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
- Finite operator and ket conventions
- Code restriction and decoded channel types
- Entanglement fidelity on a finite logical space
- Finite support square roots and generalized inverses
- QEC matrix as an error-vector Gram operator
- Support-normalized transpose recovery and completion
- Finite Kraus representation
- Partial trace over the first tensor factor
- Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison
- The Near-optimal Performance of Quantum Error Correction Codes · Accepted manuscript, Theorem 1, Eq. (4), and discussion following it
- Ordered tensor registers, their inner product, and wire permutations
- Tensor products of operator matrix units form a basis
- Finite ancilla amplification is a well-defined block map
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Kraus trace formula for maximally mixed entanglement fidelity
- Every subset of ℝ̄ has a least upper bound and a greatest lower bound in ℝ̄, agreeing with the real supremum and infimum on nonempty sets bounded in ℝ
- Completely positive maps
- Finite channel, Kraus, Choi, and Stinespring equivalence
- The Near-optimal Performance of Quantum Error Correction Codes · PDF p. 2, Eqs. (2)-(5) and Theorem 1; PDF p. 8 (supplement p. 2), Eq. (S7), Lemma 1 and Eq. (S8)
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
Independent transpose Gram fidelity formula
Lemma
Logical Kraus operators of the completed transpose decoder
Lemma
Completion independence with an entangled reference
Lemma
Feasibility and range of transpose entanglement fidelity
Lemma
Equivalence of logical and code-supported physical decoding
Theorem