# Weighted Slice Rank and a Minimax
Correspondence to Strassen’s Spectra

Matthias Christandl Department of Mathematical Sciences University of
Copenhagen<EMAIL_ADDRESS>Vladimir Lysikov Department of Mathematical
Sciences University of Copenhagen<EMAIL_ADDRESS>Jeroen Zuiddam Courant
Institute, NYU KdVI, Univ. of Amsterdam<EMAIL_ADDRESS>

###### Abstract

Structural and computational understanding of tensors is the driving force
behind faster matrix multiplication algorithms, the unraveling of quantum
entanglement, and the breakthrough on the cap set problem. Strassen’s
asymptotic spectra program (SFCS 1986) characterizes optimal matrix
multiplication algorithms through monotone functionals. Our work advances and
makes novel connections among two recent developments in the study of tensors,
namely

  * •

the _slice rank_ of tensors, a notion of rank for tensors that emerged from
the resolution of the cap set problem (Ann. of Math. 2017),

  * •

and the _quantum functionals_ of tensors (STOC 2018), monotone functionals
defined as optimizations over moment polytopes.

More precisely, we introduce an extension of slice rank that we call _weighted
slice rank_ and we develop a minimax correspondence between the asymptotic
weighted slice rank and the quantum functionals. Weighted slice rank
encapsulates different notions of bipartiteness of quantum entanglement.

The correspondence allows us to give a rank-type characterization of the
quantum functionals. Moreover, whereas the original definition of the quantum
functionals only works over the complex numbers, this new characterization can
be extended to all fields. Thereby, in addition to gaining deeper
understanding of Strassen’s theory for the complex numbers, we obtain a
proposal for quantum functionals over other fields. The finite field case is
crucial for combinatorial and algorithmic problems where the field can be
optimized over.

###### Contents

  1. 1 Introduction
    1. 1.1 Weighted slice rank
    2. 1.2 Asymptotic weighted slice rank
    3. 1.3 Quantum functionals
    4. 1.4 Asymptotic non-commutative ranks
    5. 1.5 Min-entropy quantum functionals
    6. 1.6 Organization of the paper
  2. 2 Weighted slice rank
    1. 2.1 Alternative characterization
    2. 2.2 Basic properties
    3. 2.3 Examples
  3. 3 Asymptotic weighted slice rank
    1. 3.1 Semistable tensors over arbitrary fields
    2. 3.2 Arbitrary tensors over the complex numbers
    3. 3.3 Tight tensors over arbitrary fields
  4. 4 Minimax correspondence
    1. 4.1 Definition of a dual pair
    2. 4.2 Von Neumann minimax theorem
    3. 4.3 Minimax correspondence
    4. 4.4 Preordered semirings and the minimax correspondence
  5. 5 Quantum functionals over arbitrary fields
    1. 5.1 Arbitrary tensors over the complex numbers
    2. 5.2 Tight tensors over arbitrary fields
    3. 5.3 Min-entropy quantum functionals
  6. A Invariance, polystability and semistability

##  1 Introduction

The study of the structural and computational aspects of tensors (multi-
dimensional arrays of numbers over a field) is deeply connected to key
research areas in mathematics, physics and computer science. Examples of these
are the research on sunflowers and cap sets in combinatorics [EG17, Tao16],
understanding quantum entanglement [DVC00, LQWD18], tensor networks, secant
varieties of Segre varieties in algebraic geometry [Lan12], scaling algorithms
[BGO+18, BFG+18], the complexity of matrix multiplication [BCS97, Blä13], and
the complexity of boolean functions [Raz13, Blä14]. We advance and make novel
connections among two recent developments in the study of tensors:

  * •

the _slice rank_ of tensors [Tao16, TS16, BCC+17a], which encapsulates the
technique that provided the breakthrough result on the cap set problem in
combinatorics,

  * •

the _quantum functionals_ of complex tensors [CVZ18], a family of
multiplicative monotones —similar in spirit and utility to multiplicative
monotones in other settings, like the Lovász theta function in graph theory or
the entropy function in information theory—which moved forward the research
direction initiated by Strassen on the asymptotic111In this paper, the term
“asymptotic” refers to considering large powers of tensors, which is motivated
by many applications including the complexity of matrix multiplication,
understanding quantum entanglement, and the sunflower and cap set problem in
combinatorics. The power is taken under the tensor Kronecker product, which
naturally generalizes the matrix Kronecker product. properties of tensors
[Str86, Str87, Str88, Str91].

To start this off, as a central component of this paper, we introduce a new
notion of rank for tensors called _weighted slice rank_. This notion
generalizes several existing notions of rank for tensors, most notably the
slice rank and the non-commutative rank. In this paper, we:

  * •

prove basic properties of the (asymptotic) weighted slice ranks,

  * •

prove, via a minimax argument, a correspondence between the asymptotic
weighted slice ranks and the quantum functionals, characterizing each as an
optimization problem in terms of the other (over $\mathbb{C}$),

  * •

as a consequence, obtain a characterization of asymptotic tripartite-to-
bipartite entanglement transformation222In this task, Alice, Bob and Charlie,
given many copies of a joint tripartite quantum state, together try to create
as much entanglement as possible between Alice and Bob by stochastic local
operations and classical communication (SLOCC) [DVC00]. [LQWD18]. Equivalently
we characterize the notion of asymptotic non-commutative rank and asymptotic
commutative rank (over $\mathbb{C}$),

  * •

propose a notion of quantum functionals over other fields than the complex
numbers via our minimax correspondence,

  * •

and develop the quantum functionals with respect to the min-entropy instead of
the entropy, which leads to a connection to the G-stable rank introduced in
[Der20].

The minimax correspondence between asymptotic weighted slice rank and the
quantum functionals (Theorem 7) has a general form which is of independent
interest (Section 4).

A crucial gap in previous work is that the usual construction of the quantum
functionals is not suitable for extension to other fields. The notion of
quantum functionals over finite fields and other fields that we propose is
obtained by applying our minimax correspondence to the asymptotic weighted
slice ranks, which are defined over any field. We conjecture that these new
functions indeed retain the properties of the complex quantum functionals. As
evidence of this we prove this conjecture for the subclass of tight tensors
over arbitrary fields. Finite fields have shown to be crucial in the study of
combinatorial problems like the cap set problem and algorithmic problems like
matrix multiplication barriers.

Our results give a partial resolution to the fundamental structural and
computational problem of understanding the _asymptotic tensor restriction
problem_. This problem is about the interplay of two tensor notions:

  * •

reductions between tensors, called restriction333A tensor is a _restriction_
of another tensor if, as multilinear maps, the first can be obtained from the
second by composing with linear maps. In the application of tensors to the
study of matrix multiplication this notion of restriction indeed amounts to a
reduction between computational problems. Restriction will be denoted by
$S\leq T$.—a concept similar to reductions between computational problems in
complexity theory,

  * •

large powers of tensors444We take powers under the tensor Kronecker product.
This product naturally generalizes the Kronecker product for matrices.—a
regime similar to parallel repetition problems, Shannon capacity, or self-
reducible computational problems, like matrix multiplication.

