bibkey: cirelson1980quantum
authors: B. S. Cirel'son
year: 1980
title: Quantum generalizations of Bell's inequality
doi: 10.1007/BF00417500
claim: The CHSH quantum expectation is bounded by two times square root two, and the bound is attained by a two-qubit realization.
strata_touched:
  - D5/S3/QuantumBounds/TsirelsonTightness
license: citation-only
triage: anchor

Quantum generalizations of Bell’s inequality

Cirel’son’s paper is the literature anchor for the sharp quantum CHSH bound conventionally called Tsirelson’s bound. The repository declaration specializes that classical result to one explicit four-by-four CHSH operator: it proves that the real trace expectation over positive-semidefinite trace-one matrices has greatest value 2 * sqrt 2.

The exact Lean IsGreatest packaging, the repository’s named Pauli matrices and Bell density, and the proof that transports matrix order through a positive trace pairing are not attributed to the paper. The cited result supplies the bound and its classical provenance; the formal declaration supplies this fixed finite-dimensional realization.

Search log

• 2026-08-15: Crossref DOI metadata was queried directly. It identifies B. S. Cirel’son, the exact title, Letters in Mathematical Physics 4(2), 93-100 (1980), DOI 10.1007/BF00417500.
• 2026-08-15: The Springer landing page was read. Its abstract states that quantum correlations obey weaker inequalities of Bell type and that the paper proves particular inequalities of this kind.
• 2026-08-15: The pinned mathlib source Mathlib.Algebra.Star.CHSH was read. It contains tsirelson_inequality, the upper-bound theorem used by the Lean declaration.
• 2026-08-15: The publisher records the article as closed access. A direct text-mining PDF request returned HTML rather than the article, so no equation number or claim about the paper’s exact notation is made here.

Verified locator

• DOI: https://doi.org/10.1007/BF00417500
• Crossref record: https://api.crossref.org/works/10.1007/BF00417500