We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) ${\mathcal{M}}_{A}$ with the following property. Considering a bipartite Bell test where Alice uses ${\mathcal{M}}_{A}$, then for any possible shared entangled state $\ensuremath{\rho}$ and any set of (possibly infinitely many) POVMs ${\mathcal{N}}_{B}$ performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.
We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) MA with the following property. Considering a bipartite Bell test where Alice uses MA, then for any possible shared entangled state \ensuremathρ and any set of (possibly infinitely many) POVMs NB performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.
The observation of quantum nonlocality, i.e. quantum correlations violating a Bell inequality, implies the use of incompatible local quantum measurements. Here we consider the converse question. That is, can any set of incompatible measurements be used in order to demonstrate Bell inequality violation? Our main result is to construct a local hidden variable model for an incompatible set of qubit measurements. Specifically, we show that if Alice uses this set of measurements, then for any possible shared entangled state, and any possible dichotomic measurements performed by Bob, the resulting statistics are local. This represents significant progress towards proving that measurement incompatibility does not imply Bell nonlocality in general.
Measurements in quantum theory can fail to be jointly measurable. Like entanglement, this incompatibility of measurements is necessary but not sufficient for violating Bell inequalities. The (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph whose vertices denote measurements and hyperedges denote all and only compatible sets of measurements. Since incompatibility is necessary for a Bell violation, the joint measurability structure on each wing of a Bell experiment must necessarily be non-trivial, i.e., it must admit a subset of incompatible vertices. Here we show that for any non-trivial joint measurability structure with a finite set of vertices, there exists a quantum realization with a set of measurements that enables a Bell violation, i.e., given that Alice has access to this incompatible set of measurements, there exists a set of measurements for Bob and an entangled state shared between them such that they can jointly violate a Bell inequality. Hence, a non-trivial joint measurability structure is not only necessary for a Bell violation, but also sufficient.
Two important ingredients necessary for obtaining Bell nonlocal correlations between two spatially separated parties are an entangled state shared between them and an incompatible set of measurements employed by each of them. However, not all entangled states can give rise to Bell nonlocality. Here, we are interested in a complementary question, regarding the role of the sets of incompatible measurements in enabling Bell nonlocality, for arbitrary entangled states. It is known that for some sets of more than two jointly incompatible measurements performed by one of the parties, there does not exist an entangled state and another set of incompatible measurements for the other party, which will result in Bell nonlocality. However, in the two-input and two-output setting per party, measurement incompatibility at a single party will always guarantee Bell nonlocality. Thus, it is interesting to ask whether all sets of incompatible measurements at both parties, in the two-input and two-output setting per party, will result in Bell nonlocality for some shared state. We first observe that Bell nonlocality can always be established when both parties employ any set of incompatible projective measurements. On the other hand, going beyond projective measurements, we present a class of incompatible positive operator-valued measures, employed by both observers, which can never activate Bell nonlocality. Furthermore, we find a sufficient criterion for achieving Bell nonlocal correlations given a fixed amount of pure two-qubit entanglement and a fixed amount of incompatibility of projective measurements applied by either both parties or a single party.
In the case of a pair of two-outcome measurements, incompatibility is equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome measurements can violate the Clauser–Horne–Shimony–Holt Bell inequality, which has been proven by Wolf et al (2009 Phys. Rev. Lett. 103 230402). In the case of more than two measurements the equivalence between incompatibility and Bell nonlocality is still an open problem, though partial results have recently been obtained. Here we show that the equivalence breaks for a special choice of three measurements. In particular, we present a set of three incompatible two-outcome measurements, such that if Alice measures this set, independent of the set of measurements chosen by Bob and the state shared by them, the resulting statistics cannot violate any Bell inequality. On the other hand, complementing the above result, we exhibit a set of N measurements for any that is -wise compatible, nevertheless it gives rise to Bell violation.
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 Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.