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
- Code restriction and decoded channel types
- Finite support square roots and generalized inverses
- Finite Kraus representation
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
- Fidelity
- Entanglement fidelity on a finite logical space
- Finite operator and ket conventions
- 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 instruments and conditional states
- Kraus and Choi characterizations of complete positivity
- A finite Kraus sum is completely positive
- Trace preservation is Kraus normalization
- Choi matrix with input factor first
- Finite channel, Kraus, Choi, and Stinespring equivalence
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- 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
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Exact recovery implies the Knill–Laflamme condition
- Complementary error records in a Kraus basis
- Finite-dimensional quantum erasure channel
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Tensor products of finitely many linear maps
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
Theorem
Support-normalized transpose recovery and completion
Theorem
Haar-average versus entanglement fidelity
Lemma
Qubit coherence channels and fidelity functionals
Lemma
Two orthogonal Pauli alternatives bound recovery fidelity
Lemma
Recovery optimization on orthogonal flagged branches
Lemma
Dimensional boundary of a rank-one KL obstruction
Definition
Syndrome flags and logical-state-independent environmental records
Lemma