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In this paper, we address spatially multiplexed (SM) multiple-input-multiple-output (MIMO) systems that are expected to be used in next-generation mobile communication systems that require high-speed data transmission. The data-transmission rate of an SM MIMO system increases in proportion to the number of spatial streams it transmits. However, it is hard to implement large numbers of spatial streams due to the spatial limits of mobile units and the high cost of installing many antennas. In this paper, we focus on the case of two spatial streams and propose a novel signal-detection technique. We first define the bidirectional detectability of a 2-D complex vector, and using the fact that only a small number of vectors, including the maximum-likelihood (ML) solution vector, satisfy the bidirectional detectability, we reduce the search space, thereby enabling efficient ML signal detection at the receiver.