Multipartite Quantum States and their Marginals

Chapter 1 Introduction

The pure state of a quantum system is described by a vector in a Hilbert space, or, more precisely, by a point in the corresponding projective space. Since the Hilbert space for multiple particles is given by the tensor product of the Hilbert spaces of the individual particles, its dimension grows exponentially with the number of particles. This exponential behavior is the key obstruction to the classical modeling of quantum systems. The observation is as old as quantum theory itself, and physicists ever since have tried to find ways around it. One way to address the aforementioned exponential complexity is to make use of the following simple yet powerful observation: Important physical properties often do not depend on the whole wave function but rather only on a small part, namely the reduced density matrix, or quantum marginal, of a few particles [Löw55]. For instance, the ground state energy of a spin chain is given by a minimization over nearest-neighbor reduced density matrices (Figure 1.1). In quantum chemistry, the binding energy of a molecule is similarly given by a minimization over two-electron reduced density matrices arising from many-electron wave functions. Mathematically, the reduced density matrix is the contraction of (or trace over) the indices of the projection operator onto the wave function over the remaining particles.

Not every collection of reduced density matrices can arise as the marginals of a quantum state—there are profound “kinematic” constraints that are purely due to the geometry of the quantum state space. The fundamental problem of characterizing the compatibility of reduced density matrices is known as the quantum marginal problem in quantum information theory and as the $n$-representability problem in quantum chemistry (Figure 1.2). It has been long recognized for its importance in many-body quantum physics and quantum chemistry [Col63, Rus69, CY00, Col01]. Unfortunately, the general problem is $\mathbf{QMA}$-complete and therefore $\mathbf{NP}$-hard, and so believed to be computationally intractable, even on a quantum computer [Liu06, LCV07]. However, even a partial understanding of the problem has proved to be immensely useful. Entropy inequalities such as the strong subadditivity of the von Neumann entropy [LR73], which constrain the reduced density matrices of a quantum state, are indispensable tools in quantum statistical physics and quantum information theory [OP93]. In computational quantum physics, the power of variational methods can be explained by their ability to reproduce the marginals of the ground state [VC06]. The fundamental Pauli exclusion principle [Pau25, Pau46], which states that the occupation numbers of a fermionic quantum state cannot exceed one, can be understood as a constraint on the one-body reduced density matrix.

The aim of this thesis is a systematic and rigorous study of the relation between multipartite quantum states and their marginals, which we carry out by using a diverse set of mathematical tools. It is naturally divided into two parts: Chapters 2–6 are concerned with one-body reduced density matrices and Chapters 7–9 with general marginals. Each part starts with an initial chapter that introduces background material. The subsequent chapters then present our research contributions; each begins with a summary of the main results that are obtained in the chapter and concludes with a discussion of the results presented. We now give a brief overview of the contents of the individual chapters.

In Chapter 2 we formally introduce the one-body quantum marginal problem, i.e., the problem characterizing the one-body reduced density matrices that are compatible with a global pure state. We describe the fundamental connection of this problem to geometric invariant theory, which is an appropriate mathematical framework for its study, and explain the physical consequences of the mathematical theory. For any given number of particles, local dimensions and statistics, there exists a finite set of linear inequalities that constrain the eigenvalues of compatible one-body reduced density matrices; these inequalities together cut out a convex polytope, known as a moment polytope in mathematics. The facets of this polytope acquire a physical interpretation through associated “selection rules”. We also discuss a dual, representation-theoretic description of the polytope. Many aspects are clarified greatly by using the appropriate perspective.

In Chapter 3 we first review some of the history of the one-body quantum marginal problem, which has seen some significant progress in recent years, culminating in Klyachko’s general solution. Along the way we give some concrete examples. We then present a different, geometric approach to the computation of moment polytopes, which is inspired by recent work of Ressayre. Significantly, our approach completely avoids many technicalities that have appeared in previous solutions to the problem, and it can be readily implemented algorithmically.

Chapter 4is devoted to the phenomenon of quantum entanglement, which profoundly influences the relation between a quantum system and its parts. We find that in the case of multipartite pure states, features of the entanglement can already be extracted from the local eigenvalues—the natural generalization of the Schmidt coefficients or entanglement spectrum. To study this systematically, we associate with any given class of entanglement an entanglement polytope, formed by the eigenvalues of the one-body marginals compatible with the class. In this way we obtain local witnesses for the multipartite entanglement of a global pure state. Our construction is applicable to systems of arbitrary size and statistics, and we explain how it can be adapted to states that are affected by low levels of noise.

In Chapter 5 we consider the following quantitative version of the one-body quantum marginal problem: Given a pure state chosen uniformly at random, what is the joint probability distribution of its one-body reduced density matrices? We obtain the exact probability distribution by reducing to the corresponding distribution of diagonal entries, which corresponds to a quantitative version of a classical marginal problem. This reduction is an instance of a more general “derivative principle” for Duistermaat–Heckman measures in symplectic geometry.

In Chapter 6 we digress in a brief interlude into a study of multiplicities of irreducible representations of compact, connected Lie groups. The asymptotic growth of such multiplicities in a “semiclassical limit” is directly related to the probability measures considered in the preceding chapter. We show that the ideas of the preceding chapter can be discretized, or “quantized”, to give an efficient algorithm for the branching problem, which asks for the multiplicity of an irreducible representation of a subgroup $K\subseteq K^{\prime}$ in the restriction of an irreducible representation of $K^{\prime}$. In particular, we obtain the first polynomial-time algorithm for computing Kronecker coefficients for Young diagrams of bounded height. There is a surprising connection between our results on entanglement polytopes and multiplicities to recent efforts in the geometric complexity approach to the $\mathbf{P}$ vs. $\mathbf{NP}$ problem in computer science. We sketch this connection and explain some additional observations regarding the relevance of asymptotics.

In Chapter 7 we initiate our study of general quantum marginals, motivated by the fundamental role of entropy in physics and information theory. Like the marginals themselves, these entropies are not independent; instead, they are constrained by linear entropy inequalities – the “laws of information theory” – such as the strong subadditivity of the von Neumann entropy, which is an indispensable tool in the analysis of quantum systems. A major open question is to decide if there are any further entropy inequalities satisfied by the von Neumann entropy that are not a consequence of strong subadditivity. Classically, such entropy inequalities have been found for the Shannon entropy, and the discovery of any further entropy inequality would be considered a major breakthrough.

In Chapter 8 we describe a first approach to the study of entropy inequalities. We consider two classes of quantum states – stabilizer states and Gaussian states – which are versatile enough to exhibit intrinsically quantum features, such as multipartite entanglement, but possess enough structure to allow for a concise and computationally efficient description. Quantum phase-space methods have been built around both classes of states, and we show how they can be used to construct a classical model that can be used to lift entropy inequalities for the Shannon entropy to quantum entropies. In particular, our technique immediately implies that the von Neumann entropy of stabilizer states satisfies all conjectured entropy inequalities.

In Chapter 9 we introduce a second approach, which is applicable to general quantum states. To this end, we unveil a novel connection between the existence of multipartite quantum states with given marginal eigenvalues and the representation theory of the symmetric group. We use this connection to give a new proof of the strong subadditivity and weak monotonicity of the von Neumann entropy, and propose a general approach to finding further entropy inequalities based on studying representation-theoretic symbols and their symmetry properties.

The list of symbols (pp. Multipartite Quantum States and their Marginals–List of Symbols) summarizes the most important notation used throughout this thesis. This introduction has been adapted from [CDKW14]. Earlier versions of Figures 1.1 and 1.2 have been used in several presentations by Matthias Christandl and the author. Most of the material in this thesis has been assembled from the works [WDGC13, CDKW14, CDW12, GW13, CŞW12], and we give the corresponding references at the beginning of each chapter.

Chapter 2 The One-Body Quantum Marginal Problem

In this chapter we formally introduce the one-body quantum marginal problem and discuss some fundamental properties. We recall some basic concepts from the theory of Lie groups and their representations that are used throughout this thesis. Next, we introduce the connection to geometric invariant theory, which is the appropriate mathematical framework for the study of the one-body quantum marginal problem and its variants. We then explain the physical consequences of the mathematical theory for the quantum marginal problem and conclude by discussing the dual, representation-theoretic description in terms of Kronecker coefficients. None of the results in this chapter are new; in each section we give pointers to relevant background literature.

Composite quantum systems are modeled by the tensor product of the Hilbert spaces describing their constituents. Throughout this thesis, we will assume that all Hilbert spaces are finite-dimensional unless stated otherwise. It is useful to think of the constituents as individual particles, although they can be of more general nature; for instance, the subsystems can describe different degrees of freedom such as position and spin. Depending on whether the particles are in principle distinguishable or indistinguishable, we distinguish two basic classes of composite systems, which are of fundamentally different nature.

