A converse to Lieb-Robinson bounds in one dimension using index theory

TL;DR

This paper extends index theory to approximately locality-preserving unitaries in 1D, proving that those with zero index can be generated by local Hamiltonians in constant time, providing a converse to Lieb-Robinson bounds.

Contribution

It demonstrates the robustness of index theory for ALPUs and establishes a converse to Lieb-Robinson bounds in one dimension.

Findings

Index theory extends to ALPUs with approximate locality.

Zero index ALPUs can be generated by local Hamiltonians in constant time.

Lieb-Robinson bound-satisfying unitaries can be realized by Hamiltonians in finite chains.

Abstract

Unitary dynamics with a strict causal cone (or "light cone") have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb-Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb-Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb-Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest.