Alphabeta MathQuantum channels and recovery
Develop finite-code channels, complementary and syndrome records, support calculus, canonical Choi weights, transpose recovery and fidelity bounds. Include passive-subsystem and rank-one correctability interfaces so the article can compare recoverable logical information with environmental error records.
1 · Prerequisites
- Real and complex inner product spaces, with the inner product linear in the first argument
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Ordered tensor registers, their inner product, and wire permutations
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Linear map between vector spaces over the same field
- Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- The transpose A^T of a matrix
- Quantum code as an isometric encoding and its code projector
- Finite-dimensional quantum channel
- Exact recovery of an encoded quantum code
- Completely positive maps
- An isometric quantum encoding gives an orthogonal code projector
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
- Non-negative and positive operators
- Density operator
- Partial trace over the first tensor factor
- Tensor products of operator matrix units form a basis
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- A simple formula for the average gate fidelity of a quantum dynamical operation · p. 1, paragraph defining entanglement fidelity; p. 2, paragraph preceding Eq. (11)
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Every linear map between finite-dimensional inner product spaces has a unique adjoint
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The orthogonal complement W^⊥={v:⟨ v,w⟩=0 for all w∈ W}
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Kernel and image of a linear map
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Linear subspace of a vector space
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Complementary channels and coherent information
- Finite Kraus representation
- Partial trace over a finite tensor factor
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Tensor products of finitely many linear maps
- Theory of Quantum Error Correction for General Noise · PDF p. 3, Noiseless Subsystems
- Choi matrix with input factor first
- Finite channel, Kraus, Choi, and Stinespring equivalence
- Finite instruments and conditional states
- The product basis is orthonormal and tensor dimensions multiply
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
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
Finite operator and ket conventions
Definition
Code restriction and decoded channel types
Definition
Entanglement fidelity on a finite logical space
Definition
Finite support square roots and generalized inverses
Definition
Complementary error records in a Kraus basis
Definition
Passive subspace and subsystem protection interface
Lemma
Canonical Choi eigenvectors, Kraus rank and branch weights
Lemma