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arXiv:quant-ph/0306092 · Physical Review A

Testing the bounds on quantum probabilities

Stefan Filipp, Karl Svozil

Published Mar 1, 2004• 12 citations
arXivHTMLPDF

Abstract

Bounds on quantum probabilities and expectation values are derived for experimental setups associated with Bell-type inequalities. In analogy to the classical bounds, the quantum limits are experimentally testable and therefore serve as criteria for the validity of quantum mechanics.
DOI
10.1103/physreva.69.032101

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Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.
Introduction Annus mirabilis 1964 1964 was a wonderful year for physics. In this annus mirabilis, summarising and simplifying a little, the quark model was proposed by Gell-Mann and Zweig and charm and colour properties were introduced, Higgs published his work on the scalar boson, the _ boson and CP violation were discovered, cosmic background radiation was identified and, finally, Bell inequalities were suggested. Most of these works were later subjects of a Nobel prize. Unluckily John Bell left us too early for this achievement, but, at least in my opinion, his work [1] is probably the most fruitful among all these huge advances, since now we can recognise that it was the root of a fundamental change in our vision of the physical world, providing, on one hand, the possibility of a test of quantum mechanics against a whole class of theories (substantially all the theories that we could define as “classical”) and, on the other hand, introducing a clear notion of quantum nonlocality that represents not only a challenge in understanding quantum mechanics, but also one of the most important resources paving the way to quantum technologies that were developed in the last decades based on the quantum correlations discussed in the Bell paper. In the following we will present the experimental progress in testing Bell inequalities. Rather amazingly even 50 years after Bell's paper, a conclusive test of his proposal is still missing [2, 3]. Nevertheless, huge progress has been achieved with respect to the first experiments.We will discuss these developments and the remaining problems for a conclusive test of Bell inequalities in detail. The Bell Inequalities, Their Hypotheses and the Experimental Loopholes In order to understand the experimental difficulties in achieving a final test of Bell inequalities, let us analyse a little how these inequalities are derived and the explicit and implicit hypotheses they contain. The proof of Bell inequalities usually starts by considering two separated subsystems of an entangled state, which are addressed to two independent measurement apparatuses measuring the expectation values of two dichotomic observables, Aa and B b . a, b are two parameters describing the settings of the measuring apparatus A and B , respectively.
The usual Bell-type inequalities involve correlations between observables. Inequalities of the Bell-type are presented which involve correlations between the probabilities associated with states and which can be used as a test for hidden variables in quantum mechanics independent of the Bell inequalities. The inequalities are universal in the sense that they are valid for all separable metric state spaces, as opposed to the usual Bell inequalities whose specific form is a function of the state space of the candidate classical model. The inequalities are harder to prove than the usual Bell-type inequalities because no essential restrictions on the state space are made.