In the case of distinguishable particles, the system is described by the tensor-product $\mathcal{H}=\bigotimes_{k=1}^{n}\mathcal{H}_{k}$ of the Hilbert spaces describing the individual particles. Given a density matrix $\rho$ on $\mathcal{H}$, the one-body reduced density matrices $\rho_{k}$ are defined by taking the partial trace of $\rho$ over all subsystems other than $k$. In other words,

In physical terms, (2.1) asserts that $\rho_{k}$ reproduces faithfully the expectation values of all local observables $O_{k}$. Hence $\rho_{k}$ describes the effective state of the $k$-th particle. The one-body quantum marginal problem then is the following compatibility problem:

Let $\mathcal{H}=\bigotimes_{k=1}^{n}\mathcal{H}_{k}$. Given density matrices $\rho_{k}$ on $\mathcal{H}_{k}$ for all $k=1,\dots,n$, does there exist a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$ on $\mathcal{H}$ such that $\rho_{k}$ are its one-body reduced density matrices?

We will call such density matrices $\rho_{1},\dots,\rho_{n}$ compatible (with a global pure state). The term “quantum marginal problem” has been coined by Klyachko in analogy to the classical marginal problem in probability theory, which asks for the existence of a joint probability distribution for a given set of marginal distributions [Kly04]. Its one-body version was first solved in the paper [Kly04]; cf. [DH04].

For two particles, $n=2$, the one-body quantum marginal problem is rather straightforward to solve. For this, recall that any vector $\ket{\psi}$ on a tensor product $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ can be expanded in the form

for orthonormal sets of vectors $\ket{i}$ in $\mathcal{H}_{1}$ and $\ket{\widetilde{i}}$ in $\mathcal{H}_{2}$ and positive numbers $p_{j}>0$. In quantum information theory this is called the Schmidt decomposition; it is a simple consequence of the singular value decomposition in linear algebra. Thus if $\rho=\lvert\psi\rangle\langle\psi\rvert$ is a pure state then it follows that $\rho_{1}=\sum_{i}p_{i}\lvert i\rangle\langle i\rvert$ and $\rho_{2}=\sum_{i}p_{i}\lvert\widetilde{i}\rangle\langle\widetilde{i}\rvert$ have the same non-zero eigenvalues, including multiplicities (and indeed the same spectrum if $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are of the same dimension). Conversely, for any two such density matrices $\rho_{1}$ and $\rho_{2}$, we can always use (2.2) with the respective eigenbases to define a corresponding global pure state. We record for future reference:

Any two density matrices $\rho_{1}$ and $\rho_{2}$ are compatible with a global pure state if and only if $\rho_{1}$ and $\rho_{2}$ have the same non-zero eigenvalues (including multiplicities), i.e., if and only if $\vec{\lambda}_{1}\setminus\{0\}=\vec{\lambda}_{2}\setminus\{0\}$.

In particular, any density matrix $\rho_{1}$ can be realized as the reduced density matrix of a pure state. In quantum information, such a pure state is called a purification of the density matrix $\rho_{1}$.

An important consequence of Chapter 2 is that $n+1$ spectra $\vec{\lambda}_{1}$, …, $\vec{\lambda}_{n}$, $\vec{\mu}$ are compatible with a pure state if and only if $\vec{\lambda}_{1}$, …, $\vec{\lambda}_{n}$ are compatible with a global state of spectrum $\vec{\mu}$. Therefore there is no loss of generality in restricting to pure states in our formulation of Chapter 2.

Another useful corollary is that the one-body quantum marginal problem for an arbitrary number of particles can always be reduced to the case $n=3$: A given collection of spectra $\vec{\lambda}_{1},\dots,\vec{\lambda}_{n}$ is compatible if and only if there exists a spectrum $\vec{\mu}$ such that both $\vec{\lambda}_{1},\vec{\lambda}_{2},\vec{\mu}$ as well as $\vec{\mu},\vec{\lambda}_{3},\dots,\vec{\lambda}_{n}$ are compatible, and this process can be iterated. This is immediate from the preceding and Chapter 2, which also shows that the rank of $\vec{\mu}$ can be bounded by the minimum of $\dim\mathcal{H}_{1}\times\dim\mathcal{H}_{2}$ and $\prod_{k=3}^{n}\dim\mathcal{H}_{k}$.

Identical Particles

In the case of $n$ identical particles, the system is described by the $n$-th symmetric or antisymmetric tensor power of the single-particle Hilbert space, $\mathcal{H}=\Sym^{n}\mathcal{H}_{1}$ or $\mathcal{H}=\bigwedge^{n}\mathcal{H}_{1}$ depending on whether the particles are bosons or fermions. By considering $\mathcal{H}$ as a subspace of $\mathcal{H}_{1}^{\otimes n}$, we can define the one-body reduced density matrices as in the case of distinguishable particles. Of course, $\rho_{1}=\dots=\rho_{n}$, since for bosons as well as for fermions the global state $\rho$ is permutation-invariant. In summary,

where in the last expression $a_{i}^{\dagger}$ and $a_{j}$ denote the creation and annihilation operators with respect to an arbitrary basis $\ket{i}$ of the single-particle Hilbert space.

For fermions, we thus arrive at the following variant of Chapter 2:

Given a density matrix $\rho_{1}$ on $\mathcal{H}_{1}$, does there exist a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$ on $\bigwedge^{n}\mathcal{H}_{1}$ such that $\rho_{1}$ is its one-body reduced density matrix?

We will call such a density matrix $\rho_{1}$ $n$-representable. In the context of second quantization, it is often more convenient to normalize the one-body marginal to trace $n$. Following quantum chemistry conventions, we correspondingly set $\gamma_{1}:=n\rho_{1}$ and call it the first-order density matrix [Löw55] (but remark that it is not a density matrix in the strict sense). In quantum chemistry, the diagonal entries of $\gamma_{1}$ are called occupation numbers, while its eigenvalues are called the natural occupation numbers. As in the case of distinguishable particles, Chapter 2 depends only on the eigenvalues of $\rho_{1}$, or, equivalently, on the natural occupation numbers of $\gamma_{1}$. In this language, the Pauli exclusion principle asserts that the natural occupation numbers, and hence all occupation numbers, never exceed one [Pau25]. This is obvious from second quantization, since $\braket{i|\gamma_{1}|i}=\tr\rho\,a_{i}^{\dagger}a_{i}\leq 1$. Equivalently, the largest eigenvalue of an $n$-representable one-body density matrix is at most $1/n$. However, there are many more constraints on the natural occupation numbers of a pure state of $n$ fermions [BD72, KA08].

Chapter 2can also be formulated for bosons. Here it can be shown that the resulting problem is in fact trivial: Any density matrix $\rho_{1}=\sum_{i}p_{i}\lvert i\rangle\langle i\rvert$ arises as the one-body reduced density matrix of a pure $\rho=\lvert\psi\rangle\langle\psi\rvert$ state on the symmetric subspace, e.g., $\ket{\psi}=\sum_{i}\sqrt{p_{i}}\ket{i}^{\otimes n}$ [KA08].

Further variants of the one-body quantum marginal problem may arise from physical or mathematical considerations. For instance, the analysis of fermionic systems with several internal degrees of freedom leads to the study of other irreducible representations besides the symmetric or antisymmetric subspace [KA08]. We will discuss one such example at the end of Section 3.4. We may also combine systems composed of different species of particles, some of them indistinguishable among each other. On a mathematical level, this situation also arises when the “purification trick” that we used to restrict to global pure states in the formulation of Chapter 2 is applied to systems of identical particles. The mathematical framework that we outline in the subsequent sections subsumes all these variants of the marginal problem.

1 Lie Groups and their Representations

Before we proceed it will be useful to recall some fundamental notions from the theory of Lie groups and their representations. We illustrate the general theory in the important case of the unitary groups and their complexification, the general linear groups, and summarize the notation in Table 2.1. We refer to [Kna86, FH91, CSM95, Kna02, Pro07, Bri10] for comprehensive introductions to the subject.

The Lie group $G$ acts on itself by conjugation, $g.h:=ghg^{-1}$. By taking the derivative, we obtain the adjoint representation $\Ad\colon G\rightarrow\GL(\mathfrak{g})$ of $G$ on $\mathfrak{g}$. Its differential is the representation of the Lie algebra $\mathfrak{g}$ on itself by the Lie bracket, $\ad(X)Y=[X,Y]$. By decomposing the adjoint representation into weight spaces $\mathfrak{g}_{\alpha}$ and observing that $\mathfrak{g}_{0}=\mathfrak{h}$, we obtain

where $R_{G}:=\{\alpha\neq 0:\mathfrak{g}_{\alpha}\neq 0\}\subseteq\Lambda^{*}_{G}$ is the set of non-trivial weights of the adjoint representation, called the roots. The corresponding weight spaces

are called root spaces. For an arbitrary representation $V$ we have that $\pi(\mathfrak{g}_{\alpha})V_{\omega}\subseteq V_{\omega+\alpha}$; in particular, $[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subseteq\mathfrak{g}_{\alpha+\beta}$. All root spaces $\mathfrak{g}_{\alpha}$ are one-dimensional, and for each root $\alpha$, $-\alpha$ is also a root. We can find basis vectors $E_{\alpha}\in\mathfrak{g}_{\alpha}$ and elements $Z_{\alpha}\in i\mathfrak{t}$, called co-roots, such that

The “Pauli matrices” $X_{\alpha}:=E_{\alpha}+E_{-\alpha}$ and $Y_{\alpha}:=i(E_{-\alpha}-E_{\alpha})$ are a basis of $i\mathfrak{k}_{\alpha}$; they satisfy the commutation relations

Now choose a decomposition $R_{G}=R_{G,+}\sqcup R_{G,-}$ of the set of roots into positive and negative roots. That is, $R_{G,-}=-R_{G,+}$ and each subset is strictly contained in a half-space of the (real) span of the roots (which is equal to $i\mathfrak{t}^{*}$ if $G$ is semisimple). Then we have a decomposition

where the $\mathfrak{n}_{\pm}$ are nilpotent Lie algebras; the corresponding Lie groups $N_{\pm}\subseteq G$ are called maximal unipotent subgroups.