The asymptotic tensor restriction problem asks whether, given two tensors, a
large power of the first tensor is a restriction of a marginally larger power
of the second tensor. While many tensor parameters are known to be NP-hard to
compute, there is an intriguing possibility that the asymptotic tensor
restriction problem has an efficiently computable answer. (This would have
considerable consequences, as, for example, computing the value of the matrix
multiplication exponent is a special case.) The asymptotic weighted slice
ranks as well as the quantum functionals provide limits on which asymptotic
restrictions between tensors are possible.

In the remainder of this introduction we will discuss in detail the main
concepts and results of the paper. The full results, technical lemmas and
proofs we will then discuss in the sections that follow after the
introduction.

Notation. In the text we often use minimization over expressions involving
$m_{i}^{1/\xi_{i}}$ with $m_{i}\geq 1$ or $\frac{h_{i}}{\xi_{i}}$ with
$h_{i}\geq 0$ in the situation where some (but not all) of $\xi_{i}$ may be
zero. If $\xi_{i}=0$, then these expressions are treated as $+\infty$, even if
$m_{i}=1$ or $h_{i}=0$.

###  1.1 Weighted slice rank

We introduce a new notion of rank for tensors called the _weighted slice
rank_. In this paper a _tensor_ is any element in a tensor space $V_{1}\otimes
V_{2}\otimes V_{3}$ for finite-dimensional vector spaces $V_{i}$ over some
field $K$. In coordinates, relative to the standard product basis
$e_{i}\otimes e_{j}\otimes e_{k}$ of $V_{1}\otimes V_{2}\otimes V_{3}$, a
tensor $T$ is defined by a three-dimensional array $(T_{i,j,k})_{i,j,k}$ of
field elements $T_{i,j,k}\in K$, so that we have the expansion
$T=\sum_{i,j,k}T_{i,j,k}\,e_{i}\otimes e_{j}\otimes e_{k}$. Our results also
hold for tensors of order higher than three, but for simplicity of the
exposition we will here only talk about tensors of order three.

Weighted slice rank generalizes several important notions of rank for tensors.
To define the weighted slice rank we first need the concept of flattenings and
slice decomposition of a tensor.

######  Definition 1 (Flattenings).

For any tensor $T=\sum_{i,j,k}T_{i,j,k}\,e_{i}\otimes e_{j}\otimes e_{k}\in
V_{1}\otimes V_{2}\otimes V_{3}$ we define the _flattenings_ of $T$ by

$\displaystyle T_{(1)}=\sum_{i,j,k}T_{i,j,k}\,\,e_{i}\otimes(e_{j}\otimes
e_{k})\in V_{1}\otimes(V_{2}\otimes V_{3}),$ $\displaystyle
T_{(2)}=\sum_{i,j,k}T_{i,j,k}\,\,e_{j}\otimes(e_{i}\otimes e_{k})\in
V_{2}\otimes(V_{1}\otimes V_{3}),$ $\displaystyle
T_{(3)}=\sum_{i,j,k}T_{i,j,k}\,\,e_{k}\otimes(e_{i}\otimes e_{j})\in
V_{3}\otimes(V_{1}\otimes V_{2}).$

In coordinates, for a tensor $T=(T_{i,j,k})_{i,j,k}$ the three flattenings of
$T$ are given by the three matrices $T_{(1)}=(T_{i,j,k})_{i,(j,k)}$,
$T_{(2)}=(T_{i,j,k})_{j,(i,k)}$, and $T_{(3)}=(T_{i,j,k})_{k,(i,j)}$.

######  Definition 2 (Slice decompositions).

For every $i\in[3]\coloneqq\\{1,2,3\\}$, we call $T\in V_{1}\otimes
V_{2}\otimes V_{3}$ an _$i$ -slice_ if the flattening $T_{(i)}$ has rank one
as a matrix. For every $r_{1},r_{2},r_{3}\in\mathbb{N}$ we say that _$T$ has a
slice decomposition of size $(r_{1},r_{2},r_{3})$_ if there are tensors
$T_{i,j}\in V_{1}\otimes V_{2}\otimes V_{3}$ ($i\in[3]$, $j\in[r_{i}]$) such
that

$T=\sum_{j=1}^{r_{1}}T_{1,j}+\sum_{j=1}^{r_{2}}T_{2,j}+\sum_{j=1}^{r_{3}}T_{3,j}$

and such that each tensor $T_{i,j}$ is an $i$-slice.

In other words, $T$ has a slice decomposition of size $(r_{1},r_{2},r_{3})$ if
and only if there are tensors $S_{1},S_{2},S_{3}$ such that
$T=S_{1}+S_{2}+S_{3}$ and for every $i\in[3]$ the flattening $(S_{i})_{(i)}$
has rank $r_{i}$.

The _slice rank_ of a tensor $T$, introduced by Tao [Tao16, TS16], is defined
as

$\min\\{r_{1}+r_{2}+r_{3}\mid\textnormal{$T$ has a slice decomposition of size
$(r_{1},r_{2},r_{3})$}\\}.$

We define the _weighted slice rank_ by weighing the numbers
$r_{1},r_{2},r_{3}$ that appear in the definition of slice rank, as follows.
It will become clear later why the particular kind of weighing that we use is
the appropriate one.

######  Definition 3 (Weighted slice rank).

Let $\Xi=\\{(\xi_{1},\xi_{2},\xi_{3})\mid\xi_{1},\xi_{2},\xi_{3}\geq
0,\max\\{\xi_{1},\xi_{2},\xi_{3}\\}=1\\}$. For any $\xi\in\Xi$ we define the
_$\xi$ -weighted slice rank of $T$_ as

$S_{\xi}(T)\coloneqq\min\\{r_{1}^{1/\xi_{1}}+r_{2}^{1/\xi_{2}}+r_{3}^{1/\xi_{3}}\mid\text{$T$
has a slice decomposition of size $(r_{1},r_{2},r_{3})$}\\}.$

If any of the $\xi_{i}$ equals $0$, then we require the value of the
corresponding number $r_{i}$ to be $0$ and we omit the term
$r_{i}^{1/\xi_{i}}$.

The weighted slice rank naturally generalizes the following known notions of
rank for tensors.

  * •

Slice rank: The weighted slice rank with weight $(1,1,1)$,

$S_{(1,1,1)}(T)=\min\\{r_{1}+r_{2}+r_{3}\mid\textnormal{$T$ has a slice
decomposition of size $(r_{1},r_{2},r_{3})$}\\},$

equals the ordinary slice rank of $T$.

