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.
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.
Bell’s inequality is an expression which can be employed in order to test for the existence of hidden variables in a correlated system. The idea of hidden variables was introduced as a potential explanation for the quantum mechanical phenomenon of non-deterministic results of measurements. The existence of hidden variables can be tested experimentally. First, a particular measurement of one component of a correlated system must be associated with a positive or negative integer value. There must be an equal probability density of obtaining either measurement. Following this, an ensemble average of these measurements must be performed and Bell’s inequality must subsequently be employed. This theorem was essential in settling the EPR argument against quantum mechanical indeterminism, experimentally showing through violation of the inequality that there are indeed no hidden variables in a quantum system. For a system exhibiting classical behavior, Bell’s inequality should not be violated, provided the correlated system meets criteria appropriate to the assumptions underlying Bell’s theorem. A Mathematica model of Bell’s inequality was developed utilizing a Monte Carlo approach of randomizing parameters in order to test models of correlated systems. The model appears to agree with classical determinism within one standard deviation. It is possible to test both classical and quantum mechanical systems if properly correlated, and it is of future interest to test whether this extends to chaotic systems.
Abstract It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values { 0 , 1 } . A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, the Kochen–Specker theorem, and quantum entanglement are briefly discussed.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality. We put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate particular maps, or functions to particular theories. The violation of a Bell inequality then implies the ability to perform some functions, or computations that classical, or more generally, local hidden variable (LHV) theories cannot.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.
Bell inequalities have traditionally been used to demonstrate that quantum theory is nonlocal, in the sense that there exist correlations generated from composite quantum states that cannot be explained by means of local hidden variables. With the advent of device-independent quantum information protocols, Bell inequalities have gained an additional role as certificates of relevant quantum properties. In this work, we consider the problem of designing Bell inequalities that are tailored to detect maximally entangled states. We introduce a class of Bell inequalities valid for an arbitrary number of measurements and results, derive analytically their tight classical, nonsignaling, and quantum bounds and prove that the latter is attained by maximally entangled states. Our inequalities can therefore find an application in device-independent protocols requiring maximally entangled states.
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.