Citation: Kitaev, Alexei, Dominic Mayers, and John Preskill. "Superselection rules and quantum protocols." Physical Review A—Atomic, Molecular, and Optical Physics 69.5 (2004): 052326.
Web: https://arxiv.org/abs/quant-ph/0310088
Tags: Computer-scientific, Quantum-advantage
In this lovely little paper, the authors examine the security of quantum information protocols under the presence of superselection sectors. Superselection sectors are a general concept in physics that come from conserved quantities - there is some physical principle that means you cannot change a conserved quantity, such as the representation under some global symmetry action or the global topological charge under some anyon theory. These place limitations on what sort of quantum operations are possible for people to physically perform. We know that a theory with superselection sectors is computationally equivalent to a theory without superselection sectors, but it is not immediately clear whether or not they are cryptographically equivalent. For instance, the quantum bit commitment problem is known to not have a solution for standard quantum systems but it could have a solution in the presence of ancillas. This question was asked to the authors by Popescu, and they gave a partial resolution. They showed that in a variety of cases, superselection sectors do not add any level of crytographic protection (namely, they do this very generally for Lie group symmetries, and for two-party protocols for more general superselection rules).
The results of this paper are subtle, and brush up close to some false results. In particular, if the cheaters are not allowed to use ancillas then using the superselection rules of non-abelian anyons there is a protocol for quantum bit commitment. This is in contrast to the case of abelian superselection (or no superselection) in which case not being allowed to use ancillas is not a restriction:
> DiVincenzo, David P., John A. Smolin, and Barbara M. Terhal. "Security trade-offs in ancilla-free quantum bit commitment in the presence of superselection rules." New Journal of Physics 6.1 (2004): 80.