We consider entanglement swapping, a key component of quantum network operations and entanglement distribution. Pure entangled states, which are the desired input to the swapping protocol, are typically mixed by environmental interactions, causing a reduction in their degree of entanglement. Thus an understanding of entanglement swapping with partially mixed states is of importance. Here we present a general analytical solution for entanglement swapping of arbitrary two-qubit states. Our result provides a comprehensive method for analyzing entanglement swapping in quantum networks. First, we show that the concurrence of a partially mixed state is conserved when this state is swapped with a Bell state. Then, we find upper and lower bounds on the concurrence of the state resulting from entanglement swapping for various classes of input states. Finally, we determine a general relationship between the ranks of the initial states and the rank of the final state after swapping.
While there exist theories that have states more strongly entangled than quantum theory, in the sense that they show Clauser-Horne-Shimony-Holt (CHSH) values above Tsirelson's bound, all known examples of such theories have a strictly smaller set of measurements. Therefore, in tasks that require both bipartite states and measurements, they do not perform better than quantum mechanics. One of the simplest information processing tasks involving both bipartite states and measurements is that of entanglement swapping. In this paper, we study entanglement swapping in generalized probabilistic theories (GPTs). In particular, we introduce the iterated CHSH game, which measures the power of a GPT to preserve nonclassical correlations, in terms of the largest CHSH value obtainable after
n rounds of entanglement swapping. Our main result is the construction of a GPT that achieves a CHSH value of 4 after an arbitrary number of rounds. This addresses a question about the optimality of quantum theory for such games recently raised by Weilenmann and Colbeck. One challenge faced when treating this problem is that there seems to be no general framework for constructing GPTs in which entanglement swapping is a well-defined operation. Therefore, we introduce an algorithmic construction that turns a bipartite GPT into a multipartite GPT that supports entanglement swapping, if consistently possible.