Quantum test for higher order Gowers norms

In this work, a quantum self-test which certifies that measurements of a quantum device have a large Gowers norm of order k is presented and analysed. The test protocol is described as a two-player quantum game, in which players provide answers based on measurements on subsystems of a maximally entangled bipartite state. The protocol makes use of 2^k+ 1 subtests. It is shown that strategies that succeed in the test with success rate 1− ϵ must have a Gowers norm larger than1−O(ϵ). This test generalises measurement tests that certify measurements based on the second order Gowers norm.