How to downsample 4k to 1080p using ffmpeg while maintaining the quality?
The default settings for ffmpeg are very low quality, and since you don't specify any codec or quality parameters it's just using the defaults (I don't know why the devs don't fix that because it generates a lot of questions on forums everywhere).
Edit: the defaults are now quite sane. With a recent (as in later than 2017) build of ffmpeg you don't need to specify anything more than input and output files to achieve good useable results. You can, of course tweak to your heart's desire.
Try adding -c:v libx264 -crf 20 -preset slow to the command.

-c:v libx264 tells it to use the libx264 encoder, this is the default now, no need to specify it
crf 20 uses the Constant Rate Factor quantiser (which paradoxially means variable bit rate, but constant quality) with a value of 20 (pretty good quality; lower is better quality / larger files, higher is crappier / smaller)—default is 23,
the slow preset is a shortcut for a bunch of encoder settings that means it puts a bit more effort into it than the default (medium). Note that the speed presets don't change the quality of the encoded file, just the efficiency, meaning a slower encode will result in a smaller file of the same quality and a faster encode will mean a larger file of the same quality.

You can tweak these settings, see the h.264 encoding guide for instructions on what knobs to twiddle.
And if you're using the audio as-is, add c:a copy. That will do a straight copy of the audio stream without re-encoding.