Quantum Nonlocality - Superquantum Nonlocality

Superquantum Nonlocality

Whilst the CHSH inequality gives restrictions on the CHSH value attainable by local hidden variable theories, the rules of quantum theory do not allow us to violate Tsirelson's bound of, even if we exploit measurements of entangled particles. The question remained whether this was the maximum CHSH value that can be attained without explicitly allowing instantaneous signaling. In 1994 two physicists, Sandu Popescu and Daniel Rohrlich, formulated an explicit set of non-signalling correlated measurements that give : the algebraic maximum. This demonstrated that there are apparently reasonable theories of parts of Nature that drastically violate the predictions of quantum theory. The attempt to understand what uniquely identifies quantum theory from such general theories motivated an abstraction from physical measurements of nonlocality, to the study of nonlocal boxes.

Nonlocal boxes generalize the concept of experimentalists making joint measurements from separate locations. As in the discussion above, the choice of measurement is encoded by the input to the box. A two-party nonlocal box takes an input A from Alice and an input B from Bob, and outputs two values a and b for Alice and Bob respectively and separately, where a, b, A and B take values from some finite alphabet (normally ). The box is characterized by the probability of outputting pair a, b, given the inputs A, B. This probability is denoted and obeys the normal probabilistic conditions of positivity and normalisation:

and

A box is local, or admits a local hidden variable model, if its output probabilities can be characterized in the following way:

where and describe single input/output probabilities at Alice's or Bob's system alone, and the value of is chosen at random according to some fixed probability distribution given by . Intuitively, corresponds to a hidden variable, or to a shared randomness between Alice and Bob. If a box violates this condition, it is explicitly nonlocal. However, the study of nonlocal boxes often also encapsulates local boxes.

The set of nonlocal boxes most commonly studied are the so-called non-signalling boxes, for which neither Alice nor Bob can signal their choice of input to the other. Physically, this is a reasonable restriction: setting the input is physically analogous to making a measurement, which should effectively provide a result immediately. Since there may be a large spatial separation between the parties, signalling to Bob would potentially require considerable time to elapse between measurement and result, which is a physically unrealistic scenario.

The non-signalling requirement imposes further conditions on the joint probability, in that the probability of a particular output a or b should depend only on its associated input. This allows for the notion of a reduced or marginal probability on both Alice and Bob's measurements, and is formalised by the conditions:

and

The constraints above are all linear, and so define a polytope representing the set of all non-signalling boxes with a given number of inputs and outputs. Moreover, the polytope is convex because any two boxes that exist in the polytope can be mixed (as above, according to some variable with probabilities ) to produce another box that also exists within the polytope.

Local boxes are clearly non-signalling, however nonlocal boxes may or may not be non-signalling. Since this polytope contains all possible non-signalling boxes of a given number of inputs and outputs, it has as subsets both local boxes and those boxes which can achieve Tsirelson’s bound in accord with quantum mechanical correlations. Indeed, the set of local boxes form a convex sub-polytope of the non-signalling polytope.

Popescu and Rohrlich’s maximum algebraic violation of the CHSH inequality can be reached by a non-signalling box, referred to as a standard PR box after these authors, with joint probability given by:

where denotes addition modulo two.

Various attempts have been made to explain why Nature does not allow for stronger nonlocality than quantum theory permits. For example, in a recent publication it was found that quantum mechanics cannot be more nonlocal without violating the Heisenberg uncertainty principle. Strikingly, it has been discovered that if PR boxes did exist, any distributed computation could be performed with only one bit of communication. An even stronger result is that for any nonlocal box theory which violates Tsirelson's bound, there cannot be a sensible measure of mutual information between pairs of systems. This suggests a deep link between nonlocality and the information-theoretic properties of quantum mechanics.