  * •

Flattening rank: The matrix ranks of the flattenings $T_{(1)},T_{(2)},T_{(3)}$
of $T$ are called the flattening ranks. The flattening ranks are equal to the
weighted slice rank with all weight on one of the numbers $r_{i}$,

$\displaystyle S_{(1,0,0)}(T)$ $\displaystyle=\min\\{r_{1}\mid\textnormal{$T$
has a slice decomposition of size $(r_{1},0,0)$}\\},$ $\displaystyle
S_{(0,1,0)}(T)$ $\displaystyle=\min\\{r_{2}\mid\textnormal{$T$ has a slice
decomposition of size $(0,r_{2},0)$}\\},$ $\displaystyle S_{(0,0,1)}(T)$
$\displaystyle=\min\\{r_{3}\mid\textnormal{$T$ has a slice decomposition of
size $(0,0,r_{3})$}\\}.$

  * •

Non-commutative ranks: The non-commutative ranks of a tensor are equal to the
weighted slice ranks with weight $1$ on two out of the three numbers $r_{i}$,

$\displaystyle S_{(1,1,0)}(T)$
$\displaystyle=\min\\{r_{1}+r_{2}\mid\textnormal{$T$ has a slice decomposition
of size $(r_{1},r_{2},0)$}\\},$ $\displaystyle S_{(0,1,1)}(T)$
$\displaystyle=\min\\{r_{2}+r_{3}\mid\textnormal{$T$ has a slice decomposition
of size $(0,r_{2},r_{3})$}\\},$ $\displaystyle S_{(1,0,1)}(T)$
$\displaystyle=\min\\{r_{1}+r_{3}\mid\textnormal{$T$ has a slice decomposition
of size $(r_{1},0,r_{3})$}\\}.$

Obviously, slice rank is at most the non-commutative ranks, and the non-
commutative ranks are in turn upper bounded by the flattening ranks according
to the following Hasse diagram:

$S_{(0,0,1)}$$S_{(0,1,0)}$$S_{(1,0,0)}$$S_{(0,1,1)}$$S_{(1,0,1)}$$S_{(1,1,0)}$$S_{(1,1,1)}$

Generally, for any weightings $\xi,\xi^{\prime}\in\Xi$, if the inequality
$\xi\geq\xi^{\prime}$ holds elementwise, then we have that $S_{\xi}\leq
S_{\xi^{\prime}}$ holds on all tensors.

We conclude that the weighted slice ranks naturally generalize and interpolate
between the slice rank, the flattening ranks and the non-commutative ranks.
Each of these ranks play their own specialized role in various settings.
Weighted slice rank for the other possible weightings likewise play their own
role, as we will see.

###  1.2 Asymptotic weighted slice rank

We are interested in the behaviour of weighted slice rank under taking large
powers. The product under which we take powers is the tensor Kronecker
product—the natural generalization of the matrix Kronecker product to tensors.
For tensors $S\in V_{1}\otimes V_{2}\otimes V_{3}$ and $T\in W_{1}\otimes
W_{2}\otimes W_{3}$ the tensor Kronecker product is a tensor $S\otimes
T\in(V_{1}\otimes W_{1})\otimes(V_{2}\otimes W_{3})\otimes(V_{3}\otimes
W_{3})$. In coordinates, if $S\in K^{n_{1}\times n_{2}\times n_{3}}$ and $T\in
K^{m_{1}\times m_{2}\times m_{3}}$ then the coefficients of $S\otimes T$ are
given by all pairwise products of coefficients of $S$ and coefficients of $T$,

$(S\otimes
T)_{(i_{1},i_{2}),(j_{1},j_{2}),(k_{1},k_{2})}=S_{i_{1},i_{2},i_{3}}T_{j_{1},j_{2},j_{3}}.$

The behaviour of weighted slice rank under taking large powers is captured by
the asymptotic weighted slice rank.

######  Definition 4 (Asymptotic weighted slice rank).

For $\xi\in\Xi$ we define the _asymptotic $\xi$-weighted slice rank of $T$_ as

$G_{\xi}(T)\coloneqq\limsup_{n\to\infty}S_{\xi}(T^{\otimes n})^{1/n}.$

We will frequently make use of the logarithm of $G_{\xi}(T)$ which we denote
by $g_{\xi}(T)$, that is,

$g_{\xi}(T)\coloneqq\log_{2}G_{\xi}(T)=\limsup_{n\to\infty}\frac{1}{n}\log_{2}S_{\xi}(T^{\otimes
n}).$

Since $G_{\xi}$ is defined as a lim sup, $G_{\xi}$ arguably does not capture
all of the asymptotic behaviour of the weighted slice rank. For example, we
could also consider the corresponding lim inf. However, it will turn out that
in the two settings that we will consider (namely, complex tensors and so-
called tight tensors), the lim sup and the lim inf coincide, and may
equivalently be replaced by a limit.

The weighted slice rank $S_{(1,0,0)}$ is a flattening rank and is therefore
multiplicative, and so $G_{(1,0,0)}(T)=S_{(1,0,0)}(T)$. The same is true for
$G_{(0,1,0)}$ and $G_{(0,0,1)}$. For general choices of $\xi\in\Xi$, however,
the weighted slice rank $S_{\xi}$ will be different from the asymptotic
weighted slice rank $G_{\xi}$. Interesting special cases of the asymptotic
weighted slice rank are the asymptotic slice rank $G_{(1,1,1)}$ and the
asymptotic non-commutative ranks $G_{(1,1,0)}$, $G_{(1,0,1)}$ and
$G_{(0,1,1)}$.

Our first main result is a dual characterization of the asymptotic weighted
slice ranks for complex tensors. This dual characterization is phrased as an
optimization problem over a convex polytope called the moment polytope. For
every complex tensor $T$, the moment polytope $\Pi(T)$ is defined as follows.
Recall that for any tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ we denote
the flattenings in $(V_{2}\otimes V_{3})\otimes V_{1}$, $(V_{1}\otimes
V_{3})\otimes V_{2}$, $(V_{1}\otimes V_{2})\otimes V_{3}$ by $T_{(1)}$,
$T_{(2)}$, and $T_{(3)}$, respectively. For a nonzero tensor $T$ let
$r(T_{(i)})$ be the probability vector obtained by squaring and normalizing
the singular values $\sigma_{1}(T_{(i)})\geq\sigma_{2}(T_{(i)})\geq\cdots$ of
the matrix $T_{(i)}$, that is,

$r(T_{(i)})=(\sigma_{1}(T_{(i)})^{2},\sigma_{2}(T_{(i)})^{2},\ldots)/\sum_{j}\sigma_{j}(T_{(i)})^{2}.$

The moment polytope of a nonzero tensor $T$ is defined as the set of triples
of probability vectors

$\Pi(T)=\\{(r(S_{(1)}),r(S_{(2)}),r(S_{(3)}))\mid S\in\overline{G\cdot
T},S\neq 0\\}.$

In the above definition the set $\overline{G\cdot T}$ is the orbit closure
(closure under equivalently the Zariski or Euclidean topology) of $T$ under
the action of the group
$G=\mathrm{GL}(V_{1})\times\mathrm{GL}(V_{2})\times\mathrm{GL}(V_{3})$. This
is the same as the closure of the set of all tensors $S$ such that $S$ is a
restriction of $T$. For the characterization of the asymptotic weighted slice
rank we furthermore need the Shannon entropy of a probability vector
$p=(p_{1},\ldots,p_{n})$, which is defined as
$H(p)=-\sum_{i}p_{i}\log_{2}p_{i}$.

