Abstract
We propose a geometric multiparty extension of Clauser–Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, A and B, defined as , where , satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events . We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems in a particular scenario. With our tripartite CH inequality, we investigate quantum violations by GHZ-type and W-type states. Our inequalities are compared to another type, so-called N-site CH inequality. In addition we argue how to generalize our method for more subsystems and measurement settings. Our method can be used to write down several Bell-type inequalities in a systematic manner.
Highlights
Intrinsic randomness of quantum mechanics has been a topic of a debate for many years
We find an extension of CH inequality for a tripartite system, by an extension of the separation measure to P (A ⊕ B ⊕ C)
+2P (A1, B1, C1) T2 = P (A1, B2, C2) + P (A2, B1, C2) + P (A2, B2, C1). They proposed the N -site CH inequality to find the minimum detection efficiency required for quantum mechanics to violate the inequality
Summary
Intrinsic randomness of quantum mechanics has been a topic of a debate for many years. We postulate that three (or more) Bell inequalities which are direct generalizations of the CH one should be derivable using the geometric properties of separations of three (or more) probabilistic events. In our derivation of CH-type Bell inequalities, statistical separation plays a crucial role It is defined by the probability of symmetric difference between two events [36]. Let us consider probability P (X ⊕ Y ⊕ Z) of the measure of the symmetric difference of three events, X, Y, and Z It because of the aforementioned permutation symmetry is a statistical separation between X and Y ⊕ Z, or between Y and Z ⊕ X, or between Z and X ⊕ Y. We analyze statistical separations of certain combinations of symmetric differences of probabilistic events for a specific type of three-qubit experiments
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