LemmaStatement: AI-generatedProof: 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.

Finite spanning-list compression and channel qualification

Statement

Let $H$ and $E \subseteq L (H)$ be the finite-dimensional space and error span of the tensor-span contract. Let $V : C^{2} \to H$ be any isometry and put $P = V V^{*}$ . For any linear subspace $S \subseteq L (H)$ , the pair-product condition for $(V, S)$ means that for every $E, F \in S$ there is a complex scalar $c_{E, F}$ with $P E^{*} F P = c_{E, F} P$ . For $S = E$ and $im V \subseteq V_{J}$ , this agrees with the tensor-span contract's condition; the algebraic formulation here also applies to arbitrary isometries into $H$ .

For any spanning list $T_{1}, \dots, T_{N}$ of $E$ , the pair-product condition for $(V, E)$ is equivalent to every $V^{*} T_{a}^{*} T_{b} V$ being a scalar matrix. If this condition holds, every quantum channel $N : L (H) \to L (H)$ with a finite Kraus representation whose Kraus operators lie in $E$ is exactly recoverable on this code. Conversely, exact recovery of a finite Kraus channel on $H$ implies the pair-product condition for its Kraus span; it implies that condition for $E$ when the Kraus span equals $E$ . Exact recovery means that a decoded recovery channel $R : L (H) \to L (C^{2})$ satisfies $R (N (V X V^{*})) = X$ for every $X \in L (C^{2})$ . A bare list with no trace-preserving channel is only an algebraic test.

Facts & Assumptions

[F1]

The tensor-span contract supplies a finite-dimensional $H$ and error span $E$ . Its pair-product condition on codes in the input block is the scalar-compression condition used in the Statement. Order-one absorption–emission tensor error span

[F2]

Exact recovery of a finite trace-preserving Kraus channel implies its pair-product identities. Exact recovery implies the Knill–Laflamme condition

[F3]

A finite trace-preserving Kraus channel satisfying those identities has an exact decoded recovery. An explicit syndrome recovery corrects a Knill–Laflamme error family

[F4]

A quantum channel is a completely positive trace-preserving complex-linear map on the full operator space; a finite Kraus representation is $N (X) = \sum_{a} K_{a} X K_{a}^{*}$ . Finite-dimensional quantum channel Finite Kraus representation

[F5]

Exact correction is the decoded-channel identity $R \circ N \circ {Ad}_{V} = id$ on the full logical operator space, with ${Ad}_{V} (X) = V X V^{*}$ . Exact recovery of an encoded quantum code

[F6]

Adjoints are additive and conjugate-homogeneous and reverse compatible compositions. Adjoints satisfy (S+T)^=S^+T^, (λ T)^=λ̄ T^, (ST)^=T^S^, and T^**=T

[F7]

For a finite-dimensional isometric encoding into $H$ , $P = V V^{*}$ is the code projector and $V^{*} V = I$ ; the encoding and decoded recovery have the types used here. Code restriction and decoded channel types

Proof

Given: The finite-dimensional objects, isometry, and pair-product formulation in the Statement.

Goal: Establish the spanning-list equivalence and the qualified channel implications.

1.1givenF1F6F7algebra

By [F7], $P = V V^{*}$ and $V^{*} V = I_{2}$ . Thus $P T_{a}^{*} T_{b} P = c_{a b} P$ and $V^{*} T_{a}^{*} T_{b} V = c_{a b} I_{2}$ are equivalent by multiplying respectively by $V^{*}, V$ or $V, V^{*}$ . If the latter identities hold and $E = \sum_{a} x_{a} T_{a}$ , $F = \sum_{b} y_{b} T_{b}$ , [F6] gives $V^{*} E^{*} F V = \sum_{a, b} \overline{x_{a}} y_{b} c_{a b} I_{2}$ , so multiplication by $V, V^{*}$ gives the pair-product condition for $(V, E)$ . Conversely, that condition applies to every $T_{a}, T_{b} \in E$ . This proves the spanning-list equivalence, using no restriction of $im V$ to the input spin block. The same expansion applies to the linear span of any finite operator family on $H$ .

2.1F2F3F4F5F7step 1.1∎

For a finite Kraus channel with Kraus family inside $E$ , step 1.1 gives all scalar pair-product identities. The finite-dimensional channel and encoding meet [F3]'s hypotheses: the channel is trace preserving on the full operator space by [F4], and the logical space is the nonzero space $C^{2}$ . Hence [F3] gives a completely positive trace-preserving decoded recovery, which is exact in the sense of [F5]. Conversely [F2] gives the scalar identities for the Kraus family of any exactly recoverable finite Kraus channel on $H$ . The expansion in step 1.1 extends them to its Kraus span, and hence to $E$ if the two spans agree. None of these implications assigns a channel to an arbitrary spanning list.

Depends on

Used by · 1

Finite signed-code KL product test

Dependency tree · 2 levels

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