# Gibbs Manifolds

Dmitrii Pavlov  Bernd Sturmfels and Simon Telen

###### Abstract

Gibbs manifolds are images of affine spaces of symmetric matrices under the
exponential map. They arise in applications such as optimization, statistics
and quantum physics, where they extend the ubiquitous role of toric geometry.
The Gibbs variety is the zero locus of all polynomials that vanish on the
Gibbs manifold. We compute these polynomials and show that the Gibbs variety
is low-dimensional. Our theory is applied to a wide range of scenarios,
including matrix pencils and quantum optimal transport.

##  1 Introduction

Toric varieties provide the geometric foundations for many successes in the
mathematical sciences. In statistics they appear as discrete exponential
families [23, p. 2], and their ideals reveal Markov bases for sampling from
conditional distributions [5]. In optimization, they furnish nonnegativity
certificates [7] and they govern the entropic regularization of linear
programming [21]. Notable sightings in phylogenetics, stochastic analysis,
Gaussian inference and chemical reaction networks gave us the slogan that the
world is toric [16, Section 8.3].

In all of these applications, the key player is the positive part of the toric
variety. That real manifold is identified with a convex polytope by the moment
map [16, Theorem 8.24]. The fibers of the underlying linear map are polytopes
of complementary dimension, and each fiber intersects the toric variety
uniquely, in the Birch point. This is the unique maximizer of the entropy over
the polytope [19, Theorem 1.10]. In statistical physics and computer science
[25], the Birch point is known as the Gibbs distribution. The name Gibbs
refers to the maximum entropy state in a quantum system, and this is also the
reason behind our title.

This paper initiates a non-commutative extension of applied toric geometry. In
that extension, points in $\mathbb{R}^{n}$ are replaced by real symmetric
$n\times n$ matrices, and linear programing is replaced by semidefinite
programming. There is a moment map which takes the cone of positive
semidefinite matrices onto a spectrahedral shadow, and whose fibers are
spectrahedra of complementary dimension. The Gibbs manifold plays the role of
the positive toric variety. Each spectrahedron intersects the Gibbs manifold
uniquely, in the Gibbs point. Just like in the toric case, we study these
objects algebraically by passing to the Zariski closure of our positive
manifold. The resulting analogues of toric varieties are called Gibbs
varieties.

We illustrate these concepts for the following linear space of symmetric
$3\times 3$-matrices:

$\mathcal{L}\,\,\,=\,\,\,\biggl{\\{}\,\small\begin{bmatrix}y_{1}+y_{2}+y_{3}&y_{1}&y_{2}\\\
y_{1}&y_{1}+y_{2}+y_{3}&y_{3}\\\
y_{2}&y_{3}&y_{1}+y_{2}+y_{3}\end{bmatrix}\,\,:\,\,y_{1},y_{2},y_{3}\in\mathbb{R}\,\biggr{\\}}.$
(1)

The Gibbs manifold ${\rm GM}(\mathcal{L})$ is obtained by applying the
exponential function to each matrix in $\mathcal{L}$. Since the matrix
logarithm is the inverse to the matrix exponential, it is a $3$-dimensional
manifold, contained in the $6$-dimensional cone ${\rm
int}(\mathbb{S}^{3}_{+})$ of positive definite $3\times 3$ matrices.

Consider the quotient map from the matrix space
$\mathbb{S}^{3}\simeq\mathbb{R}^{6}$ onto
$\mathbb{S}^{3}/\mathcal{L}\simeq\mathbb{R}^{3}$. This takes a positive
semidefinite matrix $X=[x_{ij}]$ to its inner products with the matrices in a
basis of $\mathcal{L}$:

$\pi:\,\mathbb{S}^{3}_{+}\rightarrow\mathbb{R}^{3}\,\,:\,X\,\mapsto\,\bigl{(}\,{\rm
trace}(X)+2x_{12},\,{\rm trace}(X)+2x_{13},\,{\rm
trace}(X)+2x_{23}\,\bigr{)}.$

Precisely this map appeared in the statistical study of Gaussian models in
[22, Example 1.1]. The fibers $\pi^{-1}(b)$ are three-dimensional
spectrahedra, and these serve as feasible regions in optimization, both for
semidefinite programming and for maximum likelihood estimation.

We here consider yet another convex optimization problem over the
spectrahedron $\pi^{-1}(b)$, namely maximizing the von Neumann entropy
$\,h(X)={\rm trace}(X-X\cdot{\rm log}(X))$. This problem has a unique local
and global maximum, at the intersection $\pi^{-1}(b)\,\cap\,{\rm
GM(\mathcal{L})}$. See Theorem 5.1. This Gibbs point is the maximizer of the
entropy over the spectrahedron. Therefore, the Gibbs manifold ${\rm
GM}(\mathcal{L})$ is the set of Gibbs points in all fibers $\pi^{-1}(b)$, as
$b$ ranges over $\mathbb{R}^{3}$.

To study these objects algebraically, we ask for the polynomials that vanish
on ${\rm GM}(\mathcal{L})$. The zeros of these polynomials form the Gibbs
variety ${\rm GV}(\mathcal{L})$. Thus, the Gibbs variety is the Zariski
closure of the Gibbs manifold. In our example, the Gibbs manifold has
dimension $3$, whereas the Gibbs variety has dimension $5$. The latter is the
cubic hypersurface $\,{\rm GV}(\mathcal{L})\,=\,$

$\bigl{\\{}X\in\mathbb{S}^{3}:(x_{11}-x_{22})(x_{11}-x_{33})(x_{22}-x_{33})=x_{33}(x_{13}^{2}-x_{23}^{2})+x_{22}(x_{23}^{2}-x_{12}^{2})+x_{11}(x_{12}^{2}-x_{13}^{2})\bigr{\\}}.$

As promised, the study of Gibbs manifolds and Gibbs varieties is a non-
commutative extension of applied toric geometry. Indeed, every toric variety
is a Gibbs variety arising from diagonal matrices. For instance, the toric
surface $\,\\{\,x\in\mathbb{R}^{3}:x_{1}x_{3}\,=\,x_{2}^{2}\,\\}\,$ is
realized as

${\rm
GV}(\mathcal{L}^{\prime})\,=\,\bigl{\\{}X\in\mathbb{S}^{3}\,:\,x_{11}x_{33}-x_{22}^{2}\,=\,x_{12}=x_{13}=x_{23}=0\,\bigr{\\}}$

for the diagonal matrix pencil

$\mathcal{L}^{\prime}\,=\,\small\biggl{\\{}\,\begin{bmatrix}2y_{1}&0&0\\\
0&y_{1}{+}y_{2}&0\\\
0&0&2y_{2}\end{bmatrix}\,\,:\,\,y_{1},y_{2}\in\mathbb{R}\,\biggr{\\}}.$ (2)

However, even for diagonal matrices, the dimension of the Gibbs variety can
exceed that of the Gibbs manifold. To see this, replace the matrix entry
$2y_{1}$ by $\sqrt{2}y_{1}$ in the definition of $\mathcal{L}^{\prime}$. This
explains why transcendental number theory will make an appearance in our
study.

Our presentation in this paper is organized as follows. Section 2 gives a more
thorough introduction to Gibbs manifolds and Gibbs varieties. Theorem 2.4
states that the dimension of the Gibbs variety is usually quite small. The
proof of this result is presented in Section 3. In that section we present
algorithms for computing the prime ideal of the Gibbs variety. This is an
implicitization problem, where the parametrization uses transcendental
functions. We compare exact symbolic methods for solving that problem with a
numerical approach. A key ingredient is the Galois group for the eigenvalues
of a linear space of symmetric matrices. We implemented our algorithms in
Julia, making use of the computer algebra package Oscar.jl [18]. Our code and
data are available at https://mathrepo.mis.mpg.de/GibbsManifolds.

In Section 4 we study the Gibbs varieties given by two-dimensional spaces of
symmetric matrices. This rests on the classical Segre-Kronecker classification
of matrix pencils [6].

In Section 5 we turn to the application that led us to start this project,
namely entropic regularization in convex optimization. That section develops
the natural generalization of the geometric results in [21] from linear
programming to semidefinite programming. We conclude in Section 6 with a study
of quantum optimal transport [3]. This is the semidefinite programming
analogue to the classical optimal transport problem [21, Section 3]. We show
that its Gibbs manifold is the positive part of a Segre variety in matrix
space.

##  2 From manifolds to varieties

We write $\mathbb{S}^{n}$ for the space of symmetric $n\times n$-matrices.
This is a real vector space of dimension $\binom{n+1}{2}$. The subset of
positive semidefinite matrices is denoted $\mathbb{S}^{n}_{+}$. This is a
full-dimensional closed semialgebraic convex cone in $\mathbb{S}^{n}$, known
as the PSD cone. The PSD cone is self-dual with respect to the trace inner
product, given by $\,\langle X,Y\rangle\,:=\,\mathrm{trace}(XY)$ for
$X,Y\in\mathbb{S}^{n}$.

The matrix exponential function is defined by the usual power series, which
converges for all real and complex $n\times n$ matrices. It maps symmetric
matrices to positive definite symmetric matrices. The zero matrix $0_{n}$ is
mapped to the identity matrix ${\rm id}_{n}$. We write

${\rm exp}\,\,:\,\mathbb{S}^{n}\rightarrow{\rm
int}(\mathbb{S}^{n}_{+})\,,\,\,X\,\mapsto\,\sum_{i=0}^{\infty}\,\frac{1}{i!}\,X^{i}.$

This map is invertible, with the inverse given by the familiar series for the
logarithm:

${\rm log}\,\,:\,{\rm
int}(\mathbb{S}^{n}_{+})\rightarrow\mathbb{S}^{n}\,,\,\,Y\,\mapsto\,\sum_{j=1}^{\infty}\frac{(-1)^{j-1}}{j}\,(\,Y-{\rm
id}_{n})^{j}.$

We next introduce the geometric objects studied in this article. We fix any
matrix $A_{0}\in\mathbb{S}^{n}$ and $d$ linearly independent matrices
$A_{1},A_{2},\ldots,A_{d}$, also in $\mathbb{S}^{n}$. We write $\mathcal{L}$
for the affine subspace $\,A_{0}+{\rm
span}_{\mathbb{R}}(A_{1},\ldots,A_{d})\,$ of the vector space
$\mathbb{S}^{n}\simeq\mathbb{R}^{\binom{n+1}{2}}$. Thus, $\mathcal{L}$ is an
affine space of symmetric matrices (ASSM) of dimension $d$. If $A_{0}=0$, then
${\cal L}$ is a linear space of symmetric matrices (LSSM). We are interested
in the image of ${\cal L}$ under the exponential map:

######  Definition 2.1.

The Gibbs manifold ${\rm GM}({\cal L})$ of $\mathcal{L}$ is the
$d$-dimensional manifold ${\rm exp}(\mathcal{L})\subset\mathbb{S}^{n}_{+}$.

This is indeed a $d$-dimensional manifold inside the convex cone
$\mathbb{S}^{n}_{+}$. It is diffeomorphic to
$\mathcal{L}\simeq\mathbb{R}^{d}$, with the identification given by the
exponential map and the logarithm map.

In notable special cases (e.g. that in Section 6), the Gibbs manifold is semi-
algebraic, namely it is the intersection of an algebraic variety with the PSD
cone. However, this fails in general, as seen in the Introduction. It is still
interesting to ask which polynomial relations hold between the entries of any
matrix in ${\rm GM}({\cal L})$. This motivates the following definition.

######  Definition 2.2.

The Gibbs variety ${\rm GV}({\cal L})$ of $\mathcal{L}$ is the Zariski closure
of ${\rm GM}({\cal L})$ in $\mathbb{C}^{\binom{n+1}{2}}$.

######  Example 2.3 ($n=4,d=2$).

Consider the 2-dimensional linear space of symmetric matrices

${\cal L}\,\,=\,\,\left\\{\,\small\begin{bmatrix}0&0&0&y_{1}\\\
0&0&y_{1}&y_{2}\\\ 0&y_{1}&y_{2}&0\\\
y_{1}&y_{2}&0&0\end{bmatrix}\,:\,\,y_{1},y_{2}\in\mathbb{R}\,\right\\}\,\,\subset\,\,\mathbb{S}^{4}.$

Its Gibbs manifold ${\rm GM}({\cal L})$ is a surface in
$\mathbb{S}^{4}\simeq\mathbb{R}^{10}$. The Gibbs variety ${\rm GV}({\cal L})$
has dimension five and degree three. It consists of all symmetric matrices
$X=(x_{ij})$ whose entries satisfy

$\begin{matrix}\qquad
x_{13}-x_{22}+x_{44}\,=\,x_{14}-x_{23}+x_{34}\,=\,\,x_{24}-x_{33}+x_{44}\,=\,0,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ {\rm and}\qquad{\rm
rank}\begin{bmatrix}x_{11}{-}x_{44}&x_{12}{-}x_{34}&x_{22}{-}x_{33}\\\
x_{12}&x_{22}{-}x_{44}&x_{23}{-}x_{34}\end{bmatrix}\,\leq\,1.\qquad\end{matrix}$
(3)

This follows from the general result on matrix pencils in Theorem 4.5.

