Levitt, Malcolm H., and Christian Bengs. “Hyperpolarization and the Physical Boundary of Liouville Space.” Magnetic Resonance 2, no. 1 (June 8, 2021): 395–407.
https://doi.org/10.5194/mr-2-395-2021.
The quantum state of a spin ensemble is described by a density operator, which corresponds to a point in the Liouville space of orthogonal spin operators. Valid density operators are confined to a particular region of Liouville space, which we call the physical region and which is bounded by multidimensional figures called simplexes. Each vertex of a simplex corresponds to a pure-state density operator. We provide examples for spins I = 1/2, I = 1, I = 3/2 and for coupled pairs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation.