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As proposed in Ref. [5], it can be experimentally advantageous to have the mobile particles (scattering centers) as embodied by photons (quantum dots). A rigorous formal analysis shows that V thus takes the form 'i=1NJ(σxSix+σySiy)δ(x-xi) [5]. Under RCs and for unpolarized spin states of e's, we get the effective representation VRC=JδRC(x) (σxSx+σySy) with Sx(y)='i=1NSix(y). Notwithstanding the differences in the form of the spin-spin interactions, the considerations drawn for the case of Eq. (1) are in excellent agreement with the results gathered through photonic polarizations
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As proposed in Ref. [5], it can be experimentally advantageous to have the mobile particles (scattering centers) as embodied by photons (quantum dots). A rigorous formal analysis shows that V thus takes the form ' i = 1 N J (σ x S ix + σ y S iy) δ (x - x i) [5]. Under RCs and for unpolarized spin states of e 's, we get the effective representation V RC = J δ RC (x) (σ x S x + σ y S y) with S x (y) = ' i = 1 N S ix (y). Notwithstanding the differences in the form of the spin-spin interactions, the considerations drawn for the case of Eq. (1) are in excellent agreement with the results gathered through photonic polarizations.
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In Refs. [6,14] this property is proved for the case N=2. The generalization to arbitrary N is straightforward.
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In Refs. [6,14] this property is proved for the case N = 2. The generalization to arbitrary N is straightforward.
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In general, when s1...n and sn+1...N are added and s1...n=sn+1...N
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In general, when s 1... n and s n + 1... N are added and s 1... n = s n + 1... N < n / 2, many singlet states may arise. For instance, for N = 6, when s 12 = 0, 1 is added to s 3 = 1 / 2 we obtain s 123 = 1 / 2, 3 / 2 with the quantum number s 123 = 1 / 2 being twofold degenerate. As a similar argument holds for s 456, it turns out that within the subspace s 123 = s 456 = 1 / 2 four orthogonal singlet states exist. This does not occur with s 123 = s 456 = 3 / 2 as these quantum numbers are not degenerate.
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