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

The central object of interest in nuclear magnetic resonance (NMR) theory is the density operator, which describes the quantum state of the ensemble of spin systems. It is defined as follows:

It is often useful to express the density operator as a superposition of orthogonal spin operators. For example, the highly influential papers by Sørensen, Bodenhausen, Ernst and co-workers advocate an expansion in terms of Cartesian product operators

The coefficients

Since such expansions are nearly universal in modern NMR theory, it seems natural to pose questions of the forms “what values may the coefficients

The expansion in Eq. (

In addition, we express some views on the nature and definition of hyperpolarization. For example, is pure parahydrogen hyperpolarized, even though it generates no NMR signal? (Spoiler: our answer is yes.)

To facilitate the discussion, the basis operators

In general,

Brief consideration shows that there must be limits to the physical region of Liouville space. Consider for example an ensemble of isolated spins-1/2. In this case, the dimension of Hilbert space is

The populations of the two Zeeman states are given by

In the case of isolated spins-1/2, the physical bounds on Liouville space are therefore defined by the fixing of one coordinate (

What about systems other than spins-1/2? Liouville space has more than three dimensions in such cases and is hard to visualize. Nevertheless, it is tempting to assume that the physical bounds are still spherical, albeit with an extension to higher dimensions. However, this turns out to be incorrect, in general. The physical bounds in some of the dimensions of Liouville space turn out not to be spheres but

The physical boundary of Liouville space is of little consequence for “conventional” NMR experiments, which are performed at or near thermal equilibrium. Under ordinary temperatures and magnetic fields, this is a region very close to the origin of Liouville space (except for the fixed projection onto the unity operator) and hence very far from the boundary. However, hyperpolarization techniques such as optical pumping

Furthermore, some familiar concepts in magnetic resonance which were originally developed in the context of near-equilibrium spin dynamics do not retain validity far from the origin. An important case is the inhomogeneous master equation

For an ensemble of isolated spins-

The normalized spherical tensor operators

A semantic objection may be raised over the use of the term

For isolated spins-

It follows from Eqs. (

For isolated spins-

The rank-1 polarization moment

Multipole expansions of the spin density operator as in Eq. (

There are also techniques for determining the polarization moments of a spin ensemble

The construction of spherical tensor operators for systems of coupled spins is a complicated affair. Extensive expositions of the technique have been given

The operator

The operator

Since the moments

The operator

The operator

The population of each individual spin state is bounded by

The spin density operator of the isolated spin-

The consequences are now explored for some common spin systems.

For isolated spins-1/2, the rank-0 polarization moment is given from Eq. (

The bounds on the polarization moments are more complicated for an ensemble of isolated spins-1. The rank-0 polarization moment is given through Eq. (

Physical bounds on the rank-1 and rank-2 polarization moments for isolated spins

The equilateral triangle in Fig.

The maximum

Physical bounds on the rank-1, rank-2, rank-3 polarization moments for isolated spins

In the case of isolated spins-3/2, the rank-0 polarization moment is given by

The tetrahedron in Fig.

The treatment above is readily extended to higher spins. For isolated spins-

Physical bounds on the rank-0 polarization moment

For spin-1/2 pairs, the symmetric

The physical bounds on the symmetrical polarization moments

The highlighted point in Fig.

Physical bounds on the rank-0 polarization moment

A projection of the tetrahedral bound in Fig.

The blue point in Fig.

The physical bounds depicted in Figs.

The geometry of the physical bounds is independent of the operator basis. Although a spherical tensor operator basis has been used in the discussion above, bounds of the same form are generated in any orthonormal operator basis, albeit with an overall rotation that depends on the relationship of the two bases. For example, if the ket–bra operator products

von Neumann entropy

Quantum statistical mechanics uses the

The von Neumann entropy

The von Neumann entropy

For isolated spins-1, the von Neumann entropy is a function of the rank-1 and rank-2 polarization moments, assuming that all polarization moments

The behaviour of the von Neumann entropy is readily anticipated for higher spin quantum numbers. The entropy vanishes at the

The coloured arc shows the set of thermal equilibrium density operators for spins-1 subjected to a dominant magnetic field. The spin temperature is indicated by colour, progressing from high (red) to low (blue). The black dot indicates the thermal equilibrium density operator at a particular temperature

Thermal equilibrium with the environment at temperature

In many cases, the coherent Hamiltonian is dominated by the Zeeman interaction with the main magnetic field,

The coloured arc in Fig.

The von Neumann entropy in thermal equilibrium at temperature

The criterion in Eq. (

Being inside the dark grey region is a sufficient but not necessary criterion of hyperpolarization. Points outside the dark grey region but within the pale blue region might also represent hyperpolarized states, in the case that polarization moments which are not represented in the diagram, i.e.

The criterion in Eq. (

Non-equilibrium spin dynamics for an ensemble of spin-1/2 pairs, with

The dynamics of the spin density operator is governed by a differential equation called the

The inhomogeneous master Eq. (

This point is reinforced by Fig.

The red dashed line shows the trajectory predicted by the IME, in the case that

The predicted trajectory of the Lindbladian master equation, as described in

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 partial guide for wanderers in this region.

The word “partial” is used deliberately. So far, we have concentrated on the aspects of Liouville space which concern populations, and in the case of spin-1/2 pairs, on operators that are exchange-symmetric. The map still needs to be completed by delineating the physical bounds on coherences and on operators for multiple-spin systems, including those that are not exchange-symmetric. There has already been significant progress in that direction

The physical bounds discussed in this article should not be confused with the bounds on the unitary transformations of density operators

The software code for the graphics shown in this paper is available from the authors on reasonable request.

No data sets were used in this article.

MHL and CB contributed theory. MHL wrote the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Geoffrey Bodenhausen Festschrift”. It is not associated with a conference.

This paper is dedicated to the memory of Kostya Ivanov: our collaboration was fruitful and inspiring, but too brief. We also thank James Eills, Jean-Nicolas Dumez, Thomas Schulte-Herbrüggen, Dima Budker, and Phil Kuchel for discussions. Malcolm H. Levitt would like to express his heartfelt thanks to Geoffrey Bodenhausen for many years of friendship, guidance, support, encouragement, illuminating insights, and furious arguments.

This research has been supported by the Research Councils UK (grant no. EP/P009980/1) and the H2020 European Research Council (grant no. 786707-FunMagResBeacons).

This paper was edited by Daniel Abergel and reviewed by two anonymous referees.