Another consequence of the choice of positive roots is the following. Consider the dual of the adjoint representation of $G$ on $\mathfrak{g}$, given by $(\Ad^{*}(g)\varphi)(X)=\varphi(\Ad(g^{-1})X)$ for all $g\in G$, $\varphi\in\mathfrak{g}^{*}$ and $X$ in $\mathfrak{g}$. It is not hard to see that its restriction to $K$ preserves the real subspace

and we shall call it the coadjoint representation of $K$ on $i\mathfrak{k}^{*}$ (our choice of factor $i$ is somewhat idiosyncratic but will be rather convenient in the sequel). We may consider $\mathfrak{h}^{*}\subseteq\mathfrak{g}^{*}$ and $i\mathfrak{t}^{*}\subseteq i\mathfrak{k}^{*}$ by extending each functional by zero on the root spaces $\mathfrak{g}_{\alpha}$. Then the positive Weyl chamber

is a cross-section for the coadjoint action of $K$. In other words, each coadjoint orbit $\Ad^{*}(K)\varphi$ intersects the positive Weyl chamber in a single point $\lambda\in i\mathfrak{t}^{*}_{+}$. We write $\mathcal{O}_{K,\lambda}:=\Ad^{*}(K)\lambda$ for the coadjoint orbit through $\lambda$. The positive Weyl chamber is a convex cone (pointed if $G$ is semisimple). Its (relative) interior is

For any $\lambda\in i\mathfrak{t}^{*}_{>0}$, the $K$-stabilizer is the maximal torus $T$, so that $\mathcal{O}_{K,\lambda}\cong K/T$.

The last piece of structure is the Weyl group $W_{K}=N_{K}(T)/T$, where $N_{K}(T)=\{k\in K:kT=Tk\}$ denotes the normalizer of the maximal torus $T\subseteq K$. It is a finite group that acts on $i\mathfrak{t}^{*}$. For any representation $V$, the action of the Weyl group leaves the set of weights invariant. In particular, the set of roots is left invariant. The Weyl group acts simply transitively on the set of Weyl chambers obtained from different choices of positive roots. In particular, every $W_{K}$-orbit in $i\mathfrak{t}^{*}$ has a unique point of intersection with the positive Weyl chamber $i\mathfrak{t}^{*}_{+}$, and there exists a Weyl group element, known as the longest Weyl group element $w_{0}$, that exchanges the positive and negative roots and hence sends the “negative Weyl chamber” $-i\mathfrak{t}^{*}_{+}$ to $i\mathfrak{t}^{*}_{+}$. More generally, one can define the length $l(w)$ of a Weyl group element as the minimal number of certain standard generators required to write $w$, but we will not need this level of generality.

Representation Theory

Let $\Pi\colon G\rightarrow\GL(V)$ be a finite-dimensional representation of $G$, with infinitesimal representation $\pi\colon\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ and weight space decomposition $V=\bigoplus_{\omega}V_{\omega}$. A weight vector $\ket{\psi}\in V_{\omega}$ is called a highest weight vector if $\Pi(N_{+})\ket{\psi}\equiv\ket{\psi}$, or, equivalently, if $\pi(\mathfrak{n}_{+})\ket{\psi}=0$. The corresponding highest weight $\omega$ is necessarily dominant, i.e., an element of $\Lambda^{*}_{G,+}:=\Lambda^{*}_{G}\cap i\mathfrak{t}^{*}_{+}$.

The fundamental theorem of the representation theory of compact connected Lie groups $K$ asserts that the irreducible representations of $G$ can be labeled by their highest weight: Any irreducible representation contains a highest weight vector $\ket{\lambda}$, unique up to multiplication by a scalar. Conversely, for every $\lambda\in\Lambda^{*}_{G,+}$ there exists a unique irreducible representation $V_{K,\lambda}$ with $\lambda$ as the highest weight. The dual representation $V^{*}_{K,\lambda}$ of an irreducible representation is again irreducible, and its highest weight is $\lambda^{*}:=-w_{0}\lambda$, where $w_{0}$ is the longest Weyl group element as defined above. Like any finite-dimensional representation of $K$, $V_{K,\lambda}$ extends to a rational representation $V_{G,\lambda}$ of the algebraic group $G$, i.e., a representation whose matrix elements are given by rational functions on the algebraic group $G$ (that is, by morphisms of algebraic varieties, which is the appropriate notion in this context). All irreducible rational representations of $G$ can be obtained in this way. Therefore, the representation theory of $K$ and of $G$ are essentially equivalent.

An arbitrary $G$-representation $V$ can always be equipped with a $K$-invariant inner product (choose an arbitrary inner product and average). In this case, $\Pi(K)\subseteq\U(V)$, and so $\pi(\mathfrak{k})$ consist of anti-Hermitian and $\pi(i\mathfrak{k})$ of Hermitian operators. Moreover, $V$ can always be decomposed into irreducible representations and the irreducible representations that occur in $V$ are in one-to-one correspondence with the highest weight vectors in $V$ (up to rescaling). In particular, the subspace $V^{G}=\{v\in V:\Pi(g)v=v\;(\forall g\in G)\}$ of invariant vectors is the sum of all trivial representations that occur in $V$.

The General Linear and Unitary Groups

The general linear group $G=\GL(d)$ of invertible $d\times d$-matrices is a connected reductive algebraic group, with Lie algebra $\mathfrak{g}=\mathfrak{gl}(d)$ the space of complex $d\times d$-matrices. The Lie bracket is the usual commutator, $[X,Y]:=XY-YX$, and the exponential map is the usual matrix exponential. The unitary group $K=\U(d)$, whose elements are unitary $d\times d$-matrices, is a maximal compact subgroup, with Lie algebra $\mathfrak{k}=\mathfrak{u}(d)$ the space of anti-Hermitian matrices. Thus $i\mathfrak{k}$ is the set of Hermitian matrices, which we may identify with $i\mathfrak{k}^{*}$ by using the Hilbert–Schmidt inner product.

The roots of $\GL(d)$ are the functionals $\alpha_{ij}(H)=H_{i,i}-H_{j,j}$, and the corresponding root spaces $\mathfrak{g}_{ij}$ are spanned by the elementary matrices $E_{ij}=\lvert i\rangle\langle j\rvert$ that have a single non-zero entry in the $i$-th row and $j$-th column. Indeed, we have that $[H,\lvert i\rangle\langle j\rvert]=(H_{i,i}-H_{j,j})\lvert i\rangle\langle j\rvert$ for any diagonal matrix $H=\sum_{j}H_{j,j}\lvert j\rangle\langle j\rvert\in\mathfrak{h}$. A choice of positive roots is given by those roots $\alpha_{ij}$ with $i<j$. Thus the nilpotent Lie algebras $\mathfrak{n}_{\pm}$ consist of the strictly upper and lower triangular matrices, respectively. The corresponding unipotent subgroups $N_{\pm}$ are upper and lower triangular with ones on the diagonal.

The adjoint action is by conjugation. If we identify $i\mathfrak{k}\cong i\mathfrak{k}^{*}$ then the coadjoint orbits of $K=\U(d)$ consist of Hermitian matrices with fixed spectrum. Then the positive Weyl chamber $i\mathfrak{t}^{*}_{+}$ can be identified with the set of Hermitian diagonal matrices whose entries are weakly decreasing, or with their spectra $\lambda_{1}\geq\dots\geq\lambda_{d}$. The assertion that $i\mathfrak{t}^{*}_{+}$ is a cross-section for the coadjoint action of $K$ on $i\mathfrak{k}^{*}$ corresponds to the plain fact that any Hermitian matrix can be diagonalized by a unitary. Its interior $i\mathfrak{t}^{*}_{>0}$ then corresponds to the set of non-degenerate spectra $\lambda_{1}>\dots>\lambda_{d}$. The claim that the $K$-stabilizer of any $\lambda\in i\mathfrak{t}^{*}_{>0}$ is $T$ amounts to the fact that the only unitaries that commute with a diagonal matrix with non-degenerate spectrum are the diagonal unitary matrices.

