Hyperpolarization and the physical boundary of Liouville space

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 inline-formula $M1inlinescrollmathmlI=\mathrm{normal\; 1}/\mathrm{normal\; 2}$ 37pt14ptsvg-formulamathimg3619aa10605188fc94ee6adb7c011929 mr-2-395-2021-ie00001.svg37pt14ptmr-2-395-2021-ie00001.png , inline-formulaI=1, inline-formula $M3inlinescrollmathmlI=\mathrm{normal\; 3}/\mathrm{normal\; 2}$ 37pt14ptsvg-formulamathimg190f45f9534dd05538c3d270deb69123 mr-2-395-2021-ie00002.svg37pt14ptmr-2-395-2021-ie00002.png 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.