How Dynamical Quantum Memories Forget

TL;DR

This paper investigates the long-term behavior of hybrid quantum dynamics involving unitaries and measurements, revealing different purification timescales for generic and free fermion systems, and implications for entanglement phases.

Contribution

It demonstrates that generic hybrid quantum systems purify exponentially fast, while free fermion systems purify quadratically, showing the limitations of volume law entanglement in free systems.

Findings

Generic systems purify in a time exponential in system size.

Free fermion systems purify in quadratic time.

Volume law entanglement cannot be sustained in free fermion systems.

Abstract

Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the "mixed" phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems -- i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears -- purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.