Supplies a Lean certificate formally checking the fixed-qubit separation theorem between qubit POVMs and qubit 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.
Problem
Naimark dilation turns any POVM into a projective measurement, but only on a larger Hilbert space. The open question was whether a Bell functional can analytically separate qubit POVMs from qubit projective strategies over arbitrary shared two-qubit states at fixed local dimension two.
Approach
The authors build a Bell functional with rational coefficients: a CHSH anchor plus a weak probe (strength 1/100) of Alice's ternary measurement. The probe uses a 3-4-5 rational deformation of a trine. An explicit rank-one three-outcome qubit POVM attains the POVM value. An analytic argument bounds every qubit-PVM strategy, showing the probe gains only O(sqrt q) while CHSH loses O(q). A Lean certificate for the separation theorem is provided, and an exact level-3 noncommutative SOS certificate handles the dimension-unrestricted optimum.
Figure 1: Bell scenario for comparing qubit POVMs with qubit projective measurements. A source distributes a two-qubit state \rho_{AB} to Alice and Bob. Alice receives x\in\{0,1,2\} , where x=0,1 label binary measurements and x=2 labels a ternary measurement. Bob receives y\in\{0,1\} and performs a binary measurement. Their outputs determine conditional probabilities P(a,b\mid x,y) , which are eva
Results
The qubit POVM attains 2√2+1/100. All qubit-PVM strategies, including those with shared randomness, are bounded by 2√2+√5/250+√2/32400, a certified gap greater than 1/1000. The SOS certificate shows 2√2+1/100 is also the unrestricted finite-dimensional quantum maximum.
