Quantum Information Primitives and Nonlocality without Entanglement

C. H. Bennett, D. P. DiVincenzo


Quantum information theory has provided a variety of primitive acts and consumable resources, such as the sending of a classical bit or qubit, the sharing of an EPR pair (or ebit), and the performance of an elementary gate operation such as XOR (controlled-NOT). Another resource, of a negative sort, is waste entropy that must be disposed of, for example the two unwanted bits left over at the end of teleportation. The remote-XOR (RXOR) is a positive resource consisting of the ability to perform a single XOR between a qubit of Alice's and a qubit of Bob's. (Imagine Bob and Alice are in love, but married to two other people. Then the ability to have an elementary private interaction would be valuable to them). Some circuits recently discovered by D. Gottesman relate the RXOR to other resources, for example a RXOR can be synthesized from an ebit plus a classical bit transmission in each direction. Generalizing the parardigm of communication complexity we ask ``what combinations of resources suffice to perform a specified initial-state to final-state transformation of a multipartite quantum system?'' In particular with Fuchs, Mor, Rains, Shor, Smolin, and Wootters (quant-ph/9804053), we have found a set of nine orthogonal product states of two 3-state particles that cannot be reliably distinguished by any sequence of local operations and classical communication. The states can, of course, be prepared locally from classical directions, but this preparation is necessarily irreversible, generating waste entropy. The proof of immeasurability of the nine states involves first showing that any bilocal processing can be made to occur continuously, i.e., as a sequence of arbitrarily small steps, and then showing that when, during such processing, one of the nine states' posterior probabilities rises significantly above 1/9 but still far from 1, then the nine residual states must be significantly non-orthogonal.