From topological to quantum entanglement

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. arXiv:1809.04574v1 [hep-th]

Entanglement on multiple S2 boundaries in Chern-Simons theory

“…Thus we get the maximum entanglement entropy: EE = ln dim(H A ), which is consistent with the result of [21] that the quantum state |α 1 is a maximally entangled or Bell state. This procedure of tracing out Hilbert space can be generalized for multiple S 2 boundaries.…”

“…Further if any of the t appears with multiplicity, the reduced density matrix is entangled and the states will have a W-like structure. As remarked earlier, the authors of [21] have studied the entanglement structure on two disjoint S 2 ∪ S 2 boundaries. Basically, they invoke the replica trick to obtain the entropy for the states in H S 2 ⊗ H S 2 .…”

“…The entanglement between two S 2 boundaries related by cobordism [21] gives a nice pictorial understanding on the entanglement entropy. The qualitative remarks given in [21] on the entanglement in the presence of the Wilson lines in Chern-Simons theory creating punctured S 2 boundaries needs to be quantified.…”

“…In this work, we will systematically perform the calculations of the entanglement entropy by explicitly writing the pure quantum state |Ψ on n copies of S 2 . The total Hilbert space can be bi-partitioned into two parts as: Figure 3: Some of the entanglement results presented in [21] using the replica trick. States (a) and (b) are maximally entangled and hence the Bell states.…”

“…Note that this entanglement entropy for two S 2 boundaries is the logarithm of the Chern-Simons partition function on S 2 ×S 1 in the presence of Wilson lines carrying representations R 1 , R 2 , R 3 and R 4 [21]. More precisely, we have:…”