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# Universal Bell Correlations Do Not Exist

By Cole A. Graham and William M. Hoza

Abstract (for specialists)

We prove that there is no finite-alphabet nonlocal box that generates exactly those correlations that can be generated using a maximally entangled pair of qubits. More generally, we prove that if some finite-alphabet nonlocal box is strong enough to simulate arbitrary local projective measurements of a maximally entangled pair of qubits, then that nonlocal box cannot itself be simulated using any finite amount of entanglement. We also give a quantitative version of this theorem for approximate simulations, along with a corresponding positive result.

Not-so-abstract (for curious outsiders)

⚠️ This summary might gloss over some important details.

Bell's theorem says, roughly, that it follows from the laws of quantum mechanics that two parties can interact instantaneously across arbitrary distances. This phenomenon, called "quantum nonlocality", has been called "the most profound discovery of science". The nature of these "interactions" is notoriously subtle. For example, the no-communication theorem says that quantum nonlocality is not useful for sending genuine signals. In this paper, we rule out one possible approach to characterizing the exact extent of the "nonlocal powers" granted by the laws of quantum mechanics. In particular, aside from quantum nonlocality, another situation in which two parties can interact in a limited way is if there is some discrete, classical device that each party is able to interact with. We prove that quantum nonlocality is not quite equivalent to any such discrete, classical device.

Manuscript posted online in December 2016; appeared in PRL in August 2017. The PRL version (pdf) is much more compact than the earlier arXiv version. The arXiv version has essentially the same results, but it has more detailed definitions and proofs and some suggested open problems. The arXiv version also uses slightly different notation and is missing some references.

Expository material:

Slides from my presentation in Scott Aaronson's course "Topics in Quantum and Classical Complexity Theory" (December 2016).