Congratulations on solving a beautiful problem. This is bringing back old memories ; I still remember first discussing this problem with Sergio Verdu when he was on sabbatical at Berkeley in the late 1990's . With a wink of an eye it's now 25 years later.
Some perspective on this problem from someone who has been around. This problem belongs to the class of problems called "multiuser or MIMO detection", pioneered by Sergio Verdu and received a lot of attention in the late 80's and early 90's, In the late 90's and early 2000's, there was a resurgence of interest in this problem via connections with random matrix theory. I myself have worked on a bunch of these problems, evaluating performance of different MIMO detectors. The problem that AI and @DimitrisPapai
have solved is the ultimate of these problems because the optimal MIMO detector is the joint ML detector. I am very happy to see it solved.
That being said, I have to say that MIMO detection as a whole has made little impact on how actual wireless communication systems are built. Because wireless communication systems, like all communication systems, use coding, while MIMO detection only talks about uncoded systems; the N information bits here in this problem are uncoded. It turns out that much lower SNR can be achieved with coding. Take this problem as an example, with coding, a constant SNR* = 1.32 is sufficient to support an information rate of 1 bit per antenna, in contrast to the growing SNR = 2 log N result just proved. That's one reason why although this problem is beautiful as a mathematical problem, it has not received the full attention of the information theory community even in those days of peak activity. Claude Shannon has already told us in his 1948 information theory paper: to achieve the ultimate limit of communication, one needs to do coding. Uncoded systems are sub-optimal. So SNR = 2 log N is only optimal among uncoded communication systems, but SNR* = 1.32 is optimal among all communication systems.
The problem that Claude Shannon solved in his 1948 paper was the capacity of point to point communication, The problem here is one example (so called point to point MIMO channel), and hence the optimal SNR* = 1.32 can already be computed by Shannon's 1948 theory (no AI needed). Most of the information theorists' attention in the hey days of wireless communications research has been to extend Shannon's theory to network information theory problems, such as the broadcast channel, the relay channel and the interference channels. In those problems , one ask what is the best information rate (or equivalently, the minimum SNR needed) that a network of users can communicate with each other. These problems have been proposed since the 60's and the 70's and to this day none have been solved in their full generality. I'd love to see AI take a crack at some of those open problems.