  We consider the detection of binary (antipodal) signals transmitted in a
spatially multiplexed fashion over a fading multiple-input multiple-output
(MIMO) channel and where the detection is done by means of semidefinite
relaxation (SDR). The SDR detector is an attractive alternative to maximum
likelihood (ML) detection since the complexity is polynomial rather than
exponential. Assuming that the channel matrix is drawn with i.i.d. real valued
Gaussian entries, we study the receiver diversity and prove that the SDR
detector achieves the maximum possible diversity. Thus, the error probability
of the receiver tends to zero at the same rate as the optimal maximum
likelihood (ML) receiver in the high signal to noise ratio (SNR) limit. This
significantly strengthens previous performance guarantees available for the
semidefinite relaxation detector. Additionally, it proves that full diversity
detection is in certain scenarios also possible when using a non-combinatorial
receiver structure.