The following dimension bounds constitute our main result on Gibbs varieties.

######  Theorem 2.4.

Let ${\cal L}\subset\mathbb{S}^{n}$ be an ASSM of dimension $d$. The dimension
of the Gibbs variety ${\rm GV}({\cal L})$ is at most $n+d$. If $A_{0}=0$, i.e.
${\cal L}$ is an LSSM, then ${\rm dim}\,{\rm GV}({\cal L})$ is at most
$n+d-1$.

These bounds are attained in many cases, including Example 2.3. Our proof of
Theorem 2.4 appears in Section 3, in the context of algorithms for computing
the ideal of ${\rm GV}({\cal L})$.

While it might be difficult to find all polynomials that vanish on the Gibbs
manifold, finding _linear_ relations is sometimes easier. Such relations are
useful for semidefinite optimization, see Remark 5.2. This brings us to the
final geometric object studied in this paper.

######  Definition 2.5.

The Gibbs plane ${\rm GP}({\cal L})$ is the smallest affine space containing
${\rm GV}({\cal L})$.

Clearly, we have the chain of inclusions $\,{\rm GM}({\cal
L})\,\subseteq\,{\rm GV}({\cal L})\,\subseteq\,{\rm GP}({\cal
L})\,\subseteq\,\mathbb{C}^{\binom{n+1}{2}}$.

######  Example 2.6.

The Gibbs plane of the LSSM ${\cal L}$ from Example 2.3 is the $7$-dimensional
linear space in $\mathbb{C}^{10}$ that is defined by the three linear
relations listed in the first row of (3).

We claimed in the Introduction that this article offers a generalization of
toric varieties. In what follows, we make that claim precise, by discussing
the case when ${\cal L}$ is a commuting family. This means that the symmetric
matrices $A_{0},A_{1},\ldots,A_{d}$ commute pairwise, i.e.
$A_{i}A_{j}=A_{j}A_{i}$ for all $i,j$. We now assume that this holds. Then the
ASSM $\mathcal{L}$ can be diagonalized, i.e. there is an orthogonal matrix $V$
such that $\Lambda_{i}=V^{\top}A_{i}V$ is a diagonal matrix, for all $i$. The
vector $\lambda_{i}\in\mathbb{R}^{n}$ of diagonal entries in $\Lambda_{i}={\rm
diag}(\lambda_{i})$ contains the eigenvalues of $A_{i}$.

The matrix exponential of any element in ${\cal L}$ can be computed as
follows:

${\rm exp}(A_{0}+y_{1}A_{1}+\cdots+y_{d}A_{d})\,\,=\,\,V\cdot{\rm
exp}(\Lambda_{0}+y_{1}\Lambda_{1}+\cdots+y_{d}\Lambda_{d})\cdot V^{\top}.$ (4)

Let $\mathcal{D}$ denote this ASSM of diagonal matrices, i.e.
$\,\mathcal{D}\,=\,\\{\Lambda_{0}+y_{1}\Lambda_{1}+\cdots+y_{d}\Lambda_{d}\,:\,y\in\mathbb{R}^{d}\\}$.
Then the linear change of coordinates given by $V$ identifies the respective
Gibbs manifolds:

${\rm GM}(\mathcal{L})\,=\,V\cdot{\rm GM}(\mathcal{D})\cdot V^{\top}.$ (5)

The same statement holds for the Gibbs varieties and the Gibbs planes:

${\rm GV}(\mathcal{L})\,=\,V\cdot{\rm GV}(\mathcal{D})\cdot V^{\top}\qquad{\rm
and}\qquad{\rm GP}(\mathcal{L})\,=\,V\cdot{\rm GP}(\mathcal{D})\cdot
V^{\top}.$ (6)

The dimensions of these objects are determined by arithmetic properties of the
eigenvalues.

Recall that $\Lambda_{i}={\rm diag}(\lambda_{i})$ where $\lambda_{i}$ is a
vector in $\mathbb{R}^{n}$. Let $\Lambda$ denote the linear subspace of
$\mathbb{R}^{n}$ that is spanned by the $d$ vectors
$\,\lambda_{1},\ldots,\lambda_{d}$. We have $\mathcal{D}=\lambda_{0}+\Lambda$,
and therefore

${\rm GM}(\mathcal{D})\,\,=\,\,{\rm exp}(\lambda_{0})\,\star\,{\rm
exp}(\Lambda)\,=\,\\{(e^{\lambda_{01}}w_{1},\ldots,e^{\lambda_{0n}}w_{n})\,:\,w\in{\rm
exp}(\Lambda)\\}\,\,\,\subset\,\,\,\mathbb{R}^{n}.$

Here $\star$ denotes coordinate-wise multiplication in $\mathbb{R}^{n}$. Let
$\Lambda_{\mathbb{Q}}$ be the smallest vector subspace of $\mathbb{R}^{n}$
spanned by elements of $\mathbb{Q}^{n}$ which contains $\Lambda$. Its
dimension $\,d_{\mathbb{Q}}={\rm dim}\,\Lambda_{\mathbb{Q}}\,$ satisfies
$\,d\leq d_{\mathbb{Q}}\leq n$. Fix lattice vectors
$\,a_{1},a_{2},\ldots,a_{d_{\mathbb{Q}}}\,$ in $\mathbb{Z}^{n}$ that form a
basis of $\Lambda_{\mathbb{Q}}$. Then, inside an $n$-dimensional linear space
defined by the diagonality condition, we have

${\rm
GV}(\mathcal{D})\,\,=\,\,\overline{\biggl{\\{}\bigl{(}\,e^{\lambda_{01}}\prod_{i=1}^{d_{\mathbb{Q}}}z_{i}^{a_{i1}},\,\,e^{\lambda_{02}}\prod_{i=1}^{d_{\mathbb{Q}}}z_{i}^{a_{i2}},\,\ldots\,,\,\,e^{\lambda_{0n}}\prod_{i=1}^{d_{\mathbb{Q}}}z_{i}^{a_{in}}\,\bigr{)}\,\,:\,\,z\in(\mathbb{C}^{*})^{d_{\mathbb{Q}}}\,\biggr{\\}}}.$
(7)

This is a toric variety of dimension $d_{\mathbb{Q}}$. Just like in [21,
Section 2], the closure is taken in $\mathbb{C}^{n}$. The Gibbs manifold ${\rm
GM}(\mathcal{D})$ is a $d$-dimensional subset of the real points in ${\rm
GV}(\mathcal{D})$ for which $z$ has strictly positive coordinates. We
summarize our discussion in the following theorem.

######  Theorem 2.7.

Let $\mathcal{L}$ be an affine space of pairwise commuting symmetric matrices.
Then the Gibbs variety ${\rm GV}(\mathcal{L})$ is a toric variety of dimension
$d_{\mathbb{Q}}$, given explicitly by (6) and (7).

For an illustration, consider the seemingly simple case $d=1$ and $A_{0}=0$.
Here, ${\rm GM}(\mathcal{L})$ is the curve formed by all powers of ${\rm
exp}(A_{1})$, and ${\rm GV}(\mathcal{L})$ is a toric variety of generally
higher dimension. This scenario is reminiscent of that studied by Galuppi and
Stanojkovski in [8].

######  Example 2.8.

Let $n=3$ and consider the LSSM $\mathcal{L}$ spanned by
$A_{1}=\small\begin{bmatrix}4&1&1\\\ 1&3&1\\\ 1&1&3\end{bmatrix}$. We have

$A_{1}\,=\,V\cdot{\rm diag}(\lambda)\cdot V^{\top},\,\,{\rm
where}\quad\lambda=\bigl{(}2,4+\sqrt{2},4-\sqrt{2}\bigr{)}\quad{\rm and}\quad
V\,=\,\frac{1}{2}\small\begin{bmatrix}0&\sqrt{2}&-\sqrt{2}\\\ -\sqrt{2}&1&1\\\
\sqrt{2}&1&1\end{bmatrix}.$

Here, $\mathcal{D}=\Lambda=\mathbb{R}\lambda$, $\,d_{\mathbb{Q}}=2$, and
$\Lambda_{\mathbb{Q}}\,=\,\mathbb{R}\\{(1,2,2),(0,1,-1)\\}\,=\,\\{p\in\mathbb{R}^{3}\,:\,4p_{1}=p_{2}+p_{3}\\}$.
Hence ${\rm GV}(\mathcal{D})$ is the toric surface
$\\{q_{11}^{4}=q_{22}q_{33}\\}$ in $\,{\rm
GP}(\mathcal{D})=\\{Q\in\mathbb{S}^{3}\,:\,q_{12}=q_{13}=q_{23}=0\\}$. We
transform that surface into the original coordinates via (6). The computation
reveals

${\rm GV}(\mathcal{L})=\\{X\in{\rm
GP}(\mathcal{L})\,:\,x_{23}^{4}-4x_{23}^{3}x_{33}+6x_{23}^{2}x_{33}^{2}-4x_{23}x_{33}^{3}+x_{33}^{4}+2x_{13}^{2}-x_{23}^{2}-2x_{23}x_{33}-x_{33}^{2}=0\\}.$

The ambient $3$-space is $\,{\rm
GP}(\mathcal{L})=\\{X\in\mathbb{S}^{3}:x_{11}-x_{23}-x_{33}=x_{12}-x_{13}=x_{22}-x_{33}=0\\}$.

This concludes our discussion of the toric Gibbs varieties that arise from
pairwise commuting matrices. In the next section we turn to the general case,
which requires new ideas.

##  3 Implicitization of Gibbs varieties

Implicitization is the computational problem of finding implicit equations for
an object that comes in the form of a parametrization. When the parametrizing
functions are rational functions, these equations are polynomials and can be
found using resultants or Gröbner bases [16, Section 4.2]. A different
approach rests on polynomial interpolation and numerical nonlinear algebra.
This section studies the implicitization problem for Gibbs varieties. The
difficulty arises from the fact that Gibbs manifolds are transcendental, since
their parametrizations involve the exponential function. We start out by
presenting our proof of Theorem 2.4.

As in Section 2, ${\cal L}=A_{0}+{\rm span}_{\mathbb{R}}(A_{1},\ldots,A_{d})$
is a $d$-dimensional affine space of symmetric $n\times n$-matrices. Its
elements are $A_{0}+y_{1}A_{1}+\cdots+y_{d}A_{d}$. We shall parametrize the
Gibbs manifold ${\rm GM}({\cal L})$ in terms of the coordinates
$y_{1},\ldots,y_{d}$ on ${\cal L}$. This uses the following formula.

######  Theorem 3.1 (Sylvester [24]).

Let $f:D\rightarrow\mathbb{R}$ be an analytic function on an open set
$D\subset\mathbb{R}$ and $M\in\mathbb{R}^{n\times n}$ a matrix that has $n$
distinct eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ in $D$. Then

$f(M)\,=\,\sum\limits_{i=1}^{n}f(\lambda_{i})M_{i},\quad\text{with}\quad
M_{i}\,=\,\prod_{j\neq
i}\dfrac{1}{\lambda_{i}-\lambda_{j}}(M-\lambda_{j}\cdot{\rm id}_{n}).$

We note that the product on the right hand side takes place in the commutative
ring $\,\mathbb{R}[M]$.

######  Proof 3.2 (Proof of Theorem 2.4).

The characteristic polynomial of $A(y)=A_{0}+y_{1}A_{1}+\cdots+y_{d}A_{d}$
equals

$P_{\cal L}(\lambda;y)\,={\rm det}(A(y)-\lambda\cdot{\rm
id}_{n})\,=\,\,c_{0}(y)+c_{1}(y)\,\lambda+\cdots+c_{n-1}(y)\,\lambda^{n-1}+(-1)^{n}\,\lambda^{n}.$

Its zeros $\lambda$ are algebraic functions of the coordinates
$y=(y_{1},\ldots,y_{d})$ on $\mathcal{L}$.

