Settling conjectures on the collapse of degrees of freedom under finite precision CSIT

A conjecture made by Lapidoth, Shamai and Wigger at Allerton 2005 (also an open problem presented at ITA 2006) states that the degrees of freedom (DoF) of a two user broadcast channel, where the transmitter is equipped with 2 antennas and each user is equipped with 1 antenna, must collapse under finite precision channel state information at the transmitter (CSIT). That this conjecture, which predates interference alignment, has remained unresolved, is emblematic of a pervasive lack of understanding of the degrees of freedom of wireless networks-including interference and X networks-under channel uncertainty at the transmitter(s). In this work we prove that the conjecture is true in all non-degenerate settings (e.g., where the probability density function of unknown channel coefficients exists and is bounded). The DoF collapse even when perfect channel knowledge for one user is available to the transmitter. This also settles a related recent conjecture by Tandon et al. Reminiscent of Korner and Marton´s work on the images of a set, the key to our proof is a bound on the number of codewords that can cast the same image (within noise distortion) at the undesired receiver, while remaining resolvable at the desired receiver. We are also able to generalize the result to arbitrary number of users, including the K user interference channel. Remarkably, for the K user interference channel, this work and the earlier work by Cadambe and Jafar reveal two contrasting sides of the same coin. Both works close a gap between the best previously known DoF inner bound of 1 and the best previously known DoF outer bound of K/2. However, while Cadambe and Jafar do so in the optimistic direction, showing that K/2 is optimal under perfect CSIT, here we close the gap in the pessimistic direction, showing that 1 DoF is optimal under finite precision CSIT.