Stellar Representation of Grassmannians

TL;DR

This paper introduces a geometric representation of degenerate subspaces in quantum systems using Majorana-like points on the sphere, extending the classical spin state visualization to higher-dimensional subspaces relevant for quantum computing.

Contribution

It generalizes Majorana's representation to characterize degenerate subspaces via points on the sphere that transform under $SU(2)$, applicable to multi-partite qudit states.

Findings

Provides a Majorana-like geometric characterization of subspaces

Demonstrates rigid rotation of points under $SU(2)$ transformations

Applicable to arbitrary totally antisymmetric qudit states

Abstract

Pure quantum spin-$s$ states can be represented by $2s$ points on the sphere, as shown by Majorana in 1932 --- the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other properties, as the points rotate rigidly on the sphere when the state undergoes an $SU(2)$ transformation in Hilbert space. At the same time, the Wilzcek-Zee effect, which involves the cyclic evolution of a degenerate $k$-dimensional linear subspace of the Hilbert space, and the associated holonomy dictated by Schroedinger's equation, have been proposed as a fault-tolerant mechanism for the implementation of logical gates, with applications in quantum computing. We show, in this paper, how to characterize such subspaces by Majorana-like sets of points on the sphere, that also rotate rigidly under $SU(2)$ transformations --- the construction is actually valid for arbitrary totally antisymmetric $k$-partite qudit states.