The LU-LC conjecture, diagonal local operations and quadratic forms over GF(2)

Abstract

We report progress on the LU-LC conjecture---an open problem in the context of entanglement in stabilizer states (or graph states). This conjecture states that every two stabilizer states which are related by a local unitary operation, must also be related by a local operation within the Clifford group. The contribution of this paper is a reduction of the LU-LC conjecture to a simpler problem---which, however, remains to date unsolved. As our main result, we show that, if the LU-LC conjecture could be proved for the restricted case of \emph{diagonal} local unitary operations, then the conjecture is correct in its totality. Furthermore, the reduced version of the problem, involving such diagonal local operations, is mapped to questions regarding quadratic forms over the finite field GF(2). Finally, we prove that correctness of the LU-LC conjecture for stabilizer states implies a similar result for the more general case of stabilizer codes.