Rank property of the MIMO Gaussian wiretap channel with an average power constraint

This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. In particular, we study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the multiple antenna MIMO Gaussian wiretap channel under an average power constraint. The rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper. Such properties are useful steps towards characterizing the general solution to the MIMO wiretap problem with an average power constraint.