Analytic quantum bounds on Bell inequalities

Full text: Can realism be combined with the quantum world? An important tool to investigate in this question are Bell's inequalities and violations thereof - they represent a cornerstone of our present understanding of quantum mechanics and therefore the description of nature. Here we present a simple algebraic method to calculate violations for any measurement arrangements that are maximal in the sense that quantum mechanics does not allow a stronger violation. Having two or more polarization analyzers available and a source producing photon-pairs in arbitrary polarization states Bell-type inequalities tell us which probabilities for measuring the polarization in particular directions are viable in a deterministic theory. Quantum mechanics does not obey these rules, but yields a violation of these inequalities. The questions is to what extent the inequalities are violated. Making use of a min-max principle analytical expressions can be found for the 'fine structure' of the maximal violations of arbitrary Bell-like inequalities, i. e. the upper bound reachable by any state when the analyzers measure in given directions. Knowing these bounds is useful for experimental tests of the validity of quantum mechanics and can serve as a prerequisite to answer the even more pressing question, why no stronger violation has been observed until now. (author)