A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory

Ranard, Daniel; Walter, Michael; Witteveen, Freek

doi:10.1007/s00023-022-01193-x

A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory

Journal Article · 26 July 2022 · Annales Henri Poincare

DOI:https://doi.org/10.1007/s00023-022-01193-x· OSTI ID:2421201

Ranard, Daniel ^[1]; Walter, Michael;

OSTI

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.

Research Organization:: Univ. of California, Oakland, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

DOE Contract Number:: SC0019380

OSTI ID:: 2421201

Journal Information:: Annales Henri Poincare, Journal Name: Annales Henri Poincare Journal Issue: 11 Vol. 23; ISSN 1424-0637

Publisher:: Springer

Country of Publication:: United States

Language:: English

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