The stabilizer for $n$-qubit symmetric states

TL;DR

This paper investigates the stabilizer groups of n-qubit symmetric states, showing nontrivial stabilizers for small n, providing examples of states with trivial stabilizers, and confirming that for large diversity numbers, the stabilizer is trivial.

Contribution

It characterizes when n-qubit symmetric states have trivial stabilizer groups, extending previous results to include states with high diversity numbers.

Findings

Stabilizer groups are nontrivial for n ≤ 4.

Constructs classes of symmetric states with trivial stabilizers.

States with diversity number > 5 have trivial stabilizers.

Abstract

The stabilizer group for an $n$-qubit state $\ket{\phi}$ is the set of all invertible local operators (ILO) $g=g_1\otimes g_2\otimes \cdots\otimes g_n,$ $ g_i\in \mathcal{GL}(2,\mathbb{C})$ such that $\ket{\phi}=g\ket{\phi}.$ Recently, G. Gour $et$ $al.$ \cite{GKW} presented that almost all $n$-qubit state $\ket{\psi}$ own a trivial stabilizer group when $n\ge 5.$ In this article, we consider the case when the stabilizer group of an $n$-qubit symmetric pure state $\ket{\psi}$ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state $\ket{\phi}$ is nontrivial when $n\le 4$. Then we present a class of $n$-qubit symmetric states $\ket{\phi}$ with the trivial stabilizer group. At last, we prove that an $n$-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5, which confirms the main result of \cite{GKW} partly.