Projective toric designs, quantum state designs, and mutually unbiased bases

Joseph T. Iosue; T. C. Mooney; Adam Ehrenberg; Alexey V. Gorshkov

doi:10.22331/q-2024-12-03-1546

Projective toric designs, quantum state designs, and mutually unbiased bases

Joseph T. Iosue, T. C. Mooney, Adam Ehrenberg, and Alexey V. Gorshkov

Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA

Published:	2024-12-03, volume 8, page 1546
Eprint:	arXiv:2311.13479v3
Doi:	https://doi.org/10.22331/q-2024-12-03-1546
Citation:	Quantum 8, 1546 (2024).

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Abstract

Toric $t$-designs, or equivalently $t$-designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree $t$ monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of $t$-designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense $B_t \bmod m$ sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state $2$-designs in dimension $d$ of size exactly $d(d+1)$ that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension $6$ (and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.

Popular summary

A t-design on a space is a measure space that exactly integrates polynomials up to degree t. From the perspective of quantum information theory, the most important and interesting spaces over which to form designs are (1) the set of quantum states, (2) the diagonal subgroup of the unitary group, (3) the set of infinite-dimensional quantum states, (4) the n-sphere, and (5) the (projective) unitary group. In this work, we add the projective torus to this list, and we show how the projective torus and (1-4) are all intimately connected.

Furthermore, we also show that projective toric designs are crucial for understanding another interesting and notoriously difficult object in quantum information theory — complete sets of mutually unbiased bases (MUBs). Complete sets of MUBs are maximal sets of orthonormal bases where measurement in one basis reveals nothing about any of the other bases. For example, measuring a qubit in the X-basis reveals nothing about the state of the qubit along the Z-basis, and vice-versa.

Despite their fundamental importance to both the theory of quantum information and to applications, their existence or non-existence in non-prime-power dimensions has been an open problem for decades. Utilizing the connection to projective toric designs, we disprove a conjecture by Zhu in 2015 regarding the structure of complete sets of MUBs. We further prove a fundamental distinction between complete sets of MUBs in prime-power dimensions (where they are better understood) and in non-prime-power dimensions (where they are notoriously difficult) in terms of corresponding projective toric 2-designs.

► BibTeX data

@article{Iosue2024projectivetoric,
  doi = {10.22331/q-2024-12-03-1546},
  url = {https://doi.org/10.22331/q-2024-12-03-1546},
  title = {Projective toric designs, quantum state designs, and mutually unbiased bases},
  author = {Iosue, Joseph T. and Mooney, T. C. and Ehrenberg, Adam and Gorshkov, Alexey V.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1546},
  month = dec,
  year = {2024}
}

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