Anyonic Partial Transpose I: Quantum Information Aspects

TL;DR

This paper extends the concept of partial transpose and logarithmic negativity to anyonic systems with non-Abelian statistics, providing a new entanglement measure applicable to fractional quantum statistics.

Contribution

It introduces a novel generalization of the partial transpose for anyons based on braiding, defining an anyonic logarithmic negativity that satisfies key entanglement criteria.

Findings

The anyonic logarithmic negativity is a valid entanglement measure.

Computed for toy models of anyon pairs, illustrating its properties.

Conjectured that states with zero negativity are measure zero in the space of anyonic states.

Abstract

A basic diagnostic of entanglement in mixed quantum states is known as the 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 was put forward. In this paper, we propose a way to generalize the partial transpose to anyons with (non-Abelian) fractional statistics based on the apparent similarity between the partial transpose and the braiding operation. We then define the anyonic version of the logarithmic negativity and show that it satisfies the standard requirements such as monotonicity to be an entanglement measure. In particular, we elucidate the properties of the anyonic logarithmic negativity by computing it for a toy density matrix of a pair of anyons within various categories. We conjecture that the subspace of states with a vanishing logarithmic negativity is a set of measure zero in the entire space of anyonic states, in contrast with the ordinary qubit systems where this subspace occupies a finite volume. We prove this conjecture for multiplicity-free categories.