Symmetry constraints on spin dynamics: Application to hyperpolarized NMR

189Bounds on the transformation of the spin density operator in a four-spin-1/2 system. The vertical axis represents the expectation value of z-magnetization for spins 1 and 2 (species A); the horizontal axis represents the expectation value of z-magnetization for spins 3 and 4 (species X). The initial density operator (black point) represents pure A-magnetization. The boundary of the pale blue circle represents the limit of unitary transformations preserving the norm; the dark blue polygon represents the Sørensen bound, constructed by using the method in M.H. Levitt, J. Magn. Reson. 99 (1992) 1-17. The yellow polygon takes into account permutation symmetry in the case of a A2X2 spin system, as described in the current article. Four trajectories calculated using random Hamiltonians of the appropriate symmetry are shown; all remain within the boundary of the yellow polygon.