A quantum linearity test for robustly verifying entanglement

We introduce a simple two-player test which certifies that the players apply tensor products of Pauli σX and σZ observables on the tensor product of n EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive ϵ of the optimal must be poly(ϵ)-close, in the appropriate distance measure, to the honest n-qubit strategy. The test involves 2n-bit questions and 2-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for n EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for verifiable delegated quantum computation with a constant number of quantum polynomial-time provers sharing entanglement.