Boundary Chaos: Exact Entanglement Dynamics

Dear Editor,

we thank the Referees for carefully evaluating our manuscript, for their positive assesment, and their suggestions. We comment on their remarks in detail below.

Reply to Referee 1, Pieter Claeys:

1) The Referee points out an inconsistency of Eq. (21) and asks for the connections with the derivation cited in Ref. [65].

The permutation defined in (new) Eq. (21) differs from the cited reference for the following reason. The impurity interaction acts in the second layer of the circuit which generates the evolution of states in the Schrödinger picture whereas the folded impurity acts in the first layer of the super-circuit generating the evolution of operators in the Heisenberg picture. This leads to slight differences in the definition of the permutations defined in Eq. (21) and (64), respectively. In particular Eq. (21) does not correspond to Ref. [65] where the evolution of operators is described. We added a corresponding clarifying comment below the respective equations and moreover provide a simpler definition of the permutations $\sigma_\delta$.

2) The Referee asks for the entanglement dynamics, when the impurity interaction both supports a vacuum state and is T-dual.

In the minimal case of qubits demanding both T-duality and the existence of a vacuum state forces the impurity interaction to be of the form $\ket{0}\bra{0} \otimes v + \ket{0}\bra{0} \otimes w$ with diagonal single-qubit gate v and arbitrary single-qubit gate w (or with the two qubits swapped). This does not lead to entangling dynamics for the initial product states considered. For larger local Hilbert space dimension there are examples leading to non-trivial entanglement dynamics. A simple example is given by folded T-dual gates, for which the vectorized identity deals as a vacuum state. This however, is a special example, as in this case the vacuum state is also conserved upon evolution in "spatial" direction (of the gates entering the construction of the transfer matrices). Whether there are T-dual gates (for $q>2$) with vacuum state, which do not originate from such a folding procedure and which give rise to non-trivial entanglement dynamics is an open question. We now mention this after introducing T-dual gates, i.e., below Eq. (43).

3) The Referee criticizes that the discussion of the entanglement dynamics of states from generic impurity interactions is too brief.

We expanded the discussion of the entanglement dynamics for this situation, including a more detailed description of the origin of the plateaus and the subleading contributions as well as comments on different R\'enyi orders $n$. We find qualitatively similar behavior as shown in Fig. 3 with positive probability, when sampling the impurity interaction Haar random. Similarly, we also find gates, for which the subleading contribution is essentially irrelevant, with (larger) positive probability, which we now also mention in the text

Moreover, the Referee asks for the fate of the exact results upon perturbing away from T-duality or the existence of a vacuum state.

For the latter case one can obtain a rough picture as follows: For an impurity which supports an vacuum state the leading part of the transfer matrices' spectra lead to the reduced density matrix being a pure state. Perturbing away from such gates with vacuum states will cause the contribution from the leading eigenvalue to slightly differ from a pure states and hence yields non-zero entanglement thereby setting the value of $R_n$ on the plateaus. The subleading contributions, which encode the full entanglement dynamics in the unperturbed case (gates with vacuum), still give an important contribution upon small perturbation. However, they now add to non-zero entanglement entropy of the leading contribution ('on top of the plateaus') instead of the zero leading entanglement entropy in the unperturbed case. In this sense, the entanglement entropies depicted in Fig. 3, in which subleading contributions are clearly visible, indicates some proximity of the chosen gate to gates with vacuum state. We incorporated this picture into the text, when explaining the observed entanglement dynamics from generic impurities qualitatively.

In contrast, the lack of control over the subleading part of the spectrum and the associated eigenvectors of transfer matrices does not allow for a similar picture for perturbations of T-dual gates both for the entanglement dynamics of states and operators.

Additionally, as suggested by the Referee, we added Appendix C, in which we list all the impurity interactions used for numerical computations.

4) The Referee points out that the transfer matrices in the tensor network (25) correspond to the rows rather than the columns.

We changed the description of the network accordingly.

5) The Referee asks about numerical results for R\'enyi entropies of higher order $n>2$.

We have checked different orders $n$ for the R\'enyi entropies numerically. For the case of gates with vacuum and states as well as generic gates and operators, the entanglement due to the leading eigenvalues of transfer matrices is that of a pure state and hence independent from $n$. Similarly the exponential scaling with $\delta$ of subleading terms is independent from $n$. The prefactor (which we cannot compute explicitly) does weakly depend on $n$. For both operators and states in the T-dual case as well as for states in the generic case we numerically find almost no differences for different $n$. In particular the value of the plateaus for fixed $\tau>0$ does depend only very weekly on $n$. Subleading contributions are qualitatively similar for different $n$ but our analytical methods do not allow for quantifying the subleading contributions. We now comment on this when discussing numerical results, i.e., below Eq. (41) and (61) as well as in the extended Sec. 3.2.3. for states and below Eq. (69) and (109) for operators.

6) We corrected the typos pointed out by the Referee.

Reply to Referee 2:

1-3) As suggested by the Referee, we corrected the typos pointed out by the Referee, stated the values for $L$, $l$ and $t$ below Eq. (25), and explicitly mention the $\tau$ dependence of the subleading eigenvalue $\lambda_0$ below Eq. (40).

4) The Referee asks for the reasons to choose the interaction parameter $J=\pi/4 - 0.05$, when discussing T-dual gates.

This is done to ensure that the system is chaotic. In fact the point $J=\pi/4$ for qubits corresponds to the most chaotic gates (in the sense of Ref [73]). We choose $J$ slightly away from this point, to be in a more generic, but still chaotic regime. We now state this below Eq. (42) and in the caption of Fig. 2.

1) We reformulated Eqs. (21) and (64) and comment on the origin of their differences.

2) We add discussion on impurity interactions which are both T-dual and support a vacuum state below Eq. (43)

3) We expand the discussion in Sec. 3.2.3. on the entanglement dynamics for generic impurity interactions.

4) We changed 'columns' to 'rows' in the description of the network (25) and added the concrete values of $L$, $l$, and $t$.

5) We added comments on numerical results for R\'enyi entropies of order $n>2$ below Eq. (41) and (61) as well as in the extended Sec. 3.2.3. for states and below Eq. (69) and (109) for operators

6) We explicitly mention the $\tau$ dependence of subleading eigenvalues of transfer matrices below Eq. (40)

7) We comment on the choice of the parameter $J$ in Eq. (42) below the equation and in the caption of Fig. 2

8) We corrected the typos pointed out by the Referees

9) We added Appendix C, in which we list all impurity interactions used for numerical computations.