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

TL;DR

This paper advances the understanding of the LU-LC conjecture in quantum information by reducing it to a problem involving quadratic forms over GF(2), linking local unitary operations to Clifford group relations.

Contribution

It reduces the LU-LC conjecture to a simpler case involving diagonal local unitaries and maps this to quadratic forms over GF(2), also extending results to stabilizer codes.

Findings

Reduction of LU-LC conjecture to diagonal local unitaries case

Mapping of the problem to quadratic forms over GF(2)

Implication for stabilizer codes

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 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.