PropositionStatement: Literature-sourcedProof: 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.

Canonical weight as maximally mixed branch probability

PDF p. 4: First paragraph

Statement

Let $H$ and $K$ be finite-dimensional complex inner-product spaces, with $d = \dim H > 0$ , and let $N : L (H) \to L (K)$ be a quantum channel, meaning a completely positive trace-preserving linear map. Choose canonical Kraus operators $E_{k} : H \to K$ from the positive-eigenvalue decomposition of the input-first, unnormalized Choi matrix as in Canonical Choi eigenvectors, Kraus rank and branch weights. Their weight is $w_{k} = Tr (E_{k}^{†} E_{k})$ , as in Effective Kraus rank and branch weight.

For the instrument whose branch maps are $X \mapsto E_{k} X E_{k}^{†}$ , the probability of branch $k$ on the maximally mixed input $I_{H} / d$ is $w_{k} / d$ . An instrument is a finite family of completely positive maps whose sum preserves trace; its branch probability is the trace of the branch output. Finite instruments and conditional states

Facts & Assumptions

Given: The spaces, channel, canonical operators and weight in the Statement.

Goal: Show that the indicated branch probability on $I_{H} / d$ equals $w_{k} / d$ .

[F1]

Under these hypotheses the canonical operators form a Kraus representation of $N$ , and their branch maps form a finite instrument. Canonical Choi eigenvectors, Kraus rank and branch weights

[F2]

For a finite instrument, the probability of a branch on a density operator is the trace of its output under that branch map. Finite instruments and conditional states

Proof

Given: The spaces, channel, canonical operators and weight in the Statement.

Goal: Show that the indicated branch probability on $I_{H} / d$ equals $w_{k} / d$ .

1.1givenF1F2algebra

The operator $I_{H} / d$ is positive and has trace one, since $d = \dim H > 0$ ; it is therefore a density operator. By [F1] the specified branch maps form an instrument, and [F2] gives $p_{k} = Tr (E_{k} (I_{H} / d) E_{k}^{†})$ .

2.1givenstep 1.1algebra∎

In orthonormal bases of $H$ and $K$ , the two traces $Tr (E_{k} E_{k}^{†})$ and $Tr (E_{k}^{†} E_{k})$ both equal the sum of the squared absolute values of the entries of $E_{k}$ . Thus step 1.1 gives $p_{k} = Tr (E_{k}^{†} E_{k}) / d = w_{k} / d$ , as claimed.

Remarks

A choice of canonical eigenbasis specifies this mathematical instrument; it does not by itself identify a unique physical measurement record.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 4, first paragraph