TheoremStatement: AI-adaptedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Knill–Laflamme criterion

PDF p. 1: Eq. (1)

Statement

Let $H_{L}$ , $H_{P}$ , and $H_{O}$ be nonzero finite-dimensional complex inner-product spaces, let $V : H_{L} \to H_{P}$ be an isometry, and let $C = im V$ be the codespace.

Write $⟨ \cdot, \cdot ⟩_{r o o t}$ for each given inner product, linear in its first argument. On each space, $∣ v ⟩$ denotes $v$ and $⟨ v ∣$ denotes the linear functional $x \mapsto ⟨ x, v ⟩_{r o o t}$ . Thus $⟨ u ∣ v ⟩ = ⟨ v, u ⟩_{r o o t}$ and $⟨ u ∣ A ∣ v ⟩ = ⟨ A v, u ⟩_{r o o t}$ for compatible vectors and linear maps. The symbol $A^{†}$ denotes the unique adjoint characterized by $⟨ A x, y ⟩_{r o o t} = ⟨ x, A^{†} y ⟩_{r o o t}$ for all compatible $x, y$ .

Let $N : L (H_{P}) \to L (H_{O})$ be a quantum channel with finite Kraus representation $N (X) = \sum_{a = 1}^{m} E_{a} X E_{a}^{†}$ . A quantum channel is a completely positive trace-preserving complex-linear map; a Kraus representation is the displayed operator sum. Exact correction means that some quantum channel $R : L (H_{O}) \to L (H_{L})$ satisfies $R (N (V X V^{†})) = X$ for every $X \in L (H_{L})$ .

Exact correction is possible if and only if there is a Hermitian matrix $(c_{a b})$ such that $⟨ ψ ∣ E_{a}^{†} E_{b} ∣ φ ⟩ = c_{a b} ⟨ ψ ∣ φ ⟩$ for every $a, b$ and every $∣ ψ ⟩, ∣ φ ⟩ \in C$ . The article attributes this criterion to its Ref. [18].

Facts & Assumptions

[F1]

The root inner product is linear in its first argument, conjugate symmetric and positive definite. The intrinsic bra in the Statement is therefore linear in its input. Adjoints are defined by the displayed pairing identity and exist uniquely in finite dimensions; they reverse products and conjugate scalar multiples. 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

[F2]

The codespace is the image of the isometry, its orthogonal projector is $P = V V^{†}$ , and exact correction is identity on the full logical operator space, as in Code restriction and decoded channel types.

[F3]

A finite Kraus representation is an operator-sum representation of the indicated channel. Finite Kraus representation

[F4]

For a finite-dimensional isometric code and a finite Kraus channel, the scalar compression conditions $P E_{a}^{†} E_{b} P = c_{a b} P$ imply a decoded recovery channel. An explicit syndrome recovery corrects a Knill–Laflamme error family

[F5]

For the same finite-dimensional code and channel, exact decoded recovery implies scalar compression conditions $P E_{a}^{†} E_{b} P = c_{a b} P$ . Exact recovery implies the Knill–Laflamme condition

[F6]

On standard coordinate spaces with specified ordered orthonormal bases, the intrinsic pairings in the Statement agree with the finite operator and bra-ket conventions: $⟨ u ∣ A ∣ v ⟩ = u^{†} A v = ⟨ A v, u ⟩_{r o o t}$ . Finite operator and ket conventions

Proof

Given: The finite-dimensional spaces, isometry, codespace, and finite Kraus channel in the statement.

Goal: Prove the equivalence between exact correction and the stated Hermitian scalar-overlap condition.

1.1givenF1F2algebra

Put $P = V V^{†}$ . Since $P$ is the orthogonal projector onto $C$ , scalar compression $P E_{a}^{†} E_{b} P = c_{a b} P$ implies the stated overlap equality by evaluation on code vectors. Conversely, the overlap equality says $⟨ (E_{a}^{†} E_{b} - c_{a b} I) φ, ψ ⟩_{r o o t} = 0$ for all $ψ, φ \in C$ , using linearity in the first argument. For fixed $φ \in C$ , put $z = P (E_{a}^{†} E_{b} - c_{a b} I) φ \in C$ . Self-adjointness of $P$ gives $⟨ z, ψ ⟩_{r o o t} = 0$ for every $ψ \in C$ . Taking $ψ = z$ and using positive definiteness gives $z = 0$ . As $φ$ ranges over $C$ , this proves the scalar compression.

2.1givenF1F5step 1.1algebra

If exact correction holds, [F5] supplies scalar compressions. Taking adjoints and interchanging $a, b$ gives $c_{b a} P = \overline{c_{a b}} P$ ; because $C$ is nonzero, $c_{b a} = \overline{c_{a b}}$ . Thus the coefficients form a Hermitian matrix and step 1.1 gives the overlap condition.

3.1givenF2F3F4step 1.1∎

Conversely, if the stated overlap condition holds, step 1.1 gives the scalar compressions. The channel and code meet the hypotheses of [F4], which provides a decoded recovery equal to the identity on all logical operators. This is exact correction.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 1, Eq. (1) and immediately preceding paragraph