Maximum stabilizer dimension for nonproduct states

TL;DR

This paper investigates the structure of stabilizer subalgebras in n-qubit pure states, revealing the maximum dimension for nonproduct states and analyzing the special case of GHZ states.

Contribution

It determines the maximum stabilizer subalgebra dimension for nonproduct n-qubit states and characterizes the structure of states achieving this maximum.

Findings

Maximum stabilizer dimension is n-1 for nonproduct states of three or more qubits.

GHZ states attain the maximum stabilizer subalgebra dimension.

Examples show different stabilizer structures can achieve the maximum dimension.

Abstract

Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for n-qubit pure states, and find its maximum dimension to be n-1 for nonproduct states of three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: we show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the n-qubit GHZ state.