Finally, the Weyl group can be identified with the symmetric group $S_{d}$; it acts on $i\mathfrak{t}$ by permuting diagonal entries. The length $l(w)$ of a permutation $w$ is the number of transpositions $(i\;i\!+\!1)$ required to write the permutation $w\in S_{d}$, and the longest Weyl group element $w_{0}$ is the “order-reversing permutation” which sends any $-\lambda\in-i\mathfrak{t}^{*}_{+}$ to $\lambda^{*}:=(-\lambda_{d},\dots,-\lambda_{1})\in i\mathfrak{t}^{*}_{+}$.

The first diagram in the example has two rows and four boxes, while the second diagram has three boxes as well as rows. The number of rows is also called the height of a Young diagram $\lambda$. We write $\lambda\vdash_{d}k$ for a Young diagram with $k$ boxes and at most $d$ rows.

2 Geometric Invariant Theory

In this section, we introduce some geometric invariant theory, which is a powerful mathematical framework for studying the one-body quantum marginal problem and its variants. We refer to [Kir84a, MFK94, Bri10, Woo10, VB11, GRS13] for further material.

where $\rho^{\prime\prime}=\lvert\psi^{\prime\prime}\rangle\langle\psi^{\prime\prime}\rvert$. In other words, the traceless part of each one-body reduced density matrix $\rho^{\prime\prime}_{k}$ vanishes. We conclude that each one-body reduced density matrix $\rho^{\prime\prime}_{k}$ is proportional to the identity matrix. In the language of quantum information theory, the quantum state $\rho^{\prime\prime}$ is locally maximally mixed. This way of reasoning establishes a first link between the existence of invariants and of pure states with prescribed marginals. In the following we will see that the above argument can be generalized to arbitrary one-body marginals and turned into an equivalence that completely characterizes the one-body quantum marginal problem and its variants. We follow along the lines of the exposition in [VB11] and take some ideas from [NM84, Bri87].

Mathematically, the set of pure states on a Hilbert space $\mathcal{H}$,

is known as a complex projective space. It is a smooth submanifold of the real vector space of Hermitian operators on $\mathcal{H}$. The unitary group $\U(\mathcal{H})$ acts transitively by conjugation, $U\lvert\psi\rangle\langle\psi\rvert U^{\dagger}$, so that the tangent space at a point $\rho=\lvert\psi\rangle\langle\psi\rvert$ is spanned by the tangent vectors $X_{\rho}=[X,\rho]$ for all $X\in\mathfrak{u}(\mathcal{H})$. Since the $X$ are anti-Hermitian, it is easy to verify that the tangent space can be equivalently written as

In this way, the tangent space acquires a complex structure, which can be written as

as well as a Hermitian inner product. The real part of the inner product is a Riemannian metric, $g(V,W)=\tr VW$, and its imaginary part is the Fubini–Study symplectic form

For tangent vectors generated by elements of the Lie algebra $\mathfrak{u}(\mathcal{H})$, this becomes

The action of $\U(\mathcal{H})$ can be extended to its complexification, the general linear group $\GL(\cal H)$ by the formula

The tangent vector generated by a Hermitian matrix $iX\in i\mathfrak{u}(\mathcal{H})$ is then given by $(iX)_{\rho}:=\{iX,\rho\}-2(\tr iX\rho)\rho$, with $\{A,B\}:=AB+BA$ the anti-commutator. It is easily verified that $(iX)_{\rho}=J[X_{\rho}]$. Thus the complex structures of projective space and of the Lie group are compatible with each other.

The Moment Map

Unfortunately, there are as many conventions for the moment map as there are textbooks on the subject. For the representation $\mathcal{H}$ that we considered at the beginning of this section, the moment map maps pure states onto the functionals evaluating (traceless) local observables; cf. (2.7). Thus the one-body quantum marginal problem is equivalent to characterizing the image of a moment map. In Section 2.3 we will explain this connection in more detail.

A crucial property of the moment map is the following relation between the differential of its components and the tangent vector $X_{\rho}=[\pi(X),\rho]$ generated by the infinitesimal action of the Lie algebra of the compact group:

This follows readily from (2.10). Since the Fubini–Study form is non-degenerate, an immediate consequence is that the component (2.13) of the differential vanishes if and only if $X_{\rho}=0$. Dually, we find that the range of the differential of the moment map at any point $\rho$ is given by the annihilator of $\mathfrak{k}_{\rho}:=\{X\in\mathfrak{k}:X_{\rho}=[\pi(X),\rho]=0\}$, the Lie algebra of the $K$-stabilizer of $\rho$ [GS82a]:

Thus its image consists of a union of coadjoint orbits, and so is characterized by its intersection with the positive Weyl chamber $i\mathfrak{t}^{*}_{+}$. In the context of the quantum marginal problem, this amounts to our previous observation that its solution depends only on the eigenvalues of the one-body reduced density matrices.

Our basic argument above showed that any non-zero vector of minimal length in an orbit closure has expectation value zero with respect to all local traceless observables (i.e., the image under the moment map is zero). The following result by Kempf and Ness shows a converse [KN79] (cf. [Kem78, NM84] and Figure 2.1 for an illustration).

Suppose for the sake of finding a contradiction that the $G$-orbit through $\ket{\psi}$ is not closed. Then the Hilbert–Mumford criterion asserts that we can reach a point in the “boundary” $\overline{\Pi(G)\ket{\psi}}\setminus\Pi(G)\ket{\psi}$ by using a single one-parameter subgroup (e.g., [Kra85, p. 171]). More formally, it states that there exists $X\in i\mathfrak{k}$ such that

Let $\ket{\psi}=\sum_{k}\ket{\psi_{k}}$ be the decomposition of $\ket{\psi}$ into eigenvectors of the Hermitian operator $\pi(X)$, with $\ket{\psi_{k}}$ an eigenvector with eigenvalue $k$. Clearly, $\ket{\psi_{k}}=0$ for all $k>0$, since otherwise the limit (2.15) cannot exist. But then

so that $\ket{\psi_{k}}=0$ also for all $k<0$. It follows that $\ket{\psi}=\ket{\psi_{0}}$, i.e. $\pi(X)\ket{\psi}=0$. Thus the one-parameter subgroup in fact leaves the vector invariant,

This is the desired contradiction to (2.15). ∎

In the following we need to study the image of the moment map not only for the set of all pure states but also for certain subsets of projective space. In the context of algebraic geometry, it is natural to consider $G$-invariant projective subvarieties, which we define in the following way (e.g., [Har77]):

To generalize Section 2.2 to arbitrary points in the image of the moment map, we need as the last ingredient the Borel–Weil theorem (see, e.g., [VB11, Lemma 94]).

The Moment Polytope

The following proposition formalizes the fundamental link between the image of the moment map and the decomposition of the ring of regular functions into irreducible representations [GS82b, NM84, Bri87].

Fix $\lambda\in\Lambda^{*}_{G,+}$ and $k>0$. Let $\widetilde{\mathcal{H}}:=\Sym^{k}(\mathcal{H})\otimes V_{G,\lambda^{*}}$. Then

It follows by using the second assertion in (2.16) and $\lambda^{*}=-w_{0}\lambda$ that

It is instructive to apply Section 2.2 to the situation of Section 2.2.

It follows as an immediate consequence that

is a convex polytope with rational vertices, whose rational points are given by

The study of moment maps and their convexity properties has a long history in mathematics. Among the well-known special cases are: The Schur–Horn theorem concerning the diagonal entries of Hermitian matrices with fixed spectrum [Sch23, Hor54]; Kostant’s convexity theorem, which is the generalization to general coadjoint orbits [Kos73]; the Atiyah–Guillemin–Sternberg convexity theorem for torus actions [Ati82, GS82a]; Heckman’s convexity theorem, which considers projections of coadjoint orbits [Hec82]; and Kirwan’s convexity theorem, which is the symplectic analogue of Theorem 2.11 [Kir84b] (cf. [Sja98, Bri99] and the recent monograph [GS05]).

3 Consequences for the Quantum Marginal Problem

We now describe the precise connection between the geometry of the moment map and the one-body quantum marginal problem and draw some general consequences.

where we identify diagonal matrices with non-increasing entries with their spectrum (as in Section 2.1 and Table 2.1).

In practice, the above modeling of the quantum marginal problem has the disadvantage that the moment polytope is always of positive codimension: since $\tr\rho_{k}\equiv\tr\rho=1$, $\Delta_{K}$ is contained in the affine subspace $\sum_{j}\lambda_{k,j}=1$ for all $k=1,\dots,n$. It will usually be more convenient to instead use the special linear and unitary groups, $G=\SL(\mathcal{H}_{1})\times\dots\times\SL(\mathcal{H}_{n})$ and $K=\SU(\mathcal{H}_{1})\times\dots\times\SU(\mathcal{H}_{n})$, so that $i\mathfrak{k}$ consists of tuples of traceless Hermitian matrices. Then the moment map preserves only the traceless part of the one-body reduced density matrices, which avoids the above degeneracy.

