PhysicsLogic / Hastings-Koma 2006 States

Quantum State Positivity

Positive probabilities and separately available measurements do not by themselves supply a joint experiment.

Definitions and assumptions

Scope and physical limitations

The shared state predicate now requires each expectation of a* a to be real and nonnegative, as well as normalization at the identity. The independent algebra C x C supplies a normalized complex-linear counterexample to the old real-part-only predicate and an ordinary equal-mixture control. The repaired predicate gives real probabilities in [0,1] for every projection. Vector states, normalized finite ground averages, and nontrivial-filter weak limits retain full complex positivity. This predicate constrains the expectation, not the auxiliary ground term; it does not establish a ground-state realization, a spectral gap, or a signalling bound. A separate noncommutative qubit diagnostic uses the actual sharp Z and X yes-effects. Each effect and its complement is positive, but no four positive effects have both exact binary marginals. The shared availability sector is nonempty and all its members are available; this does not quantify over a common measurement. Availability here is ideal POVM admissibility, not laboratory implementation. Identical sharp measurements and an explicitly half-strength unsharp pair do admit joint effects. The obstruction neither forbids approximate joint measurement nor asserts a measurement-disturbance inequality.

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Examples and distinctions

Constructed examples of this model; each answer includes the Lean statement.

Exact root statement

(distorted.expectation 1 = 1 ∧
    ∀ (observable : Algebra), 0 ≤ (distorted.expectation (star observable * observable)).re) ∧
  IsStarProjection yes ∧
    yes ≠ 0 ∧
      yes ≠ 1 ∧
        distorted.expectation yes = 1 / 2 + Complex.I ∧
          ¬distorted.NormalizedPositiveAt ∧ mixture.NormalizedPositiveAt ∧ mixture.expectation yes = 1 / 2
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Checked statements: parameters, hypotheses, and interpretation

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Constructors, projections, and other meaning dependencies

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External mathematical declarations

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