Extendible quantum measurements and limitations on classical communication

Unextendibility of quantum states and channels is inextricably linked to the no-cloning theorem of quantum mechanics, it has played an important role in understanding and quantifying entanglement, and more recently it has found applications in providing limitations on quantum error correction and entanglement distillation. Here we generalize the framework of unextendibility to quantum measurements and define $k$-extendible measurements for every integer $k\ge 2$. Our definition provides a hierarchy of semidefinite constraints that specify a set of measurements containing every measurement that can be realized by local operations and one-way classical communication. Furthermore, the set of $k$-extendible measurements converges to the set of measurements that can be realized by local operations and one-way classical communication as $k\to \infty$. To illustrate the utility of $k$-extendible measurements, we establish a semidefinite programming upper bound on the one-shot classical capacity of a channel, which outperforms the best known efficiently computable bound from [Matthews and Wehner, IEEE Trans. Inf. Theory 60, pp. 7317-7329 (2014)] and also leads to efficiently computable upper bounds on the $n$-shot classical capacity of a channel.