User:Reinhard F Werner/Proposed/Bell's Theorem, Steering and Entanglement
Outline:
Following the classic 1935 article of Einstein Podolsky and Rosen, and Schrödinger's analysis of the same year we study quantum correlations in terms of the decomposition of a local state into conditional states according to a remote measurement. The core of the historic argument is what Schrödinger called _steering_, formalized as the statement that different such decompositions may not have a common refinement, which would be furnished by a hypothetical "true, although hidden" decomposition of the mixed local state into pure states.
The steering property relates to classical models for the quantum data, which are based on quantum detectors on one side. Making this assumption also on the other side, one comes from steering to the weaker notion of _entanglement_, taken as synonymous with non-separability in modern usage. If one drops such quantum assumptions altogether, one arrives at the setting of _Bell's Theorem_.
This shows in what sense Einstein's lifelong quest for local hidden variables was a rational research program, refuted only after his death by Bell, and identifies exactly the missing part of the argument. It is shown that by going outside the quantum description the classical modeling of Nature can indeed sometimes be restored. However, for states violating the standard (CHSH) Bell inequality this necessarily leads to non-causal signaling.
Here one might add a "hidden variable construction kit", showing that it easy to the point of triviality to produce non-local and "contextual" hidden variable models, even deterministic ones. In this context the status of Bohmian and Nelsonian mechanics is discussed.
Taking non-signalling seriously is the starting point for a hierarchy of impossible machines. These are hypothetical devices which must be impossible in any statistical theory which on the one hand describes Bell inequality violating correlations, and on the other avoids non-local signaling. Since this description includes quantum mechanics, these impossibility claims are also features of quantum theory. (1) Bell's telephone, the device using quantum correlations for signaling could be built from (2) a joint measuring device, which jointly determines the outcomes for both devices used in a CHSH-violation. This could be built from (3) a quantum copier (or cloning machine, or "broadcaster") taking one quantum input and producing two equivalent outputs. This could be built from (4) a faithful converter of quantum information to classical information and back.
The overall conclusion is that in a world with CHSH correlations but obeying causality, quantum information cannot be reduced to classical.
The article is accompanied by an appendix in which the salient mathematical background is briefly summarized and put in context (mostly by citation) .
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References: EPR, Schrödinger, Bell. Werner1989, Werner: Quantum information theory - An invitation, quant-ph/0101061 Recnt papers by Wiseman and Doherty
