Analytic Qubit Separation between POVMs and Projective Measurements

• URL: http://arxiv.org/abs/2608.01317v1
• Date: Sun, 02 Aug 2026 15:37:41 GMT
• Title: Analytic Qubit Separation between POVMs and Projective Measurements
• Abstract summary: We construct a Bell functional with rational coefficients that separates the two measurement models at local dimension two.<n>An explicit three-outcome qubit positive-operator-valued measure with rational matrix entries attains $2sqrt2+1/100$.<n>This is the first fully analytic Bell-functional separation between qubit POVMs and qubit projective measurements over arbitrary shared two-qubit states.
• Score: 11.317423514829237
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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