Would a symmetric cipher with a keylength a big as the data length be information theoretically secure?
No, using any keyed permutation with key length equal to the block size reduces the number of possible plain texts by ~1/3

Assume first that AES reasonably approximates a pseudorandom permutation, for each block of the message, the attacker (assuming unbounded computing power) can calculate the plain text for all keys. This is equivalent to throwing 2^128 balls into 2^128 baskets randomly. There will be collisions. The chance that any plain text is not a valid solution is approximately 1/e 

filled fraction = 1-((n-1)/n)^n , n=2^128

this converges to 1-1/e

this reduces the effective number of plaintexts by  about 1/3 (=0.63 ~= -0.66 bits). The reduction also tells the adversary about the probablity of any ciphertext being correct since about half the remaining plain texts got two hits instead of one with 2x the probability of being correct.

A problem with this scheme is that fooling the observer into accepting a plausible fake message requires solving for the key K which would connect the plaintexts and ciphertexts, fortunately, thanks to the recent SHA3 competition the cryptanalysis to do so has been mostly done.