Symmetric Multiqudit States: Stars, Entanglement, Rotosensors

TL;DR

This paper introduces a generalized stellar representation for symmetric multiqudit states using Majorana constellations, enabling analysis of entanglement and the design of highly sensitive quantum rotosensors.

Contribution

It extends Majorana's stellar representation to symmetric states of multiqudits with multiple subsystems, providing new tools for entanglement analysis and quantum sensing.

Findings

Representations of symmetric states of four qutrits and two spin-3/2 systems.

Revisiting Hermite and Murnaghan isomorphisms for multipartite states.

Identification of states optimal for quantum rotation sensing.

Abstract

A constellation of $N=d-1$ Majorana stars represents an arbitrary pure quantum state of dimension $d$ or a permutation-symmetric state of a system consisting of $n$ qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of $k$ subsystems with $d$ levels each. For $d\geq 3$, such states are equivalent, as far as rotations are concerned, to a collection of various spin states, with definite relative complex weights. Following Majorana's lead, we introduce a multiconstellation, consisting of the Majorana constellations of the above spin states, augmented by an auxiliary, "spectator" constellation, encoding the complex weights. Examples of stellar representations of symmetric states of four qutrits, and two spin-$3/2$ systems, are presented. We revisit the Hermite and Murnaghan isomorphisms, which relate multipartite states of various spins, number of parties, and even symmetries. We show how the tools introduced can be used to analyze multipartite entanglement and to identify optimal quantum rotosensors, i.e., pure states which are maximally sensitive to rotations around a specified axis, or averaged over all axes.