Anyonic partial transpose

A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose and the corresponding entanglement measure is called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the partial transpose to fermionic systems remained a technical challenge until recently when a new definition that accounts for the Fermi statistics has been put forward. In this talk, I will present our attempts to generalize partial transpose to anyons with fractional statistics. It turns out that there is a connection between the partial transpose and braiding statistics. As an application, I will show how the braiding matrix can be used to reproduce many-body topological invariants of 1D time-reversal symmetric topological superconductors.