This chapter is aimed to dissociate nonlocality fromquantum theory. We indicate that the tests on violation of the Bell inequalitiescan be interpreted as statistical tests of observables local incompatibility.In fact, these are tests on violation of the Bohr complementarityprinciple. Thus, the attempts to couple experimental violations of the Bell-type inequalities with “quantum nonlocality” are misleading. These violationsare explained by the standard quantum theory as exhibitions of observablesincompatibility even for a single quantum system. Mathematically this chapter is based on the Landau equality. Thequantum CHSH-inequality is considered withoutcoupling to the tensor product, We point out that the notion of local realism isambiguous. The main impact of the Bohm–Bell experiments is on the developmentof quantum technology: creation of efficient sources of entangledsystems and photodetectors.
This chapter is aimed to dissociate nonlocality fromquantum theory. We indicate that the tests on violation of the Bell inequalitiescan be interpreted as statistical tests of observables local incompatibility.In fact, these are tests on violation of the Bohr complementarityprinciple. Thus, the attempts to couple experimental violations of the Bell-type inequalities with “quantum nonlocality” are misleading. These violationsare explained by the standard quantum theory as exhibitions of observablesincompatibility even for a single quantum system. Mathematically this chapter is based on the Landau equality. Thequantum CHSH-inequality is considered withoutcoupling to the tensor product, We point out that the notion of local realism isambiguous. The main impact of the Bohm–Bell experiments is on the developmentof quantum technology: creation of efficient sources of entangledsystems and photodetectors.
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.
Abstract By filtering out the philosophic component we can be said that the EPR-paper was directed against the straightforward interpretation of the Heisenberg’s uncertainty principle or more generally the Bohr’s complementarity principle. The latter expresses contextuality of quantum measurements: dependence of measurement’s output on the complete experimental arrangement. However, Bell restructured the EPR-argument against complementarity to justify nonlocal theories with hidden variables of the Bohmian mechanics’ type. Then this Bell’s kind of nonlocality - subquantum nonlocality - was lifted to the level of quantum theory - up to the terminology “quantum nonlocality”. The aim of this short note is to explain that Bell’s test is simply a special test of local incompatibility of quantum observables, similar to interference experiments, e.g., the two-slit experiment.
One of the major concepts introduced by quantum theory is the compatibility of quantum measurements. There are certain types of measurements that cannot be made at the same time. Thus,we say that measurements are compatible if they can be measured at the same time and others are incompatible. The other major concept of quantum mechanics is that of nonlocality which is one of the most counterintuitive concepts of quantum physics. This major concept is due to John Bell who showed that quantum mechanics is intrinsically non-local. Thus, we speak of violation of Bell's inequalities by quantum mechanics. Today, nonlocality is understood through nonlocal games. A nonlocal game consists of two or more players Alice and Bob playing against a referee. The referee will ask a number of questions to the players who will have to generate a number of answers using a classical or quantum strategy. It turns out that the maximum number of answers that Alice and Bob can generate is intrinsically linked to a tensor norm characterizing the game. In this formalism, the use of classical strategies is related to the norm of the matrix of the game itself, so the violation of Bell's inequalities results in a strict inequality between the tensor norms. The aim of this thesis is to understand the incompatibility of quantum measures and the link with Bell's inequalities. First, we introduced the compatibility of quantum measures from a new point of view, and analyzed the types of noise that can be made to make the system compatible. This new point of view consists in understanding and analyzing the effect of the dimension of the Hilbert space on the incompatibility of measurements. Moreover, in order to make the measurements compatible, we can introduce the effect of a noise. As an application, some states known as MUB are incompatible in nature, we show that even if we add noise to the MUB it remains incompatible, there is an isometry and a Hilbert space of smaller dimension making the MUB compatible. In a second step, we have analyzed the intrinsic link between the incompatibility of quantum measurements and the violation of Bell's inequalities. For this purpose, we considered the framework of non-local games, where Alice's measurements are fixed. It is known that a violation of Bell's inequalities requires the use of incompatible measurements. On the other hand, if Alice wants to know if she will observe a violation of Bell's inequalities if she uses incompatible measures. To do this, she must compute two tensor norms of a tensor constructed from her measurements. These tensorial norms will characterize on the one hand the compatibility of Alice's measurements and on the other hand the violation of Bell's inequalities. In this natural framework, to understand the link between the incompatibility of quantum measurements and the violation of Bell's inequalities, we have to compare the two tensorial norms. Now, it turns out that for the CHSH game these two norms are equal, but it can be shown generally that they are not. We can ask ourselves if there are other types of games satisfying this equality of the tensorial norms? It turns out that we have shown that with sufficient conditions, only the CHSH game with a multiplicative constant gives the equality between the tensor norms.
Abstract One of the Bell’s assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell’s inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell’s inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality in canonical notatons reads 〈AB〉 + 〈A’B〉 + 〈AB’〉 − 〈A’B’〉 ⩽ 6.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.