######  Theorem 5.

Let $T$ be a tensor over $\mathbb{C}$. For every $\xi\in\Xi$, the limsup in
the definition of $G_{\xi}(T)$ is a limit and

$G_{\xi}(T)=\max_{p\in\Pi(T)}\min_{i\in[3]}2^{H(p_{i})/\xi_{i}}.$

This result extends the result in [CVZ18] that gave the same dual description
only for the asymptotic slice rank (i.e., for $\xi=(1,1,1)$).

For tensors over arbitrary fields (i.e. different from $\mathbb{C}$) we prove
a dual description for the asymptotic weighted slice rank for an important
subclass of all tensors, namely tensors with a _tight support_ 555The
_support_ of a tensor is the set of all triples of coordinates with nonzero
coefficient. We call such a support
$S\subseteq[n_{1}]\times[n_{2}]\times[n_{3}]$ _tight_ if there are injective
maps $u_{i}\mathrel{\mathop{\mathchar 58\relax}}[n_{i}]\to\mathbb{Z}$ such
that for every $s\in S$ it holds that $\sum_{i}u_{i}(s_{i})=0$. A tensor in
$V_{1}\otimes V_{2}\otimes V_{3}$ is called tight if its support is tight for
some choice of bases for the vector spaces $V_{i}$.. Examples of tensors with
a tight support have appeared in the applications to matrix multiplication and
combinatorics. In this version of the dual description, the moment polytope
$\Pi(T)$ (which is only available over $\mathbb{C}$) is replaced by the
polytope $W(T)$ of triples of marginals of probability vectors on the support
of the tensor.

######  Theorem 6.

Let $T$ be a tensor with tight support. For every $\xi\in\Xi$, the limsup in
the definition of $G_{\xi}(T)$ is a limit and

$G_{\xi}(T)=\max_{p\in W(T)}\min_{i\in[3]}2^{H(p_{i})/\xi_{i}}.$

This result extends the result in [TS16] and [CVZ18] that gave the same dual
description only for the weighting $\xi=(1,1,1)$, albeit for the broader class
of tensors with _oblique_ support.

###  1.3 Quantum functionals

The quantum functionals were introduced in [CVZ18] to study the asymptotic
behaviour of complex tensors, advancing a line of research initiated by
Strassen in the context of the arithmetic complexity of matrix multiplication.
The quantum functionals are a family of functions parametrized by the
probability simplex
$\Theta\coloneqq\\{(\theta_{1},\theta_{2},\theta_{3})\in\mathbb{R}^{3}\mid\theta_{1},\theta_{2},\theta_{3}\geq
0,\,\theta_{1}+\theta_{2}+\theta_{3}=1\\}.$ Each quantum functional is defined
as an optimization problem over the moment polytope that we defined
previously. Let $\theta\in\Theta$. For any nonzero complex tensor $T$ we
define

$F_{\theta}(T)=\max_{p\in\Pi(T)}2^{\langle\theta,(H(p_{1}),H(p_{2}),H(p_{3}))\rangle},$

where, as before, $H$ denotes the Shannon entropy of a probability vector. The
functions $F_{\theta}$ are called the _quantum functionals_.

The main theorem on the quantum functionals that was proved in [CVZ18] is that
for any two complex tensors $S$ and $T$ the following properties hold:

  * •

Monotonicity: $S\leq T\Rightarrow F_{\theta}(S)\leq F_{\theta}(T)$,666Recall
that $S\leq T$ means that $S$ is a restriction of $T$, i.e. that $S$ can be
obtained from $T$ by applying linear maps to the tensor legs.

  * •

Additivity: $F_{\theta}(S\oplus T)=F_{\theta}(S)+F_{\theta}(T)$,

  * •

Multiplicativity: $F_{\theta}(S\otimes T)=F_{\theta}(S)\cdot F_{\theta}(T)$,

  * •

Normalization: $F_{\theta}(I_{n})=n$ for any $n\in\mathbb{N}$ where
$I_{n}=\sum_{i=1}^{n}e_{i}\otimes e_{i}\otimes
e_{i}\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}\otimes\mathbb{C}^{n}$ is the unit
tensor.

Tensors parameters that satisfy the above four properties were called
_universal spectral points_ in [Str88]. In particular, the quantum functionals
(and universal spetral points in general) have the following useful
application: if an asymptotic inequality $S^{\otimes n}\leq T^{\otimes\alpha
n+o(n)}$ holds for some rate $\alpha\in\mathbb{R}_{\geq 0}$, then
$F_{\theta}(S)\leq F_{\theta}(T)^{\alpha}$ for all $\theta\in\Theta$.
Therefore, contrapositively, if there exists a $\theta$ such that
$F_{\theta}(S)>F_{\theta}(T)^{\alpha}$ then the asymptotic inequality
$S^{\otimes n}\leq T^{\otimes\alpha n+o(n)}$ cannot hold. (Considering all
currently available results and examples, it is possible that $S^{\otimes
n}\leq T^{\otimes\alpha n+o(n)}$ holds if and only if $F_{\theta}(S)\leq
F_{\theta}(T)^{\alpha}$ for all $\theta\in\Theta$.)

The above properties of the quantum functionals were used to prove barriers
for square matrix multiplication in [CVZ19], which used the quantum functional
with the uniform $\theta=(1/3,1/3,1/3)$. These barriers were concurrently
obtained in [Alm19] using slice rank in an important line of work [AW18a,
AW18b, BCC+17b, BCC+17a] improving and extending the earliest barriers in
[AFLG15]. Barriers for rectangular matrix multiplication were obtained in
[CGLZ20], which used the quantum functionals with non-uniform
$\theta\in\Theta$.

Our second main result is a correspondence between the quantum functionals and
the asymptotic weighted slice ranks. This correspondence allows us to compute
the asymptotic weighted slice ranks in terms of the quantum functionals and
vice versa.

######  Theorem 7.

For every complex tensor $T$ and every $\xi\in\Xi$ we have

$G_{\xi}(T)=\min_{\theta\in\Theta(\xi)}{F_{\theta}(T)}^{1/\langle\theta,\xi\rangle}.$

where $\Theta(\xi)=\\{\theta\in\Theta\mid\theta_{i}=0\text{ if
$\xi_{i}=0$}\\}$.

Conversely, for every complex tensor $T$ and every $\theta\in\Theta$ we have

$F_{\theta}(T)=\max_{\xi\in\Xi}G_{\xi}(T)^{\langle\theta,\xi\rangle}.$

While the quantum functionals are only defined over the complex numbers, the
asymptotic weighted slice ranks are defined over any field. We conjecture that
via the correspondence in Theorem 7 we can extend the quantum functionals to
other fields, as follows.

Define for every $\theta\in\Theta$ the function $F^{K}_{\theta}$ on tensors
$T$ over a field $K$ via

$F^{K}_{\theta}(T)\coloneqq\max_{\xi\in\Xi}G_{\xi}(T)^{\langle\theta,\xi\rangle}.$

The following properties follow directly from properties of $G_{\xi}$ as we
will see:

######  Proposition 8.

For every field $K$ and every $\theta\in\Theta$, the function $F^{K}_{\theta}$
is monotone and normalized.

We make the following conjecture:

######  Conjecture 9.

For every field $K$ and every $\theta\in\Theta$, the function $F^{K}_{\theta}$
is additive and multiplicative.

