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Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits

by Bruno Bertini, Pavel Kos, Tomaz Prosen

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Bruno Bertini · Pavel Kos
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-03-27
Date submitted: 2020-03-13 01:00
Submitted by: Bertini, Bruno
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously prove that if a circuit has an ultralocal soliton moving to the left (right), the entanglement of local operators initially supported on even (odd) sites saturates to a constant value and its dynamics can be computed exactly. Importantly, this does not bound the growth of complexity in chiral circuits, where solitons move only in one direction, say to the left. Indeed, in this case we observe numerically that operators on the odd sublattice have unbounded entanglement. Finally, we present a closed-form expression for the local-operator entanglement entropies in circuits with ultralocal solitons moving in both directions. Our results hold irrespectively of integrability.

Published as SciPost Phys. 8, 068 (2020)

Author comments upon resubmission

Dear Editor,

We thank the referee for her/his careful reading of our manuscript, for her/his positive assessment.

Here is a response to her/his only query.

"1: A precise comparison between the eigenvalues/eigenvector properties of transfer matrices yielding respectively linear behavior in I and log or constant behavior in II."

In the revised version of Paper I we now discuss how the “completely chaotic” class is incompatible with conservation laws, exhibiting exponentially many “additional” (to the x+1 given in Paper I) eigenvectors corresponding to eigenvalue 1 of the horizontal and vertical transfer matrices that one can construct in the presence of conserved quantities. Here (in the revised Section 3) we now refer to that discussion.

List of changes

- Sentence added after Eq 60

- Typos fixed in Eq 89 and 95

Reports on this Submission

Anonymous Report 1 on 2020-3-16 (Invited Report)


All questions suitably answered. Paper accepted for publication.

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