  In this paper, we introduce the resource theory of unextendibility as a
relaxation of the resource theory of entanglement. The free states in this
resource theory are the k-extendible states, associated with the inability to
extend quantum entanglement in a given quantum state to multiple parties. The
free channels are k-extendible channels, which preserve the class of
k-extendible states. We define several quantifiers of unextendibility by means
of generalized divergences and establish their properties. By utilizing this
resource theory, we obtain non-asymptotic upper bounds on the rate at which
quantum communication or entanglement preservation is possible over a finite
number of uses of an arbitrary quantum channel assisted by k-extendible
channels at no cost. These bounds are significantly tighter than previously
known bounds for both the depolarizing and erasure channels. Finally, we
revisit the pretty strong converse for the quantum capacity of antidegradable
channels and establish an upper bound on the non-asymptotic quantum capacity of
these channels.