We first assume that $\mathcal{L}$ has distinct eigenvalues, i.e. there is a
Zariski open subset $U\subset\mathbb{R}^{d}$ such that $P_{\cal
L}(\lambda;y^{*})$ has $n$ distinct real roots $\lambda$ for all $y^{*}\in U$.
Sylvester’s formula writes the entries of ${\rm exp}(A(y))$ as rational
functions of $y,\lambda_{i}(y)$ and $e^{\lambda_{i}(y)}$ for $y\in U$. These
functions are multisymmetric in the pairs $(\lambda_{i},e^{\lambda_{i}})$.
They evaluate to convergent power series on $\mathbb{R}^{d}$.

Let $V$ be the subvariety of $U\times\mathbb{R}^{n}$ that is defined by the
equations

$c_{i}(y)\,=\,(-1)^{i}\sigma_{n-i}(\lambda)\quad{\rm for}\quad
i=0,\ldots,n-1,$ (8)

where $\sigma_{t}(\lambda)$ is the $t^{\rm th}$ elementary symmetric
polynomial evaluated at $(\lambda_{1},\ldots,\lambda_{n})$. We have $\dim
V=d$. Define a map $\phi:V\times\mathbb{R}^{n}\rightarrow\mathbb{S}^{n}$,
using coordinates $z_{1},\ldots,z_{n}$ on $\mathbb{R}^{n}$, as follows:

$(y_{1},\ldots,y_{d},\lambda_{1},\ldots,\lambda_{n},z_{1},\ldots,z_{n})\,\longmapsto\,\sum_{i=1}^{n}z_{i}\,\prod_{j\neq
i}\frac{1}{\lambda_{i}-\lambda_{j}}(A(y)-\lambda_{j}\cdot{\rm id}_{n}).$ (9)

The closure $\overline{\phi(V\times\mathbb{R}^{n})}$ of the image of this map
is a variety. It contains the Gibbs variety: setting $z_{i}=e^{\lambda_{i}}$
parametrizes a dense subset of the Gibbs manifold, by Theorem 3.1.

The Gibbs variety of the LSSM $\mathbb{R}{\cal L}$ spanned by the ASSM ${\cal
L}$ also lies in $\overline{\phi(V\times\mathbb{R}^{n})}$, because ${\rm
exp}(y_{0}A(y))=\phi(y_{0}\cdot y,y_{0}\cdot\lambda,e^{y_{0}\cdot\lambda})$
for any $y\in U$ and $y_{0}\in\mathbb{R}\backslash\\{0\\}$. We thus have

$\dim{\rm GV}({\cal
L})\,\leq\,\dim\overline{\phi(V\times\mathbb{R}^{n})}\,\leq\,d+n\quad\text{and}\quad\dim{\rm
GV}(\mathbb{R}{\cal L})\,\leq\,d+n.$

Finally, suppose that ${\cal L}$ is an LSSM, i.e. $A_{0}=0$. Then
$\mathcal{L}$ is the linear span of an ASSM of dimension $d-1$ in
$\mathbb{S}^{n}$. The second inequality therefore gives $\,\dim{\rm GV}({\cal
L})\leq d+n-1$.

We finally consider the case when ${\cal L}$ has $m<n$ distinct eigenvalues.
Since symmetric matrices are diagonalizable, Sylvester’s formula can easily be
adapted to this case: it suffices to sum over the distinct eigenvalues of $M$,
and to adjust the parametrization (9) accordingly. That is, we replace $n$ by
$m$. See [12, Chapter 6.1, Problem 14] for details.

######  Remark 3.3.

If the points ${\rm
exp}(\lambda(y))=(e^{\lambda_{1}(y)},\ldots,e^{\lambda_{n}(y)})$, $y\in U$,
lie in a lower-dimensional subvariety $W\subset\mathbb{R}^{n}$, then the proof
of Theorem 2.4 gives the better bound $\dim{\rm GV}({\cal L})\leq d+\dim W$.
We saw this in Example 2.8. In general, no such subvariety $W$ exists, i.e.
one expects $W=\mathbb{R}^{n}$. This is an issue of Galois theory, to be
discussed at the end of this section.

For ease of exposition, we work only with LSSMs in the rest of this section.
That is, we set $A_{0}=0$. We comment on the generalization to ASSMs in Remark
3.9. Our discussion and the proof of Theorem 2.4 suggest Algorithm 1, for
computing the ideal of the Gibbs variety of an LSSM ${\cal L}$.

Algorithm 1 Implicitization of the Gibbs variety of an LSSM ${\cal L}$,
defined over $\mathbb{Q}$

Input: Linearly independent matrices $A_{1},\ldots,A_{d}\in\mathbb{S}^{n}$
with rational entries
Output: Polynomials that define ${\rm GV}({\cal L})$, where ${\cal L}={\rm
span}_{\mathbb{R}}(A_{1},\ldots,A_{d})$

1: $\text{Compute the characteristic polynomial }\,P_{\cal
L}(\lambda;y)=c_{0}(y)+c_{1}(y)\lambda+\cdots+c_{n}(y)\lambda^{n}$

2:$P_{\cal L}(\lambda;y)$ has $n$ distinct roots in $\overline{\mathbb{R}(y)}$

3: $E_{1}^{\prime}\leftarrow\\{\text{the $n$ polynomials
$(-1)^{i}\sigma_{n-i}(\lambda)-c_{i}(y)$ in \eqref{eq:symmeq}}\\}$

4: $E_{1}\leftarrow\\{\text{generators of any associated prime over
}\mathbb{Q}\text{ of }\langle E_{1}^{\prime}\rangle\\}$

5: $E_{2}\leftarrow\\{\text{the entries of }\phi(y,\lambda,z)-X\\},\text{ with
$X=(x_{ij})$ a symmetric matrix of variables}$

6: $E_{2},D\leftarrow\text{ clear denominators in $E_{2}$ and record the least
common denominator $D$}$

7:if the roots $\lambda_{1},\ldots,\lambda_{n}$ of $P_{\cal L}(\lambda;y)$ are
$\mathbb{Q}$-linearly dependent then

8:
$E_{3}\leftarrow\\{z^{\alpha}-z^{\beta}\,:\,\sum\alpha_{i}\lambda_{i}=\sum\beta_{j}\lambda_{j},\,\alpha,\beta\in\mathbb{Z}_{\geqslant
0}^{n}\\}$

9:else

10: $E_{3}\leftarrow\emptyset$

11: $I\leftarrow\text{ the ideal generated by $E_{1},E_{2},E_{3}$ in the
polynomial ring $\mathbb{R}[y,\lambda,z,X]$}$

12: $I\leftarrow I:D^{\infty}$

13: $J\leftarrow\text{ elimination ideal obtained by eliminating $y,\lambda,z$
from $I$}$

14:return $\text{a set of generators of }J$

That ideal lives in a polynomial ring $\mathbb{R}[X]$ whose variables are the
entries of a symmetric $n\times n$ matrix. The algorithm builds three subsets
$E_{1},E_{2},E_{3}$ of the larger polynomial ring $\mathbb{R}[y,\lambda,z,X]$.
After the saturation (step 12), the auxiliary variables $y,\lambda,z$ are
eliminated. The equations $E_{1}^{\prime}$ come from (8). They constrain
$(y,\lambda)$ to lie in $V$. The set $E_{1}$ generates an associated prime of
$\langle E_{1}^{\prime}\rangle$ (step 4), see the discussion preceding Theorem
3.7. The equations $E_{2}$ come from the parametrization (9). Note that, if
${\cal L}$ has $m<n$ distinct eigenvalues, this formula can be adjusted as in
the end of the proof of Theorem 2.4, and the requirement after step 1 can be
dropped. Later in the algorithm, one replaces $n$ with $m$. It is necessary to
clear denominators in order to obtain polynomials (step 6). The saturation by
the LCD $D$ avoids spurious components arising from this step. Finally,
$E_{3}$ accounts for toric relations between the $z_{i}$ arising from
$\mathbb{Q}$-linear relations among the $\lambda_{i}$. If no such relations
exist, then Theorem 3.6 ensures that the assignment $E_{3}\leftarrow\emptyset$
in step 10 is correct.

Steps 7 and 8 in Algorithm 1 require a detailed discussion. Further below we
shall explain the $\mathbb{Q}$-linear independence of eigenvalues, how to
check this, and how to compute $E_{3}$. Ignoring this for now, one can also
run Algorithm 1 with $E_{3}=\emptyset$. Then step 14 still returns polynomials
that vanish on the Gibbs variety ${\rm GV}(\mathcal{L})$ but these may cut out
a larger variety.

We implemented Algorithm 1 in Julia (v1.8.3), using Oscar.jl [18], and tested
it on many examples. The code is available at
https://mathrepo.mis.mpg.de/GibbsManifolds.

######  Example 3.4.

The Gibbs variety ${\rm GV}({\cal L})$ for the LSSM ${\cal L}$ in (1) has the
parametrization

$\phi\,=\,\,\sum_{i=1}^{3}\dfrac{z_{i}}{q(\lambda_{i},y_{1},y_{2},y_{3})}\small\begin{bmatrix}p_{11}(\lambda_{i},y_{1},y_{2},y_{3})&p_{12}(\lambda_{i},y_{1},y_{2},y_{3})&p_{13}(\lambda_{i},y_{1},y_{2},y_{3})\\\
p_{12}(\lambda_{i},y_{1},y_{2},y_{3})&p_{22}(\lambda_{i},y_{1},y_{2},y_{3})&p_{23}(\lambda_{i},y_{1},y_{2},y_{3})\\\
p_{13}(\lambda_{i},y_{1},y_{2},y_{3})&p_{23}(\lambda_{i},y_{1},y_{2},y_{3})&p_{33}(\lambda_{i},y_{1},y_{2},y_{3})\\\
\end{bmatrix},\,\,\,{\rm where}$
$\small\begin{matrix}&q&=&2y_{1}^{2}+6y_{1}y_{2}+2y_{2}^{2}+6y_{1}y_{3}+6y_{2}y_{3}+2y_{3}^{2}-6y_{1}\lambda-6y_{2}\lambda-6y_{3}\lambda+3\lambda^{2},\\\
&p_{11}&=&y_{1}^{2}+2y_{1}y_{2}+y_{2}^{2}+2y_{1}y_{3}+2y_{2}y_{3}-2y_{1}\lambda-2y_{2}\lambda-2y_{3}\lambda+\lambda^{2},\\\
&p_{12}&=&-y_{1}^{2}-y_{1}y_{2}-y_{1}y_{3}+y_{2}y_{3}+y_{1}\lambda,\\\
&p_{13}&=&-y_{1}y_{2}-y_{2}^{2}+y_{1}y_{3}-y_{2}y_{3}+y_{2}\lambda,\\\
&p_{22}&=&y_{1}^{2}+2y_{1}y_{2}+2y_{1}y_{3}+2y_{2}y_{3}+y_{3}^{2}-2y_{1}\lambda-2y_{2}\lambda-2y_{3}\lambda+\lambda^{2},\\\
&p_{23}&=&y_{1}y_{2}-y_{1}y_{3}-y_{2}y_{3}-y_{3}^{2}+y_{3}\lambda,\\\
&p_{33}&=&2y_{1}y_{2}+y_{2}^{2}+2y_{1}y_{3}+2y_{2}y_{3}+y_{3}^{2}-2y_{1}\lambda-2y_{2}\lambda-2y_{3}\lambda+\lambda^{2}.\end{matrix}$

Our Julia code for Algorithm 1 easily finds the cubic polynomial defining
${\rm GV}(\mathcal{L)}$.

In spite of such successes, symbolic implicitization is limited to small $n$
and $d$. Numerical computations can help, in some cases, to find equations for
more challenging Gibbs varieties.

######  Example 3.5.

We consider the LSSM of $4\times 4$ Hankel matrices with upper left entry
zero:

${\cal L}\,=\,\left\\{\begin{bmatrix}0&y_{2}&y_{3}&y_{4}\\\
y_{2}&y_{3}&y_{4}&y_{5}\\\ y_{3}&y_{4}&y_{5}&y_{6}\\\
y_{4}&y_{5}&y_{6}&y_{7}\end{bmatrix}\,:\,(y_{2},\ldots,y_{7})\in\mathbb{R}^{6}\right\\}.$

Algorithm 1 failed to compute its Gibbs variety. We proceed using numerics as
follows. Fix a degree $D>0$ and let $N=\binom{9+D}{D}$ be the number of
monomials in the 10 coordinates $x_{11},\ldots,x_{44}$ on $\mathbb{S}^{4}$. We
create $M\geq N$ samples on ${\rm GM}({\cal L})$ by plugging in random values
for the six parameters $y_{i}$ and applying the matrix exponential. Finding
all vanishing equations of degree $D$ on these samples amounts to computing
the kernel of an $M\times N$ Vandermonde matrix. If this matrix has full rank,
then there are no relations of degree $D$. We implemented this procedure in
Julia. In our example, Theorem 2.4 suggests that ${\rm GV}({\cal L})$ is a
hypersurface, and this is indeed the case. Its defining equation has degree
$D=6$. We found it using $M=5205\geq N=5005$ samples. Our denary sextic has
$853$ terms with integer coefficients:

$x_{11}^{3}x_{22}x_{24}x_{34}-x_{11}^{3}x_{23}^{2}x_{34}-x_{11}^{3}x_{23}x_{24}^{2}+x_{11}^{3}x_{23}x_{24}x_{33}+\cdots+3x_{23}x_{24}^{2}x_{33}x_{34}^{2}+x_{24}^{4}x_{33}x_{34}-x_{24}^{3}x_{33}^{2}x_{34}.$

Its Newton polytope has the f-vector
$(456,5538,21560,41172,44707,29088,11236,2370,211)$. Note that the package
Oscar.jl conveniently allows to perform symbolic and numerical implicitization
and polyhedral computations in the same programming environment.

We emphasize that our numerical Julia code is set up to find _exact_ integer
coefficients. For this, we first normalize the numerical approximation of the
coefficient vector by setting its first (numerically) nonzero entry to one.
Then we rationalize the coefficients using the built in command rationalize in
Julia, with error tolerance tol = 1e-7. Correctness of the result is proved by
checking that the resulting polynomial vanishes on the parametrization.

We now turn to $\mathbb{Q}$-linear relations among eigenvalues of
$\mathcal{L}$. Our arithmetic discussion begins with a version of [1, (SP)],
which is well-known in transcendental number theory:

######  Theorem 3.6 (Ax-Schanuel).

If the eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ of the LSSM
$\,\mathcal{L}$ are $\mathbb{Q}$-linearly independent, then
$e^{\lambda_{1}},\ldots,e^{\lambda_{n}}$ are algebraically independent over
the field $\,\mathbb{C}(y_{1},\ldots,y_{d})$.

On the other hand, suppose that the eigenvalues of $\mathcal{L}$ satisfy some
non-trivial linear relation over $\mathbb{Q}$. We can then find nonnegative
integers $\alpha_{i}$ and $\beta_{j}$, not all zero, such that

$\sum_{i=1}^{n}\alpha_{i}\lambda_{i}\,\,=\,\,\sum_{j=1}^{n}\beta_{j}\lambda_{j}.$
(10)

This implies that the exponentials of the eigenvalues satisfy the toric
relations

$\prod_{i=1}^{n}z_{i}^{\alpha_{i}}\,\,=\,\,\prod_{j=1}^{n}z_{j}^{\beta_{j}}.$
(11)

The linear relations (10) can be found from the ideal $\langle
E_{1}^{\prime}\rangle$ in step 3 which specifies that the $\lambda_{i}$ are
the eigenvalues of $A(y)$. This ideal is radical if we assume that
$\mathcal{L}$ has distinct eigenvalues. We compute the prime decomposition of
the ideal over $\mathbb{Q}$. All prime components are equivalent under
permuting the $\lambda_{i}$, so we replace $\langle E_{1}^{\prime}\rangle$ by
any of these prime ideals in step 4. We compute (10) as the linear forms in
that prime ideal. Using (11), we compute the toric ideal $\langle
E_{3}\rangle$ in step 8, which is also prime. This ideal defines a toric
variety $W^{\prime}$, whose $S_{n}$-orbit is the variety $W$ in Remark 3.3. We
arrive at the following result.

######  Theorem 3.7.

Let $\mathcal{L}\subset\mathbb{S}^{n}$ be an LSSM with distinct eigenvalues.
The Gibbs variety ${\rm GV}(\mathcal{L})$ is irreducible and unirational, and
the ideal $J$ found in Algorithm 1 is its prime ideal.

######  Proof 3.8.

Sylvester’s formula yields a rational parametrization $\psi$ of ${\rm
GV}(\mathcal{L})$ with parameters $y_{1},\ldots,y_{d},z_{1},\ldots,z_{n}$. The
parameters $\lambda_{i}$ in (9) can be omitted: the entries in the image are
multisymmetric in $(\lambda_{i},z_{i})$, so that they can be expressed in
terms of elementary symmetric polynomials of the $\lambda_{i}$ [2, Theorem 1].
The point $(z_{1},\ldots,z_{n})$ lies on the toric variety $W^{\prime}$
defined above. The domain $\mathbb{C}^{d}\times W^{\prime}$ of $\psi$ is an
irreducible variety, and it is also rational. The image of $\psi$ is the Gibbs
variety ${\rm GV}(\mathcal{L})$, which is therefore unirational and
irreducible. The ideals given by $E_{1}$ and $E_{2}$ in Algorithm 1 are prime,
after saturation, and elimination in step 13 preserves primality. Hence the
output in $J$ in step 14 is the desired prime ideal.

We define the Galois group $G_{\mathcal{L}}$ of an LSSM $\mathcal{L}$ to be
the Galois group of the characteristic polynomial $P_{\mathcal{L}}(\lambda,y)$
over the field $\mathbb{Q}(y_{1},\ldots,y_{d})$. Note that $G_{\mathcal{L}}$
is the subgroup of the symmetric group $S_{n}$ whose elements are permutations
that fix each associated prime of $\langle E_{1}^{\prime}\rangle$. Hence the
index of the Galois group $G_{\mathcal{L}}$ in $S_{n}$ is the number of
associated primes. In particular, the Galois group equals $S_{n}$ if and only
if the ideal $\langle E_{1}^{\prime}\rangle$ formed in step 3 is prime.

The existence of linear relations (10) depends on the Galois group
$G_{\mathcal{L}}$. If the Galois group is small then the primes of $\langle
E_{1}\rangle$ are large, and more likely to contain linear forms. There is a
substantial literature in number theory on this topic. See [9, 10] and the
references therein. For instance, by Kitaoka [14, Proposition 2], there are no
linear relations if $n$ is prime, or if $n\geq 6$ and the Galois group is
$S_{n}$ or $A_{n}$. If this holds, $E_{3}=\emptyset$ in step 10 of Algorithm
1.

The computation of Galois groups is a well-studied topic in symbolic
computation and number theory. Especially promising are methods based on
numerical algebraic geometry (e.g. in [11]). These fit well with the approach
to implicitization in Example 3.5. For a future theoretical project, it would
be very interesting to classify LSSMs by their Galois groups.

######  Remark 3.9.

We briefly comment on how to adjust Algorithm 1 to compute the Gibbs variety
of an ASSM ${\cal L}$ with $A_{0}\neq 0$. In this case, algebraic relations
between $e^{\lambda_{1}},\ldots,e^{\lambda_{n}}$ come from $\mathbb{Q}$-linear
relations between the eigenvalues of ${\cal L}$, but this time modulo
$\mathbb{C}$: an affine relation
$\sum\alpha_{i}\lambda_{i}=\sum\beta_{j}\lambda_{j}+\gamma$ gives
$z^{\alpha}-e^{\gamma}\cdot z^{\beta}=0$, where $z_{i}=e^{\lambda_{i}}$,
$\alpha_{i},\beta_{j}\in\mathbb{Z}_{\geq 0}$, $\gamma\in\mathbb{C}$. Here
$\gamma$ is a $\mathbb{Q}$-linear combination of eigenvalues of $A_{0}$.
Theorem 3.7 holds for ASSMs as well, provided that these $\mathbb{Q}$-linear
relations modulo $\mathbb{C}$ can be computed in practice. This can usually
not be done over $\mathbb{Q}$. We leave this algorithmic challenge for future
research.

##  4 Pencils of quadrics

In this section we study the Gibbs variety ${\rm GV}(\mathcal{L})$ where
$\mathcal{L}\subset\mathbb{S}^{n}$ is a pencil of quadrics, i.e. an LSSM of
dimension $d=2$. We follow the exposition in [6], where pencils $\mathcal{L}$
are classified by Segre symbols. The Segre symbol
$\,\sigma=\sigma(\mathcal{L})$ is a multiset of partitions that sum up to $n$.
It is computed as follows: Pick a basis $\\{A_{1},A_{2}\\}$ of $\mathcal{L}$,
where $A_{2}$ is invertible, and find the Jordan canonical form of
$A_{1}A_{2}^{-1}$. Each eigenvalue determines a partition, according to the
sizes of the corresponding Jordan blocks. The multiset of these partitions is
the Segre symbol $\sigma$.

We use the canonical form in [6, Section 2]. Suppose the Segre symbol is
$\sigma=[\sigma_{1},\ldots,\sigma_{r}]$, where the $i$th partition
$\sigma_{i}$ equals
$(\sigma_{i,1}\geq\sigma_{i,2}\geq\cdots\geq\sigma_{i,n}\geq 0)$. There are
$r$ groups of blocks, one for each eigenvalue $\alpha_{i}$ of
$A_{1}A_{2}^{-1}$. The $j$th matrix in the $i$th group is the
$\sigma_{i,j}\times\sigma_{i,j}$ matrix

$y_{1}\cdot\small\begin{bmatrix}0&0&\ldots&0&\alpha_{i}\\\
0&0&\ldots&\alpha_{i}&1\\\ \vdots&\vdots&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&\vdots\\\
0&\alpha_{i}&1&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&0\\\ \alpha_{i}&1&\ldots&0&0\\\
\end{bmatrix}\,+\,\,y_{2}\cdot\small\begin{bmatrix}0&\ldots&0&0&1\\\
0&\ldots&0&1&0\\\ 0&\ldots&1&0&0\\\ \vdots&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&\vdots&\vdots&\vdots\\\
1&\ldots&0&0&0\\\ \end{bmatrix}.$

There are $13$ Segre symbols for $n=4$; see [6, Example 3.1]. It is
instructive to compute their Gibbs varieties. All possible dimensions, $2,3,4$
and $5$, are attained. Dimension $2$ arises for the diagonal pencil
$\mathcal{L}_{\sigma}={\rm
diag}(\alpha_{1}y_{1}{+}y_{2},\alpha_{2}y_{1}{+}y_{2},\alpha_{3}y_{1}{+}y_{2},\alpha_{4}y_{1}{+}y_{2})$,
with Segre symbol $\sigma=[1,1,1,1]$. When the $\alpha_{i}$ are distinct
integers, ${\rm GV}(\mathcal{L}_{\sigma})={\rm GM}(\mathcal{L}_{\sigma})$ is a
toric surface. This is similar to (2). Dimension $5$ arises for $\sigma=[4]$,
which was presented in Example 2.3.

The following examples, also computed with Algorithm 1, exhibit the dimensions
$5,4,3$.

######  Example 4.1.

Consider the Segre symbol $\sigma=[3,1]$. The canonical pencil ${\cal
L}_{[3,1]}$ is spanned by

$\begin{bmatrix}0&0&\alpha_{1}&0\\\ 0&\alpha_{1}&1&0\\\ \alpha_{1}&1&0&0\\\
0&0&0&\alpha_{2}\\\ \end{bmatrix}\quad\text{and}\quad\begin{bmatrix}0&0&1&0\\\
0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ \end{bmatrix},\qquad{\rm
for}\,\,\,\,\alpha_{1},\alpha_{2}\in\mathbb{R}\,\,\hbox{distinct}.$

Here, $\dim{\rm GV}({\cal L}_{[3,1]})=5$, the upper bound in Theorem 2.4.
Algorithm 1 produces the ideal

$J\,=\,\bigl{\langle}\,x_{14},x_{24},x_{34},\,x_{13}-x_{22}+x_{33},\,x_{12}^{2}-x_{11}x_{22}-x_{12}x_{23}+x_{11}x_{33}+x_{22}x_{33}-x_{33}^{2}\,\bigr{\rangle}.$

If $\alpha_{1}=\alpha_{2}$, then the Segre symbol changes to $\sigma=[(3,1)]$.
We now find the additional cubic

$x_{11}x_{22}x_{33}+2x_{12}x_{13}x_{23}-x_{13}^{2}x_{22}-x_{11}x_{23}^{2}-x_{12}^{2}x_{33}\,-\,x_{44}\quad\in\,\,\,J.$
(12)

This cuts down the dimension by one, and we now have $\dim{\rm GV}({\cal
L}_{[(3,1)]})=4$.

