DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-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.

The outcome law and outputs of a measured circuit

Definition

For a finite instrument $(I_{a})_{a \in Ω}$ and input density operator $ρ$ , define the unnormalized branch output $τ_{a} : = I_{a} (ρ)$ and probability $p_{a} : = tr (τ_{a})$ . The family $(p_{a})_{a \in Ω}$ is a probability distribution by Instrument outcome weights normalize. Its uniquely induced POVM $(E_{a})$ satisfies $p_{a} = tr (ρ E_{a})$ by Every finite instrument induces a unique POVM and the outcome-law definition in Finite POVMs and their outcome law.

The conditional branch output is $τ_{a} / p_{a}$ when $p_{a} > 0$ and is $⊥$ when $p_{a} = 0$ , exactly as specified by Finite instruments and conditional states. No normalized state is assigned to a zero-probability branch. A Kraus representation $I_{a} (X) = \sum_{j} M_{a, j} X M_{a, j}^{†}$ is only a representation of the observed branch map: $a$ is the outcome and $j$ is not an additional observed outcome.

Facts & Assumptions

Given: A finite instrument, a density operator $ρ$ on its input space, and its finite outcome set $Ω$ .

[F1]

Finite instruments and conditional states: $τ_{a} = I_{a} (ρ)$ , $p_{a} = tr (τ_{a})$ , and a conditional output is assigned only for $p_{a} > 0$ .

[F2]

Instrument outcome weights normalize: the weights $p_{a}$ are nonnegative and sum to one.

[F3]

Every finite instrument induces a unique POVM: each branch weight equals the induced POVM weight $tr (ρ E_{a})$ .

[F4]

Finite POVMs and their outcome law: a POVM assigns weights $tr (ρ E_{a})$ and does not specify a conditional output state.

Proof

1.1givenF1

The branch output is the unnormalized operator $τ_{a} = I_{a} (ρ)$ , and its trace is the branch weight $p_{a}$ , by [F1].

1.2givenF2F3F4

The weights form a finite probability law by [F2], and the same law is $tr (ρ E_{a})$ by [F3] and [F4]. This establishes consistency of the instrument and POVM descriptions.

2.1givenF1step 1.2∎

If $p_{a} > 0$ , the conditional output is defined by division by that positive scalar; if $p_{a} = 0$ , [F1] assigns $⊥$ and no division is made. The finite Kraus index remains internal to $I_{a}$ and is not part of $Ω$ .

Depends on

Used by

All measurements may be moved to terminal readout Theorem
Finite instruments can be coherently deferred Theorem

Dependency tree · two levels

22 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 Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)