In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to complementarity, by which we mean that consistently assigning values to different observables at the same time is not possible. Nonlocality can only occur when some of the relevant observables do not commute, and this noncommutativity makes the observables complementary. Beyond quantum mechanics, the concept of complementarity can be formalized in several distinct ways. Here we describe some of these possible formalizations and ask how they relate to nonlocality. We partially answer this question by describing two toy theories which display nonlocality and obey the no-signaling principle, although each of them does not display a certain kind of complementarity. The first toy theory has the property that it maximally violates the Clauser, Horn, Shimony, and Holt (CHSH) inequality, although the corresponding local observables are pairwise jointly measurable. The second toy theory also maximally violates the CHSH inequality, although its state space is classical and all measurements are mutually nondisturbing: if a measurement sequence contains some measurement twice with any number of other measurements in between, then these two measurements give the same outcome with certainty.
In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to complementarity, by which we mean that consistently assigning values to different observables at the same time is not possible. Nonlocality can only occur when some of the relevant observables do not commute, and this noncommutativity makes the observables complementary. Beyond quantum mechanics, the concept of complementarity can be formalized in several distinct ways. Here we describe some of these possible formalizations and ask how they relate to nonlocality. We partially answer this question by describing two toy theories which display nonlocality and obey the no-signaling principle, although each of them does not display a certain kind of complementarity. The first toy theory has the property that it maximally violates the Clauser, Horn, Shimony, and Holt (CHSH) inequality, although the corresponding local observables are pairwise jointly measurable. The second toy theory also maximally violates the CHSH inequality, although its state space is classical and all measurements are mutually nondisturbing: if a measurement sequence contains some measurement twice with any number of other measurements in between, then these two measurements give the same outcome with certainty.
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 The Clauser–Horne–Shimony-Holt (CHSH) inequality tests locality in quantum theory and is recognized as one of Bell’s inequalities. In contrast, the Klyachko–Can–Binicioğlu–Shumovsky (KCBS) inequality tests noncontextuality, which refers to the idea that measurement outcomes are determined by intrinsic properties independent of other measurements that could be performed simultaneously. While individual quantum states can violate these inequalities separately, it was believed that no single state could violate both simultaneously. This relationship suggests a trade-off between nonlocality and contextuality: if one is violated, the other cannot be. Such a relationship is known as a monogamy relation. Xue et al demonstrated, however, that simultaneous violations of both the CHSH and KCBS inequalities are possible with specific choices of observables. This finding challenges the universal validity of the monogamy relation. They further showed that numerous scenarios exist in which both the CHSH inequality and a noncontextuality inequality involving more than five observables are violated. Nevertheless, the number of such scenarios is finite. In this paper, we extend these findings in two significant ways. First, we show that infinitely many scenarios exist where both the CHSH inequality and a noncontextuality inequality involving an odd number of observables are violated. This demonstrates that the monogamy relation between the CHSH inequality and the noncontextuality inequality fails in infinitely many cases. Second, we identify quantum states that violate both the KCBS inequality and a locality inequality distinct from the CHSH inequality. Our results highlight that the so-called ‘monogamy relation’ between nonlocality and contextuality is not universally valid. Instead, it appears that both phenomena can coexist under broader conditions than initially expected, thus challenging the conventional viewpoint and paving the way for new theoretical and experimental research.
Bell nonlocality and Kochen-Specker contextuality are two remarkable nonclassical features of quantum theory, related to strong correlations between outcomes of measurements performed on quantum systems. Both phenomena can be witnessed by the violation of certain inequalities, the simplest and most important of which are the Clauser-Horne-Shimony-Holt (CHSH) and the Klyachko-Can-Binicioğlu-Shumovski (KCBS), for Bell nonlocality and Kochen-Specker contextuality, respectively. It has been shown that, using the most common interpretation of Bell scenarios, quantum systems cannot violate both inequalities concomitantly, thus suggesting a monogamous relation between the two phenomena. In this Letter, we show that the joint consideration of the CHSH and KCBS inequalities naturally calls for the so-called generalized Bell scenarios, which, contrary to the previous results, allows for joint violation of them. In fact, this result is not a special feature of such inequalities: We provide very strong evidence that there is no monogamy between nonlocality and contextuality in any scenario where both phenomena can be observed. We also implement a photonic experiment to test the synchronous violation of both CHSH and KCBS inequalities. Our results agree with the theoretical predictions, thereby providing experimental proof of the coexistence of Bell nonlocality and contextuality in the simplest scenario, and lead to novel possibilities where both concepts could be jointly employed for quantum information processing protocols.
Nonlocality is one of the main characteristic features of quantum systems involving more than one spatially separated subsystem. It is manifested theoretically as well as experimentally through violation of some local realistic inequality. On the other hand, classical behavior of all physical phenomena in the macroscopic limit gives a general intuition that any physical theory for describing microscopic phenomena should resemble classical physics in the macroscopic regime, the so-called macrorealism. In the 2-2-2 scenario (two parties, with each performing two measurements and each measurement having two outcomes), contemplating all the no-signaling correlations, we characterize which of them would exhibit classical (local realistic) behavior in the macroscopic limit. Interestingly, we find correlations which at the single-copy level violate the Bell-Clauser-Horne-Shimony-Holt inequality by an amount less than the optimal quantum violation (i.e., Cirel'son bound 22), but in the macroscopic limit gives rise to a value which is higher than 22. Such correlations are therefore not considered physical. Our study thus provides a sufficient criterion to identify some of unphysical correlations.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.