Thus 9 (if true), together with Proposition 8, implies that the
$F^{K}_{\theta}$ are universal spectral points over $K$. We provide two pieces
of evidence for 9, namely that the conjecture is true for $K=\mathbb{C}$
(since then $F_{\theta}^{\mathbb{C}}$ equals the quantum functionals
$F_{\theta}$ for tensors over $\mathbb{C}$ by Theorem 7) and that the
conjecture is true for the subclass of tight tensors over arbitrary fields
$K$.

As another consequence of the dual description in Theorem 7 and the super-
multiplicativity and super-additivity of the quantum functionals, we will
obtain via a general argument the following properties of the asymptotic
weighted slice rank.

######  Corollary 10.

Over the complex numbers the asymptotic $\xi$-weighted slice rank is:

  * •

Super-multiplicative: $G_{\xi}(T_{1}\otimes T_{2})\geq G_{\xi}(T_{1})\cdot
G_{\xi}(T_{2})$

  * •

Super-additive: $G_{\xi}(T_{1}\oplus T_{2})\geq
G_{\xi}(T_{1})+G_{\xi}(T_{2})$.

We prove the statement of Corollary 10 also for all tight tensors over
arbitrary fields via the description in Theorem 6 and the minimax
correspondence.

In the process of proving the above results we obtain a characterization of an
important class of tensors in terms of the asymptotic weighted slice rank. A
tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ is defined to be _semistable_ ,
under the action of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$, if the
closure of the orbit
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})\cdot T$
does not contain the element 0.

######  Theorem 11.

A tensor
$T\in\mathbb{C}^{m_{1}}\otimes\mathbb{C}^{m_{2}}\otimes\mathbb{C}^{m_{3}}$ is
semistable under the action of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ if and
only if $G_{\xi}(T)=\min_{i}m_{i}^{1/\xi_{i}}$ for every $\xi\in\Xi$.

We also prove a version of Theorem 11 for tensors over arbitrary fields, with
the condition that $T$ is semistable replaced by the condition that all powers
of $T$ are semistable.

###  1.4 Asymptotic non-commutative ranks

A tensor $T\in K^{m_{1}\times m_{2}\times m_{3}}$ can be seen as a linear map
from $(K^{m_{3}})^{*}$ to $K^{m_{1}\times m_{2}}$ or, equivalently, a matrix
in $K[x_{1},\dots,x_{m_{3}}]^{m_{1}\times m_{2}}$, each entry of which is a
linear form in variables $X=(x_{1},\dots,x_{m_{3}})$. Two notions of rank for
such matrices are defined in [FR07]: the _commutative rank_ $\textrm{crk}(T)$,
which is the usual rank of a matrix of linear forms over the field $K(X)$, and
the _non-commutative rank_ , which is the minimal $r$ such that a matrix of
linear forms can be decomposed as a product $AB$ with $A\in K[X]^{m_{1}\times
r}$ and $B\in K[X]^{r\times m_{2}}$. The weighted slice rank $S_{(1,1,0)}$ of
a tensor $T$ is equal to the noncommutative rank of the corresponding matrix
of linear forms [FR07, Thm. 1].

Commutative rank is supermultiplicative [LQWD18], so the limit
$\lim_{n\to\infty}\mathrm{crk}(T^{\otimes n})^{1/n}$ exists by Fekete’s lemma.
Additionally, it is known that $\mathrm{crk}(T)\leq S_{(1,1,0)}(T)\leq
2\mathrm{crk}(T)$ [FR07, Cor.2], so asymptotically

$G_{(1,1,0)}(T)=\lim_{n\to\infty}S_{(1,1,0)}(T^{\otimes
n})^{1/n}=\lim_{n\to\infty}\mathrm{crk}(T^{\otimes n})^{1/n}.$

As a special case of Theorem 5 and Theorem 7 we obtain new dual descriptions
of the asymptotic non-commutative ranks in terms of the quantum functionals:

######  Corollary 12.

For every complex tensor $T$ we have that

$\displaystyle G_{(1,1,0)}(T)$
$\displaystyle=\max_{p\in\Pi(T)}\min\\{2^{H(p_{1})},\,2^{H(p_{2})}\\}=\min_{\begin{subarray}{c}\theta_{1},\theta_{2}\geq
0\\\
\theta_{1}+\theta_{2}=1\end{subarray}}F_{(\theta_{1},\theta_{2},0)}(T)^{\frac{1}{\theta_{1}+\theta_{2}}}$

Non-commutative rank has received much interest in recent years [IQS15, DM18].
In quantum information theory, the study of asymptotic non-commutative rank is
motivated by the fact that commutative rank is precisely the same as
tripartite-to-bipartite entanglement transformations under Stochastic Local
Operations and Classical Communication (SLOCC) [CDS10], and formulas for
asymptotic non-commutative rank have previously been obtained for special
cases [LQWD18], [Jen19, §3.4].

###  1.5 Min-entropy quantum functionals

Finally, we explore a new variation on the quantum functionals by replacing
the Shannon entropy $H$ with a smaller notion of entropy called the min-
entropy. The min-entropy of a probability vector $p=(p_{1},\ldots,p_{n})$ is
defined as $H_{\infty}(p)=-\log_{2}\max_{i}p_{i}$. Thus the min-entropy is the
number of bits required to describe the largest coefficient in a probability
vector. It is the smallest among all of the Rényi entropies and in particular
satisfies $H_{\infty}(p)\leq H(p)$ for every $p$. For $\theta\in\Theta$, we
define the _min-entropy quantum functional_ as

$F_{\theta,\infty}(T)\coloneqq\max_{p\in\Pi(T)}2^{\langle\theta,(H_{\infty}(p_{1}),H_{\infty}(p_{2}),H_{\infty}(p_{3}))\rangle}.$

It follows directly from $H_{\infty}(p)\leq H(p)$ that
$F_{\theta,\infty}(T)\leq F_{\theta}(T)$. Expanding the singular value
definition of the moment polytope $\Pi(T)$ we have

$F_{\theta,\infty}(T)=\sup_{S\in G\cdot
T}\prod_{i=1}^{3}\Biggl{(}\frac{\sum_{j}\sigma_{j}(S_{(i)})^{2}}{\sigma_{1}(S_{(i)})^{2}}\Biggr{)}^{\\!\\!\theta_{i}},$

where, as before, $\sigma_{1}(S_{(i)})\geq\sigma_{2}(S_{(i)})\geq\cdots$
denote the ordered singular values of the matrix $S_{(i)}$. In other words, we
may write $F_{\theta,\infty}(T)$ in terms of the Frobenius norm
$\mathinner{\lVert\cdot\rVert}_{F}$ and the spectral norm
$\mathinner{\lVert\cdot\rVert}_{2}$ of matrices as

$F_{\theta,\infty}(T)=\sup_{S\in G\cdot
T}\prod_{i=1}^{3}\Biggl{(}\frac{\mathinner{\lVert
S_{(i)}\rVert}_{F}^{2}}{\mathinner{\lVert
S_{(i)}\rVert}_{2}^{2}}\Biggr{)}^{\\!\\!\theta_{i}}.$

We prove the following basic properties of the min-entropy quantum
functionals.