######  Example 4.2.

Consider the Segre symbol $\sigma=[(2,2)]$. The pencil $\mathcal{L}_{[(2,2)]}$
is spanned by

$\begin{bmatrix}0&\alpha&0&0\\\ \alpha&1&0&0\\\ 0&0&0&\alpha\\\
0&0&\alpha&1\\\ \end{bmatrix}\quad\text{and}\quad\begin{bmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{bmatrix},\qquad\hbox{for
some}\,\,\,\,\alpha\in\mathbb{R}.\,\ $

A version of Algorithm 1 for LSSMs with multiple eigenvalues produces the
ideal

$J\,=\,\langle\,x_{11}-x_{33},\,x_{12}-x_{34},\,x_{22}-x_{44},\,x_{13},\,x_{14},\,x_{23},\,x_{24}\,\rangle.$

The Gibbs variety ${\rm GV}({\cal L}_{[(2,2)]})$ is $3$-dimensional and equals
the Gibbs plane ${\rm GP}({\cal L}_{[(2,2)]})$.

The cubic (12) which distinguishes the Segre symbols $[3,1]$ and $[(3,1)]$ is
explained by the following result. This applies not just to pencils but to all
ASSMs with block structure.

######  Proposition 4.3.

Let ${\cal L}$ be a block-diagonal ASSM with $r$ blocks $X_{i}(y)$ of size
$\tau_{i}$, where $\tau_{1}+\cdots+\tau_{r}=n$. The Gibbs plane ${\rm
GP}({\cal L})$ is contained in
$\mathbb{S}^{\tau_{1}}\times\cdots\times\mathbb{S}^{\tau_{r}}\subset\mathbb{S}^{n}$.
Moreover, with the notation ${\cal J}=\\{\\{i,j\\}\in\binom{[r]}{2}:{\rm
trace}(X_{i}(y))={\rm trace}(X_{j}(y))\\}$, we have

${\rm GV}({\cal L})\,\subseteq\,\\{(X_{1},\ldots,X_{r})\in{\rm GP}({\cal
L})\,:\,\det(X_{i})=\det(X_{j})\text{ for all }\,\\{i,j\\}\in{\cal J}\\}.$

######  Proof 4.4.

Block-diagonal matrices are exponentiated block-wise. The entries outside the
diagonal blocks are zero. The statement follows from $\,{\rm det}({\rm
exp}(X_{i}(y)))={\rm exp}({\rm trace}(X_{i}(y)))$.

Proposition 4.3 holds for the canonical pencil $\mathcal{L}_{\sigma}$ of any
Segre symbol $\sigma$. First of all, for all indices $(i,j)$ outside the
diagonal blocks, we have $x_{ij}=0$ on the Gibbs plane ${\rm GP}({\cal
L}_{\sigma})$. Next, one has equations for the exponential of a single block,
like those in Theorem 4.5 below. Finally, there are equations that link the
blocks corresponding to entries $\sigma_{ij}$ of the same partition
$\sigma_{i}$. Some of these come from trace equalities between blocks of
${\cal L}_{\sigma}$, and this is the scope of Proposition 4.3. In particular,
blocks $ij$ and $ik$ for which $\sigma_{ij}=\sigma_{ik}\,{\rm mod}\,2$
exponentiate to $X_{ij}\in\mathbb{S}^{\sigma_{ij}}_{+}$ and
$X_{ik}\in\mathbb{S}^{\sigma_{ik}}_{+}$ with equal determinant. We saw this in
(12). In all examples we computed, the three classes of equations above
determine the Gibbs variety.

We now derive the equations that hold for the exponential of a single block.
To this end, we fix $\sigma=[n]$ with $\alpha_{1}=0$. The canonical LSSM
${\cal L}_{[n]}$ consists of the symmetric matrices

$Y\,\,=\,\,\begin{bmatrix}0&0&\ldots&0&y_{1}\\\ 0&0&\ldots&y_{1}&y_{2}\\\
\vdots&\vdots&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&\vdots\\\ 0&y_{1}&y_{2}&\vdots&0\\\
y_{1}&y_{2}&\vdots&0&0\\\ \end{bmatrix}.$

The case $n=4$ was featured in Example 2.3. In what follows we generalize that
example.

######  Theorem 4.5.

The following linear equations hold on the Gibbs plane ${\rm
GP}(\mathcal{L}_{[n]})$:

$x_{i-1,j}+x_{i+1,j}\,\,=\,\,x_{i,j-1}+x_{i,j+1}\quad\hbox{for~{}$\,2\leqslant
i<j\leqslant n$.}$ (13)

The $2\times 2$-minors of the following $2\times(n-1)$ matrix vanish on the
Gibbs variety ${\rm GV}(\mathcal{L}_{[n]})$:

$D(X)\,\,=\,\,\,\begin{bmatrix}x_{11}&x_{12}&x_{22}&\ldots&\\\
x_{12}&x_{22}&x_{23}&\ldots&\\\
\end{bmatrix}-\begin{bmatrix}x_{n,n}&x_{n-1,n}&x_{n-1,n-1}&\ldots&\\\
0&x_{n,n}&x_{n-1,n}&\ldots&\\\ \end{bmatrix}.$ (14)

If the Galois group $G_{\mathcal{L}_{[n]}}$ is the symmetric group $S_{n}$,
then the prime ideal of ${\rm GV}(\mathcal{L}_{[n]})$ is generated by (13) and
(14), and we have ${\rm dim}\,{\rm GP}(\mathcal{L}_{n})=2n-1$ and ${\rm
dim}\,{\rm GV}(\mathcal{L}_{[n]})=n+1$.

######  Remark 4.6.

We conjecture that $G_{\mathcal{L}_{[n]}}=S_{n}$. This was verified
computationally for many values of $n$, but we currently do not have a proof
that works for all $n$. This gap underscores the need, pointed out at the end
of Section 3, for a study of the Galois groups of LSSMs.

######  Proof 4.7.

We claim that the linear equations (13) hold for every non-negative integer
power of $Y$. This implies that they hold for ${\rm exp}(Y)$. We will show
this by induction. The equations clearly hold for $Y^{0}={\rm id}_{n}$.
Suppose they hold for $(m_{ij})=M=Y^{k}$. Write $(b_{ij})=B:=Y^{k+1}=MY$.

The two-banded structure of $Y$ implies $b_{i,j}=y_{1}\cdot
m_{i,n-j+1}+y_{2}\cdot m_{i,n-j+2}$ for $1\leqslant i<j$. The following
identity holds for $2\leqslant i<j$, and it shows that ${\rm exp}(Y)$
satisfies the equations (13):

$b_{i-1,j}-b_{i,j-1}-b_{i,j+1}+b_{i+1,j}\,\,=\,\,y_{1}\cdot
m_{i-1,n-j+1}+y_{2}\cdot m_{i-1,n-j+2}-y_{1}\cdot m_{i,n-j+2}\,-\\\ y_{2}\cdot
m_{i,n-j+3}-y_{1}\cdot m_{i,n-j}-y_{2}\cdot m_{i,n-j+1}+y_{1}\cdot
m_{i+1,n-j+1}+y_{2}\cdot m_{i+1,n-j+2}\\\
=\,\,y_{1}\cdot(m_{i-1,n-j+1}-m_{i,n-j+2}-m_{i,n-j}+m_{i+1,n-j+1})\,+\\\
y_{2}\cdot(m_{i-1,n-j+2}-m_{i,n-j+3}-m_{i,n-j+1}+m_{i+1,n-j+2})\,\,=\,\,0.$

We next consider the matrix $D(X)$ in (14). We must show that $D(X)$ has rank
$\leq 1$ for $X\in{\rm GV}(\mathcal{L}_{[n]})$. We claim that the rows of
$D(Y^{k})$ are proportional with the same coefficient for all
$k\in\mathbb{Z}_{\geq 0}$. This will imply that the rows of $D(\exp{(Y}))$ are
proportional. For the proof, let $\vec{v}_{1}$ and $\vec{v}_{2}$ be the rows
of $D(B)$, where $B=Y^{k}$. We will show that
${y_{1}\vec{v}_{1}+y_{2}\vec{v}_{2}=0}$.

First note that $D({\rm id}_{n})=0$. Also note that each column of $D(B)$ has
the form

$\begin{bmatrix}b_{i,i}-b_{n+1-i,n+1-i}\\\ b_{i,i+1}-b_{n+1-i,n+2-i}\\\
\end{bmatrix}\quad{\rm or}\quad\begin{bmatrix}b_{i,i+1}-b_{n-i,n+1-i}\\\
b_{i+1,i+1}-b_{n-i+1,n-i+1}\\\ \end{bmatrix}.$

We start with the left case. We must show
$y_{1}(b_{i,i}-b_{n+1-i,n+1-i})+y_{2}(b_{i,i+1}-b_{n+1-i,n+2-i})=0$.

Recall from above that $b_{i,j}=y_{1}\cdot m_{i,n-j+1}+y_{2}\cdot
m_{i,n-j+2}$, where $(m_{i,j})=M=Y^{k-1}$ for $i<j$. Using this and the fact
that the powers of $Y$ are symmetric, we write

$y_{1}(b_{i,i}-b_{n+1-i,n+1-i})+y_{2}(b_{i,i+1}-b_{n+1-i,n+2-i})=\\\
=y_{1}((y_{1}\cdot m_{i,n-i+1}+y_{2}\cdot m_{i,n-i+2})-(y_{1}\cdot
m_{n+1-i,i}+y_{2}\cdot m_{n+1-i,i+1}))\,+\\\ y_{2}((y_{1}\cdot
m_{i,n-i}+y_{2}\cdot m_{i,n-i+1})-(y_{1}\cdot m_{n+1-i,i-1}+y_{2}\cdot
m_{n+1-i,i}))\\\
=y_{1}y_{2}(m_{i,n-i+2}-m_{i+1,n+1-i}+m_{i,n-i}-m_{n+1-i,i-1})=0,$

where the last equality follows from (13). Now, for the second case we have

$y_{1}(b_{i,i+1}-b_{n-i,n+1-i})+y_{2}(b_{i+1,i+1}-b_{n-i+1,n-i+1})=\\\
y_{1}(y_{1}\cdot m_{i,n-i}+y_{2}\cdot m_{i,n-i+1}-y_{1}\cdot
m_{n-i,i}-y_{2}\cdot m_{n-i,i+1})\,+\\\ y_{2}(y_{1}\cdot
m_{i+1,n-i}+y_{2}\cdot m_{i+1,n-i+1}-y_{1}\cdot m_{n-i+1,i}-y_{2}\cdot
m_{n-i+1,i+1})=0.$

This proves that the $2\times 2$ minors of $D(X)$ vanish on the Gibbs variety
$\,{\rm GV}(\mathcal{L}_{[n]})$.

Suppose now that the eigenvalues of $Y$ are $\mathbb{Q}$-linearly independent.
We can check this directly for $n\leq 5$. For $n\geq 6$ it follows from our
hypothesis $G_{\mathcal{L}_{[n]}}=S_{n}$, by [14, Proposition 2]. That
hypothesis implies ${\rm dim}\,{\rm GV}(\mathcal{L}_{[n]})=n+1$, by Theorems
2.4 and 3.6. The $2\times 2$-minors of $D(X)$ generate a prime ideal of
codimension $n-2$ in the coordinates of the $(2n-1)$-dimensional space given
by (13). The equality of dimensions yields ${\rm GP}(\mathcal{L}_{[n]})=2n-1$,
and we conclude that our linear and quadratic constraints generate the prime
ideal of ${\rm GV}(\mathcal{L}_{[n]})$.

Theorem 2.4 and its refinement in Remark 3.3 furnish an upper bound on the
dimension of any Gibbs variety. This raises the question when this bound is
attained. In what follows, we offer a complete answer for $d=2$. Let
$\mathcal{L}$ be a pencil with eigenvalues $\lambda_{i}(y)$, and let $W$
denote the Zariski closure in $\mathbb{R}^{n}$ of the set of points ${\rm
exp}(\lambda(y))=(e^{\lambda_{1}(y)},\ldots,e^{\lambda_{n}(y)})$,
$y\in\mathbb{R}^{2}$.

######  Theorem 4.8.

Let $\mathcal{L}={\rm span}_{\mathbb{R}}(A_{1},A_{2})$, where
$A_{1}A_{2}\not=A_{2}A_{1}$. Then ${\rm dim}\,{\rm GV}(\mathcal{L})={\rm
dim}(W)+1$. In particular, if the Galois group $G_{\mathcal{L}}$ is the
symmetric group $S_{n}$ then ${\rm dim}\,{\rm GV}(\mathcal{L})=n+1$.

