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

   https://doi.org/10.22331/q-2024-12-03-1546  

   Iosue, Joseph T; Mooney, T C; Ehrenberg, Adam; Gorshkov, Alexey V (December 2024, Quantum)

   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 $B_{t}modm$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. 

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2. Students’ Conceptual Understanding of Normalization of Vectors from ℝ2 to ℂ2

   Schermerhorn, B. P.; Wawro, M. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A. (Ed.)

   Interdisciplinary studies illuminate ways mathematics is incorporated into core STEM courses. Vector normalization is a crosscutting idea that appears in several mathematics and physics courses. The research question pursued in this study is: how do quantum physics students reason about normalization of vectors from ℝ2 and ℂ2, before and after quantum mechanics instruction? The data are analyzed using the theory of coordination classes (diSessa & Sherin, 1998). Results focus on students’ thinking as they normalize different types of vectors: (A) a real vector and (B) a complex vector before instruction; and (C) a complex vector after instruction. Analysis identifies the ideas students coordinate when problem solving, which problem aspects students attend to, and how students take up or disregard ideas while they problem solve. 

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3. An alternative formalism for modeling spin

   https://doi.org/10.1140/epjc/s10052-022-10652-y  

   Powers, Sam; Stojkovic, Dejan (August 2022, The European Physical Journal C)

   Abstract We present an alternative formalism for modeling spin. The ontological elements of this formalism are base-2 sequences of length n . The machinery necessary to model physics is then developed by considering correlations between base-2 sequences. Upon choosing a reference base-2 sequence, a relational system of numbers can be defined, which we interpret as quantum numbers. Based on the properties of these relational quantum numbers, the selection rules governing interacting spin systems are derived from first principles. A tool for calculating the associated probabilities, which are the squared Clebsch–Gordan coefficients in quantum mechanics, is also presented. The resulting model offers a vivid information theoretic picture of spin and interacting spin systems. Importantly, this model is developed without making any assumptions about the nature of space-time, which presents an interesting opportunity to study emergent space-time models. 

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4. Sequential learning on a tensor network Born machine with trainable token embedding

   https://doi.org/10.1088/2632-2153/adde29  

   Hou, Wanda; Li, Miao; You, Yi-Zhuang (June 2025, Machine Learning: Science and Technology)

   Abstract Generative models aim to learn the probability distributions underlying data, enabling the generation of new, realistic samples. Quantum-inspired generative models, such as Born machines based on the matrix product state (MPS) framework, have demonstrated remarkable capabilities in unsupervised learning tasks. This study advances the Born machine paradigm by introducing trainable token embeddings through positive operator-valued measurements (POVMs), replacing the traditional approach of static tensor indices. Key technical innovations include encoding tokens as quantum measurement operators with trainable parameters and leveraging QR decomposition to adjust the physical dimensions of the MPS. This approach maximizes the utilization of operator space and enhances the model’s expressiveness. Empirical results on RNA data demonstrate that the proposed method significantly reduces negative log-likelihood compared to one-hot embeddings, with higher physical dimensions further enhancing single-site probabilities and multi-site correlations. The model also outperforms GPT-2 in single-site estimation and achieves competitive correlation modeling, showcasing the potential of trainable POVM embeddings for complex data correlations in quantum-inspired sequence modeling. 

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5. A non-abelian analogue of DBI from $$T \overline{T}$$

   https://doi.org/10.21468/SciPostPhys.8.4.052  

   Brennan, T. Daniel; Ferko, Christian; Sethi, Savdeep (January 2020, SciPost Physics)

   The Dirac action describes the physics of the Nambu-Goldstone scalarsfound on branes. The Born-Infeld action defines a non-linear theory ofelectrodynamics. The combined Dirac-Born-Infeld (DBI) action describesthe leading interactions supported on a single D-brane in string theory.We define a non-abelian analogue of DBI using the T\bar{T} T T ‾ deformation in two dimensions. The resulting quantum theory iscompatible with maximal supersymmetry and such theories are quiterare. 

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