######  Theorem 13.

The min-entropy quantum functionals are monotone, normalized and super-
multiplicative.

Next we prove a minimax correspondence to a second family of functions, just
like we did for the quantum functionals. We define for $\xi\in\Xi$ the
function

$G_{\xi,\infty}(T)=\max_{p\in\Pi(T)}\min_{i\in[3]}2^{H_{\infty}(p_{i})/\xi_{i}}.$

######  Theorem 14.

For every $\xi\in\Xi$ and every tensor $T$ we have

$G_{\xi,\infty}(T)=\min_{\theta\in\Theta}{F_{\theta,\infty}(T)}^{1/\langle\theta,\xi\rangle}.$

For every $\theta\in\Theta$ and every tensor $T$ we have

$F_{\theta,\infty}(T)=\max_{\xi\in\Xi}G_{\xi,\infty}(T)^{\langle\theta,\xi\rangle}.$

The function $G_{\xi,\infty}(T)$ is similar to the G-stable rank that was
introduced in [Der20], but subtly different, as our kind of weighing by $\xi$
is different.

From Theorem 13 and Theorem 14 we prove using a general argument that:

######  Corollary 15.

For every $\xi\in\Xi$ the function $G_{\xi,\infty}$ is monotone, normalized
and super-multiplicative.

###  1.6 Organization of the paper

This concludes the introduction. In the remaining sections we discuss the
results and technical lemmas in detail and provide the proofs for all claims.
In Section 2 we discuss the basic properties of the weighted slice rank. In
Section 3 we characterize the asymptotic weighted slice rank for arbitrary
complex tensors and for tight tensors over arbitrary fields. In Section 4 we
discuss a general correspondence for what we call dual pair families, of which
the quantum functionals and asymptotic weighted slice ranks are an example. In
Section 5 we discuss the correspondence between the quantum functionals and
the asymptotic weighted slice ranks, and we discuss the min-entropy quantum
functionals.

##  2 Weighted slice rank

We introduced in Section 1.1 for
$\xi\in\Xi=\\{(\xi_{1},\xi_{2},\xi_{3})\mid\xi_{1},\xi_{2},\xi_{3}\geq
0,\max\\{\xi_{1},\xi_{2},\xi_{3}\\}=1\\}$ the $\xi$-weighted slice rank of a
tensor $T$ as the minimum value of
$r_{1}^{1/\xi_{1}}+r_{2}^{1/\xi_{2}}+r_{3}^{\xi_{3}}$ such that there is a
slice decomposition of $T$ of size $(r_{1},r_{2},r_{3})$. In this section we
will discuss its most basic properties.

###  2.1 Alternative characterization

We begin with an alternative characterization of the notion of a slice
decomposition that we defined in Definition 2, the straightforward proof of
which we leave to the reader.

######  Lemma 16.

A tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ has a slice decomposition of
size $(r_{1},r_{2},r_{3})$ if and only if there are subspaces $W_{i}\subseteq
V_{i}$ of dimension $\dim(W_{i})=r_{i}$ such that

$T\in W_{1}\otimes V_{2}\otimes V_{3}+V_{1}\otimes W_{2}\otimes
V_{3}+V_{1}\otimes V_{2}\otimes W_{3}.$

In other words, a tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ has a slice
decomposition of size $(r_{1},r_{2},r_{3})$ if and only if there is a choice
of bases for the $V_{i}$ such that the support of $T$ has a $2\times 2\times
2$ block structure of the form

where the missing block has dimensions
$(\dim(V_{1})-r_{1})\times(\dim(V_{2})-r_{2})\times(\dim(V_{3})-r_{3})$. Thus
a slice decomposition corresponds to a block triangularization for tensors.

As a direct consequence of Lemma 16 we have the following alternative
characterization of weighted slice rank:

######  Lemma 17.

For every tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ and $\xi\in\Xi$ the
$\xi$-weighted slice rank $G_{\xi}(T)$ is the minimum over
$r_{1}^{1/\xi_{1}}+r_{2}^{1/\xi_{2}}+r_{3}^{1/\xi_{3}}$ such that there are
subspaces $W_{i}\subseteq V_{i}$ such that $\dim(W_{i})=r_{i}$ and

$T\in W_{1}\otimes V_{2}\otimes V_{3}+V_{1}\otimes W_{2}\otimes
V_{3}+V_{1}\otimes V_{2}\otimes W_{3}.$

###  2.2 Basic properties

There are three basic properties of slice decompositions which we will discuss
now, followed by a discussion of basic properties of the weighted slice ranks.
Recall that we defined the restriction preorder $\leq$ on tensors by saying
that $T^{\prime}\leq T$ if and only if there are (not necessarily invertible)
matrices $A_{i}$ such that $T^{\prime}=(A_{1}\otimes A_{2}\otimes A_{3})\cdot
T$.

######  Lemma 18.

Let $T$ and $T^{\prime}$ be tensors.

  1. (i)

If $T^{\prime}\leq T$ and $T$ has a slice decomposition of size
$(r_{1},r_{2},r_{3})$, then there are nonnegative integers $r^{\prime}_{i}\leq
r_{i}$ such that $T^{\prime}$ has a slice decomposition of size
$(r^{\prime}_{1},r^{\prime}_{2},r^{\prime}_{3})$.

  2. (ii)

If $T$ has a slice decomposition of size $(r_{1},r_{2},r_{3})$ and
$T^{\prime}$ has a slice decomposition of size
$(r_{1}^{\prime},r_{2}^{\prime},r_{3}^{\prime})$, then $T+T^{\prime}$ has a
slice decomposition of size
$(r_{1}+r_{1}^{\prime},r_{2}+r_{2}^{\prime},r_{3}+r_{3}^{\prime})$.

  3. (iii)

Every tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ has a slice decomposition
of size $(d_{1},0,0)$, of size $(0,d_{2},0)$ and of size $(0,0,d_{3})$, where
$d_{i}=\dim(V_{i})$.

###### Proof.

(i) If $S$ is an $i$-slice in the slice decomposition of $T$, and
$S^{\prime}\leq S$, then $S^{\prime}$ is either an $i$-slice or zero.

(ii) This follows from adding the two slice decompositions.

(iii) This follows from the characterization of slice decompositions in Lemma
16. ∎

######  Lemma 19.

Let $T$ and $T^{\prime}$ be tensors and let $\xi\in\Xi$.

  1. (i)

If $T^{\prime}\leq T$, then $S_{\xi}(T^{\prime})\leq S_{\xi}(T)$.

  2. (ii)

For every tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ we have
$S_{\xi}(T)\leq\min_{i}d_{i}^{1/\xi_{i}}$ where $d_{i}=\dim(V_{i})$.

  3. (iii)

If $\xi\leq\xi^{\prime}$ elementwise, then $S_{\xi}(T)\geq
S_{\xi^{\prime}}(T)$.

  4. (iv)

If a tensor $T\in K^{m\times m\times m}$ has slice rank $S_{(1,1,1)}(T)$ equal
to $m$, then $S_{\xi}(T)=m$ for all $\xi\in\Xi$.

