Quoth Shaogo:
Hoagy Carmichael and Johnny Mercer wrote this great song for a movie. Sadly, I can't remember the name of the movie; it was about a girl, a guy, and a maniacal scientist but despite my best tries on web and at library I can't find the disc that attributes Sinatra's singing of the song to the movie.
You can find the download of "How Little We Know" peddled by a variety of vendors of music on the web. I prefer the Sinatra version, although Carmen McRae does it justice. Don't make a mistake and download the Vivian Leigh version, though; you might be disappointed. That song was recorded far fewer times than the Carmichael version.
There are two songs commonly known by the name How Little We Know. One, as shaogo mentioned, was written by Hoagy Carmichael. Its first movie appearance was in 1944 in To Have and Have Not and was sung by Lauren Bacall - this is the movie that featured Bacall's now-famous line,
"You know how to whistle, don't you, Steve? You just put your lips together and blow."
This is not, however, the song that Shaogo is thinking of - it was never recorded by Sinatra and never, so far as I can tell, appeared in a movie like the one he remembers.
There's another song, though, a song titled (How Little it Matters) How Little We Know written not by Carmichael but by Carolyn Leigh and Philip Springer - for reference, Leigh wrote lyrics for some of the songs in the Peter Pan musical, and Philip Springer wrote "Santa Baby". THIS is the song that Sinatra recorded and released as a single in 1956, though so far as I can tell it never appeared in a movie, or if it did I can't figure out when or where.
There is supposedly a text floating around library systems specifically about these two songs, a text I can't get my hands on because 25 bucks for 8 pages is a bit outrageous.
I'm gonna keep digging.
(after some more digging, I've found):
How Little We Know, (How Little It Matters)
m-Philip Springer, w-Carolyn Leigh
Pop Song: Frank Sinatra (1956)
04/05/56 Arr: Nelson Riddle
Capitol, Studio
CD: Complete Capital Singles
04/30/63 Arr: Nelson Riddle
Reprise, Studio
CD: Sinatra's Sinatra