A Separation between Full-Rank PVM and Assumption-free Self-Testing
Abstract
We construct a nonlocal game that self-tests a maximally entangled qubit strategy among pure full-Schmidt-rank projective strategies, but admits an inequivalent optimum using one nonprojective measurement. This resolves a conjecture of Baptista et al. on imposing full rank and projectivity simultaneously. The construction combines CHSH with an auxiliary game G that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of G as a line segment parametrized by the tracial states of ℂ⊕ M2(ℂ). Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.
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BibTeX
@misc{chen2026separation,
title = {{A Separation between Full-Rank PVM and Assumption-free Self-Testing}},
author = {Ranyiliu Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10013},
eprint = {2609.10013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Separation between Full-Rank PVM and Assumption-free Self-Testing
AU - Ranyiliu Chen
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10013
UR - https://arxiv.org/abs/2609.10013
AB - We construct a nonlocal game that self-tests a maximally entangled qubit strategy among pure full-Schmidt-rank projective strategies, but admits an inequivalent optimum using one nonprojective measurement. This resolves a conjecture of Baptista et al. on imposing full rank and projectivity simultaneously. The construction combines CHSH with an auxiliary game G that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of G as a line segment parametrized by the tracial states of ℂ⊕ M_2(ℂ). Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.
ER -