No convoluted time travel logic is required to explain the events in Arrival.
The two keys to understanding the movie are that the alien language rewires the brain so it perceives all of time happening at once and the movie makers are playing with the chronology and linearity of the script (either to mess with the audience or to enhance the oddness of the alien perception of time).
Once you accept that your mind can see all of time at once, you no longer need any time travel to explain knowledge of the future. Louise knows the wife's dying words because of this, not because anyone has travelled in time.
But the director plays with the audience's understanding of time by unveiling the story in a non-linear sequence (which makes a point using movie technique about just how odd it is to see time this way). The family scenes with Louise and her daughter, for example, are seen at the start of the movie. The audience assumes they are in the past: they are not; they are from the future. It isn't clear whether the scene with Louise and the Chinese general at the book launch (which the audience assumes happens long after the main action) is a future scene or Louise's memory in the present of a future event (there are odd features in it about what she remembers from the past). I suspect the ambiguity is deliberate (again to highlight just how odd the alien perception of time is in terms that an observant cinema audience can feel and perceive).
All of this makes sense given the nature of the alien perception of time and none of it requires time travel. The director has masterfully used to art of cinema to muddle our normal perception of linear narrative to create some of the same oddness in our heads as an audience.