AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Informationally complete linear tomography

Definition

Input integers $d, m \geq 1$ , a finite POVM $(F_{a})_{a = 1}^{m}$ on $C^{d}$ that spans the real Hermitian matrices, and real frequencies $f_{a} \geq 0$ with $\sum_{a} f_{a} = 1$ . Form the Gram matrix $G_{a b} = Tr (F_{a} F_{b})$ , compute its Moore–Penrose inverse $G^{+}$ , set $D_{a} = \sum_{b} (G^{+})_{b a} F_{b}$ , and return

X = \sum_{a = 1}^{m} f_{a} D_{a} .

All sums and matrices are finite. The output is Hermitian. The procedure does not project $X$ onto the positive cone, and it makes no positivity claim for finite frequency inputs.

Definition references

Finite Hermitian frames and their Gram pseudoinverse

Finite effects and instruments: normalization and non-determination of the update

Depends on

Dependency tree · 2 levels

Sources

Xhek Turkeshi and Markus Heinrich, Quantum Information Methods for Many-Body Physics, §3.1, equations (3.7)–(3.9)