bibkey: mozrzymas2017irreduciblycovariant
authors: Marek Mozrzymas; Michał Studziński; Nilanjana Datta
year: 2017
title: Structure of irreducibly covariant quantum channels for finite groups
doi: 10.1063/1.4983710
url: https://arxiv.org/abs/1610.05657v2
claim: For a finite-group irreducible representation whose tensor product with its contragredient is simply reducible, representation data characterize covariant linear maps and quantum channels; the quaternion example gives explicit representation matrices and characters.
strata_touched: []
license: citation-only
triage: anchor

Finite-group representation methods for covariant channels

Source and locator

Literature status: literature-attested.

Mozrzymas, Studziński and Datta, Journal of Mathematical Physics 58, 052204 (2017), DOI. The primary version arXiv:1610.05657v2 supplies the finite-representation background in §4 and the quaternion example in §8.2. Equations (123)–(125) give the group, a faithful two-dimensional representation and its tensor decomposition; Table 1 gives the five irreducible characters.

RT use and boundary

RT IV §108 uses finite-group representation methods as intermediate tools. Its central-partner D8/Q8 argument explicitly proves the eight-coordinate regular decomposition, invariant tensor projection and product-state overlap needed for an original-channel bound. The source’s irreducibly covariant channels act on the irreducible representation dimension; their channel formulas are not substituted for the original C8-to-C4 product channel or its fibre-block complete norm.

The RT theorem assumes that one correct-output Gram has the specified regular translation symmetry and that the original partner operation is the central involution. It permits arbitrary original leakage and proves a sharp half-error bound with physical attainers. The noncentral-reflection action, nonsymmetric residual classes and unrestricted quarter optimum are outside that conclusion. No general representation theorem or exhaustive literature-priority claim is attributed to the RT derivation.