For fermions, we similarly choose $\mathcal{H}=\bigwedge^{n}\mathcal{H}_{1}$ and $G=\SL(\mathcal{H}_{1})$ acting by $\Pi(g_{1})=g_{1}^{\otimes n}$. With $K=\SU(\mathcal{H}_{1})$ and using (2.3) we obtain that

where $\gamma_{1}:=n\rho_{1}$ is the first-order density matrix from quantum chemistry with trace $n$ that we had defined below (2.3). Again we find that the one-body $n$-representability problem, Chapter 2, is precisely equivalent to determining the moment polytope.

We can similarly model the other variants of the one-body quantum marginal problem alluded to at the end of the introduction of this chapter by considering different representations of unitary groups or their composition. For example, if $\mathcal{H}_{0}$ is a $K_{0}$-representation describing a pure-state problem then the corresponding mixed-state problem can be studied by taking $\mathcal{H}=\mathcal{H}_{0}\otimes\mathcal{H}_{0}$ and $K=K_{0}\times\SU(\mathcal{H}_{0})$; e.g., the mixed-state problem for fermions amounts to the moment polytope for the $\SU(\mathcal{H}_{1})\otimes\SU(\bigwedge^{n}\mathcal{H}_{1})$-representation $\bigwedge^{n}\mathcal{H}_{1}\otimes\bigwedge^{n}\mathcal{H}_{1}$. In Section 3.4 we discuss another example that involves the marginal problem for the spin and orbital degrees of freedom of a fermionic system. In Table 2.3 we summarize the mathematical modeling of the scenarios of main physical interest.

For all these variants of the one-body quantum marginal problem, Theorem 2.11 immediately implies that the solution is given by a convex polytope, i.e., by linear inequalities on the eigenvalues of the one-body reduced density matrices. For example, Pauli’s original exclusion principle is one such inequality—but in general there are many further constraints. As we will see in several concrete examples in Chapter 3, there is a rich variety of subtle kinematic constraints on the one-body marginals of a multipartite quantum state.

To compute the actual linear inequalities for a given number of particles, statistics and local dimensions is in general a difficult problem that we will study in the next chapter. All known general solutions rely in one way or the other on the invariant-theoretic description of the moment polytope given by Theorem 2.11, including the original solution by Klyachko [Kly04] and the solution that we present in Chapter 3. In the remainder of this section we discuss the physical significance of the facets of the moment polytope, and we then describe more explicitly the representation-theoretic content of Theorem 2.11.

Pinning

An important consequence of the general theory is that the facets of the polytope have a rather particular structure. Before we show this, we record the following useful lemma for future reference.

Consider the symplectic cross section $Y:=\mu_{K}^{-1}(i\mathfrak{t}^{*}_{>0})$. By $K$-equivariance of the moment map, $\mu_{K}$ meets $i\mathfrak{t}^{*}_{>0}$ transversally, so that $Y$ is a smooth manifold with tangent space $T_{\rho}Y=d\mu_{K}^{-1}\big|_{\rho}(i\mathfrak{t}^{*})$ [GS84b, Theorem 26.7]. Thus we may choose any curve in $Y$ that starts with $\rho_{0}=0$ and $\dot{\rho}_{0}=V$. ∎

Let $\rho=\lvert\psi\rangle\langle\psi\rvert$ be a pure state such that $\mu_{K}(\rho)\in i\mathfrak{t}^{*}_{>0}$ is a point on a facet of the moment polytope corresponding to the inequality $(-,H)\geq c$. Then

Since $(\mu_{K}(\rho_{t}),H)\geq c$ is an inequality for the moment polytope, it follows that $(\omega,H)=0$—for otherwise we could walk through the facet! On the other hand, (2.14) shows that

Therefore, $H\in i\mathfrak{t}$ is necessarily an element of the Lie algebra of the $T$-stabilizer of $\rho$, i.e., $H_{\rho}=[\pi(H),\rho]=0$. This implies that $d(\mu_{K},H)\big|_{\rho}=0$ by (2.13), but also that $\ket{\psi}$ is an eigenvector of $\pi(H)$, with corresponding eigenvalue $\braket{\psi|\pi(H)|\psi}=(\mu_{K}(\rho),H)=c$. ∎

In the language of Klyachko, the eigenvalue equation (2.19) is called the selection rule which is satisfied by a quantum state that is pinned to a facet of the moment polytope [Kly09]. Thus pinned states live on a potentially much lower-dimensional subspace of the Hilbert space, with potential implications on the physics.

It is an interesting question if and under which circumstances states in concrete systems are pinned. For example, it is an empirical fact that many molecules are well-explained by assuming that the natural occupation numbers are close to 0 and 1 (pinning), so that the global state can be well-approximated by a Slater determinant (the corresponding selection rule), which is a first step to Hartree–Fock theory and the Aufbau principle. Thus it is not be unreasonable to wonder if approximate pinning might hold for some of the other defining inequalities of the moment polytope. See [Kly09, Kly13] for preliminary investigations in the context of small molecules and magnetism and [SGC13] for a study of pinning in a model with small harmonic interactions.

Crucially, the selection rule is stable at least in an elementary sense. We phrase the following result in terms of the trace norm $\lVert X\rVert_{1}:=\tr\lvert X\rvert$, which has a useful operational meaning (but this choice is completely arbitrary since the proof is based on a purely topological argument):

and $\delta(\varepsilon)\searrow 0$ as $\varepsilon\searrow 0$.

is well-defined. Clearly, $\delta(\varepsilon)$ is monotonic in $\varepsilon$, and $\delta(0)=0$ by Section 2.3.

For concrete applications, it might be interesting to obtain explicit bounds of the form $\delta(\varepsilon)\leq L\varepsilon$. So far this has only been achieved in rather special situations [SGC13, BRGBS13]. It might be possible to obtain a general solution by carefully analyzing the local model for symplectic group actions [GS82a, GS84a, Mar85].

4 Kronecker coefficients, Schur–Weyl duality, and Plethysms

In this section we describe more explicitly the representation-theoretic content of Theorem 2.11 for Problems 2 and 2.

Thus the Kronecker coefficients can be equivalently defined as the dimension of the invariant subspace in a triple tensor product of irreducible representations of the symmetric group:

In particular, we find that each Kronecker coefficient only depends on the triple of Young diagrams rather than the concrete values chosen for $a$, $b$ and $c$ (but of course $a$, $b$ and $c$ have to be chosen at least as large as the number of rows of the Young diagrams). The role of the Kronecker coefficients for the one-body quantum marginal problem has first been observed in [CM06] by using the spectrum estimation theorem (cf. [Kly04, CHM07] and the proof of Theorem 9.6). They also play a fundamental role in representation theory [Ful97] and in Mulmuley and Sohoni’s geometric complexity theory approach to the $\mathbf{P}$ vs. $\mathbf{NP}$ problem in computer science [MS01, MS08, Mul07, BLMW11] (see Section 6.1), and they occur in the “quantum method of types” [Har05]. In Chapter 6 we will give an efficient algorithm for their computation.

There is a different, asymmetric way of defining the Kronecker coefficients that is also quite useful. For this, we recall that the irreducible representations of the symmetric group are self-dual, i.e., $[\lambda]\cong[\lambda]^{*}$ [JK81, §2.1]. Therefore,

for the space of $G$-equivariant linear maps, or $G$-linear maps between two representations $V$ and $W$. For irreducible $V$ and $W$, Schur’s lemma asserts that

It follows that $g_{\alpha,\beta,\gamma}$ can also be defined as the multiplicity of $[\alpha]$ in the tensor product $[\beta]\otimes[\gamma]$ of two irreducible representations of the symmetric group:

In particular, for $[\gamma]=\mathbf{1}$ the trivial representation of $S_{k}$ we obtain that $g_{\alpha,\beta,\mathbf{1}}=\delta_{\alpha,\beta}$. In view of Schur–Weyl duality, this implies that

This equation corresponds to the one-body quantum marginal problem for $n=2$ particles and is therefore the bipartite counterpart of (2.21). The fact that the irreducible representations of the two factors are perfectly paired is the representation-theoretic version of the fact that the marginals of a bipartite pure state are isospectral (Chapter 2)—indeed, the latter is a direct consequence of (2.25) and Theorem 2.11.

Geometric Quantization

Before we proceed, we offer a word of caution for people acquainted with the theory of geometric quantization [GS77, GS84b, Woo92]. Although we formally use a similar mathematical framework as in geometric quantization, the physical interpretation is markedly different. Unlike in geometric quantization, our quantum states do not arise via some quantization procedure from a classical symplectic phase space. On the contrary, in the mathematical modeling of the quantum marginal problem the projective space of pure states corresponds to the classical phase space, while its description in terms of the representations that occur in the ring of regular functions can be seen as its ‘‘quantization’’. The ‘‘semiclassical limit’’ $k\rightarrow\infty$ in which we recover the description of the moment polytope plays a purely purely mathematical role (cf. Section 6.6).

