Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators

TL;DR

This paper introduces a continuous-variable lattice model of quantum harmonic oscillators that allows for the detection of topological entanglement entropy through simple quadrature measurements, offering a practical approach to study topological order.

Contribution

It presents a novel continuous-variable analog of the surface code that enables measurement of topological entanglement entropy with only two-body interactions and simple measurements.

Findings

Topological entanglement entropy grows linearly with squeezing.

The topological mutual information is robust to certain errors.

Detection can be achieved via single-mode quadrature measurements.

Abstract

The Kitaev surface-code model is the most studied example of a topologically ordered phase and typically involves four-spin interactions on a two-dimensional surface. A universal signature of this phase is topological entanglement entropy (TEE), but due to low signal to noise, it is extremely difficult to observe in these systems, and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to reveal the order. We describe a continuous-variable analog to the surface code using quantum harmonic oscillators on a two-dimensional lattice, which has the distinctive property of needing only two-body nearest-neighbor interactions for its creation. Though such a model is gapless, satisfies an area law, and the ground state can be simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster state, which does not have topological order. Asymptotically, the TEE grows linearly with the squeezing parameter, and we show that its mixed-state generalization, the topological mutual information, is robust to some forms of state preparation error and can be detected simply using single-mode quadrature measurements. Finally, we discuss scalable implementation of these methods using optical and circuit-QED technology.