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arXiv:2607.18050

New bound on $S_{1}\times S_{2}$-setting Bell locality of a nonseparable Werner state

Elena R. Loubenets

quant-phmath-phSubmitted Jul 20, 2026• Announced Jul 21, 2026• 0 citations• Revised Oct 25, 2026 (v3)
v1 · Jul 20, 2026v2 · Jul 27, 2026v3 · Oct 25, 2026
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Abstract

In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers $S_{1},S_{2}\geq1$ of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension $d\leq\min\{S_{1},S_{2}\}$ to satisfy all Bell inequalities under every $S_{1}\times S_{2}$-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous $-$ that is, to be $S_{1}\times S_{2}$-setting Bell local, for short. For a variety of measurement settings $S_{1}, S_{2}$, this new general bound is beyond (i) Werner's and Barrett's locality bounds for a nonseparable Werner state and (ii) the related result by Johnson and Viola [Phys. Rev. A 88, 032323 (2013)] on "sharability" of this state. When specialized to $d = 2$ and $\min\{S_{1},S_{2}\} = 2, 3$, our new general bound improves upon the corresponding results by Terhal et. al. in [Phys. Rev. Lett. 90, 157903 (2003)], moreover, for $d=\min\{S_{1},S_{2}\} = 2$, it reduces exactly to the bound derived for this case via the CHSH inequality. We prove also explicitly in operator terms the result of Terhal et. el. that every nonseparable Werner state with a dimension $d >\min\{S_{1},S_{2}\}$ is $S_{1}\times S_{2}$-setting Bell local. The new results presented in this article are important for both the theory of Bell nonlocality and quantum applications based on this fundamental concept.

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9 pages

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New bound on S1×S2S_{1}\times S_{2}S1​×S2​-setting Bell locality of a nonseparable Werner state
In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers S1,S2≥1S_{1},S_{2}\geq1S1​,S2​≥1 of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension d≤min⁡{S1,S2}d\leq\min\{S_{1},S_{2}\}d≤min{S1​,S2​} to satisfy all Bell inequalities under every S1×S2S_{1}\times S_{2}S1​×S2​-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous −-− that is, to be S1×S2S_{1}\times S_{2}S1​×S2​-setting Bell local, for short. For a variety of measurement settings S1,S2S_{1}, S_{2}S1​,S2​, this new general bound is beyond (i) Werner's and Barrett's locality bounds for a nonseparable Werner state and (ii) the related result by Johnson and Viola [Phys. Rev. A 88, 032323 (2013)] on "sharability" of this state. When specialized to d=2d = 2d=2 and min⁡{S1,S2}=2,3\min\{S_{1},S_{2}\} = 2, 3min{S1​,S2​}=2,3, our new general bound improves upon the corresponding results by Terhal et. al. in [Phys. Rev. Lett. 90, 157903 (2003)], moreover, for d=min⁡{S1,S2}=2d=\min\{S_{1},S_{2}\} = 2d=min{S1​,S2​}=2, it reduces exactly to the bound derived for this case via the CHSH inequality. We prove also explicitly in operator terms the result of Terhal et. el. that every nonseparable Werner state with a dimension d>min⁡{S1,S2}d >\min\{S_{1},S_{2}\}d>min{S1​,S2​} is S1×S2S_{1}\times S_{2}S1​×S2​-setting Bell local. The new results presented in this article are important for both the theory of Bell nonlocality and quantum applications based on this fundamental concept.
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.
We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented.
Abstract We introduce a geometric framework for studying Bell nonlocality in Hilbert space, where, for a given quantum state, nonlocality is quantified by the distance between the state and the set of local states. This approach applies to any Bell inequality and any measurement scenario. Whenever the local set is characterized, the proposed nonlocality measure can be computed explicitly. As a general result, we prove that for any scenario in arbitrary dimension the closest local state to a Werner state is itself a Werner state, and analogously, the closest local state to an isotropic state is again isotropic. In the two-qubit case, we further show that the closest local state to a Bell-diagonal state is Bell-diagonal as well. These structural results are independent of the specific Bell inequality considered, thus revealing intrinsic geometric features of these families of states and providing significant simplifications for computing the proposed measures. For the Clauser–Horne–Shimony–Holt inequality in two-qubit systems and the Collins-Gisin-Linden-Massar-Popescu inequality for two qudits of arbitrary finite dimension, we derive explicit geometric measures of nonlocality for Bell-diagonal, Werner, and isotropic states using various distance metrics, including the trace, Hellinger, Hilbert–Schmidt distances, and relative entropy. Furthermore, we prove in all generality that for all scenarios in which the local set is not fully characterized, the geometric measures provide rigorous lower bounds on nonlocality.
In a general setting, we introduce a new bipartite state property sufficient for the validity of the perfect correlation form of the original Bell inequality for any three bounded quantum observables. A bipartite quantum state with this property does not necessarily exhibit perfect correlations. The class of bipartite states specified by this property includes both separable and nonseparable states. We prove analytically that, for any dimension d ⩾ 3, every Werner state, separable or nonseparable, belongs to this class.
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.
We find two two-qubit bipartite states ${\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}$ such that arbitrarily many copies of one or the other cannot exhibit nonlocal correlations in a two-setting--two-outcome Bell scenario. However, the bipartite state ${\ensuremath{\rho}}_{1}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{2}$ violates the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).] by an amount of 2.023. We also identify a CHSH-local state $\ensuremath{\rho}$ such that ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}2}$ is CHSH inequality-violating. The tools employed can be easily adapted to find instances of nonlocality activation in arbitrary Bell scenarios.