A dynamical algebra of protocol-induced transformations on Dicke states

TL;DR

This paper characterizes the algebraic structure of protocols transforming symmetric Dicke states in quantum information, revealing fixed points and connections to algebraic schemes, with implications for quantum state manipulation.

Contribution

It provides an algebraic framework for understanding protocol-induced transformations on Dicke states, including fixed points and connections to classical algebraic structures.

Findings

Algebraic characterization using Weyl algebra and su(2)

Explicit fixed points under protocol combinations

Connections to Hamming scheme, Hadamard transform, and Krawtchouk polynomials

Abstract

Quantum $n$-qubit states that are totally symmetric under the permutation of qubits are essential ingredients of important algorithms and applications in quantum information. Consequently, there is significant interest in developing methods to prepare and manipulate Dicke states, which form a basis for the subspace of fully symmetric states. Two simple protocols for transforming Dicke states are considered. An algebraic characterization of the operations that these protocols induce is obtained in terms of the Weyl algebra $W(2)$ and $\mathfrak{su}(2)$. Fixed points under the application of the combination of both protocols are explicitly determined. Connections with the binary Hamming scheme, the Hadamard transform, and Krawtchouk polynomials are highlighted.