  The multiple-input multiple-output (MIMO) detection problem, a fundamental
problem in modern digital communications, is to detect a vector of transmitted
symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood
detector can be formulated as a complex least-squares problem with discrete
variables, which is NP-hard in general. Various semidefinite relaxation (SDR)
methods have been proposed in the literature to solve the problem due to their
polynomial-time worst-case complexity and good detection error rate
performance. In this paper, we consider two popular classes of SDR-based
detectors and study the conditions under which the SDRs are tight and the
relationship between different SDR models. For the enhanced complex and real
SDRs proposed recently by Lu et al., we refine their analysis and derive the
necessary and sufficient condition for the complex SDR to be tight, as well as
a necessary condition for the real SDR to be tight. In contrast, we also show
that another SDR proposed by Mobasher et al. is not tight with high probability
under mild conditions. Moreover, we establish a general theorem that shows the
equivalence between two subsets of positive semidefinite matrices in different
dimensions by exploiting a special "separable" structure in the constraints.
Our theorem recovers two existing equivalence results of SDRs defined in
different settings and has the potential to find other applications due to its
generality.
