Almost all multipartite qubit quantum states have trivial stabilizer

TL;DR

This paper proves that for five or more qubits, almost all multipartite entangled states have trivial stabilizers, implying they are isolated and cannot be converted into other states via deterministic LOCC, with implications for quantum state transformations.

Contribution

The paper demonstrates that almost all n-qubit states with n>4 have trivial stabilizers, showing they are isolated and cannot be transformed into other states via deterministic LOCC.

Findings

Almost all multipartite entangled states with n>4 have trivial stabilizers.

Deterministic LOCC can almost always be simulated by local unitaries for these states.

Provides a simple expression for the maximal conversion probability between states.

Abstract

The stabilizer group of an n-qubit state \psi is the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,...,g_n being any 2x2 invertible complex matrices, that satisfy g\psi=\psi. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial; that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves that almost all n-qubit states with n>4 are isolated, that is they can neither be reached nor converted into any other (n-partite entangled), LU-inequivalent state via deterministic LOCC. We also find a simple and elegant expression for the maximal probability to convert one multi-qubit entangled state to another for this generic set of states.