A note on tight projective 2-designs

• URL: http://arxiv.org/abs/2101.11756v2
• Date: Thu, 11 Feb 2021 22:27:47 GMT
• Title: A note on tight projective 2-designs
• Abstract summary: We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. We show that every tight projective 2-design for Hd determines an equi-isoclinic tight fusion frame of d (2d-1) subspaces of Rd (2d+1) of dimension 3.
• Score: 6.488575826304023
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.

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