DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Finite projective measurements

Definition

Let $H$ be a nonzero finite-dimensional complex inner-product space and let $Ω$ be a finite nonempty outcome set (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Real and complex inner-product spaces and their induced length). Its endomorphisms are its complex-linear maps $H \to H$ (The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication); for an endomorphism $P$ , $P^{*}$ denotes its adjoint, which exists uniquely in finite dimension (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Every linear map between finite-dimensional inner product spaces has a unique adjoint). A finite projective measurement on $H$ with outcomes $Ω$ is a family $(P_{a})_{a \in Ω}$ of endomorphisms such that

$P_{a}^{*} = P_{a}, P_{a}^{2} = P_{a}, P_{a} P_{b} = 0 (a \neq b), \sum_{a \in Ω} P_{a} = I_{H} .$

The products above are compositions, the sum is pointwise addition of endomorphisms, and $I_{H}$ is the identity map; these operations preserve linear maps ( $L (V, W)$ is a vector space over the common scalar field, Identity maps and composites of linear maps are linear). The projections $P_{a} = 0$ are permitted. For a density operator $ρ$ on $H$ (nonnegative and trace one, Density operator), the associated outcome weights are $p_{ρ} (a) = tr (ρ P_{a})$ ; their nonnegativity and normalization are proved in Projective measurement outcome weights are probabilities ↗.

Depends on

Used by

Amplitude amplification with a fixed integer iteration count Algorithm
Amplitude amplification with independently chosen phase rotations Algorithm
Coherent SDP matrix-expectation oracle Algorithm
Complete-graph continuous-time search Algorithm
Dense-coding procedure Algorithm
Exact success amplification by attenuating the good branch Algorithm
Finite amplification from a supplied success lower bound Algorithm
Finite capped quantum quantile minimum Algorithm
Hypercube continuous-time state transfer Algorithm
Iterative no-QFT amplitude estimation by finite-grid likelihood Algorithm
Known-amplitude amplification with integer rounding Algorithm
MNRS finite reversible-walk finding Algorithm
One finite abelian hidden-subgroup Fourier sample Algorithm
One Simon Fourier-sampling sample Algorithm
Power-of-two dihedral hidden-subgroup sieve Algorithm
Quantum phase estimation Algorithm
Quantum phase-estimation spectral sampler Algorithm
Qubit teleportation procedure Algorithm
Szegedy-walk marked-set detection Algorithm
Unknown-amplitude randomized search Algorithm
Amplitude estimation and counting input/output contract Definition
Coherent finite sampler and observable value Definition
Coherent variable-time algorithm model Definition
Eigenphase estimation access model Definition
Finite adiabatic path model Definition
Finite open-graph adaptive measurement pattern Definition
Fusion-channel measurement access Definition
Marked sets, stationary mass, and walk-search tasks Definition
Pauli measurement branch semantics Definition
Prepared unitary and success subspace for amplitude amplification Definition
A projective measurement branch with probability zero Example
One-bit feed-forward and deferred measurement Example
A finite instrument has an isometric coherent record Lemma
Capped randomized amplitude amplification preserves a flagged output Lemma
Computational-basis measurement weights are squared amplitudes Lemma
Dihedral hidden-state pair combination Lemma
Fourier sample from a generalized-dihedral reflection coset state Lemma
Measurement law of a coherent sampler Lemma
One-bit J-gate teleportation branch identity Lemma
Phase-estimation finite outcome law Lemma

…and 19 more results.

Dependency tree · two levels

32 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), §2.3.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution, §3.1.1 (standard reference, not scraped)