Complete product-state contact set and optimality of the canonical three-qubit Shifts witness
Abstract
For the three-qubit Shifts unextendible product basis (UPB), the minimum product-state expectation value λShifts=1−3√6/8 of the UPB projector is known, and four attaining product states were exhibited previously. We determine the complete equality set over all complex product states and prove that it consists of exactly eight product rays: two rays invariant under cyclic permutation of the three qubits and two cyclic orbits of size three. The proof combines exact phase analysis, a Gröbner-basis classification of all 28 finite real stationary points, and a complete treatment of the projective boundaries. Crucially, the eight contact rays span the full three-qubit Hilbert space, whereas the four previously known rays do not. The complete equality classification therefore establishes that the associated projector witness satisfies the spanning criterion and is optimal in the sense that no nonzero positive semidefinite operator can be subtracted while preserving block positivity. As a corollary, for the white-noise family ρ(p)=(1−p)ρShifts+p𝕀/8, where ρShifts is the complementary-projector Shifts state, the same witness detects entanglement for p<2−3√6/4≈16.29%.
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BibTeX
@misc{wan2026complete,
title = {{Complete product-state contact set and optimality of the canonical three-qubit Shifts witness}},
author = {Bing-Dong Wan and Yu-Chen Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39332},
eprint = {2609.39332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete product-state contact set and optimality of the canonical three-qubit Shifts witness
AU - Bing-Dong Wan
AU - Yu-Chen Guo
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39332
UR - https://arxiv.org/abs/2609.39332
AB - For the three-qubit Shifts unextendible product basis (UPB), the minimum product-state expectation value λ_Shifts=1−3√6/8 of the UPB projector is known, and four attaining product states were exhibited previously. We determine the complete equality set over all complex product states and prove that it consists of exactly eight product rays: two rays invariant under cyclic permutation of the three qubits and two cyclic orbits of size three. The proof combines exact phase analysis, a Gröbner-basis classification of all 28 finite real stationary points, and a complete treatment of the projective boundaries. Crucially, the eight contact rays span the full three-qubit Hilbert space, whereas the four previously known rays do not. The complete equality classification therefore establishes that the associated projector witness satisfies the spanning criterion and is optimal in the sense that no nonzero positive semidefinite operator can be subtracted while preserving block positivity. As a corollary, for the white-noise family ρ(p)=(1−p)ρ_Shifts+p𝕀/8, where ρ_Shifts is the complementary-projector Shifts state, the same witness detects entanglement for p<2−3√6/4≈16.29%.
ER -