Abstract For an even qudit dimension we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell–Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value 3/2—the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger–Horne–Zeilinger (GHZ) state with an arbitrary even exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the 3/2 upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.
Abstract For an even qudit dimension we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell–Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value 3/2—the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger–Horne–Zeilinger (GHZ) state with an arbitrary even exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the 3/2 upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.
Abstract We formulate and prove the main properties of the generalized Gell–Mann representation for traceless qudit observables with eigenvalues in and analyze via this representation violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension , this allows us to find two new bounds, lower and upper , expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify a class of two-qudit states where the new upper bound improves the upper bound of Tsirelson. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH result of Horodeckis, and also, for the Greenberger–Horne–Zeilinger (GHZ) state with an odd where the new upper bound is lower than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in is equal to if is even and to if d > 2 is odd.
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3 2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3 2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.
In this chapter we introduce the general class of symmetric two-qubit statesguaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomeswhenever some spin observable is measured at both sites. It is proventhat, for all states from this class, the maximal violation of the original Bellinequality (OB) is upper bounded by 3/2 and specify the two-qubit stateswhere this quantum upper bound is attained. The case of two-qutrit statesis more complicated. Here, for all two-qutrit states, we obtain the same upperbound 3/2 for violation of the original Bell inequality under Alice and Bobspin measurements. But it has not yet been shown that this quantum upperbound is the least one. The experimental consequences of this mathematicalstudy are discussed.
Abstract Violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements is still known only for s = 1 / 2 . In the present article, for an arbitrary state of two spin-s qudits, each of dimension d = 2 s + 1 ⩾ 2 , we introduce the notion of the spin-s correlation matrix, which has dimension 3 × 3 for all s ⩾ 1 2 ; establish its relation to the general correlation ( d 2 − 1 ) × ( d 2 − 1 ) matrix of this state within the generalized Pauli representation and derive in terms of the spin-s correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements in this state. Specifying this general expression for the two-qudit Greenberger–Horne–Zeilinger state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ( s ⩾ 1 ) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin- 1 / 2 measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different—for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.