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√2+1/100. On the other hand, all qubit-projective strategies are bounded by 2√2+√5/250+√2/32400, 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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BibTeX
@misc{zhu2026analytic,
title = {{Analytic Qubit Separation between POVMs and Projective Measurements}},
author = {Lin Zhu and Ranyiliu Chen and Xin Wang and Shenggen Zheng},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.01317},
eprint = {2608.01317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01317},
}
RIS
TY - UNPB
TI - Analytic Qubit Separation between POVMs and Projective Measurements
AU - Lin Zhu
AU - Ranyiliu Chen
AU - Xin Wang
AU - Shenggen Zheng
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.01317
UR - https://arxiv.org/abs/2608.01317
AB - 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√2+1/100. On the other hand, all qubit-projective strategies are bounded by 2√2+√5/250+√2/32400, 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.
ER -