A2/The Bell theorem
We are now ready to formulate the quintessence of our story, namely the Bell theorem. This theorem states, that three properties of any theory cannot be true at the same time:
(E) The Bell inequality is violated in an experiment.
(C) The theory admits a classical description. This means, that any individual system can be described with a series of (possibly hidden) parameters.
(L) Locality holds in the theory. This means, that any action performed on Alice’s system can only travel with the speed of light and cannot influence Bob’s system instantaneously.
Connection with the story
In the story, Bob derived the inequality assuming C and K, concluding that E cannot hold. Our discussion of the Bell telephone assumed C and E, where L was violated and superluminal signalling was possible. In quantum theory the properties E and L hold, thus C cannot. This implies that there are certain quantities whose behaviour is in general not explainable with a local hidden variable model.
No classical translation
A consequence of the theorem is, that there is a genuine distinction between classical and quantum information. In classical mechanics it is in principle possible to extract all information about a given system in a single measurement. Even though it would be potentially hard to perform in practice, there are no limits on how much information can be extracted from a system. In quantum physics, such a measurement cannot exist, as this would allow superluminal signalling and violate the principle of relativity.
As quantum physics is not compatible with the idea of local hidden variables, the statistical behaviour of quantum measurements cannot be only the result of ignorance, but must be objective randomness. This is in contrast to classical physics, which is a deterministic theory and where all randomness is a result of ignorance about initial parameters of a system. For instance would the behaviour of any classical gas be predictable for all times, once the position and momenta of all particles constituting the gas were known. For quantum physics this especially implies, that the observed randomness is not a result of an interaction with a somehow “clumsy” measurement apparatus.
In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen published a paper on quantum correlations, which is today famous as the EPR-paper, in which they questioned some of the at that time orthodox interpretations of quantum physics. They asked the question, in which sense the wave function can be considered a complete description of a single quantum particle. In modern terminology this interpretation would simply say, that the wave function is a hidden variable for the behaviour of the system. They argued, that certain quantum states (similar to the one described in the story) would allow one party (say Alice) to make predictions for measurement results of a second party (say Bob) for of either position or momentum, even when they are far apart. EPR used this observation to argue, that the wave function itself was not a good candidate for an individual description of quantum particles. They also speculated, whether there might be a possibility to find a better, i.e., more complete, description of physics using new parameters. It was not before the work of Bell, that this idea of enriching quantum physics could be ruled out.
One should note here, that one crucial assumption in the EPR argument was, that a description of individual particles should be found. If one considers the wave function as a description of an ensemble of particles, no such contradiction arises.
The notion of entanglement
As a result of the EPR-paper, Schrödinger wrote a series of articles dedicated to fundamental aspects of quantum physics. In these he coined the term “entanglement” for the special property of quantum systems that was described in the EPR-paper. Also the correlations encountered in our story can only arise with entangled quantum systems. Today, we call two systems entangled if the correlations between them cannot be explained by preparing individual quantum states in any (possibly randomized way) and sending them to Alice and Bob. States that obey such a model are also called “classically correlated” or “separable”. The idea of this definition is similar to the story: Bob assumed that the lottery tickets were drawn at random from the bag but permitted an individual description. If this had been true, the lottery tickets would not have been able to surprise him.