Chapter 3 Solving The One-Body Quantum Marginal Problem

In this chapter we review some of the history of the one-body quantum marginal problem that culminated in Klyachko’s general solution and give some concrete examples. We then present a different approach to the problem of computing moment polytopes for projective space, which we have seen subsumes the one-body quantum marginal problem and its variants. Significantly, our geometric approach completely avoids many technicalities that have appeared in previous solutions to the problem, such as Schubert calculus, and it can be readily implemented algorithmically. We illustrate our method with a number of illustrative examples.

The results in this chapter are based on unpublished joint work in progress with Michèle Vergne.

The history of the quantum marginal problem goes back at least to the late 1950s, where it had been observed that the ground state energy of a two-body Hamiltonian is a function of the two-body reduced density matrices only [Löw55, May55]. The main focus was therefore on the two-body $n$-representability problem—given a two-body density matrix, is it compatible with a state of $n$ fermions [Col63, Rus69, CY00, Col01]? Some results had also been obtained for the one-body marginals. For instance, Coleman proved that a first-order density matrix $\gamma_{1}$ is compatible with a (not necessarily pure) state of $n$ fermions if and only if the natural occupation numbers do not exceed $1$—that is, if and only if the Pauli principle is satisfied [Col63, Theorem 9.3].

(see Figure 3.3). This was perhaps the first non-trivial solution of Chapter 2. Remarkably, the resulting polytope is only three-dimensional; this coincides with the fact that any pure state can be written as a linear combination of only 8 Slater determinants as was proved by Ruskai and Kingsley (while $\dim i\mathfrak{t}^{*}=5$ and $\dim\mathcal{H}=20$). It is interesting to observe that the equality $\lambda_{1}+\lambda_{6}=1$ strengthens the Pauli principle $\lambda_{1}\leq 1$.

(see Figure 3.3). In fact, these inequalities hold for any multipartite quantum state, but they are in general not sufficient for compatibility (see Section 9.7 for an elementary proof based on the variational principle). In the meanwhile, the three-qutrit polytope had already been computed by Franz [Fra02], as was only later recognized.

Subsequently, Bravyi solved the case of mixed states of two qubits by a remarkable explicit argument [Bra04]. Here, the necessary and sufficient conditions are given by

The connection of the one-body quantum marginal problem to representation theory was first observed in [CM06] by using quantum information methods rather than the theory of Section 2.2 (cf. [CHM07]). Shortly after, a completely general solution was given by Klyachko both for distinguishable particles [Kly04] and for fermions [KA08, Alt08]. Almost simultaneously, Daftuar and Hayden had published a solution to the “one-sided” problem that concerns the constraints between the eigenvalues of $\rho_{AB}$ and $\rho_{A}$ [DH04]. Both results build on previous work by Berenstein and Sjamaar [BS00], who used geometric invariant theory to study the moment polytope for projections of coadjoint orbits; this latter work in turn generalizes techniques from Klyachko’s seminal paper on Weyl’s problem [Kly98] (see the discussion at the end of the preceding chapter). We refer to [Kly04, Knu09] for eloquent expositions of the method. More recently, Ressayre has refined the result of Berenstein and Sjamaar to give an irredundant set of necessary and sufficient inequalities in a very general mathematical setup [Res10b, Res10a]. We remark that a variant of the quantum marginal problem for Gaussian states has been considered in [EG08] (mathematically, this scenario is covered by a more general convexity theorem for non-compact manifolds with proper moment maps).

1 Summary of Results

Our approach in the following is based on analyzing the non-trivial facets in terms of the local differential geometry of their preimages up to second order. Conceptually, such an analysis should be sufficient since the moment map is locally quadratic.

In Section 3.2 we start by studying the moment map to first order. It is well-known that interior points of non-trivial facets are critical values for $(\mu_{K},H)$, where $H$ is the normal vector of the facet. Indeed, this is equivalent to the selection rule from Section 2.3. We show that as a consequence any non-trivial facet of $\Delta_{K}$ is necessarily contained in a hyperplane spanned by weights of the representation $\mathcal{H}$. This already reduces the problem to a finite set of candidates.

In Section 3.3 we then consider the Hessian of the moment map. If a state is mapped into the interior of a non-trivial facet of the polytope then this implies a positive semidefiniteness of the corresponding Hessian in certain tangent directions. We exploit this fact to obtain another necessary condition that is satisfied by non-trivial facets $(-,H)\geq c$ of the moment polytope. To state it, let $\mathfrak{n}_{-}(H<0)$ denote the direct sum of negative root spaces $\mathfrak{g}_{\alpha}$ with $(\alpha,H)<0$ and $\mathcal{H}(H<c)$ the sum of eigenspaces of $\pi(H)$ with eigenvalue smaller than $c$. Then we show that there necessarily exists an eigenvector $\ket{\psi}\in\mathcal{H}$ of $\pi(H)$ with eigenvalue $c$ such that the map

is an isomorphism. Conversely, we prove that any inequality $(-,H)\geq c$ that satisfies the above two necessary conditions is a valid inequality of the moment polytope. We thus obtain a complete description of the moment polytope $\Delta_{K}$ in terms of what we call inequalities of Ressayre type (Section 3.3 and Theorem 3.14).

Our notion of a Ressayre-type inequality is closely related to Ressayre’s notion of a dominant pair, and our approach is inspired by his ideas [Res10b, Res10a]. Our description of the moment polytope is also related to a result by Brion [Bri99], as we explain further below. However, while the more refined results of [Bri99, Res10b, Res10a] are established using high-powered algebraic geometry, we proceed in essence by a straightforward differential-geometric analysis, combined with Theorem 2.11.

In Section 3.4, we illustrate the method with some examples. We remark that, crucially, our description of the moment polytope obtained in Theorem 3.14 can be completely automatized: It is straightforward to determine all inequalities of Ressayre type in a mechanical fashion, and hence also on a computer.

2 The Torus Action

A facet of the moment polytope is trivial if it is of the form $(-,Z_{\alpha})\geq 0$ for some positive root $\alpha\in R_{G,+}$. Otherwise, the facet is called non-trivial.

Non-trivial facets have also been called “general” in the literature [Bri99]. We record the following straightforward observation:

Any non-trivial facet of $\Delta_{K}$ meets the relative interior $i\mathfrak{t}^{*}_{>0}$ of the positive Weyl chamber.

Any facet of $\Delta_{K}$ that does not intersect the relative interior of the positive Weyl chamber is fully contained in

which is a finite arrangement of hyperplanes. Since by assumption the facet is of codimension one, it has to be contained in a single one of these hyperplanes. Thus its normal vector is either $\pm Z_{\alpha}$ for some positive root $\alpha$. If it was $-Z_{\alpha}$ then the moment polytope would be strictly contained in the hyperplane $(-,Z_{\alpha})=0$, and therefore not of maximal dimension, in contradiction with our assumption. We conclude that the facet is of the form $(-,Z_{\alpha})\geq 0$, and therefore trivial. ∎

We now consider the moment map for the action of the maximal torus $T\subseteq K$,

For the quantum marginal problem, this amounts to considering diagonal entries rather than eigenvalues. Let $\mathcal{H}=\bigoplus_{\omega\in\Omega}\mathcal{H}_{\omega}$ be the decomposition of $\mathcal{H}$ into weight spaces, and $\rho=\lvert\psi\rangle\langle\psi\rvert$ a pure state with $\ket{\psi}=\sum_{\omega}\psi_{\omega}\ket{\omega}$ decomposed accordingly. Then $\mu_{T}$ has the following concrete description:

The set of critical values of $\mu_{T}$ is equal to the union of the codimension-one convex hulls of subsets of weights.

We now derive a basic necessary condition that cuts down the defining inequalities of the moment polytope to a finite set of candidates.

Any non-trivial facet of $\Delta_{K}$ is contained in an affine hyperplane spanned by a subset of weights.

By Section 3.2, the intersection of any non-trivial facet with the interior of the positive Weyl chamber $i\mathfrak{t}^{*}_{>0}$ is non-empty. Each point in this intersection is a critical value for $(\mu_{K},H)=(\mu_{T},H)$ by selection rule (Section 2.3), hence of $\mu_{T}$, and therefore contained in an affine hyperplane spanned by a subset of weights (Section 3.2). Since this is true for all points in the intersection, which contains the relative interior of the facet, it follows that the facet is in fact contained in a single such hyperplane. ∎

3 Facets of the Moment Polytope

Throughout this section, let $\rho$ be the preimage of a point $\mu_{K}(\rho)\in i\mathfrak{t}^{*}_{>0}$ on the facet $(-,H)\geq c$ of the moment polytope. We have seen that the selection rule Section 2.3 shows that $\rho$ is a critical point of the component $(\mu_{K},H)$, and we have used this in Section 3.2 to gain information on the set of possible facets.

