TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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Trace preservation is Kraus normalization

Statement

Let $H, K$ be finite-dimensional complex inner-product spaces and let $Φ (X) = \sum_{a \in A} K_{a} X K_{a}^{†}$ be a finite Kraus representation, where each $K_{a} : H \to K$ . The map $Φ$ is trace-preserving on all of $L (H)$ , meaning ${tr}_{K} (Φ (X)) = {tr}_{H} (X)$ for every $X \in L (H)$ , if and only if $\sum_{a \in A} K_{a}^{†} K_{a} = I_{H} .$

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H, K$ , a finite family $K_{a} : H \to K$ , and the indicated Kraus formula for every $X \in L (H)$ .

[F1]

Finite Kraus representation: a finite Kraus representation satisfies $Φ (X) = \sum_{a} K_{a} X K_{a}^{†}$ on every input operator.

[F2]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : for $A : H \to K$ and $B : K \to H$ , ${tr}_{K} (A B) = {tr}_{H} (B A)$ , including zero dimensions.

[F3]

Trace is a linear functional on $M_{n} (F)$ : trace is linear over finite sums.

[F4]

The trace pairing on a finite endomorphism space is nondegenerate: if ${tr}_{H} (A X) = 0$ for every $X \in L (H)$ , then $A = 0$ , including $H = 0$ .

Proof

1.1givenF1F2F3algebra

Put $S = \sum_{a} K_{a}^{†} K_{a} \in L (H)$ . For every $X \in L (H)$ , [F1], trace linearity [F3], and rectangular cyclicity [F2] give ${tr}_{K} (Φ (X)) = \sum_{a} {tr}_{K} (K_{a} X K_{a}^{†}) = \sum_{a} {tr}_{H} (K_{a}^{†} K_{a} X) = {tr}_{H} (S X)$ . If $S = I_{H}$ , this equals ${tr}_{H} (X)$ for every $X$ , so $Φ$ is trace-preserving by the Statement. This includes an empty family when $H = 0$ , where $I_{H} = 0$ .

2.1givenstep 1.1F4algebra∎

Conversely, suppose $Φ$ is trace-preserving on all of $L (H)$ . The identity in step 1.1 gives ${tr}_{H} ((S - I_{H}) X) = 0$ for every $X \in L (H)$ . Apply [F4] with $A = S - I_{H}$ to obtain $S - I_{H} = 0$ . Thus the normalization holds. In particular, if $H = 0$ both sides are the unique zero endomorphism and the argument still applies.

Depends on

Used by

Exact correction of a protected subsystem Definition
Exact recovery of an encoded quantum code Definition
A degenerate code with a non-diagonal Knill–Laflamme matrix Example
Orthogonal error spaces and a recovery branch Example
Qubit dephasing channel in Kraus and Choi forms Example
A finite POVM defines a classical-output quantum channel Lemma
Coherent normal form for finite quantum query algorithms Lemma
Complement and degradation maps of the erasure channel Lemma
Compression dimension bounds entanglement fidelity Lemma
Optimal Cloning Worst Case Upper Bound Lemma
Preserving the logical marginal forces a gauge-only channel Lemma
Unitary conjugation is a channel Proposition
An explicit syndrome recovery corrects a Knill–Laflamme error family Theorem
Channel data processing for finite quantum relative entropy Theorem
Channel Diamond Norm One Theorem
Every finite-dimensional channel has a finite isometric dilation Theorem
Exact recovery implies the Knill–Laflamme condition Theorem
Finite channel, Kraus, Choi, and Stinespring equivalence Theorem
Finite-dimensional channels compose and tensor Theorem
Finite-dimensional Schumacher quantum source coding theorem Theorem
Full Rank Pair No Broadcasting Theorem
Singular Pair No Broadcasting Theorem
The Choi partial trace characterizes trace preservation Theorem
The Knill–Laflamme criterion for a protected subsystem Theorem
Trace Distance Channel Contractivity Theorem

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information (2018), Theorem 2.26 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3, §3.2.1 (standard reference, not scraped)