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We consider a sensor network, where an access point (AP) communicates with many sensor nodes (SN), which are simple, cheap, low-complexity and low-power communication nodes. Such systems typically use nonlinear modulation and detection, due to their low power consumption. Increasing their performance by means of multiple antennas at the AP and the SNs has not been considered, since this would violate the stringent power and cost constraints at the SN. We consider SNs with MIMO receivers that perform a nonlinear operation on the complex-valued received signal (amplitude or phase detection). These receivers enjoy the low-cost, low-power, low-complexity characteristics that are crucial for a sensor network. Such nonlinear MIMO systems are first introduced and studied here. They bring the high-rate, high-performance world of MIMO systems and the low-cost, low-complexity world of sensor networks together. We only consider the single-user MIMO system between the AP and one SN, and study the fundamental limits of such systems. We compute achievable rates under perfect and noisy CSI at the SN, and observe that these systems also achieve spatial multiplexing gain, albeit different to legacy linear MIMO systems. We quantify and analyze these gains using numerical means, and give insight into the effect of the nonlinearity on the information theoretic limits of nonlinear MIMO systems.