It is thus natural to continue by studying the Hessian of $(\mu_{K},H)$, which is a quadratic form $Q(-,-)$ on the tangent space at $\rho$. Since $\rho$ is a critical point, we can compute it by

for all anti-Hermitian operators $X\in\mathfrak{u}(\mathcal{H})$. It follows that [GS84b, (32.8)]

for all $X,Y\in\mathfrak{u}(\mathcal{H})$ (which is indeed symmetric in $X$ and $Y$). We now decompose

where $\mathcal{H}(H<c)=\bigoplus_{\omega:(\omega,H)<c}\mathcal{H}_{\omega}$ is the sum of the eigenspaces of the Hermitian operator $\pi(H)$ with eigenvalue less than $c$, etc. Then we have the following interpretation of the index of the Hessian (the dimension of a maximal subspace on which the quadratic form $Q$ is negative definite):

The index of the Hessian at $\rho$ is equal to the real dimension of $\mathcal{H}(H<c)$.

Since $\psi$ itself is in $\mathcal{H}(H=c)$ by the selection rule, the claim follows. ∎

Since $(\mu_{K}(\rho),H)=c$ and $d(\mu_{K},H)\big|_{\rho}=0$, the Hessian is necessarily positive semidefinite on the subspace of those tangent vectors that get mapped to $i\mathfrak{t}^{*}$. Indeed, let $V\in d\mu_{K}^{-1}\big|_{\rho}(i\mathfrak{t}^{*})$ and consider the curve $\rho_{t}$ from Section 2.3. Then,

which shows that, indeed, $Q(V,V)\geq 0$. The subspace of all such $V$ can be computed in a different way. For this, let $\mathfrak{r}=\bigoplus_{\alpha\in R_{G,+}}\mathfrak{k}_{\alpha}$ denote the sum of the root spaces of the compact Lie algebra as in (2.5) and

the corresponding subspace of tangent vectors. Then,

The Hessian at $\rho$ is positive semidefinite on the subspace

The first claim is a reformulation of the fact that the $K$-stabilizer of any $\lambda\in i\mathfrak{t}^{*}_{>0}$ is $T$, while $\mathfrak{k}=\mathfrak{t}\oplus\mathfrak{r}$. The second claim follows from the first, since $\mu_{K}(\rho)\in i\mathfrak{t}^{*}_{>0}$ and $d\mu_{K}(R_{\rho})=\ad^{*}(R)\mu_{K}(\rho)$ by equivariance of the moment map. ∎

The tangent space at $\rho$ decomposes as a direct sum

and the Hessian $Q$ is block-diagonal with respect to this decomposition.

For the first claim we only need to show that $M\cap M^{\omega}=0$, which is standard (see, e.g., [GS82a, Lemma 6.7]): Let $R\in\mathfrak{r}$. Suppose that $R_{\rho}\in M^{\omega}$. That is,

For the second claim, observe that (3.6) immediately implies that

for all $R_{\rho}\in M$ and $V\in M^{\omega}$, since $[-iH,R]\in\mathfrak{r}$ and hence $[-iH,R]_{\rho}\in M$. ∎

The index of the Hessian at $\rho$ is equal to twice the number of positive roots $\alpha\in R_{G,+}$ such that $(\alpha,H)>0$.

Since the tangent map $R\mapsto R_{\rho}$ is injective (Section 3.3), we may instead consider the form

on $\mathfrak{r}$. For this, observe that $\widetilde{Q}$ is block-diagonal with respect to $\mathfrak{r}=\bigoplus_{\alpha\in R_{G,+}}\mathfrak{k}_{\alpha}$, since for all $R\in\mathfrak{k}_{\alpha}$ and $S\in\mathfrak{k}_{\beta}$, $[[H,R],S]\in i\mathfrak{k}_{\alpha\pm\beta}$, while $\mu_{K}(\rho)\in i\mathfrak{t}^{*}$. And for each root space $\mathfrak{k}_{\alpha}$, we have that

where we have used the “Pauli matrices” $X_{\alpha},Y_{\alpha},Z_{\alpha}$ and their commutation relations (2.6). Likewise, $\widetilde{Q}(iY_{\alpha},iY_{\alpha})=-2(\alpha,H)(\mu_{K}(\rho),Z_{\alpha})$, while $\widetilde{Q}(iX_{\alpha},iY_{\alpha})=0$. Since $(\mu_{K}(\rho),Z_{\alpha})>0$, we conclude that the index of $Q$ is equal to twice the number of positive roots $\alpha$ with $(\alpha,H)>0$. ∎

Ressayre Elements

So far we have used the Lie algebra $\mathfrak{k}$ of the compact Lie group in our analysis. We will now translate the preceding to the complexified setting. To this end, we consider the Lie algebra $\mathfrak{n}_{-}=\bigoplus_{\alpha\in R_{G,-}}\mathfrak{g}_{\alpha}$ of the negative unipotent subgroup, which plays a role analogous to $\mathfrak{r}$ for states $\rho$ that are mapped into the positive Weyl chamber (compare the following with Section 3.3).

The tangent map $\mathfrak{n}_{-}\rightarrow\mathcal{H},X\to\pi(X)\ket{\psi}$ is injective.

Let $E_{-}:=\sum_{\alpha\in R_{G,+}}z_{\alpha}E_{-\alpha}$ be an arbitrary element in $\mathfrak{n}_{-}$. Since $\pi(E_{\pm\alpha})^{\dagger}=\pi(E_{\mp\alpha})$, we find that $\pi(E_{-})^{\dagger}=\pi(E_{+})$ with $E_{+}:=\sum_{\alpha\in R_{G,+}}\bar{z}_{\alpha}E_{\alpha}$ (the Cartan involution of $E_{-}$). Therefore,

so that by using $\mu_{K}(\rho)\in i\mathfrak{t}^{*}_{>0}$ we find that

In contrast to Section 3.3, which continues to hold true if $\mu_{K}(\rho)$ is mapped to the relative interior of a different Weyl chamber (since $K$ is invariant under the Weyl group), it is important in Section 3.3 to choose the negative unipotent subgroup (relative to the choice of positive Weyl chamber). For example, consider an irreducible $G$-representation $\mathcal{H}=V_{G,\lambda}$ with highest weight $\lambda\in i\mathfrak{t}^{*}_{>0}$ and highest weight vector $\ket{\lambda}$. Then $\mu_{K}(\lvert\lambda\rangle\langle\lambda\rvert)=\lambda\in i\mathfrak{t}^{*}_{>0}$ by Section 2.2, and the “lowering operators” in $\mathfrak{n}_{-}$ act indeed injectively (all $\pi(E_{-\alpha})\ket{\lambda}$ live in different weight spaces and are non-zero, since $\pi(E_{\alpha})\pi(E_{-\alpha})\ket{\lambda}=\pi(Z_{\alpha})\ket{\lambda}=(\lambda,Z_{\alpha})\ket{\lambda}\neq 0$). On the other hand, the “raising operators” in the positive nilpotent Lie algebra $\mathfrak{n}_{+}$ annihilate the highest weight vector (by definition).

We now decompose the Lie algebra $\mathfrak{n}_{-}$ similarly to (3.7),

where $\mathfrak{n}_{-}(H<0)=\bigoplus_{\alpha\in R_{G,-}:(\alpha,H)<0}\mathfrak{g}_{\alpha}$ is the sum of the complex root spaces with negative $H$-weight $(\alpha,H)<0$, etc. By combining Lemmas 3.3 and 3.3 and using $R_{G,-}=-R_{G,+}$, we observe that

Note that $\pi(\mathfrak{n}_{-}(H<0))\mathcal{H}(H=c)\subseteq\mathcal{H}(H<c)$, since for any $\ket{\psi}\in\mathcal{H}(H=c)$ and $X\in\mathfrak{g}_{\alpha}$ we have that

Thus we obtain the following important result:

The fact that $\ket{\psi}\in\mathcal{H}(H=c)$ is just a reformulation of the selection rule. By the preceding discussion, the tangent map is well-defined as a map from $\mathfrak{n}_{-}(H<0)$ to $\mathcal{H}(H<c)$; it is injective by Section 3.3 and surjective since the dimensions agree according to (3.10). ∎

We now prove a partial converse to Section 3.3. Our proof is inspired by the argument of Ressayre [Res10b].

Suppose there exists $\ket{\psi}\in\mathcal{H}(H=c)$ such that the tangent map

is surjective. Then $(-,H)\geq c$ is a valid inequality for the moment polytope.

Its differential at $(1,\ket{\psi})$ is the linear map

The assumption implies that this map is surjective. It follows that $\Pi(N_{-})\mathcal{H}(H\geq c)\subseteq\mathcal{H}$ contains a small Euclidean ball around $\ket{\psi}$. In particular, any polynomial that is zero on $\Pi(N_{-})\mathcal{H}(H\geq c)$ is zero everywhere on $\mathcal{H}$.

