SICs and the Triangle Group $(3,3,3)$

• URL: http://arxiv.org/abs/2312.13400v2
• Date: Wed, 29 May 2024 05:58:06 GMT
• Title: SICs and the Triangle Group $(3,3,3)$
• Authors: Danylo Yakymenko,
• Abstract summary: We show that order-three unitaries appear in projective unitary representations of the triangle group $ (3,3,3)$. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group $(3,3,3)$. We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.

Related papers

• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids [48.225436651971805]
  We employ the SU(n)_k Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way.
  arXiv  Detail & Related papers  (2025-01-16T14:42:00Z)
• A Constructive Approach to Zauner's Conjecture via the Stark Conjectures [0.0]
  We present a construction of $d2$ complex equiangular lines in $mathbbCd$, also known as SICPOVMs. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. We prove that our construction gives a valid SIC in every case assuming two conjectures.
  arXiv  Detail & Related papers  (2025-01-07T18:16:43Z)
• Lattice random walks and quantum A-period conjecture [0.0]
  enumeration problem is mapped to the trace of powers of anisotropic Hofstadter-like Hamiltonian. We propose a conjecture connecting the above signed area enumeration $C_N(A)$ in statistical mechanics to the quantum A-period of associated toric Calabi-Yau threefold in topological string theory.
  arXiv  Detail & Related papers  (2024-12-30T18:00:02Z)
• Entanglement renormalization of fractonic anisotropic $\mathbb{Z}_N$ Laplacian models [4.68169911641046]
  Gapped fracton phases constitute a new class of quantum states of matter which connects to topological orders but does not fit easily into existing paradigms. We investigate the anisotropic $mathbbZ_N$ Laplacian model which can describe a family of fracton phases defined on arbitrary graphs.
  arXiv  Detail & Related papers  (2024-09-26T18:36:23Z)
• Wigner's Theorem for stabilizer states and quantum designs [0.6374763930914523]
  We describe the symmetry group of the stabilizer polytope for any number $n$ of systems and any prime local dimension $d$. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. We extend an observation of Heinrich and Gross and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments.
  arXiv  Detail & Related papers  (2024-05-27T18:00:13Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Enriching Diagrams with Algebraic Operations [49.1574468325115]
  We extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.
  arXiv  Detail & Related papers  (2023-10-17T14:12:39Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions [77.34726150561087]
  We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $mathcal H$ can be constructed. We show that the threshold value $gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $gammagamma_c$.
  arXiv  Detail & Related papers  (2022-07-01T10:01:14Z)
• Threshold size for the emergence of a classical-like behaviour [68.8204255655161]
  We design a procedure to estimate the minimum size beyond which a system is amenable to a classical-like description. The specific case of a magnetic system is considered, with details of a gedanken experiment presented and thoroughly commented.
  arXiv  Detail & Related papers  (2022-03-25T11:31:14Z)
• Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors [0.0]
  We prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and denoteity. This, in particular, extends a result of Moln'ar to the wider setting of atomic JBW$*$-triples not containing rank-one Cartan factors.
  arXiv  Detail & Related papers  (2021-01-03T17:21:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.