###### Proof.

(i) This follows from Lemma 18 (i).

(ii) This follows from Lemma 18 (iii).

(iii) This is clear from the definition of weighted slice rank.

(iv) From (ii) and (iii) it follows that $m=S_{(1,1,1)}(T)\leq
S_{\xi}(T)\leq\min_{i}m^{1/\xi_{i}}=m$. ∎

###  2.3 Examples

######  Example 20 (Diagonal tensors and the square matrix multiplication
tensor).

The diagonal tensors $I_{n}=\sum_{i=1}^{n}e_{i}\otimes e_{i}\otimes e_{i}$
have slice rank $n$ [Tao16], so $S_{\xi}(I_{n})=n$ for every $\xi\in\Xi$.
Similarly, the slice rank of the square matrix multiplication tensors $\langle
n,n,n\rangle=\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}e_{ij}\otimes
e_{jk}\otimes e_{ki}$ is $n^{2}$ [BCC+17a, Rem. 4.9], and so $S_{\xi}(\langle
n,n,n\rangle)=n^{2}$ for every $\xi\in\Xi$.

######  Example 21 (Rectangular matrix multiplication tensor).

Consider the rectangular matrix multiplication tensor $\langle
n,n,m\rangle=\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{m}e_{ij}\otimes
e_{jk}\otimes e_{ki}$. The decomposition

$\langle
n,n,m\rangle=\sum_{i=1}^{r}\sum_{j=1}^{n}e_{ij}\otimes\Bigl{(}\sum_{k=1}^{m}e_{jk}\otimes
e_{ki}\Bigr{)}+\sum_{i=r+1}^{n}\sum_{k=1}^{m}\Bigl{(}\sum_{j=1}^{n}e_{ij}\otimes
e_{jk}\Bigr{)}\otimes e_{ki}$

gives the upper bound $S_{(\xi,1,1)}(\langle
n,n,m\rangle)\leq(nr)^{1/\xi}+m(n-r)$ for every $r$ such that $0\leq r\leq n$.
If the inequality $n^{1/\xi}<m<n^{1/\xi}(n^{1/\xi}-(n-1)^{1/\xi})$ holds, then
this bound is stronger than the general upper bound of Lemma 19 (ii). For
example, taking $r=1$ we get $S_{(0.5,1,1)}(\langle 2,2,5\rangle)\leq 9$,
which is less than $\min\\{4^{1/0.5},10,10\\}$.

##  3 Asymptotic weighted slice rank

In this section we analyze the asymptotic behaviour of the weighted slice rank
when we take large powers of a tensor. More precisely, we study the asymptotic
weighted slice ranks $G_{\xi}$ defined in Definition 4. This section is
divided into three parts corresponding to three regimes:

  1. 1.

In Section 3.1 we give a characterization of the tensors for which the
asymptotic weighted slice rank is maximal. This characterization is in terms
of semistable tensors. This part works for tensors over arbitrary fields. An
important tool that we introduce in this part is a semistability test based on
work of Kempf, which we will use in the other parts of this section.

  2. 2.

In Section 3.2 we go deeper than just characterizing maximality and give a
description of the value of the asymptotic weighted slice rank in terms of
moment polytopes. This part works for tensors over the complex numbers.

  3. 3.

Finally, in Section 3.3 we work over an arbitrary field again and give an
upper bound on the value of the asymptotic weighted slice rank in terms of the
polytope of probability distributions on the support of the tensors, the torus
version of the moment polytope. For the subclass of tensors with tight support
(still over arbitrary fields) we prove that this upper bound is tight.

Section 3.2 and Section 3.3 each rely on a decomposition for powers of
tensors, namely

  * •

the Schur–Weyl decomposition and

  * •

the weight decomposition.

The first is precisely described by the moment polytope and the second by its
torus version.

###  3.1 Semistable tensors over arbitrary fields

In this section we characterize when the asymptotic weighted slice rank is
maximal. Let $K$ be an arbitrary field and let $V_{1},V_{2},V_{3}$ be vector
spaces over $K$. For any tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ it
holds that $G_{\xi}(T)\leq\min_{i}m_{i}^{1/\xi_{i}}$ where
$m_{i}=\dim(V_{i})$, by the upper bound on the weighted slice rank (Lemma
19(ii)). We will characterize for what tensors this upper bound is an equality
in terms of a notion called semistability from the field of geometric
invariant theory. This characterization generalizes the connection between
semistability and slice rank for cubic tensors (i.e., tensors of format
$n\times n\times n$) explored in [BCC+17a, BGO+18] to tensors of non-cubic
format (i.e., tensors of format $m_{1}\times m_{2}\times m_{3}$ with the
$m_{i}$’s potentially different).

Semistability can be defined in high generality (for actions of reductive
algebraic groups on schemes), but we are only interested in the natural action
of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ on the
tensor space $V_{1}\otimes V_{2}\otimes V_{3}$.

######  Definition 22.

Let $V_{1}$, $V_{2}$, $V_{3}$ be vector spaces over an algebraically closed
field $K$. A tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$ is called
_semistable_ if the Zariski closure777Over $\mathbb{C}$ this coincides with
the closure in the Euclidean topology. (See, e.g., [Wal17, Lem. 3.1].) of the
orbit $\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})\cdot
T$ does not contain $0$. Otherwise the tensor is called _unstable_. 888When
$K$ is not algebraically closed, we call a tensor semistable (unstable) if it
becomes semistable (unstable) after extending the field to the algebraic
closure $\overline{K}$.

The following semistability test, which follows from the results of Kempf
[Kem78], and the proof of which we defer to Appendix A, gives a simple but
powerful necessary condition for a tensor to be semistable.

######  Lemma 23 (Semistability test).

Let $V_{1},V_{2},V_{3}$ be irreducible representations of a group $\Gamma$.
Let $\Gamma$ act diagonally on the tensor product $V_{1}\otimes V_{2}\otimes
V_{3}$. For any nonzero tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$, if $T$
is invariant under the action of $\Gamma$, then $T$ is semistable under the
action of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$.999We
stress that the semistability of tensors is always taken with respect to the
action of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ and we
will not mention this explicitly anymore. The group $\Gamma$ in Lemma 23 could
be any group, as long as its action on the $V_{i}$ is irreducible.101010The
condition in the semistabiliy test is _sufficient_ for semistability but not
_necessary_ as the following example shows. It can be shown that the direct
sum $\langle 2,2,2\rangle\oplus\langle 1\rangle$ is semistable using the fact
that this tensor is tight and the ideas of Section 3.3. However, by
considering the stabilizer it follows that this tensor does not satisfy the
condition of the semistability test.

We will use the semistability test in proofs in Section 3.2 and Section 3.3.
For now we use the test to give simple explicit examples of semistable
tensors, so that the reader has some tensors to work with in the rest of this
subsection.

