Entanglement and interaction in a topological quantum walk

1) Interesting discussion about edge states in a topological quantum walk.
2) Detailed study of localization in relation to interaction and topology.

1) The effect of the symmetry of the initial state is not clear to me, since the interaction is spin-independent.
2) The authors have not been careful with notations, specially in the figure captions.

The authors study the quantum walk of two interacting particles on a line with an interface separating two topologically distinct regions, and analyze the localization and entanglement properties of the edge state as a function of the interaction strength and topology interface.

In my opinion, the paper is interesting. However, there is a couple of questions that should be addressed before I can recommend publication.

- The interaction operator Eq. (6) is spin independent (this equation can in fact be written in terms of position projectors only). Then, it is not intuitive how the dynamics can depend on whether the initial state is spin-symmetric or antisymmetric, while keeping the rest of parameters unchanged. The authors should clarify this point in connection with Figs. (5) and (6).

- The authors should give a justification for Eq. (24).

The authors have not been very careful about the notations introduced in the paper, specially in the figure captions. Here are some comments:

- Regarding Fig. 2: What represent black dots in the upper panel? Are there quasienergies that do not correspond to eigenvalues of P? The discussion about this figure is confusing.

- The authors introduce a position dependence of \theta_+(x) characterized by \theta_L and \theta_R. Is this dependence kept throughout the rest of the paper? For example, in Fig. 2 they write specific values for \theta_+= and \theta_.

- Same for Fig. 4. Also, the definition of \Lambda in the caption does not agree with Eq. (16).

- In Fig. 6: What is \theta? The caption of this figure has to be carefully checked. Is this figure calculated for a specific value of N?

- How is \omega appearing after Eq. (15) related to E?

- The introduction of the reduced density matrix \rho_x(t) in Eq. (21) is confusing. What does it mean \bar{x}=(c_1,x_2,c_2)?

- The use of P_E(x,log t) before Eq. (26) is misleading.

- The notation (c,b,g,i) used to label different sets of parameters is not transparent. I recommend using a more explicit notation.

1) Clarify the role of symmetry of the initial state.
2) Check notations.