Are you sure about the CRT type? I thought that Admiral 7" TVs of that vintage used a 7JP4 tube, not 7QP4. But there are plenty of things that I don't know
Do you mean that the thin white line stays at exactly the same place, even if you move the raster up or down in available screen area?
I thought that the classic "ion burn" mentioned by other responders made a darkened shape in the screen: either a horizontal line (created by failure of vertical deflection) or a central dark spot (created by a mis-adjusted ion trap magnet). But the line in the OP's photo is
lighter than the rest of the CRT face, not darker . . . .
Phil Nelson
Phil's Old Radios
https://antiqueradio.org/index.html