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

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Instrument branch Kraus families satisfy total normalization

Statement

Let $(I_{a})_{a \in Ω}$ be a finite instrument from a nonzero finite-dimensional complex inner-product space $H$ to a finite-dimensional complex inner-product space $K$ . Suppose that for every $a$ a finite Kraus family $(M_{a, j})_{j}$ represents $I_{a}$ . Then $\sum_{a \in Ω} \sum_{j} M_{a, j}^{†} M_{a, j} = I_{H} .$

Facts & Assumptions

Given: The finite instrument and, for each observed outcome $a$ , a finite Kraus representation of its branch map.

[F1]

Finite instruments and conditional states: $\sum_{a} {tr}_{K} (I_{a} (X)) = {tr}_{H} (X)$ for every $X \in L (H)$ .

[F2]

Finite Kraus representation: $I_{a} (X) = \sum_{j} M_{a, j} X M_{a, j}^{†}$ for every input operator $X$ .

[F3]

Trace is a linear functional on $M_{n} (F)$ and For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : trace is linear and ${tr}_{K} (A B) = {tr}_{H} (B A)$ for compatible rectangular maps $A : H \to K$ , $B : K \to H$ .

[F4]

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

Proof

1.1givenF1F2F3algebra

Put $S = \sum_{a, j} M_{a, j}^{†} M_{a, j} \in L (H)$ . For every $X \in L (H)$ , [F1]–[F3] give ${tr}_{H} (X) = \sum_{a} {tr}_{K} (I_{a} (X)) = \sum_{a, j} {tr}_{K} (M_{a, j} X M_{a, j}^{†}) = \sum_{a, j} {tr}_{H} (X M_{a, j}^{†} M_{a, j}) = {tr}_{H} (X S)$ .

2.1givenstep 1.1F3F4algebra∎

Cyclicity in [F3] gives ${tr}_{H} ((S - I_{H}) X) = {tr}_{H} (X (S - I_{H})) = 0$ for every $X$ . Apply [F4] to $D = S - I_{H}$ to obtain $S - I_{H} = 0$ , which is the stated normalization. This proof uses the exact trace-pairing lemma The trace pairing on a finite endomorphism space is nondegenerate and does not use a channel characterization.

Remarks

The result is specific to the sum of the branches of an instrument. Its proof uses the instrument trace identity directly and does not depend on the separate QC05 trace-preserving Kraus theorem.

Depends on

Used by

Dependency tree · two levels

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Sources

John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution (standard reference, not scraped)
John Watrous, The Theory of Quantum Information (2018 author-hosted draft), §§2.2.2 and 2.3.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution (standard reference, not scraped)