######  Proof 4.9.

We claim that the fibers of the map $\phi:V\times W\to\mathbb{S}^{n}$ defined
by (9) are one-dimensional. Let $B\in\phi(V\times W)$ and consider any point
$p=(y_{1},y_{2},\lambda_{1},\ldots,\lambda_{n},z_{1},\ldots,z_{n})\in\phi^{-1}(B)$.
The condition that $p$ lies in the fiber $\phi^{-1}(B)$ is equivalent to

  1. (1)

$z_{1},\ldots,z_{n}$ are the eigenvalues of $B$, and

  2. (2)

$X=y_{1}A_{1}+y_{2}A_{2}$ and $B$ have the same eigenvectors, and

  3. (3)

$\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $X$.

Condition (1) follows from Theorem 3.1 for $f={\rm exp}$. It implies that
there are only finitely many possibilities for the $z$-coordinates of the
point $p$ in the fiber: up to permutations, they are the eigenvalues of $B$.
Condition (3) follows from $(y_{1},y_{2},\lambda_{1},\ldots,\lambda_{n})\in
V$. It says that the $\lambda$-coordinates are determined, up to permutation,
by $y_{1},y_{2}$. Therefore, it suffices to show that the matrices in ${\cal
L}$ whose eigenvectors are those of $B$ form a one-dimensional subvariety.

Symmetric matrices have common eigenvectors if and only if they commute.
Define $S=\\{X=y_{1}A_{1}+y_{2}A_{2}\in{\cal L}\,:X\cdot B=B\cdot
X\\}\subset{\cal L}$. This is a pairwise commuting linear subspace. Note that
$S$ contains a nonzero matrix $X$, since there is a point in $\phi^{-1}(B)$
whose $y$-coordinates define a nonzero matrix in ${\cal L}$. Therefore $\dim
S\geq 1$. Since $A_{1}A_{2}\neq A_{2}A_{1}$, we also have $\dim S\leq 1$.
Hence $\dim S=\dim\phi^{-1}(B)=1$ and the upper bound $\dim W+1$ for the
dimension of ${\rm GV}({\cal L})$, which is given by Remark 3.3, is attained
in our situation.

##  5 Convex optimization

In this section we show how Gibbs manifolds arise from entropic regularization
in optimization (cf. [21]). We fix an arbitrary linear map
$\pi:\mathbb{S}^{n}\rightarrow\mathbb{R}^{d}$. This can be written in the form

$\pi(X)=\bigl{(}\langle A_{1},X\rangle,\langle A_{2},X\rangle,\ldots,\langle
A_{d},X\rangle\bigr{)}.$

Here the $A_{i}\in\mathbb{S}^{n}$, and $\langle A_{i},X\rangle:={\rm
trace}(A_{i}X)$. The image $\pi(\mathbb{S}^{n}_{+})$ of the PSD cone
$\mathbb{S}^{n}_{+}$ under our linear map $\pi$ is a _spectrahedral shadow_.
Here it is a full-dimensional semialgebraic convex cone in $\mathbb{R}^{d}$.
Interestingly, $\pi(\mathbb{S}^{n}_{+})$ can fail to be closed, as explained
in [13].

Semidefinite programming (SDP) is the following convex optimization problem:

${\rm Minimize}\quad\langle C,X\rangle\quad\hbox{subject to}\quad
X\in\mathbb{S}^{n}_{+}\,\text{ and }\,\pi(X)=b.$ (15)

See e.g. [16, Chapter 12]. The instance (15) is specified by the cost matrix
$C\in\mathbb{S}^{n}$ and the right hand side vector $b\in\mathbb{R}^{d}$. The
feasible region $\mathbb{S}^{n}_{+}\cap\pi^{-1}(b)$ is a _spectrahedron_. The
SDP problem (15) is feasible, i.e. the spectrahedron is non-empty, if and only
if $\,b$ is in $\pi(\mathbb{S}^{n}_{+})$.

Consider the LSSM $\mathcal{L}={\rm span}_{\mathbb{R}}(A_{1},\ldots,A_{d})$.
We usually assume that $\mathcal{L}$ contains a positive definite matrix. This
hypothesis ensures that each spectrahedron $\pi^{-1}(b)$ is compact.

As a natural extension of [21, eqn (2)], we now define the entropic
regularization of SDP:

${\rm Minimize}\quad\langle C,X\rangle\,-\,\epsilon\cdot
h(X)\quad\hbox{subject to}\quad X\in\mathbb{S}^{n}_{+}\,\text{ and
}\,\pi(X)=b.$ (16)

Here $\epsilon>0$ is a small parameter, and $h$ denotes the von Neumann
entropy, here defined as

$h\,:\,\mathbb{S}^{n}_{+}\,\rightarrow\,\mathbb{R}\,,\,\,X\,\mapsto\,{\rm
trace}\bigl{(}X-X\cdot{\rm log}(X)\bigr{)}.$

We note that $h$ is invariant under the action of the orthogonal group on
$\mathbb{S}^{n}_{+}$. This implies
$h(X)=\sum_{i=1}^{n}(\lambda_{i}-\lambda_{i}{\rm log}(\lambda_{i}))$, where
$\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $X$. Hence the von
Neumann entropy $h$ is the matrix version of the entropy function on
$\mathbb{R}^{n}_{+}$ used in [21].

Our next result makes the role of Gibbs manifolds in semidefinite programming
explicit. The following ASSM is obtained by incorporating $\epsilon$ and the
cost matrix $C$ into the LSSM:

$\mathcal{L}_{\epsilon}\,\,:=\,\,\mathcal{L}-\frac{1}{\epsilon}C\quad\hbox{for
any}\,\,\epsilon>0.$

Here we allow the case $\epsilon=\infty$, where the dependency on $C$
disappears and the ASSM is simply the LSSM, i.e.
$\mathcal{L}_{\infty}=\mathcal{L}$. The following theorem is the main result
in this section.

######  Theorem 5.1.

For $b\in\pi(\mathbb{S}^{n}_{+})$, the intersection of $\pi^{-1}(b)$ with the
Gibbs manifold ${\rm GM}(\mathcal{L}_{\epsilon})$ consists of a single point
$X_{\epsilon}^{*}$. This point is the optimal solution to the regularized SDP
(16). For $\epsilon=\infty$, it is the unique maximizer of von Neumann entropy
on the spectrahedron $\pi^{-1}(b)$.

The importance of this result for semidefinite programming lies in taking the
limit as $\epsilon$ tends to zero. This limit $\,{\rm
lim}_{\epsilon\rightarrow 0}\,X^{*}_{\epsilon}\,$ exists and it is an optimal
solution to (15). The optimal solution is unique for generic $C$. Entropic
regularization is about approximating that limit.

######  Remark 5.2.

Theorem 5.1 implies that adding the condition $X\in{\rm GV}({\cal
L}_{\epsilon})$ to (16) leaves the optimizer unchanged. Hence, if we know
equations for the Gibbs variety, we may shrink the feasible region by adding
polynomial constraints. Most practical are the affine-linear equations:
imposing $X\in{\rm GP}({\cal L}_{\epsilon})$ allows to solve (16) on a
spectrahedron of lower dimension.

To prove Theorem 5.1, we derive two key properties of the von Neumann entropy:

######  Proposition 5.3.

The function $h$ satisfies:

  * (a)

$h$ is strictly concave on the PSD cone $\mathbb{S}^{n}_{+}$, and

  * (b)

the gradient of $h$ is the negative matrix logarithm: $\,\nabla(h)(X)=-{\rm
log}(X)$.

######  Proof 5.4.

For (a), we use a classical result by Davis [4]. The function $h$ is invariant
in the sense that its value $h(X)$ depends on the eigenvalues of $X$. In fact,
it is a symmetric function of the $n$ eigenvalues
$\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$. This function equals
$h(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})=\sum_{i=1}^{n}(\lambda_{i}-\lambda_{i}{\rm
log}(\lambda_{i}))$, and this is a concave function
$\mathbb{R}^{n}_{+}\rightarrow\mathbb{R}$. The assertion hence follows from
the theorem in [4].

For (b) we prove a more general result. For convenience, we change variables
$Y=X-{\rm id}_{n}$ so that $f(Y)=h(Y+{\rm id}_{n})$ is analytic at $Y=0$. Fix
any function $f:\mathbb{R}\rightarrow\mathbb{R}$ that is analytic in a
neighborhood of the origin. Then $Y\mapsto{\rm trace}(f(Y))$ is a well-defined
real-valued analytic function of $n\times n$ matrices $Y=(y_{ij})$ that are
close to zero. The gradient of this function is the $n\times n$ matrix whose
entries are the partial derivatives $\partial{\rm trace}(f(Y))/\partial
y_{ij}$. We claim that

$\nabla{\rm trace}(f(Y))\,=\,f^{\prime}(Y^{\top}).$ (17)

Both sides are linear in $f$, and $f$ is analytic, so it suffices to prove
this for monomials, i.e.

$\nabla{\rm trace}(Y^{k})\,=\,k\cdot(Y^{\top})^{k-1}\qquad\hbox{for all
integers}\,\,k\geq 1.$ (18)

Note that ${\rm trace}(Y^{k})$ is a homogeneous polynomial of degree $k$ in
the matrix entries $y_{ij}$, namely it is the sum over all products
$y_{i_{1}i_{2}}y_{i_{2}i_{3}}\cdots y_{i_{k-2}i_{k-1}}y_{i_{k-1}i_{1}}$ that
represent closed walks in the complete graph on $k$ nodes. When taking the
derivative $\partial/\partial y_{ij}$ of that sum, we obtain $k$ times the sum
over all walks that start at node $j$ and end at node $i$. Here each walk
occurs with the factor $k$ because $y_{ij}$ can be inserted in $k$ different
ways to create one of the closed walks above. This polynomial of degree $k-1$
is the entry of the matrix power $Y^{k-1}$ in row $j$ and column $i$, so it is
the entry of its transpose $(Y^{\top})^{k-1}$ is row $i$ and column $j$. To
prove the proposition, we now apply (17) to the function
$f(y)=(y+1)-(y+1)\cdot{\rm log}(y+1)$.

If $\mathcal{L}=\mathcal{D}$ consists of diagonal matrices then the Gibbs
manifold ${\rm GM}(\mathcal{D})$ is a discrete exponential family [23, §6.2],
and $\pi({\rm GM}(\mathcal{D}))$ is the associated convex polytope. This uses
the moment map from toric geometry [16, Theorem 8.24]. In particular, if the
linear space $\mathcal{D}$ is defined over $\mathbb{Q}$ then the polytope is
rational and the Zariski closure of ${\rm GM}(\mathcal{D})$ is the toric
variety of that polytope. If the space $\mathcal{D}$ is not defined over
$\mathbb{Q}$ then ${\rm GM}(\mathcal{D})$ is an analytic toric manifold, whose
Zariski closure is the larger toric variety ${\rm GV}(\mathcal{D})={\rm
GM}(\mathcal{D}_{\mathbb{Q}})$ seen in (7).

The key step to proving Theorem 5.1 is a non-abelian version of the toric
moment map.

######  Theorem 5.5.

The restriction of the linear map
$\pi:\mathbb{S}^{n}_{+}\rightarrow\mathbb{R}^{d}$ to the Gibbs manifold ${\rm
GM}(\mathcal{L})$ defines a bijection between $\,{\rm GM}(\mathcal{L})$ and
the open spectrahedral shadow $\,{\rm int}(\pi(\mathbb{S}^{n}_{+}))\,$ in
$\,\mathbb{R}^{d}$.

######  Proof 5.6.

Fix an arbitrary positive definite matrix $X\in{\rm int}(\mathbb{S}^{n}_{+})$
and set $b=\pi(X)$. We must show that the spectrahedron $\pi^{-1}(b)$ contains
precisely one point that lies in ${\rm GM}(\mathcal{L})$.

