Optimality of the Proper Gaussian Signal in Complex MIMO Wiretap Channels

The multiple-input multiple-output (MIMO) wiretap channel (WTC) serves as a fundamental model for exploring information-theoretic secrecy in wireless communication systems, involving a transmitter, a legitimate user, and an eavesdropper. This paper investigates the optimality of proper complex signals in complex WTCs. Our primary contribution lies in the derivation of a determinant inequality, which establishes that the secrecy rate of degraded complex MIMO WTCs is maximized when the signal is proper, meaning that its pseudo-covariance matrix is a zero matrix. Remarkably, we extend this result beyond the degraded scenario to the general complex WTC by leveraging a min-max reformulation of the secrecy capacity. Thus, we demonstrate that focusing on proper signals is sufficient when examining the secrecy capacity of the complex WTC. Overall, this work highlights the significance of the determinant inequality we derive and its implications for optimizing secrecy rates in the complex WTC.