Multipartite entanglement arising from dense Euclidean lattices in dimensions 4–24

Abstract

The group of automorphisms of Euclidean (embedded in ) dense lattices such as the root lattices D4 and E8, the Barnes–Wall lattice BW16, the unimodular lattice D12+ and the Leech lattice Λ24 may be generated by entangled quantum gates of the corresponding dimension. These (real) gates/lattices are useful for quantum error correction: for instance, the two- and four-qubit real Clifford groups are the automorphism groups of the lattices D4 and BW16, respectively, and the three-qubit real Clifford group is maximal in the Weyl group W(E8). Technically, the automorphism group Aut(Λ) of the lattice Λ is the set of orthogonal matrices B such that, following the conjugation action by the generating matrix of the lattice, the output matrix is unimodular (of determinant ±1, with integer entries). When the degree n is equal to the number of basis elements of Λ, Aut(Λ) also acts on basis vectors and is generated with matrices B such that the sum of squared entries in a row is 1, i.e. B may be seen as a quantum gate. For the dense lattices listed above, maximal multipartite entanglement arises. In particular, one finds a balanced tripartite entanglement in E8 (the two- and three-tangles have the same magnitude 1/4) and a Greenberger–Horne–Zeilinger-type entanglement in BW16. In this paper, we also investigate the entangled gates from D12+ and Λ24, by seeing them as systems coupling a qutrit to two- and three-qubits, respectively. In addition to quantum computing, the work may be related to particle physics in the spirit of Planat et al (2011 Rep. Math. Phys.66 39–51).