Analytic Qubit Separation between POVMs and Projective Measurements

Abstract:Generalized measurements can be implemented projectively after enlarging the Hilbert space, but this dilation changes the available local dimension. We construct a Bell functional with rational coefficients that separates the two measurement models at local dimension two. An explicit three-outcome qubit positive-operator-valued measure with rational matrix entries attains $2\sqrt2+1/100$. On the other hand, all qubit-projective strategies are bounded by $2\sqrt2+\sqrt5/250+\sqrt2/32400$, giving a fully analytic certified gap greater than $1/1000$. To our knowledge, this is the first fully analytic Bell-functional separation between qubit POVMs and qubit projective measurements over arbitrary shared two-qubit states. Lean certificate for the separation theorem is provided for completeness. Separately, an exact level-3 noncommutative sum-of-squares certificate proves that the explicit qubit strategy attains the unrestricted finite-dimensional tensor-product quantum optimum.

9 pages. Comments are welcome

Quantum Physics (quant-ph); Operator Algebras (math.OA)

arXiv:2608.01317 [quant-ph]

(or arXiv:2608.01317v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.01317 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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