Topological view on entanglement and complexity

Abstract

Topological Quantum Field Theories are examples of quantum field theories with a discrete and even finite-dimensional Hilbert space. In this respect they are an intermediate step between quantum mechanics and quantum field theory. Such a special position allows one to study some non-trivial aspects of quantum field theories in more accessible and familiar quantum-mechanical terms. In particular, since the topological theories do not possess local dynamical degrees of freedom, one can study, in such theories, various interesting non-local correlations. Quantum entanglement is an example of non-classical, non-local correlation between parts of a quantum system. Description of entanglement in topological terms is a familiar idea in quantum information theory. In this note I will review some recent developments of this idea bearing explicit connections of entanglement with knots and their topological invariants. The new formulation implies a somewhat updated view on quantum computation. I will illustrate some new aspects by reviewing the notion of complexity.