Alphabeta MathArticle items · 05 — Trace Norm Rank One Formula
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Trace Norm
- The singular values of a linear map as the eigenvalues of the positive square root of T^*T
- A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S
- A non-negative operator has a unique non-negative square root
- Dirac Notation Conversion
- 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
- Real and complex inner-product spaces and their induced length
- Linear map between vector spaces over the same field
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Finite Kraus representation
- The trace pairing on a finite endomorphism space is nondegenerate
- Trace is a linear functional on Mₙ(F)
- Density operator
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Non-negative and positive operators
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis
- Pi as twice the smallest positive zero of cosine
- Cosine has a smallest positive zero, lying strictly between zero and two
- The addition formulas for sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3
- Real powers for positive bases, with the zero-base positive-exponent convention
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- The exponential function is smooth and (exp)'=exp
- 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)
- 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
- A function differentiable at c is continuous at c
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The exponential addition formula exp(x+y)=exp(x)exp(y)
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- Sine and cosine defined by their real power series
- The sine and cosine power series converge absolutely for every real argument
- Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius
- 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)
- Pi is the first positive zero of sine
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- The principle of mathematical induction
- Integer powers aᵐ
- Laws of integer exponents
- Singular values are well defined because the positive square root of T^*T is unique
- For a subspace W of a finite-dimensional inner product space, V=W⊕ W^⊥
- The generator of a unitary group is closed and skew-adjoint
- A self-adjoint operator generates a strongly continuous unitary group
- Infinitesimal generator of a unitary group
- Strongly continuous one-parameter unitary group
- Symmetric, self-adjoint and essentially self-adjoint operators
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Hilbert space
- Laplace resolvents of a unitary group
- Unbounded Borel functional calculus: domains, products, spectral mapping
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
- Every section of a product-measurable function is measurable
- For sigma-finite measures, the two section-measure integrals of a measurable set agree
- Every nonnegative measurable function admits an explicit increasing sequence of simple approximations
- Monotone convergence for the integral
- Additivity of the nonnegative Lebesgue integral
- Nuclear series characterizes trace norm
- Trace class operator
- Absolute value and singular values of a compact operator
- Bounded finite rank operators are compact
- Norm limit of compact operators is compact
- Hilbert-adjoint identities
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- A bounded linear operator between normed spaces
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Banach space
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Greatest lower bound (infimum)
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Trace of a trace class operator
- Singular value decomposition for compact operators
- A Hilbert space with a dense sequence has a finite or countable orthonormal basis
- Parseval equivalences for an orthonormal family
- Fourier expansion in a Hilbert space
- The finite Bessel inequality and best approximation by a finite orthonormal family
- Square-summable families on an arbitrary index set and the space ℓ²(I)
- Orthonormal families, complete orthonormal systems and Hilbert bases
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
- Finite, countably infinite, countable, uncountable
- The trace tr(A) as the sum of the diagonal entries
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule
- Field
- Spectral theorem for compact self adjoint operators
- Positive square root of a compact positive operator
- Hilbert-adjoint identities
- The Hilbert-space adjoint of a bounded operator
- Self-adjoint, positive, unitary and normal operators
- Compositions with a compact operator are compact
- Compact linear operator
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Orthogonality and the orthogonal complement
- Hilbert space
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- A bounded linear operator between normed spaces
- Real and complex inner-product spaces and their induced length
- Finite, countably infinite, countable, uncountable
- The Axiom of Countable Choice (AC_ω)
- State vectors, unit vectors, rays, normalization and phases of a Hilbert space
- Rays, unit vectors and rank-one Hilbert projections
- Density operators and scalar spectral probability
- Rank-one operator calculus
- Trace of a trace class operator
