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تسجيل مستخدم جديدDiscretisation of the Bloch Sphere, Fractal Invariant Sets and Bells Theorem
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T. N. Palmer
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An arbitrarily dense discretisation of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretised representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretised spheres) entanglement. Unlike Meyers earlier discretisation of the Bloch Sphere, there are no orthonormal triples, hence the Kocken-Specker theorem is not nullified. A physical interpretation of points on the discretised Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorisation, where these conditions are defined solely from states on the invariant set. In this finite representation there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.
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(A) Bells theorem rests on a conjunction of three assumptions: realism, locality and ``free will. A discussion of these assumptions will be presented. It will be also shown that, if one adds to the assumptions the principle or rotational symmetry o
f physical laws, a stronger version of the theorem emerges. (B) A link between Bells theorem and communication complexity problems will be presented. This also includes experimental realizations, which surprisingly do not involve entanglement. (C) A new sufficient and necessary criterion for entanglement of general (mixed) states is be presented. It is derived using the same geometric starting point as the inclusion of the symmetry in (A). The set of entanglement identifiers (EIs) emerging via this method contains entanglement witnesses (EWs), but they form only a subset of all EIs. Thus the method is more powerful than the one based on EWs.
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