An Operational Definition of Topological Order

Abstract

The unrivaled robustness of topologically ordered states of matter against perturbations has immediate applications in quantum computing and quantum metrology, yet their very existence poses a challenge to our understanding of phase transitions. In particular, topological phase transitions cannot be characterized in terms of local order parameters, as it is the case with conventional symmetry-breaking phase transitions. Currently, topological order is mostly discussed in the context of nonlocal topological invariants or indirect signatures like the topological entanglement entropy. However, a comprehensive understanding of what actually constitutes topological order is still lacking. Here we show that one can interpret topological order as the ability of a system to perform topological error correction. We find that this operational approach corresponding to a measurable observable does not only lay the conceptual foundations for previous classifications of topological order, but it can also be applied to hitherto inaccessible problems, such as the question of topological order for mixed quantum states arising in open quantum systems. We demonstrate the existence of topological order in open systems and their phase transitions to topologically trivial states, including topological criticality. Our results demonstrate the viability of topological order in nonequilibrium quantum systems and thus substantially broaden the scope of possible technological applications. We therefore expect our work to be a starting point for many future theoretical and experimental investigations, such as the application of our approach to fracton or Floquet topological order, or the direct experimental realization of the error correction protocol presented in our work for the development of future quantum technological devices.Comment: 7 pages, 5 figure