In this report, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this report, we first show that the conventional SDR is generically not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap.