Quantum communication on closed time-like curves

We define a new quantity we call a ctcbit that provides a means for quantifying a qubit on a closed time-like curve (CTC) as a shared resource. We describe a simple protocol for the sharing of information that is similar to quantum teleportation but does not require an entangled particle pair or ebit. While there is the appearance that the given resource is free, we employ a non-Hausdorff topology to prevent any limitless information exchanges. Nonetheless, the resource does limit the type of information that may be transferred, though this limitation can be reduced through the use of weak measurements. While the reality of CTCs is highly speculative, the present paper provides a manner by which quantum informational methods may be employed to study such problems and may ultimately prove useful in studying quantum gravity.