Abstract
Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are “strings” connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.
Highlights
JHEP05(2019)116 expressed in terms of topological invariants of links in different three-dimensional manifolds
We show that in some cases, to be entangling, the cobordisms need to be endowed with an additional structure supported by Wilson lines
This description gives a graphical idea of quantum entanglement between two subsystems: they need to be tied by strings of Wilson line operators
Summary
Topological quantum field theories provide an interesting class of field theories with no local propagating degrees of freedom. We may consider a disjoint set of circles, which should be mapped to a direct product of Hilbert spaces. Since manifolds Σ are considered to be orientable, we associate with Σ , which differs from Σ by the choice of orientation, a dual vector space Z(Σ ) = V ∗. This allows us to define the scalar product in terms of topological spaces. We have defined some quantum mechanics (Hilbert space, states, scalar product and operators) in terms of topological spaces and cobordisms.. Witten in [17], where Z was the functional integral of a non-Abelian Chern-Simons theory in three dimensions
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