Stabilizer states constitute a set of pure states which plays a dominant role in quantum error correction, measurement-based quantum computation, and quantum communication. Central in these applications are the local symmetries of these states. We characterize all local, invertible (unitary and nonunitary) symmetries of arbitrary stabilizer states and provide an algorithm which determines them. We demonstrate the usefulness of these results by showing that the additional local symmetries find applications in entanglement theory and quantum error correction. More precisely, we study a central problem in entanglement theory, which is concerned with the existence of transformations via local operations among pure states.We demonstrate that the identified symmetries enable additional transformations from a stabilizer state to some other multipartite pure state. Furthermore, we demonstrate how the identified symmetries can be used to construct stabilizer codes with diagonal transversal gates.

We acknowledge preliminary research of D. Sauerwein and R. Brieger [41], especially their results on the transformations via SEP1 used in Sec. VB. Furthermore, we thank R. Raussendorf for pointing out the additional symmetries of the Reed-Muller codes. We acknowledge financial support from the Austrian Science Fund (FWF) Grant No. DK-ALM: W1259-N27 and the SFB BeyondC (Grant No. F7107-N38) and of the Austrian Academy of Sciences via the Innovation Fund "Research, Science and Society."