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Positive refinement of a coarse qubit map

Abstract

Ordinary positive records of a general CP coarse map can fail complete positivity.

Theorem 1.1 (The transpose record is a counterexample).

Lean statement: D5/S3/Quantum/Reduction/PositiveCoarseRefinementRefutation.result

Proof. Machine-checked in Lean as D5/S3/Quantum/Reduction/PositiveCoarseRefinementRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The claim asserts that for all complex linear maps S,T on M_2(C), ordinary positivity of both maps, the trace identities Tr(S(X))=Tr(T(X))=Tr(X)/2 for every X, and complete positivity of S+T imply complete positivity of S. The theorem is the literal negation of that claim.

Take S(X)=transpose(X)/2 and T(X)=(Tr(X) I-transpose(X))/2. Transposition preserves positivity. The second map is one half of conjugation by the real matrix with rows (0,1) and (-1,0), so it is positive as well. Both effects are I/2. Their sum is the completely depolarizing map X maps to Tr(X) I/2, with a four-matrix-unit Kraus representation.

Applying the transpose branch tensored with the two-dimensional identity to the unnormalized Bell projector gives half the swap matrix. Its quadratic form on the antisymmetric vector is negative, contradicting positivity of that amplification. The example therefore concerns a general coarse CP map; it does not contradict scalar rigidity for positive refinements of a single canonical filter.

References