8. Quantum description of angles in the plane

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Context: This paper is the direct sequence of the previous one on integral quantisation on the real plane. We extend our results to formulate entanglement and spin one-half coherent states in this formalism.

Method: I define Bell states and their quantum correlations, and investigate under which conditions the classical and quantum expectation values differ. We then use the Bell states to map the Euclidean plane to the complex plane, and then prove there is an isomorphism between this plane and the space resulting from the entanglement between Bell states. This isomorphism requires the introduction of a “flip” operator, which appears in the construction of spin one-half coherent states.

Results:

• We recover the usual violation of Bell inequalities with only real quantities…
• … which allows us to write the Bell states in real space, a necessary step for constructing spin one-half coherent states by a set of orientations in \(\mathbb{R}^3\).