Consider the restriction of the von Neumann entropy $h$ to the spectrahedron
$\pi^{-1}(b)$. This restriction is strictly concave on the convex body
$\pi^{-1}(b)$ by Proposition 5.3. Therefore $h$ attains a unique maximum
$X^{*}$ in the relative interior of $\pi^{-1}(b)$. The first order condition
at this maximum tells us that $\nabla(h)(X^{*})=-{\rm log}(X^{*})$ lies in
$\mathcal{L}$, which is the span of the gradients of the constraints $\langle
A_{i},X\rangle=b_{i}$. Hence, the optimal matrix $X^{*}$ lies in the Gibbs
manifold

${\rm GM}(\mathcal{L})\,\,=\,\,\bigl{\\{}X\in\mathbb{S}^{n}_{+}:{\rm
log}(X)\in\mathcal{L}\bigr{\\}}.$

The assignment $b\mapsto X^{*}=X^{*}(b)$ is well defined and continuous on the
interior of the cone $\pi(\mathbb{S}^{n}_{+})$. We have shown that it is a
section of the linear map $\pi$, which means $\pi(X^{*}(b))=b$. We conclude
that $\pi$ defines a homeomorphism between ${\rm GM}(\mathcal{L})$ and $\,{\rm
int}(\pi(\mathbb{S}^{n}_{+}))$.

######  Proof 5.7 (Proof of Theorem 5.1).

For any fixed $\epsilon>0$, any minimizer $X^{*}=X^{*}_{\epsilon}$ of the
regularized problem (16) in the spectrahedron $\pi^{-1}(b)$ must satisfy
$\,C+\epsilon\cdot{\rm log}(X^{*})\in\mathcal{L}$. This is equivalent to
$X^{*}\in{\rm GM}(\mathcal{L}_{\epsilon})$, and it follows from the first
order optimality conditions. By the same convexity argument as in the proof of
Theorem 5.5, the objective function in (16) has only one critical point in the
spectrahedron $\pi^{-1}(b)$. This implies $\,\pi^{-1}(b)\cap{\rm
GM}(\mathcal{L}_{\epsilon})\,=\,\\{X^{*}_{\epsilon}\\}$.

We can now turn the discussion around and offer a definition of Gibbs
manifolds and Gibbs varieties purely in terms of convex optimization. Fix any
LSSM $\mathcal{L}$ of dimension $d$ in $\mathbb{S}^{n}$. This defines a
canonical linear map
$\pi:\mathbb{S}^{n}_{+}\rightarrow\mathbb{S}^{n}/\mathcal{L}^{\perp}\simeq\mathbb{R}^{d}$.
Each fiber $\pi^{-1}(b)$ is a spectrahedron. If this is non-empty then the
entropy $h(X)$ has a unique maximizer $X^{*}(b)$ in $\pi^{-1}(b)$. The Gibbs
manifold ${\rm GM}(\mathcal{L})$ is the set of these entropy maximizers
$X^{*}(b)$ for $b\in\mathbb{R}^{d}$. The Gibbs variety ${\rm GV}(\mathcal{L})$
is defined by all polynomial constraints satisfied by these $X^{*}(b)$.

This extends naturally to any ASSM $A_{0}+\mathcal{L}$. We now maximize the
concave function $h(X)+\langle A_{0},X\rangle$ over the spectrahedra
$\pi^{-1}(b)$. The Gibbs manifold ${\rm GM}(A_{0}+\mathcal{L})$ collects all
maximizers, and the Gibbs variety ${\rm GV}(A_{0}+\mathcal{L})$ is defined by
their polynomial constraints.

######  Example 5.8.

Let $\mathcal{L}$ denote the space of all Hankel matrices $[y_{i+j-1}]_{1\leq
i,j\leq n}$ in $\mathbb{S}^{n}$. This LSSM has dimension $d=2n-1$. The linear
map $\pi:\mathbb{S}^{n}_{+}\rightarrow\mathbb{R}^{d}$ takes any positive
definite matrix $X$ to a nonnegative polynomial $b=b(t)$ in one variable $t$
of degree $2n-2$. We have
$b(t)=(1,t,\ldots,t^{n-1})X(1,t,\ldots,t^{n-1})^{\top}$, so the matrix $X$
gives a sum-of-squares (SOS) representation of $b(t)$. The fiber $\pi^{-1}(b)$
is the Gram spectrahedron [20] of the polynomial $b$. The entropy maximizer
$X^{*}(b)$ in the Gram spectrahedron is a favorite SOS representation of $b$.
The Gibbs manifold ${\rm GM}(\mathcal{L})$ gathers the favorite SOS
representations for all non-negative polynomials $b$. The Gibbs variety ${\rm
GV}(\mathcal{L})$, which has dimension $\leq 3n-2$, is the tightest outer
approximation of ${\rm GM}(\mathcal{L})$ that is definable by polynomials in
the matrix entries.

In Example 3.5 we saw a variant of $\mathcal{L}$, namely the sub-LSSM where
the upper left entry of the Hankel matrix was fixed to be zero. If $C=-E_{11}$
is the corresponding negated matrix unit, then (15) is the problem of
minimizing $b(t)$ over $t\in\mathbb{R}$. See [16, Section 12.3] for a first
introduction to polynomial optimization via SOS representations. It would be
interesting to explore the potential of the entropic regularization (16) for
polynomial optimization.

One of the topics of [21] was a scaling algorithm for solving the optimization
problem (16) for linear programming (LP), i.e. the case when
$A_{1},\ldots,A_{d}$ are diagonal matrices. This algorithm extends the
Darroch-Ratcliff algorithm for Iterative Proportional Fitting in statistics.
Combining this with a method for driving $\epsilon$ to zero leads to a
numerical algorithm for large-scale LP problems, such as the optimal transport
problems in [21, Section 3].

We are hopeful that the scaling algorithm can be extended to the problem (16)
in full generality. By combining this with a method for driving $\epsilon$ to
zero, one obtains a numerical framework for solving SDP problems such as
quantum optimal transport in Section 6.

One important geometric object for SDP is the limiting Gibbs manifold, ${\rm
lim}_{\epsilon\rightarrow 0}\,{\rm GM}(\mathcal{L}_{\epsilon})$. This is the
set of optimal solutions, as $b$ ranges over $\mathbb{R}^{d}$. In the case of
LP, with $C$ generic, it is the simplicial complex which forms the regular
triangulation given by $C$. This reveals the combinatorial essence of entropic
regularization of LP, as explained in [21, Theorem 7]. From the perspective of
positive geometry, it would be worthwhile to study ${\rm
lim}_{\epsilon\rightarrow 0}\,{\rm GM}(\mathcal{L}_{\epsilon})$ for SDP. This
set is semialgebraic, and it defines a nonlinear subdivision of the
spectrahedral shadow $\pi(\mathbb{S}^{n}_{+})$. If we vary the cost matrix
$C$, the theory of fiber bodies in [15] becomes relevant.

##  6 Quantum optimal transport

In this section we examine a semidefinite programming analogue of the
classical optimal transport problem, known as quantum optimal transport (QOT).
We follow the presentation by Cole, Eckstein, Friedland, and Zyczkowski in
[3]. Our notation for the dimensions is as in [21, Section 3.1]. We consider
the space $\mathbb{S}^{d_{1}d_{2}}$ of real symmetric matrices $X$ of size
$d_{1}d_{2}\times d_{1}d_{2}$. Rows and columns are indexed by
$[d_{1}]\times[d_{2}]$. Thus, we write $X=(x_{ijkl})$, where $(i,j)$ and
$(k,l)$ are in $[d_{1}]\times[d_{2}]$. The matrix being symmetric means that
$x_{ijkl}=x_{klij}$ for all indices. Each such matrix is mapped to a pair of
two partial traces by the following linear map:

$\mathbb{S}^{d_{1}d_{2}}\,\rightarrow\,\mathbb{S}^{d_{1}}\times\mathbb{S}^{d_{2}}\,,\,\,\,X\mapsto(Y,Z),$

where the $d_{1}{\times}d_{1}$ matrix $Y=(y_{ik})$ satisfies
$y_{ik}=\sum_{j=1}^{d_{2}}x_{ijkj}$, and the $d_{2}{\times}d_{2}$ matrix
$Z=(z_{jl})$ satisfies $z_{jl}=\sum_{i=1}^{d_{1}}x_{ijil}$. If $X$ is positive
semidefinite then so are its partial traces $Y$ and $Z$. Hence our
marginalization map restricts to a linear projection of closed convex cones,
denoted

$\mu\,:\,\mathbb{S}_{+}^{d_{1}d_{2}}\,\rightarrow\,\mathbb{S}_{+}^{d_{1}}\times\mathbb{S}_{+}^{d_{2}}\,,\,\,\,X\mapsto(Y,Z).$
(19)

Diagonal matrices in $\mathbb{S}_{+}^{d_{1}d_{2}}$ can be identified with
rectangular matrices of format $d_{1}\times d_{2}$ whose entries are
nonnegative. The map $\mu$ takes such a rectangular matrix to its row sums and
column sums. Hence the restriction of $\mu$ to diagonal matrices in
$\mathbb{S}_{+}^{d_{1}d_{2}}$ is precisely the linear map that defines
classical optimal transport in the discrete setting of [21, Section 3.1].

The quantum optimal transportation problem (QOT) is the task of minimizing a
linear function $X\mapsto\langle C,X\rangle$ over any transportation
spectrahedron $\mu^{-1}(Y,Z)$. This is an SDP. Our main theorem in this
section states that its Gibbs manifold is semialgebraic.

######  Theorem 6.1.

The Gibbs manifold ${\rm GM}(\mathcal{L})$ for QOT is a semialgebraic subset
of $\mathbb{S}^{d_{1}d_{2}}_{+}$. It consists of all symmetric matrices
$Y\otimes Z$, where $Y\in\mathbb{S}^{d_{1}}_{+}$ and
$Z\in\mathbb{S}^{d_{2}}_{+}$. The Gibbs variety ${\rm GV}({\cal
L})\subset\mathbb{S}^{d_{1}d_{2}}$ is linearly isomorphic to the cone over the
Segre variety
$\,\mathbb{P}^{\binom{d_{1}+1}{2}-1}\times\mathbb{P}^{\binom{d_{2}+1}{2}-1}$.

The image of the marginalization map $\mu$ generalizes the polytope
$\Delta_{d_{1}-1}\times\Delta_{d_{2}-1}$, and the fibers of $\mu$ are quantum
versions of transportation polytopes. These shapes are now nonlinear.

######  Lemma 6.2.

The image of the map $\mu$ is a convex cone of dimension
$\binom{d_{1}+1}{2}+\binom{d_{2}+1}{2}-1$:

${\rm
image}(\mu)\,\,\,=\,\,\,\bigl{\\{}(Y,Z)\in\mathbb{S}_{+}^{d_{1}}\times\mathbb{S}_{+}^{d_{2}}\,:\,{\rm
trace}(Y)={\rm trace}(Z)\bigr{\\}}.$ (20)

For any point $(Y,Z)$ in the relative interior of this cone, the
transportation spectrahedron $\mu^{-1}(Y,Z)$ is a compact convex body of
dimension $\frac{1}{2}(d_{1}-1)(d_{2}-1)(d_{1}d_{2}+d_{1}+d_{2}+2)$.

######  Proof 6.3 (Proof of Lemma 6.2).

The partial trace map $\mu$ in (19) restricts to tensor products as follows:

$\mu(Y\otimes Z)\,=\,\bigl{(}\,{\rm trace}(Z)\cdot Y\,,\,{\rm trace}(Y)\cdot
Z\,\bigr{)}.$ (21)

Hence, if $Y\in\mathbb{S}^{d_{1}}_{+}$ and $Z\in\mathbb{S}^{d_{2}}_{+}$
satisfy $t={\rm trace}(Y)={\rm trace}(Z)$ then $\frac{1}{t}Y\otimes Z$ is a
positive semidefinite matrix in the fiber $\mu^{-1}(Y,Z)$. This shows that the
image is as claimed on the right hand side of (20). The image is a
spectrahedral cone of dimension $\binom{d_{1}+1}{2}{+}\binom{d_{2}+1}{2}{-}1$.
Subtracting this from ${\rm
dim}\,\mathbb{S}^{d_{1}d_{2}}_{+}=\binom{d_{1}d_{2}+1}{2}$ yields the
dimension of the interior fibers.

######  Example 6.4 ($d_{1}{=}d_{2}{=}2$).

The map $\mu$ projects positive semidefinite $4\times 4$ symmetric matrices

$X\,=\,\small\begin{bmatrix}x_{1111}&x_{1112}&x_{1121}&x_{1122}\\\
x_{1112}&x_{1212}&x_{1221}&x_{1222}\\\ x_{1121}&x_{1221}&x_{2121}&x_{2122}\\\
x_{1122}&x_{1222}&x_{2122}&x_{2222}\end{bmatrix}$

onto a $5$-dimensional convex cone, given by the direct product of two disks.
The formula is

$Y\,=\,\begin{bmatrix}x_{1111}+x_{1212}&x_{1121}+x_{1222}\\\
x_{1121}+x_{1222}&x_{2121}+x_{2222}\end{bmatrix}\quad{\rm and}\quad
Z\,=\,\begin{bmatrix}x_{1111}+x_{2121}&x_{1112}+x_{2122}\\\
x_{1112}+x_{2122}&x_{1212}+x_{2222}\end{bmatrix}.$

The fibers of this map $\mu$ are the $5$-dimensional transportation
spectrahedra $\mu^{-1}(Y,Z)$.

