12 Mar 2021
12 Mar 2021
Hyperpolarization and the Physical Boundary of Liouville Space
 School of Chemistry, University of Southampton, SO17 1BJ, UK
 School of Chemistry, University of Southampton, SO17 1BJ, UK
Abstract. 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 purestate density operator. We provide examples for spins I = 1 / 2, I = 1, I = 3 / 2, and for coupled pairs of spins1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to nonphysical results in some cases, a problem that may be avoided by using the Lindbladian master equation.
Malcolm H. Levitt and Christian Bengs
Status: open (until 15 Apr 2021)

RC1: 'Comment on mr202126', Anonymous Referee #1, 25 Mar 2021
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I we take spin 1/2 and express density matrix in terms of pauli matrices then the coordinates lie inside a Bloch sphere. The paper studies how does this result generalize to higher spins.
The paper studies density operator of a spin ensemble. Valid density operator are confined to reigons of Liouville space which authors call physical region. It is shown the phyical region is bounded by multidimensional figures called simplexes. vertex of which corresponds to pure state. Examples are given for spins 1/2, 1, 3/2 and coupled spins 1/2. Von Neumann entropy is used as a criterion for hyper polarization. It is shown that inhomogeneous master equation for spin dynamics leads to non physical results in some cases, a problem that may be avoided by using Lindbladian master equation.
In line 58, the other three operators should be Q2 Q3 Q4
In line 75, it is not known what zero quantum parts mean

AC1: 'Reply on RC1', Malcolm Levitt, 25 Mar 2021
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 The referee is correct about the misnumbering of the terms in line 58. This will be corrected.
 The term "zeroquantum" in line 75 does require explanation. The term is used here to indicate that only operators that may be expressed as combinations of populations for the Zeeman states are considered here. This does require clarification and that will be done in the revised manuscript.

AC1: 'Reply on RC1', Malcolm Levitt, 25 Mar 2021
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CC1: 'Comment on mr202126', Tom Barbara, 25 Mar 2021
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Spin 3/2 and two spin 1/2 are both SU(4) systems. Are there notable differences in behavior?

AC2: 'Reply on CC1', Malcolm Levitt, 25 Mar 2021
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The physical boundaries have the same form of a tetrahedron as shown in the figures. But as far as the spin dynamics is concerned there are of course big differences. For example the long lived nature of the singlet state of a spin1/2 pair has no counterpart for spins3/2. However this paper is more about the general landscape and in that respect the two systems are similar.

AC2: 'Reply on CC1', Malcolm Levitt, 25 Mar 2021
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RC2: 'Comment on mr202126', Anonymous Referee #2, 13 Apr 2021
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In their paper, Levitt and Bengs show that physical states characterizing nuclear spin systems are confined within a boundary of multidimensional simplexes in the Liouville space. The paper is very interesting and well written; there are only a few very minor comments (see below) with respect to the content. I recommend publication in OMR after minor revision.
The main issue, in my opinion, lies in the terminology rather than science and it is about the term “hyperpolarization”.
The authors start in the beginning by asking “is pure parahydrogen hyperpolarized?” and later replying “our answer is yes”. They later admit (page 17) that their “… definition of hyperpolarization makes no explicit mention of population difference, or the existence of a net magnetic moment in a certain direction.”
Let me be very clear. My concern is not about science, it is about terminology. I worry that the definition of hyperpolarization given by the authors goes against the semantics of the word “polarization” (as well as its origins, i.e., etymology).
Where does the term “polarization” come from? For a twospin system, the situation is clear. Equilibrium polarization is a normalized population difference, Peq = (p_a  p_b)/(p_a + p_b); where p_a and p_b are populations of levels for spin states “alpha” and “beta”, respectively and, in NMR, Peq is a very small number, typically below 10^(5). Strong polarization in NMR is usually termed “hyperpolarization”; the border of what % polarization is considered “hyper” is now beautifully connected to the von Neumann entropy in this paper. So, clearly, in a specific case of isolated spins 1/2, overpopulation of one level with respect to another creates oriented magnetization which can be measured. Both words “magnetization” and “polarization” usually imply a vector that has a specific orientation.
So, it totally makes sense for the spin1/2 case to call deviation from zero population difference “polarization” because it means a specific orientation of some vector quantity (magnetization) in space. The term “polarization” beyond physics is often used to imply a “division into two sharply distinct opposites” (see https://dictionary.cambridge.org/dictionary/english/polarization, https://www.merriamwebster.com/dictionary/polarization or https://www.thefreedictionary.com/polarization) which connects well with this idea. In almost all forms of common NMR, we measure magnetization or, more fundamentally, oscillating singlequantum coherences (i.e., onespinflipatatime allowed transitions). Calling other overpopulated multipole moments “hyperpolarization” is not the best idea, especially ones with rank zero! The phrase “singlet polarization” is an oxymoron, in my opinion; it is an NMR jargon which, more likely, will make many young minds studying this subject very confused, at least at the begging of their “wondering in this region”. I note, once the definition of “polarization moments” is introduced (eq. 9), everything else follows from that. This is probably where the root of my dissatisfaction is.
There is a big asymmetry here, though. While I am in a position to criticize such a definition, I do not necessarily have a good alternative... If we go deep into semantics, then probably equating polarization with orientation (i.e., only rank 1 tensors in the multipole extension of the density matrix) would be a correct choice. Obviously, this would probably cause many disagreements, since a huge body of NMR literature already uses this term in a much broader sense. One should also note that hyperpolarization is usually defined with respect to the specific groups of spins (because in highfield NMR we can selectively measure signals from different types of spins), this corresponds to the enhanced magnetization corresponding to those spins.
If we accept for a moment that hyperpolarization refers only to NMRobservable states, other states that satisfy equations 38 and 39 and not magnetization can be called “latent" or “hidden" hyperpolarization. This would be useful in the context of parahydrogen experiments. Indeed, there, it makes sense to talk about the conversion of nonobservable spin orders into observable spin orders. Thus, defining hyperpolarization only as enhanced magnetization (which can be converted from latent nonobservable spin orders via a chemical reaction or by using special pulse sequences) would make sense. Otherwise, everything in PHIP/SABRE is hyperpolarization and the discussion is over: the term becomes too broad to be useful.
Minor comments:
 Could the author add more information on the condition in the eq. (23). Where exactly does it come from (i.e., why a sum of lambda from 1 to 2I)?
 In Figure 4, 1/3 is listed as a lower bound but one can clearly see that the intersect with the xaxis (representing a contribution of the rank 0 polarization moment) is lower than 0.3 (while it should cross at 0.33). Is it a representation error?
 A similar analysis of the absence of information in the nuclear spin system was recently performed and could be mentioned with respect to the eq. (35): https://doi.org/10.1038/s41467019107879
 Page 17. “being inside the red region” – what exactly is the red region?
 In Figure 7, “hyperpolarized states” are mentioned. This is clearly NMR jargon. States can be overpopulated or depleted but not hyperpolarized.
 I am not sure I fully understand Figure 8. Why does not polarization come to the equilibrium polarization eventually, even in the case of the Lindbladian equation? From the graph, it looks like it cannot ever come there..

CC2: 'Reply on RC2', Tom Barbara, 13 Apr 2021
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One semantic style approach can invoke the iterminology of "order". So called "Zeeman order", "Quadrupolar order", "Dipolar order" etc. Certainly these are commonly used, or at least they were when NMR was a younger field. Dipolar order will contain a term in Iz1*Iz2. The identity matrix can then be referred to as the "orderfree" state, or something similar.
Malcolm H. Levitt and Christian Bengs
Malcolm H. Levitt and Christian Bengs
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