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.

The Bell vectors

Definition

Let $Q_{2} = C^{2}$ be the ordered qubit register with computational basis $(∣ 0 ⟩, ∣ 1 ⟩)$ , and use the ordered two-factor register $Q_{2} \otimes Q_{2}$ from Ordered tensor registers, their inner product, and wire permutations. Define the four Bell vectors by

$∣ Φ^{+} ⟩ : = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{\sqrt{2}}, ∣ Φ^{-} ⟩ : = \frac{∣ 00 ⟩ - ∣ 11 ⟩}{\sqrt{2}}, ∣ Ψ^{+} ⟩ : = \frac{∣ 01 ⟩ + ∣ 10 ⟩}{\sqrt{2}}, ∣ Ψ^{-} ⟩ : = \frac{∣ 01 ⟩ - ∣ 10 ⟩}{\sqrt{2}} .$

The ordered label convention is $∣ 00 ⟩ \mapsto ∣ Φ^{+} ⟩$ , $∣ 01 ⟩ \mapsto ∣ Ψ^{+} ⟩$ , $∣ 10 ⟩ \mapsto ∣ Φ^{-} ⟩$ , and $∣ 11 ⟩ \mapsto ∣ Ψ^{-} ⟩$ . Thus the first label is the phase/sign bit and the second is the bit-flip bit. The next theorem verifies that these vectors form an orthonormal basis and gives the corresponding preparation and decoding maps.

Depends on

Used by

Dense-coding procedure Algorithm
Qubit teleportation procedure Algorithm
Entanglement-assisted capacity of a noiseless qubit Example
Bell preparation transform and inverse Theorem
Dense-coding correctness and resources Theorem
Teleportation Bell-basis expansion Theorem

Dependency tree · two levels

24 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), §6.3.1 (standard reference, not scraped)
Richard Cleve, Quantum Information Processing course notes for QIC 710 / CS 768 / PHYS 767 / CO 681 / AMATH 871 / PMATH 871, version dated October 13, 2026 (standard reference, not scraped)