Theorem
Finite GKSL generates CPTP semigroups
Theorem
Schwarz and trace-norm channel contraction
Theorem
Self-adjoint trace-Gibbs normalization and form-energy optimization
Lemma
Distinct complex exponentials have full interval Gram rank
Lemma
Support of an integrated finite trajectory family
Lemma