Article items · 08 — Knill–Laflamme criterion

• Finite operator and ket conventions
• Code restriction and decoded channel types
• Finite Kraus representation
• An explicit syndrome recovery corrects a Knill–Laflamme error family
• Exact recovery implies the Knill–Laflamme condition
• Real and complex inner-product spaces and their induced length
• 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
• Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T
• Entanglement fidelity on a finite logical space
• Finite-dimensional quantum channel
• Normalized Haar probability on a compact group
• Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
• The Axiom of Choice
• Integrable real and complex functions, and their integrals
• The Borel sigma-algebra of a topological space
• The nonnegative Lebesgue integral
• The integral of a nonnegative simple function
• Measure-preserving transformations and systems
• Integral invariance under measure-preserving maps
• The Lebesgue integral is linear on L¹(μ)
• Law or distribution of a random element
• Finite support square roots and generalized inverses
• For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
• Fidelity
• Finite spin blocks, spherical tensors and Clebsch–Gordan phases
• Tensor spin composition
• The Wigner–Eckart Theorem, Physics 221A Notes 20 · Sections 6–7, equations (43)–(44) and the selection rules following equation (45)
• Four-level spontaneous-decay model
• Finite instruments and conditional states
• Finite projective measurements
• Instrument outcome weights normalize
• Positive-probability instrument branches are density operators
• Calibrated diagonal lifts and transition phase differences
• The induced length is a norm
• A linear map from a finite-dimensional normed space is bounded
• The operator norm as the least bound and as the unit-sphere or unit-ball supremum
• The operator norm is a norm on the space of bounded linear operators
• Angular-momentum basis and tensor labels
• Hilbert–Schmidt operator and Hilbert–Schmidt norm
• The basis-independent trace of an endomorphism of a finite-dimensional vector space
• Choi matrix with input factor first
• Canonical Choi eigenvectors, Kraus rank and branch weights
• QEC matrix as an error-vector Gram operator
• Partial trace over the first tensor factor
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices
• Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
• Telescoping error budget for repeated finite channels
• Probability measures and probability spaces
• Filtration and filtered probability space
• Sigma-algebras
• Measures are monotone
• The finite union bound
• Finite matrix trace-norm ideal inequalities
• 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 spin-block decomposition
• Order-one absorption–emission tensor error span
• The selected spin-four and spin-five codes have nontrivial cubic moments
• Reference-uniform trace-norm sandwich estimates
• A Knill–Laflamme decoder retaining the code syndrome
• Uniform expansion of calibrated finite channels and null branches
• Finite no-jump and ordered-jump expansion
• Schwarz and trace-norm channel contraction
• Finite GKSL evolution and the QEC channel interface
• A finite Kraus sum is completely positive
• Continuous finite-dimensional ordered-simplex integrals
• The exponential series of a bounded operator
• Every finite-dimensional normed space is Banach
• Finite matrix spectral and state calculus
• Downward dipole coefficient normalization contract
• Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ
• The trace tr(A) as the sum of the diagonal entries
• Invariant finite truncation for lowering generators
• Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius
• The exponential function is smooth and (exp)'=exp
• 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
• 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)
• 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)
• The exponential dominates every fixed nonnegative integer power at +∞
• Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
• Conditional jump-time phases form a positive Gram matrix
• 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
• exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
• Qubit coherence channels and fidelity functionals
• The direct sum of an indexed family of modules
• Regularized transverse photon dipole bath model
• Hilbert direct sums of unitary representations
• The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
• Complementary error records in a Kraus basis
• Syndrome flags and logical-state-independent environmental records
• Every finite-dimensional real or complex inner product space has an orthonormal basis
• Dirac Notation Conversion
• Two orthogonal Pauli alternatives bound recovery fidelity
• Dimensional boundary of a rank-one KL obstruction
• Finite channel, Kraus, Choi, and Stinespring equivalence

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