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

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 $B_t\mathrm{mod}m$ 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.

17+8 pages, 1 figure. V3: match journal version

References in corpus (29)

• Black holes as mirrors: quantum information in random subsystems
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Fast Scramblers
• Symmetric Informationally Complete Quantum Measurements
• Randomized Benchmarking of Quantum Gates
• Robust randomized benchmarking of quantum processes
• On mutually unbiased bases
• A simple formula for the average gate fidelity of a quantum dynamical operation
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Scalable Noise Estimation with Random Unitary Operators
• Chaos and complexity by design
• Randomizing quantum states: Constructions and applications
• Information Scrambling in Computationally Complex Quantum Circuits
• Tight informationally complete quantum measurements
• Provably efficient machine learning for quantum many-body problems
• The power of random measurements: measuring Tr(ρ^n) on single copies of ρ
• Mutually Unbiased Bases and Finite Projective Planes
• Scalable randomized benchmarking of non-Clifford gates
• Experimental Estimation of Average Fidelity of a Clifford Gate on a 7-qubit Quantum Processor
• Gate fidelity fluctuations and quantum process invariants
• On the complexity of quantum partition functions
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Mutually unbiased bases as minimal Clifford covariant 2-designs
• The curious nonexistence of Gaussian 2-designs
• Phase Retrieval Using Unitary 2-Designs
• Continuous-variable quantum state designs: theory and applications
• The apparent structure of dense Sidon sets
• Hypergeometric heritage of W.N. Bailey. With an appendix: Bailey's letters to F. Dyson
• A rigidity property of complete systems of mutually unbiased bases

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