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In this paper, 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 independent identically distributed (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 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 also possible in certain scenarios when using a noncombinatorial receiver structure.