Terra incognita of the quantum dimension – Local and nonlocal dimensions

One qubit: Of these four parameters, only 3 can be freely chosen, as the sum of both probabilities is normalized to 1. These 3 parameters are locally assigned to one chessboard field.

Two qubits: In the case of two chessboard fields, there are four possible combinations. The corresponding 4 probabilities are generalised, in the quantum dimension, to amplitudes with a total of 8 real parameters. As the sum of the probabilities is 1, in general, a 7-dimensional space arises in the quantum dimension. Three dimensions are assigned locally to the first or second chessboard field, respectively. The four Bell states can be found in the non-local seventh dimension c, irrespective of whether we are looking at the polarisation of a photon pair, the spin of an electron pair or any other quantum states.

Three qubits: We dive deeper into the quantum dimension and consider at three chessboard fields. In this case, 3 detectors, A, B and C, would record eight possible measurement results. How many non-local dimensions can be found in the quantum dimension behind 3 chessboard fields? And how large is the quantum dimension associated to a quantum computer with 64 qubits?