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.

Logical Kraus operators of the completed transpose decoder

Statement

Let $d \geq 1$ be an integer, let $H_{P}, H_{O}$ be nonzero finite-dimensional complex inner-product spaces, and let $V : C^{d} \to H_{P}$ be a linear isometry. Set $P = V V^{†}$ . Let $N : L (H_{P}) \to L (H_{O})$ be a quantum channel with finite Kraus representation $N (X) = \sum_{b} E_{b} X E_{b}^{†}$ , where $E_{b} : H_{P} \to H_{O}$ . Here $L (H)$ denotes the complex-linear endomorphisms of $H$ .

Set $A = N (P)$ and let $S$ be the orthogonal projector onto $supp A = (\ker A)^{⊥}$ . The operator $A^{- 1 / 2}$ is the inverse square root on this support and zero on $\ker A$ . For any code state $σ$ , meaning $σ \geq 0$ , $tr σ = 1$ and $P σ P = σ$ , define $R_{a} = P E_{a}^{†} A^{- 1 / 2}$ and the completed transpose recovery $T (Y) = \sum_{a} R_{a} Y R_{a}^{†} + tr ((I_{H_{O}} - S) Y) σ$ .

The logical decoder $D : L (H_{O}) \to L (C^{d})$ given by $D (Y) = V^{†} T (Y) V$ is completely positive and trace-preserving (CPTP). On encoded noisy inputs its composition has Kraus operators $K_{a b} = V^{†} E_{a}^{†} A^{- 1 / 2} E_{b} V$ , so $D N (V X V^{†}) = \sum_{a, b} K_{a b} X K_{a b}^{†}$ for every $X \in L (C^{d})$ .

Statement references

Support-normalized transpose recovery and completion

Code restriction and decoded channel types

Finite Kraus representation

Facts & Assumptions

Given: The spaces, maps and operators in the Statement.

Goal: Prove the asserted identities and conclusions.

[F1]

The completed transpose map is CPTP, code-supported, and agrees with its support-normalized part on every noisy code-supported operator. Support-normalized transpose recovery and completion

[F2]

An isometric encoding is $X \mapsto V X V^{†}$ , with $V^{†} V = I$ and $V V^{†} = P$ . Compressing any code-supported quantum channel $T$ by $V^{†} (\cdot) V$ gives a CPTP logical decoder, as justified by ancilla conjugation and composition. Code restriction and decoded channel types

[F3]

A finite Kraus representation expresses a complex-linear map as a finite sum of conjugations, with the identity valid for every input operator. Finite Kraus representation

Proof

Given: The spaces, maps and operators in the Statement.

Goal: Prove the asserted identities and conclusions.

Proof technique: direct.

1.1givenF1F2

Compression $Z \mapsto V^{†} Z V$ is completely positive. Every output $Z$ of $T$ satisfies $P Z P = Z$ , so $tr (V^{†} Z V) = tr (P Z) = tr Z$ ; the trace equality follows by expanding both traces in orthonormal bases adapted to the code. Thus $D$ is trace preserving and completely positive.

2.1F1F2F3step 1.1∎

For $V X V^{†} = P (V X V^{†}) P$ , the completion term vanishes after the noise. Substitute $R_{a} = P E_{a}^{†} A^{- 1 / 2}$ into the composition, and use $V^{†} P = V^{†}$ and $P V = V$ . This gives exactly the displayed $K_{a b}$ and channel formula.

Depends on

Used by · 2

Dependency tree · 2 levels

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