We now prove the inequality. By the description of the moment polytope of Theorem 2.11, it suffices to show that $(\lambda/k,H)\geq c$ for all highest weights $\lambda$ such that $V^{*}_{G,\lambda}\subseteq R_{k}(\mathcal{H})$. Recall that the highest weight of $V^{*}_{G,\lambda}$ is $\lambda^{*}=-w_{0}\lambda$, where $w_{0}$ is the longest Weyl group element that flips the positive and negative roots. Consider a lowest weight vector, i.e., a homogeneous polynomial $P\in R_{k}(\mathcal{H})$ that is a weight vector of weight $-\lambda$ and stabilized by $N_{-}$. Then $\pi(H)P=-(\lambda,H)P$ and the restriction of $P$ to $\mathcal{H}(H\geq c)$ is non-zero by our discussion above. But this restriction is an element of $R_{k}(\mathcal{H}(H\geq c))=\Sym^{k}(\mathcal{H}(H\geq c))^{*}$, the space of homogeneous polynomials of degree $k$ on $\mathcal{H}(H\geq c)$. Thus all $H$-weights in $R_{k}(\mathcal{H}(H\geq c))$ are less or equal to $-kc$. It follows that $-(\lambda,H)\leq-kc$, hence $(\lambda/k,H)\geq c$, as we set out to prove. ∎

We remark that Section 3.3 holds unconditionally without any assumption on the dimension of the moment polytope $\Delta_{K}$. We summarize our findings in the following definition and theorem:

An inequality $(-,H)\geq c$ is said to be of Ressayre type if

$(-,H)=c$ is an affine hyperplane spanned by a subset of weights.

There exists $\ket{\psi}\in\mathcal{H}(H=c)$ such that the map

We note that the first condition is invariant under the action of the Weyl group.

This follows directly from Section 3.2, Section 3.3 and Section 3.3. ∎

Theorem 3.14gives a complete description of the moment polytope of the projective space of an arbitrary $G$-representation $\mathcal{H}$ (under the assumption that $\Delta_{K}$ is of maximal dimension). The set of inequalities thus obtained may still be redundant (i.e., not all inequalities necessarily correspond to facets of the moment polytope). In contrast, Ressayre’s well-covering pairs [Res10b, Res10a] characterize the facets of the moment polytope precisely. Our characterization is also related to [Bri99, Theorem 2], which uses algebraic geometry to characterize non-trivial faces of arbitrary codimension. Unlike Section 3.3, it relies on an assumption about lower-dimensional moment polytopes, which can in principle be obtained recursively.

It is straightforward to enumerate all inequalities of Ressayre type. Since there are only finitely many weights, the first condition in Section 3.3 cuts down the number of possible inequalities down to a finite list of candidates, and for each such candidate $(-,H)\geq c$, the second condition can be easily checked: Indeed, we only need to verify that $\dim\mathfrak{n}_{-}(H<0)=\dim\mathcal{H}(H<c)$ and that the determinant polynomial

is non-zero (take the determinant with respect to any fixed pair of bases). Both steps can easily be implemented in a short computer program.

4 Examples

We now illustrate the method by considering some of the examples discussed at the beginning of the chapter.

We will start by showing that the solution of the one-body quantum marginal problem for $n$ qubits is indeed given by the polygonal inequalities (3.2), which we recall are given by

By using that $\lambda_{k,1}+\lambda_{k,2}=1$ for each qubit, we find that (3.12) is equivalent to the following inequality for the moment polytope $\Delta_{K}$,

which we will now prove by using the criterion of Section 3.3.

On the other hand, there is precisely one negative root with $(-,H)<0$, namely $-\alpha_{n}$. Therefore,

where the generator $E_{-n}$ acts as the “lowering operator” $\lvert 1\rangle\langle 0\rvert$ on the $n$-th tensor factor of $\mathcal{H}$. But then $\pi(E_{-n})\ket{0\dots{}0}=\ket{0\dots{}01}$, so that the map

is indeed an isomorphism for $\ket{\psi}=\ket{0\dots{}0}\in\mathcal{H}(H=2-n)$. Thus Section 3.3 shows that the polygonal inequality (3.12) is indeed valid. We remark that the $n$ weights corresponding to (3.13) span the hyperplane $(-,H)=2-n$ and hence the inequality is of Ressayre type—but we did not need this to apply the proposition. The other polygonal inequalities follow from the above since the solution to the one-body quantum marginal problem is symmetric under permutation of the qubits.

To see that there are no further constraints, we could now verify that the polygonal inequalities imply all other Ressayre-type inequalities (which, according to Theorem 3.14, characterize the moment polytope completely). In the case at hand, we observe instead that for each $k\leq n-2$ as well as for $k=n$ the quantum states

and likewise for their permutations. These points are just the vertices of the convex polytope cut out by the polygonal inequalities (3.2), which is therefore equal to the moment polytope.

Mixed States of Two Qubits

We now consider the mixed-state one-body quantum marginal problem for two qubits. We will focus on the last constraint in (3.3),

which by symmetry can be reduced to the following two linear inequalities

In [Bra04], the inequalities are proved by a “formidable” two-page calculation that is tailored towards the two-qubit scenario. In contrast we will obtain (3.14) completely mechanically by using the general machinery developed in Section 3.3.

is an isomorphism for some $\ket{\psi}\in\mathcal{H}(H=0)$. We can do so completely mechanically by computing the determinant polynomial (3.11) with respect to the bases in (3.15) and (3.16). Using that each $E_{X,ij}$ acts by $\lvert i\rangle\langle j\rvert$ on the corresponding tensor factor, we readily find that it is given by

and thus is indeed non-zero. For example, $\ket{\psi}=\ket{121}+\ket{112}+\ket{213}\in\mathcal{H}(H=0)$ is a choice for which (3.17) is an isomorphism. Thus Section 3.3 shows that the first inequality in (3.14) is indeed a valid inequality for the one-body marginal problem for mixed states of two qubits. The second inequality can be established in a completely identical fashion.

Three Qutrits

corresponding to the facet $h_{7}$ in [Fra02, Proposition 5.1].

The significance of this facet is that it is neither trivial nor does it contain the vertex $\lambda_{A,1}=\lambda_{B,1}=\lambda_{C,1}=1$ of the moment polytope that corresponds to product states $\ket{000}$ – unlike in the case of $n$ qubits, where the polygonal inequalities (3.2) are saturated for product states (cf. Figure 3.3). As was pointed out in [Fra02, Remark 5.3], this shows that in general the moment polytope is not determined by the trivial facets together with the local cone at the point corresponding to the product state (more generally, the local cone at the highest weight of an irreducible representation).

Let $K=\SU(3)\times\SU(3)\times\SU(3)$ and $G=\SL(3)\times\SL(3)\times\SL(3)$ with their tensor product action on the Hilbert space $\mathcal{H}$. Using the same notation as in the preceding example, we find that (3.18) can be written in the form

and thus (3.18) is indeed a valid inequality.

Three Fermions with Total Spin 1/21/2

(In analogy to (2.25), each Young diagram is paired with its transpose.) The different summands in the decomposition correspond to different sectors of total spin $J=1/2,3/2$. Now suppose that we know in addition that the global state is a pure state in the sector for total spin $J=1/2$:

Then the problem of determining the relation between the orbital and spin occupation numbers amounts to computing the moment polytope for the $G$-representation $\mathcal{H}$. This illustrates how representations other than those in Table 2.3 may enter the picture due to physical considerations. In [KA08, §6.1] it was observed that there are five inequalities which are “apparently independent” of the orbital dimension $d$, as they verified for $d=4,5$. Their first equation is

where $\lambda_{1}\geq\dots\geq\lambda_{d}$ denote the natural occupation numbers of the first-order orbital density matrix $\gamma_{O}$, normalized to trace $3$, and $\mu_{1}\geq\mu_{2}$ the eigenvalues of the spin density matrix $\rho_{S}=\gamma_{S}/3$, normalized to trace $1$.

We will now prove (3.19) for arbitrary $d$. For this, we will use the following description of the irreducible $\GL(d)$-representation $V^{d}_{\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}$ (see, e.g., [Ful97]): Let $V$ denote the vector space with one basis vector ⟩\ket{\text{\scriptsize\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfila\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilb\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilc\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}} for each filling of the Young diagram with numbers $a,b,c\in\{1,\dots,d\}$. Then $V^{d}_{\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}$ can be identified with the quotient of $V$ by the relations

It is not hard to see that a basis of $V^{d}_{\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}$ is given by those ⟩\ket{\text{\scriptsize\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfila\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilb\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilc\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}} with $a\leq b$, $a<c$ (in the literature, these fillings are known as the semistandard Young tableaux of shape ). Each vector ⟩\ket{\text{\scriptsize\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfila\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilb\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilc\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}} is a weight vector of weight $\omega(H)=H_{a,a}+H_{b,b}+H_{c,c}$, and the generators $E_{ij}$ of $\mathfrak{g}_{ij}$ send ⟩\ket{\text{\scriptsize\hskip 0.0pt\vbox{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfila\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilb\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt\vbox{\hrule height=0.3pt\hbox{\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt\hbox to7.4pt{\hfilc\hfil}\vrule height=5.92001pt,width=0.3pt,depth=1.47997pt}\hrule height=0.3pt}\vskip-0.3pt}}\hskip 0.0pt}} to the sum of all vectors that arise by replacing a single $j$ by $i$. For example,