We derive new cut-set bounds on the achievable rates in a general multi-terminal network with finite number of states. Multiple states are common in communication networks in the form of multiple channel and nodes' states. Our results are broadly applicable and provide much tighter upper bounds than the known single state min-cut max-flow theorem, and hence form an important new tool to bound the performance of multi-node networks.

Two examples are presented to illustrate the tightness and the utility of the new bounds. In each of the example applications, the known single-state max-flow min-cut theorem provides a bound strictly looser than the new cut-set bounds. The first illustrative example is single-user compound channels, where both the transmitter and receiver have channel state information. The example of compound channels represents the smallest possible network with only two nodes, but has multiple states due to channel variations. The upper bound derived using the proposed bounds turns out to be the capacity of the compound channel, which implies that the bound is tight in this case. The second example is from a contemporary network problem. Here, we demonstrate the application of new bounds to characterize the limits on rate of information transfer in `cheap' relay networks, where the relay nodes can either transmit or receive, but not both simultaneously. In this case, each constituent channel has a single state but relay nodes can be in one of the two states, transmit or receive mode, giving rise to multiple network states. Here, again, the upper bound coincide with the capacity of the channel if the relay channel is degraded.