SchurWeyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
Abstract
SchurWeyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the tfold tensor powers U^{⊗t} of all unitaries U ∈U (d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of faulttolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that "stabilizerness" is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2021
 DOI:
 10.1007/s00220021041187
 arXiv:
 arXiv:1712.08628
 Bibcode:
 2021CMaPh.385.1325G
 Keywords:

 Quantum Physics;
 Mathematical Physics;
 Mathematics  Representation Theory
 EPrint:
 60 pages, 2 figures, accepted at Communications in Mathematical Physics