Classical capacity of Weyl channel invariant with respect to the action of a unitary representation of the cyclic group

Grigori Gennadievich Amosov

Grigori Gennadievich Amosov

Grigori Gennadievich Amosov, L. A. Ryskin

Abstract We study quantum channels generated by irreducible projective unitary representations of finite groups. It is shown that to calculate a classical capacity of channel it is not necessary to claim that it is covariant with respect to a representation. It helps to consider non-Abelian groups because in this case such channels are not covariant respect to the generating representation.

Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała

Twirling, i.e. averaging over symmetry actions, is a standard tool for reducing quantum states and channels to a symmetry-invariant form. We study channel twirling from the perspective of the channel-state duality and provide a constructive Choi-level description of the averaging map induced by arbitrary input/output representations. Our main technical result concerns the collective setting: for $π^{\textrm{in}}(U)=U^{\otimes t_{\textrm{in}}}$ and $π^{\textrm{out}}(U)=U^{\otimes t_{\textrm{out}}}$, we introduce a partial-transpose reduction that removes the contragredient action and converts the mixed (walled Brauer) channel twirl into an ordinary Schur-Weyl twirl of the partially transposed Choi operator under $U^{\otimes(t_{\textrm{in}}+t_{\textrm{out}})}$, enabling explicit permutation-based formulas without constructing walled Brauer idempotents or mixed Schur transforms. Beyond compact symmetries, we extend channel twirling to reductive, generally non-unitary groups via Cartan decomposition and obtain an invariant-sector decomposition of the averaged Choi operator with weights determined solely by the Abelian Cartan component. Finally, we provide two finite realizations of channel averaging: a "dual" implementation as a convex mixture of unitary-$1$-design channels acting on invariant sectors, and a design-like reconstruction showing that weighted group $t$-designs induce channel $t$-designs for $t=t_{\textrm{in}}+t_{\textrm{out}}$.

Andrew Stewart Caldin

FINDING: Jones polynomial knot invariants are directly realizable as quantum computing circuits, linking A_n root system representations to topological quantum gates. | MATH: Jones polynomial \( V_L(t) \) evaluated at roots of unity (e.g., \( t = e^{2\pi i / r} \)) yields unitary representations of braid groups; A_n root system has Cartan matrix \( A_{ij} = 2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j) \), with simple roots \( \alpha_i \) in \( \mathbb{R}^n \), Weyl group order \( |W(A_n)| = (n+1)! \). | CONNECTION: A_n root lattice is the \( n \)-dimensional hypercubic lattice \( \mathbb{Z}^n \) with all permutations; the golden ratio \( \varphi = 1.618 \) appears in the quantum dimension \( d = [2]_q = q + q^{-1} \) for \( q = e^{i\pi/5} \) (Fibonacci anyons), related to \( A_1 \) at level 3; base-60 emerges in the cyclic structure of \( \mathbb{Z}_n \) actions on braid groups. | DEPTH: 8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Danylo Yakymenko

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.