- Trace class is a two sided Banach operator ideal
- The complex exponential by its power series
- The complex exponential series converges absolutely for every complex argument
- The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts
- Finite dimensional norm equivalence over a complete valued field
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Zonal and quadratic sphere moments for the polar surface measure on S²
- Agreement with the existing polar sphere measure
- The mass-shell weighted transverse space matches the Lebesgue transverse space
- A smooth bump between concentric Euclidean balls
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ
- Finite completely positive trace-preserving maps
- Singular value decomposition for compact operators
- Bounded finite rank operators are compact
- If a series converges then its terms tend to 0
- Series and absolute convergence in a normed space
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Greatest lower bound (infimum)
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complete metric space: every Cauchy sequence converges in the space
- Cauchy sequence in a metric space
- Uniform continuity of a map of metric spaces: one δ serving every point
- Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- The reverse triangle inequality in a normed space
- Vector space over a field
- The induced length is a norm
- Algebraic multiplicity of a nonzero compact-operator eigenvalue
- The Axiom of Choice
- Banach space
- The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- 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
- Orthonormal families, complete orthonormal systems and Hilbert bases
- Square-summable families on an arbitrary index set and the space ℓ²(I)
- Nuclear series characterizes trace norm
- Squaring is monotone on the nonnegatives
- Square-summable orthogonal families have norm-convergent finite sums
- Weyl product and sum inequalities for compact operators
- Algebra of limits: sums, scalar multiples, products and quotients
- The Bessel inequality for an arbitrary orthonormal family
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- AC implies DC implies countable choice
- A topological space is Hausdorff if and only if every net has at most one limit
- Fourier expansion in a Hilbert space
- Norm limit of compact operators is compact
- Trace is absolutely convergent and basis independent
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- The norm completion of an inner-product space is a Hilbert space
- Completion of a normed space
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Parseval equivalences for an orthonormal family
- Schwartz/distribution spaces and Hilbert Fock conventions
- Unbounded linear operators: domain, graph and extension
- Densely defined, closed and closable operators, and cores
- Adjoint of a densely defined operator
- Symmetric, self-adjoint and essentially self-adjoint operators
- Permutation sector projections on the n-fold tensor power
- Finite matrix spectral and state calculus
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
TheoremLibrary prerequisite
Trace Norm Rank One Formula
TheoremLibrary prerequisite
Trace preservation is Kraus normalization
TheoremLibrary prerequisite
A unit-vector outer product is a density operator
TheoremLibrary prerequisite
Quarter-turn values and shifts by pi/2 and pi
TheoremLibrary prerequisite
Continuity and derivatives of positive-base real powers
TheoremLibrary prerequisite
The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
TheoremLibrary prerequisite
The derivatives of sine and cosine are cosine and minus sine
TheoremLibrary prerequisite
Signs, monotonicity intervals, and ranges of sine and cosine
TheoremLibrary prerequisite
The zero sets of sine and cosine and the least positive common period 2 pi
TheoremLibrary prerequisite
Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition
TheoremLibrary prerequisite
Stone's theorem: unitary groups and self-adjoint generators
TheoremLibrary prerequisite
Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
TheoremLibrary prerequisite
Trace class is a two sided Banach operator ideal
TheoremLibrary prerequisite
Trace is absolutely convergent and basis independent
TheoremLibrary prerequisite
For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
DefinitionLibrary prerequisite
Absolute value and singular values of a compact operator
DefinitionLibrary prerequisite
The state space: density operators, pure and mixed states, and purity
DefinitionLibrary prerequisite
Finite-dimensional matrix exponential
DefinitionLibrary prerequisite
Transverse one-photon space and helicity projectors
DefinitionLibrary prerequisite
Finite GKSL generators and units
DefinitionLibrary prerequisite
Trace class operator
LemmaLibrary prerequisite
Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals
LemmaLibrary prerequisite
Diagonal trace-class operators on ℓ²(ℕ,ℂ)
LemmaLibrary prerequisite
Bounded tensor maps: extension, norm estimate, adjoint and commutation
LemmaLibrary prerequisite
Hilbert tensor construction: inner product, completion and product bases
LemmaLibrary prerequisite
Nuclear bounded cyclicity for trace-class operators
LemmaLibrary prerequisite
Bosonic and fermionic Fock operators with exact domains
LemmaLibrary prerequisite