The performance of millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems is limited by the sparse nature of propagation channels and the restricted number of radio frequency (RF) chains at transceivers. The introduction of reconfigurable antennas offers an additional degree of freedom on designing mmWave MIMO systems. This paper provides a theoretical framework for studying the mmWave MIMO with reconfigurable antennas. Based on the virtual channel model, we present an architecture of reconfigurable mmWave MIMO with beamspace hybrid analog-digital beamformers and reconfigurable antennas at both the transmitter and the receiver. We show that employing reconfigurable antennas can provide throughput gain for the mmWave MIMO. We derive the expression for the average throughput gain of using reconfigurable antennas in the system, and further derive the expression for the outage throughput gain for the scenarios where the channels are (quasi) static. Moreover, we propose a low-complexity algorithm for reconfiguration state selection and beam selection. Our numerical results verify the derived expressions for the throughput gains and demonstrate the near-optimal throughput performance of the proposed low-complexity algorithm.