Nonlocal operations: Purification, storage, compression, tomography, and probabilistic implementation

We provide several applications of a previously introduced isomorphism between physical operations acting on two systems and entangled states [Phys. Rev. Lett. 86, 544 (2001)]. We show (i) how to implement (weakly) nonlocal two-qubit unitary operations with a small amount of entanglement; (ii) that a known, noisy, nonlocal unitary operation as well as an unknown, noisy, local unitary operation can be purified; (iii) how to perform the tomography of arbitrary, unknown, nonlocal operations; (iv) that a set of local unitary operations as well as a set of nonlocal unitary operations can be stored and compressed; and (v) how to implement probabilistically two-qubit gates for photons. We also show how to compress a set of bipartite entangled states locally, as well as how to implement certain nonlocal measurements using a small amount of entanglement. Finally, we generalize some of our results to multiparty systems.