To illustrate the QOT problem, we fix the margins and the cost matrix as
follows:

$Y\,=\,\begin{bmatrix}5&1\\\ 1&6\end{bmatrix}\quad{\rm and}\quad
Z\,=\,\begin{bmatrix}7&2\\\ 2&4\end{bmatrix}\quad{\rm and}\quad
C\,\,=\,\,\small\begin{bmatrix}2&3&5&7\\\ 3&11&13&17\\\ 5&13&23&29\\\
7&17&29&31\end{bmatrix}.$ (22)

We wish to minimize $\langle C,X\rangle$ subject to $\mu(X)=(Y,Z)$. The
optimal solution $X^{*}$ is equal to

$\\!\\!\\!\tiny\begin{bmatrix}3.579128995196972555885181314&2.148103387337332721011731020&2.671254991031789281229265149&-2.07566204542024789990696017\\\
2.148103387337332721011731020&1.420871004803027444114818686&1.16978382139276763200240537&\\!\\!\\!-1.671254991031789281229265149\\\
2.671254991031789281229265149&1.169783821392767632002405371&3.420871004803027444114818686&-0.14810338733733272101173102\\\
-2.07566204542024789990696017&-1.671254991031789281229265149&-0.14810338733733272101173102&2.579128995196972555885181314\\\
\end{bmatrix}\\!.$

This matrix has rank $2$. The optimal value equals
$v=156.964485798827271035367539305...$. This is an algebraic number of degree
$12$. Its exact representation is the minimal polynomial

$\small\begin{matrix}\\!125v^{12}-465480v^{11}+770321646v^{10}{-}744236670798v^{9}{+}463560077206539v^{8}{-}193865445786866004v^{7}\\\
+54901023652716544539v^{6}-10330064181552258647604v^{5}+1219620644420527588643307v^{4}\\\
-77994100149206862070472310v^{3}+1395374211380010273312826701v^{2}\\\
+83502957914204004050312708316v-2047417613706778627978564647804\,\,=\,\,0.\end{matrix}$

This was derived from the KKT equations in [17, Theorem 3]. We conclude that
the algebraic degree of QOT for $d_{1}=d_{2}=2$ is equal to $12$. This is
smaller than the algebraic degree of semidefinite programming, which is $42$.
That is the entry for $m{=}5$ and $n{=}4$ in [17, Table 2].

This drop arises because QOT is a very special SDP. The LSSM for our QOT
problem is

$\mathcal{L}\,\,=\,\,\left\\{\small\begin{bmatrix}y_{1}+y_{3}&y_{5}&y_{4}&0\\\
y_{5}&y_{1}&0&y_{4}\\\ y_{4}&0&y_{2}+y_{3}&y_{5}\,\\\
0&y_{4}&y_{5}&y_{2}\,\end{bmatrix}\,:\,\,y_{1},y_{2},y_{3},y_{4},y_{5}\in\mathbb{R}\,\right\\}.$
(23)

This defines our $5$-dimensional Gibbs manifold ${\rm GM}(\mathcal{L})$ in the
$10$-dimensional cone $\mathbb{S}^{4}_{+}$. Theorem 6.1 states that it equals
the positive part of the Gibbs variety, i.e. ${\rm GM}(\mathcal{L})={\rm
GV}(\mathcal{L})\cap\mathbb{S}^{4}_{+}$.

We compute the entropy maximizer inside the $5$-dimensional transportation
spectrahedron $\mu^{-1}(Y,Z)$ for the marginal matrices $Y$ and $Z$ in (22).
Notably, its entries are rational:

$\mu^{-1}(Y,Z)\,\cap\,{\rm
GV}(\mathcal{L})\quad=\quad\mu^{-1}(Y,Z)\,\cap\,{\rm
GM}(\mathcal{L})\quad=\quad\left\\{\,\frac{1}{11}\small\begin{bmatrix}35&10&7&2\\\
10&20&2&4\\\ 7&2&42&12\\\ 2&4&12&24\end{bmatrix}\,\right\\}.$

######  Proof 6.5 (Proof of Theorem 6.1).

By linear extension, the equation (21) serves as a definition of the
marginalization map $\mu$ on $\mathbb{S}^{d_{1}d_{2}}$. We observe the
following for the trace inner product on $\mathbb{S}^{d_{1}d_{2}}$:

$\begin{matrix}&{\rm trace}\bigl{(}(A\otimes{\rm id}_{d_{2}})(Y\otimes
Z)\bigr{)}\,=\,{\rm trace}(Z)\cdot{\rm trace}(AY)&\hbox{for
all}\,\,\,A\in\mathbb{S}^{d_{1}}\\\ {\rm and}&{\rm trace}\bigl{(}({\rm
id}_{d_{1}}\otimes B)(Y\otimes Z)\bigr{)}\,=\,{\rm trace}(Y)\cdot{\rm
trace}(BZ)&\hbox{for all}\,\,\,B\in\mathbb{S}^{d_{2}}.\\\ \end{matrix}$

Therefore, the $(i,j)$ entry of ${\rm trace}(Z)\cdot Y$ is obtained as
$\frac{1}{2}\langle(E_{ij}+E_{ji})\otimes{\rm id}_{d_{2}},Y\otimes Z\rangle$,
where $E_{ij}$ is the $(i,j)$-th matrix unit. A similar observation holds for
the entries of ${\rm trace}(Y)\cdot Z$. This means that $\mu(X)$ is computed
by evaluating ${\rm trace}\bigl{(}(A\otimes{\rm id}_{d_{2}})X\bigr{)}$ and
${\rm trace}\bigl{(}({\rm id}_{d_{1}}\otimes B)X\bigr{)}$, where $A$ ranges
over a basis of $\mathbb{S}^{d_{1}}$ and $B$ ranges over a basis of
$\mathbb{S}^{d_{2}}$. Therefore, we have

$\mathcal{L}\,\,=\,\,\bigl{\\{}\,A\otimes{\rm id}_{d_{2}}\,+\,{\rm
id}_{d_{1}}\otimes B\,\,:\,A\in\mathbb{S}^{d_{1}}\,\,{\rm
and}\,\,B\in\mathbb{S}^{d_{2}}\,\bigr{\\}}.$ (24)

Now, the key step in the proof consists of the following formula for the
matrix logarithm

${\rm log}(Y\otimes Z)\,=\,{\rm log}(Y)\otimes{\rm id}_{d_{2}}\,+\,{\rm
id}_{d_{1}}\otimes{\rm log}(Z).$

This holds for positive semidefinite matrices $Y$ and $Z$, and it is verified
by diagonalizing these matrices. By setting $Y={\rm exp}(A)$ and $Z={\rm
exp}(B)$, we now conclude that the Gibbs manifold ${\rm GM}(\mathcal{L})$
consists of all tensor products $Y\otimes Z$ where
$Y\in\mathbb{S}^{d_{1}}_{+}$ and $Z\in\mathbb{S}^{d_{2}}_{+}$.

We have shown that ${\rm GM}(\mathcal{L})$ is the intersection of a variety
with $\mathbb{S}^{d_{1}d_{2}}_{+}$. This variety must be the Gibbs variety
${\rm GV}(\mathcal{L})$. More precisely, ${\rm GV}(\mathcal{L})$ consists of
all tensor products $Y\otimes Z$ where $Y,Z$ are complex symmetric. This is
the cone over the Segre variety, which is the projective variety in
$\mathbb{P}^{\binom{d_{1}d_{2}+1}{2}-1}$ whose points are the tensor products
${X=Y\otimes Z}$.

We have the following immediate consequence of the proof of Theorem 6.1. The
entropy maximizers have rational entries. This explains the matrix at the end
of Example 6.4

######  Corollary 6.6.

The Gibbs point for QOT is given by $\frac{Y\otimes Z}{{\rm trace}(Y)}$, with
$Y,Z$ the given margins.

At this point, it pays off to revisit Section 3 and to study its thread for
the LSSM in (24).

######  Example 6.7.

We apply Algorithm 1 to the LSSM $\mathcal{L}$ in (23). The eigenvalues of
$\mathcal{L}$ are distinct, and the ideal $\langle E_{1}^{\prime}\rangle$ in
step 3 is the intersection of six prime ideals. One of them is

$\begin{matrix}\langle\,\lambda_{1}+\lambda_{2}-y_{1}-y_{2}-y_{3}\,,\,\,\lambda_{3}+\lambda_{4}-y_{1}-y_{2}-y_{3}\,,\qquad\qquad\qquad\\\
2\lambda_{2}\lambda_{4}-\lambda_{2}y_{1}-\lambda_{4}y_{1}-\lambda_{2}y_{2}-\lambda_{4}y_{2}+2y_{1}y_{2}-\lambda_{2}y_{3}-\lambda_{4}y_{3}+y_{1}y_{3}+y_{2}y_{3}+y_{3}^{2}-2y_{4}^{2}+2y_{5}^{2},\qquad\\\
\qquad\lambda_{2}^{2}+\lambda_{4}^{2}-\lambda_{2}y_{1}-\lambda_{4}y_{1}-\lambda_{2}y_{2}-\lambda_{4}y_{2}+2y_{1}y_{2}-\lambda_{2}y_{3}-\lambda_{4}y_{3}+y_{1}y_{3}+y_{2}y_{3}-2y_{4}^{2}-2y_{5}^{2}\,\rangle.\end{matrix}$

The other five associated primes are found by permuting indices of
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$. Hence, the Galois group
$G_{\mathcal{L}}$ is the Klein four-group $S_{2}\times S_{2}$ in $S_{4}$, and
we infer the linear relation
$\lambda_{1}+\lambda_{2}-\lambda_{3}-\lambda_{4}$. The set $E_{3}$ in step 8
is the singleton $\\{z_{1}z_{2}-z_{3}z_{4}\\}$. The elimination in step 13
reveals the prime ideal in $\mathbb{R}[X]$ that is shown for arbitrary
$d_{1},d_{2}$ in Corollary 6.8.

Our final result is derived from Theorem 6.1 using tools of toric algebra [16,
Chapter 8].

######  Corollary 6.8.

The Gibbs variety for QOT is parametrized by monomials $x_{ijkl}=y_{ik}z_{jl}$
that are not all distinct. Its prime ideal in $\mathbb{R}[X]$ is minimally
generated by the $2\times 2$ minors of a matrix of format
$\binom{d_{1}+1}{2}\times\binom{d_{2}+1}{2}$, together with
$\binom{d_{1}}{2}\binom{d_{2}}{2}$ linear forms in the entries of $X$.

We propose to extend QOT to quantum graphical models [26]. In statistics,
every undirected graph $G$ on $s$ vertices defines such a model [23, Section
13.2]. The graphical model lives in the probability simplex
$\Delta_{d_{1}d_{2}\cdots d_{s}-1}$. Its points are nonnegative tensors of
format $d_{1}\times d_{2}\times\cdots\times d_{s}$ whose entries sum to $1$.
The quantum graphical model lives in the high-dimensional PSD cone
$\mathbb{S}^{d_{1}d_{2}\cdots d_{s}}_{+}$, where the marginalization records
the partial trace for every clique in $G$. It would be interesting to study
the Gibbs manifold and the Gibbs varieties for these models. One may ask
whether they agree for all graphs $G$ that are decomposable. By Theorem 6.1,
this holds for QOT, where $G$ is the graph with two nodes and no edges.

## Acknowledgements

We are grateful to François-Xavier Vialard and Max von Renesse for inspiring
discussions about Gibbs manifolds in optimization and optimal transport, and
we thank Benjamin Bakker for answering our questions about transcendental
number theory. Simon Telen was supported by a Veni grant from the Netherlands
Organisation for Scientific Research (NWO).

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Authors’ addresses:

Dmitrii Pavlov, MPI-MiS Leipzig<EMAIL_ADDRESS>
Bernd Sturmfels, MPI-MiS Leipzig and UC Berkeley<EMAIL_ADDRESS>
Simon Telen, CWI Amsterdam and MPI-MiS Leipzig<EMAIL_ADDRESS>