Reply on RC1
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.
... airs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for 5 spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation. This point is reinforced by figure 8, which compares the predictions of the inhomogeneous and Lindbladian master equations when applied to spin-1/2 pairs in a low-entropy state of pure singlet polarization. The initial state of pure singlet order is shown 315 by the red dot. The plot shows an expanded view of Liouville space, in the vicinity of the initial condition. The physical bounds of Liouville space are indicated by the blue triangle, as in figure 4 . The red dashed line shows the trajectory predicted by the IME, in the case that T S T 1 , where T S is the relaxation time constant for singlet order (Levitt (2019)), and T 1 is the relaxation time constant for z-polarization. Since T 1 is relatively short, the z-polarization rapidly assumes its thermal equilibrium value ρ eq z , which is finite in the presence of a strong magnetic field. 320 However, as shown in figure 8 , this leads the density operator into a forbidden region outside the physical boundary of Liouville space. This proves that the inhomogeneous master equation must be invalid in this regime. The predicted trajectory of the Lindbladian master equation, as described in Bengs and Levitt (2020) is shown by the blue line. This uneventful trajectory always stays well within the physical boundary of Liouville space. Conclusions 325 This article has been an exploration of the geometry and physical boundary of Liouville space, the home territory of all spin density operators. In the past, most NMR experiments have only explored a tiny region of this space, very close to the origin (except for the fixed projection onto the unity operator). However, NMR experiments are increasingly performed on highly non-equilibrium spin states, which are sometimes located on or near the physical Liouville space boundary. We hope that this article is useful as a guide for wanderers in this region. 330