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.
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Kraus and Choi characterizations of complete positivity

Statement

Let $H, K$ be finite-dimensional complex inner-product spaces, fix an ordered orthonormal basis $B$ of $H$ , and let $Φ : L (H) \to L (K)$ be complex-linear. The following are equivalent: $Φ$ is completely positive; $J_{B} (Φ)$ is non-negative; and $Φ$ has a finite Kraus representation.

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H, K$ , an ordered orthonormal input basis $B$ , and a complex-linear map $Φ$ between their endomorphism spaces.

[F1]

Complete positivity implies Choi positivity (same-page result): complete positivity implies $J_{B} (Φ) ⪰ 0$ .

[F2]

A positive Choi matrix yields finite Kraus operators (same-page result): $J_{B} (Φ) ⪰ 0$ implies a finite Kraus representation.

[F3]

A finite Kraus sum is completely positive (same-page result): a finite Kraus representation implies complete positivity.

[A1]

Completely positive maps: complete positivity is the positivity-preserving property of every finite ancillary block amplification.

[L1]

Choi matrix with input factor first (same-page definition): the Choi convention is $J_{B} (Φ) = \sum_{i, j} E_{i j} \otimes Φ (E_{i j})$ , input factor first and output factor second.

[L2]

Finite Kraus representation (same-page definition): a finite family $(K_{a})$ represents $Φ$ when $Φ (X) = \sum_{a} K_{a} X K_{a}^{†}$ for every $X$ .

Proof

1.1givenF1F2A1L1

Suppose $Φ$ is completely positive. By [F1], its Choi matrix is non-negative. Suppose instead that its Choi matrix is non-negative. By [F2], $Φ$ has a finite Kraus representation. These give the first two implications in the cycle.

2.1givenF3L1L2step 1.1∎

If $Φ$ has a finite Kraus representation in the sense of [L2], [F3] implies that $Φ$ is completely positive. This closes the cycle, so each of the three stated conditions implies the other two. All statements use the same fixed input basis and the input-first/output-second Choi convention of [L1].

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information (2018), §2.2.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3, §§3.2.6 and 3.3.1–3.3.2 (standard reference, not scraped)