We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
This paper discusses some tests of Bell-like inequalities not requiring entangled states. The proposed tests are based on consecutive measurements on a single qubit. Available hidden-variable models for a single qubit [see, e.g., J. S. Bell, Rev. Mod. Phys. 38, 447 (1966)] reproduce the predictions of quantum mechanics and hence violate the Bell-like inequalities addressed in this paper. It is shown how this fact is connected with the state ``collapse'' and with its random nature. Thus, it becomes possible to test truly realistic and deterministic hidden-variable models. In this way, it can be shown that a hidden-variable model should entail at least one of the following features: (i) nonlocality, (ii) contextuality, or (iii) discontinuous measurement-dependent probability functions. The last two features are put to the test with the experiments proposed in this paper. A hidden-variable model that is noncontextual and deterministic would be at variance with some predictions of quantum mechanics. Furthermore, the proposed tests are more likely to be loophole-free, as compared to former ones.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such distribution experimentally, as set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to quantum probe, however, produces minimal back-action, and so enables weak measurement of the projector of one observable, followed by projective measurement of non-commuting observable. By this technique it is possible to empirically measure probabilities for all of the values of the observables relevant to Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0, 1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within a [local] hidden variable view of nature ... everything works fine if we permit negative probabilities. However, there are infinitely many such theories. Our method, involving weak-valued singles out unique set of probabilities, and moreover does so empirically.
Cooperative multiparticle quantum states in an Einstein-Podolsky-Rosen type of experiment are shown to violate a Bell inequality. Thus correlations can occur between wave packets each having N bosons which cannot be explained within local hidden-variable theories. This provides a way to test quantum measurement theory for multiparticle states. In these states, a local, positive probability distribution exists for all N. However, the representation of correlations with this probability distribution is different from that in standard (Bell-type) hidden-variable theories.
The conceptually puzzling features of quantum mechanics as a statistical theory all have their source in the impossibility of relating the probability assignments defined by the quantum state to distributions over determinate values of the physical magnitudes. While some interpretations of quantum mechanics, as well as “hidden variable” modifications of the theory, have proposed several constructions for assigning values to magnitudes (beyond the value assignments defined by the quantum state alone), a variety of theorems exist which impose severe restrictions on such theoretical constructions. This note will deal with Bell's proof (in [2]), that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. Bell's result has initiated a series of experiments designed to refute the entire class of local hidden variable theories. We show that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated.
Quantum theory is inconsistent with any local hidden variable model as was first shown by Bell. To test Bell inequalities two separated observers extract correlations from a common ensemble of identical systems. Since quantum theory does not allow simultaneous measurements of noncommuting observables, on each system every party measures a single randomly chosen observable out of a given set. Here we suggest a different approach for testing Bell inequalities that is experimentally realizable by current methods. We show that Bell inequalities can be maximally violated even when all observables are measured on each member of the ensemble. This is possible by using weak measurements that produce small disturbance, at the expense of accuracy. However, our approach does not constitute an independent test of quantum nonlocality since the local hidden variables may correlate the noise of the measurement instruments. Nevertheless, by adding a randomly chosen precise measurement at the end of every cycle of weak measurements, the parties can verify that the hidden variables were not interfering with the noise, and thus validate the suggested test.
This paper is aimed to dissociate nonlocality from quantum theory. We demonstrate that the tests on violation of the Bell type inequalities are simply statistical tests of local incompatibility of observables. In fact, these are tests on violation of the Bohr complementarity principle. Thus, the attempts to couple experimental violations of the Bell type inequalities with “quantum nonlocality” is really misleading. These violations are explained in the quantum theory as exhibitions of incompatibility of observables for a single quantum system, e.g., the spin projections for a single electron or the polarization projections for a single photon. Of course, one can go beyond quantum theory with the hidden variables models (as was suggested by Bell) and then discuss their possible nonlocal features. However, conventional quantum theory is local.