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Register a new userExclusivity principle and the quantum bound of the Bell inequality
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Authors
Adan Cabello
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We show that, for general probabilistic theories admitting sharp measurements, the exclusivity principle together with two assumptions exactly singles out the Tsirelson bound of the Clauser-Horne-Shimony-Holt Bell inequality.
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Adopting the graph-theoretic approach to the correlation experiments, we analyze the origin of monogamy and prove that it can be recognised as a consequence of exclusivity principle(EP). We provide an operational criterion for monogamy: if the fractional packing number of the graph corresponding to the union of event sets of several physical experiments does not exceed the sum of independence numbers of each individual experiment graph, then these experiments are monogamous. As applications of this observation, several examples are provided, including the monogamy for experiments of Clauser-Horne-Shimony-Holt (CHSH) type, Klyachko-Can-Biniciou{g}lu-Shumovsky (KCBS) type, and for the first time we give some monogamy relations of Swetlichnys genuine nonlocality. We also give the necessary and sufficient condition for several experiments to be monogamous: several experiments are monogamous if and only if the Lovasz number the union exclusive graph is less than or equal to the sum of independence numbers of each exclusive graph.
Entanglement is a critical resource used in many current quantum information schemes. As such entanglement has been extensively studied in two qubit systems and its entanglement nature has been exhibited by violations of the Bell inequality. Can the amount of violation of the Bell inequality be used to quantify the degree of entanglement. What do Bell inequalities indicate about the nature of entanglement?
A finite non-classical framework for physical theory is described which challenges the conclusion that the Bell Inequality has been shown to have been violated experimentally, even approximately. This framework postulates the universe as a deterministic locally causal system evolving on a measure-zero fractal-like geometry $I_U$ in cosmological state space. Consistent with the assumed primacy of $I_U$, and $p$-adic number theory, a non-Euclidean (and hence non-classical) metric $g_p$ is defined on cosmological state space, where $p$ is a large but finite Pythagorean prime. Using number-theoretic properties of spherical triangles, the inequalities violated experimentally are shown to be $g_p$-distant from the CHSH inequality, whose violation would rule out local realism. This result fails in the singular limit $p=infty$, at which $g_p$ is Euclidean. Broader implications are discussed.
The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by $P_{B}$ is always less than or at most equal to one for the local realistic model ($P_{B}^{lc}leq1$) regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as $P_{B}^{max}$ $=2$, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.
For an even qudit dimension $dgeq 2,$ 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 $dgeq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $frac{3}{2}$ 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.
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