A converse to Lieb-Robinson bounds in one dimension using index theory
- URL: http://arxiv.org/abs/2012.00741v2
- Date: Tue, 15 Mar 2022 08:34:31 GMT
- Title: A converse to Lieb-Robinson bounds in one dimension using index theory
- Authors: Daniel Ranard, Michael Walter, Freek Witteveen
- Abstract summary: Unitary dynamics with a strict causal cone (or "light cone") have been studied extensively, under the name of quantum cellular automata (QCAs)
We show that the index theory is robust and completely extends to one-dimensional ALPUs.
For the special case of finite chains with open boundaries, any unitary satisfying the Lieb-Robinson bound may be generated by such a Hamiltonian.
- Score: 1.3807918535446089
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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