Inseparability of the multiple access wiretap channel

We examine the separability of the parallel multiple access wiretap channel. Separability, when exists, is useful as it enables us to code separately over parallel channels, and still achieve the optimum overall performance. It is well-known that the parallel single-user channel, parallel multiple access channel (MAC) and parallel broadcast channel (BC) are all separable, however, the parallel interference channel (IC) is not separable in general. In this paper, we show that, while MAC is separable MAC wiretap channel is not separable in general. We prove this via a specific linear deterministic MAC wiretap channel. We then show that even the Gaussian MAC wiretap channel is inseparable in general. Finally, we show that, when the channel gains are drawn from continuous distributions, and when the secure degrees of freedom (s.d.o.f.) region is considered, then the Gaussian MAC wiretap channel is almost surely separable.