Bell's Theorem

Bell's Theorem
First published Wed Jul 21, 2004; substantive revision Thu Jun 11, 2009



"Bell's Theorem is the collective name for a family of results, all showing the impossibility of a Local Realistic interpretation of quantum mechanics. There are variants of the Theorem with different meanings of “Local Realistic.” In John S. Bell's pioneering paper of 1964 the realism consisted in postulating in addition to the quantum state a “complete state”, which determines the results of measurements on the system, either by assigning a value to the measured quantity that is revealed by the measurement regardless of the details of the measurement procedure, or by enabling the system to elicit a definite response whenever it is measured, but a response which may depend on the macroscopic features of the experimental arrangement or even on the complete state of the system together with that arrangement. Locality is a condition on composite systems with spatially separated constituents, requiring an operator which is the product of operators associated with the individual constituents to be assigned a value which is the product of the values assigned to the factors, and requiring the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent. From his assumptions Bell proved an inequality (the prototype of “Bell's Inequality”) which is violated by the Quantum Mechanical predictions made from an entangled state of the composite system. In other variants the complete state assigns probabilities to the possible results of measurements of the operators rather than determining which result will be obtained, and nevertheless inequalities are derivable; and still other variants dispense with inequalities. The incompatibility of Local Realistic Theories with Quantum Mechanics permits adjudication by experiments, some of which are described here. Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisively because of the “detection loophole” or the “communication loophole.” The latter has been nearly decisively blocked by a recent experiment and there is a good prospect for blocking the former. The refutation of the family of Local Realistic Theories would imply that certain peculiarities of Quantum Mechanics will remain part of our physical worldview: notably, the objective indefiniteness of properties, the indeterminacy of measurement results, and the tension between quantum nonlocality and the locality of Relativity Theory."

see: http://plato.stanford.edu/entries/bell-theorem/