Entanglement and interaction in a topological quantum walk

Response to the referee Report 1:

We added two paragraphs to address the questions about the symmetry of the interaction and the self-similar form of the probability in (24) (now equation 25).

We revised the notation, figure captions and errors.

* (p4. before section 3) Dependence of the dynamics on the symmetry of the initial state:

> In summary, the model consists in two particles moving in a one dimensional lattice with an interface at the origin; their evolution is ensured by a one time step operator $U$, composed of a coin (defined with different rotation angles in the two regions), which couples to the shift operator for each particle, and a local on-site interaction between the two particles. Although the interaction is spin independent, the global unitary operator cannot be splitted into a spin dependent part and a position dependent part. Coin and shift entangle spin and position in the one particle sector, and the interaction the positions in the two particle sector; as a consequence, the interaction can modify differently the evolution of a given initial condition depending on its symmetry: symmetric and antisymmetric Bell states \eqref{e:bell}. In other words, $U = V \,(U_0 \otimes U_0)$ entangles spin and position ($U_0$) of each particle and the two particle positions ($V$), resulting in entanglement of all degrees of freedom labeled by the basis vectors $|x_1s_1x_2s_2\rangle$ of the Hilbert space.

* (p.11 after the equation for $S_0$) About equation (24) (now 25); we also changed the notation $p_E$ to $p_S$ to avoid confusion with the energy $E$:

> The self-similar form \eqref{e:pe} is borrowed from the one particle quantum walk, for which the ballistic law \(x \sim t\) applies \cite{Nayak-2000qv,Venegas-Andraca-2012zr}; this is certainly appropriated when the probability distribution is dominated by the motion of the bounded state of the two particles, and then we have an effective one particle walk, or in the other limit, when there is repulsion (as in the antisymmetric case) and the two particles are almost independent. The fact that the characteristic exponent varies with the interaction can be related to the leak of one particle position probability in correlations with the other degrees of freedom: the effective motion of one particle is no more ballistic.

### Notation, figure captions, errors, etc.

In response to the inconsistencies in the notation and mistakes in the formulas or figure captions, we addressed the following points:

1. We changed the text and the figure caption of Fig. 2 to improve its presentation.
2. We changed the text about the definition of the left and right regions, and added when necessary reference to $\theta$.
3. Corrected (definition of $\Lambda$ in caption 6.) Modification of the text to improve the reading.
4. We changed $\omega$ by $E$ (we used omega in the by hand computation with sympy...)
5. We changed the text before (22) (former 21) to clarify the notation and the definition of the reduced density matrix
6. Corrected: $P_S(x/\log t)$
7. We completed the explanation of the code (cbgi) by an example, and slightly changed the notation of the labels.

### Other modifications

We added a reference to the paper of Bisio et al. who computed the spectrum of a similar operator $U$.

We added references to one particle quantum walks in relation with the discussion of the self-similar probability form.

We corrected errors in the text and expanded some comments on the figures in the main text.