A near maximum a posteriori (MAP)-optimal soft detector that outputs a posteriori probabilities (APP's) for multiple-input multiple-output (MIMO) systems in flat fading channels is proposed. This is referred to as 'reduced state maximum a posteriori (RSMAP)' algorithm. This detection algorithm is based on BCJR algorithm and also uses ideas from reduced state sequence estimation (RSSE) and set partitioning. This algorithm is shown to be near optimal and the computational complexity to implement it is estimated. Finally we also show that applying the well known max-log approximation on this algorithm results in nearly same performance at much lower complexity. © 2012 IEEE.