| Summary: | <p>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 <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>I</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="37pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="3619aa10605188fc94ee6adb7c011929"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-2-395-2021-ie00001.svg" width="37pt" height="14pt" src="mr-2-395-2021-ie00001.png"/></svg:svg></span></span>, <span class="inline-formula"><i>I</i>=1</span>, <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>I</mi><mo>=</mo><mn mathvariant="normal">3</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="37pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="190f45f9534dd05538c3d270deb69123"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-2-395-2021-ie00002.svg" width="37pt" height="14pt" src="mr-2-395-2021-ie00002.png"/></svg:svg></span></span> 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.</p>
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