######  Example 24.

The matrix multiplication tensor $\langle
n_{1},n_{2},n_{3}\rangle\in(V_{1}^{*}\otimes V_{2})\otimes(V_{2}^{*}\otimes
V_{3})\otimes(V_{3}^{*}\otimes V_{1})$, where $V_{i}=\mathbb{F}^{n_{i}}$, is
semistable, because the matrix spaces $V_{1}^{*}\otimes V_{2}$,
$V_{2}^{*}\otimes V_{3}$, $V_{3}^{*}\otimes V_{1}$ are irreducible
representations of the group
$\mathrm{GL}(V_{1})\times\mathrm{GL}(V_{2})\times\mathrm{GL}(V_{3})$ and the
matrix multiplication tensor is invariant under the resulting action of this
group, which is sometimes called the sandwiching action. Finally, we apply the
semistability test (Lemma 23) to find that the matrix multiplication tensor is
semistable.111111In fact, from the stronger Lemma 63 that we discuss in
Appendix A it follows that this tensor is _polystable_ , meaning that its
$\mathrm{SL}$ orbit is closed. For the connection to slice rank we need
semistability and therefore we will discuss polystability only in Appendix A.

######  Example 25.

The diagonal tensor $\langle n\rangle\coloneqq\sum_{i=1}^{n}e_{i}\otimes
e_{i}\otimes e_{i}\in\mathbb{F}^{n}\otimes\mathbb{F}^{n}\otimes\mathbb{F}^{n}$
is semistable. An argument using the semistability test is as follows. Let $p$
be an integer. Denote by $V_{p}$ the representation of
$(K^{\times})^{n}\rtimes S_{n}$ on $K^{n}$ given by
$(t_{1},\dots,t_{n},\sigma)\cdot e_{i}=t_{i}^{p}e_{\sigma i}$. If $p\neq 0$,
then this representation is irreducible, since every subspace invariant under
$(K^{\times})^{n}$ is spanned by a subset of the standard basis and the only
such subspaces invariant under $S_{n}$ are $\\{0\\}$ and $V_{p}$. The diagonal
tensor $\sum_{i=1}^{n}e_{i}\otimes e_{i}\otimes e_{i}\in V_{1}\otimes
V_{1}\otimes V_{-2}$ is invariant under $(K^{\times})^{n}\rtimes S_{n}$, so it
is semistable by the semistability test (Lemma 23).121212Also here it follows
from the stronger Lemma 63 in Appendix A that this tensor is in fact
polystable, which means that its $\mathrm{SL}$ orbit is closed.

The main result of this section is the following characterization of
maximality of the asymptotic weighted slice ranks in terms of semistability.

######  Theorem 26.

Let $T\in K^{m_{1}\times m_{2}\times m_{3}}$ be a tensor. All Kronecker powers
$T^{\otimes k}$ are semistable if and only if $G_{\xi}(T)$ has the maximal
possible value $\min_{i}m_{i}^{1/\xi_{i}}$ for every $\xi\in\Xi$.

Generally, if $T^{\otimes k}$ is semistable, then $T$ is semistable. Over
$\mathbb{C}$, it can be proven that semistability of $T$ is equivalent to the
semistability of all Kronecker powers of $T$ [BGO+18, Lem. 6.4 (full
version)]. It is not known whether this property also holds over arbitrary
fields. In any case, over the complex numbers this fact together with Theorem
26 gives:

######  Corollary 27.

A tensor $T\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}}$ is semistable if and
only if $G_{\xi}(T)=\min_{i}m_{i}^{1/\xi_{i}}$ for every $\xi\in\Xi$.

We will now work towards proving the main result of this section, Theorem 26.
The crucial connection between slice decompositions and semistability that we
will use to prove Theorem 26 is the following sufficient condition for a
tensor to be unstable and the corresponding necessary condition for a tensor
to be semistable.

######  Lemma 28.

For any tensor $T\in V_{1}\otimes V_{2}\otimes V_{3}$, if $T$ has a slice
decomposition of size $(r_{1},r_{2},r_{3})$ such that $r_{i}<\frac{1}{3}\dim
V_{i}$ for every $i\in[3]$, then $T$ is unstable.

Phrased contrapositively, for any tensor $T\in V_{1}\otimes V_{2}\otimes
V_{3}$, if $T$ is semistable, then for every slice decomposition of $T$ of
size $(r_{1},r_{2},r_{3})$, there is an $i\in[3]$ such that
$r_{i}\geq\tfrac{1}{3}\dim V_{i}$.

###### Proof.

By Lemma 16 there are subspaces $W_{i}\subset V_{i}$ with dimensions $\dim
W_{i}=r_{i}<\frac{\dim V_{i}}{3}$ such that the tensor $T$ lies in
$W_{1}\otimes V_{2}\otimes V_{3}+V_{1}\otimes W_{2}\otimes V_{3}+V_{1}\otimes
V_{2}\otimes W_{3}$. For each of these subspaces choose a complement $U_{i}$,
so that $V_{i}=W_{i}\oplus U_{i}$. Denote $q_{i}=\dim U_{i}$. Note that
$q_{i}>2r_{i}$.

We first assume that all $r_{i}\neq 0$. Consider a $1$-parameter subgroup
$\lambda(t)=(\lambda_{1}(t),\lambda_{2}(t),\lambda_{3}(t))$ in
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ where
$\lambda_{i}(t)=t^{\frac{q_{i}}{r_{i}}r_{1}r_{2}r_{3}}\mathrm{id}\oplus
t^{-r_{1}r_{2}r_{3}}\mathrm{id}$ with respect to the decomposition
$V_{i}=W_{i}\oplus U_{i}$.

The decompositions $V_{i}=W_{i}\oplus U_{i}$ induce a direct sum decomposition
of $V_{1}\otimes V_{2}\otimes V_{3}$ into eight summands. The space
$W_{1}\otimes V_{2}\otimes V_{3}+V_{1}\otimes W_{2}\otimes V_{3}+V_{1}\otimes
V_{2}\otimes W_{3}$ containing $T$ is the direct sum of seven out of these
eight summands, the missing summand being $U_{1}\otimes U_{2}\otimes U_{3}$.
Because $q_{i}>2r_{i}$, on each of these seven summands $\lambda(t)$ acts with
a positive power of $t$. Therefore the curve $\lambda(t)\cdot T$ contains zero
in its closure and $T$ is unstable.

If some of the $r_{i}$ are zero, then we can give a similar construction where
$\lambda(t)$ does not act on factors for which $r_{i}=0$. For example, if
$r_{3}=0$ we consider the $1$-parameter subgroup
$\lambda(t)=(t^{q_{1}r_{2}}\mathrm{id}\oplus
t^{-r_{1}r_{2}}\mathrm{id},t^{r_{1}q_{2}}\mathrm{id}\oplus
t^{-r_{1}r_{2}}\mathrm{id},\mathrm{id})$. ∎

Using the necessary condition for semistability in Lemma 28 we prove the
following lower bound on the weighted slice rank.

######  Lemma 29.

Let $\xi\in\Xi$. For $T\in K^{m_{1}\times m_{2}\times m_{3}}$, if $T$ is
semistable, then $S_{\xi}(T)\geq c_{\xi}\min_{i}m_{i}^{1/\xi_{i}}$ for some
constant $c_{\xi}$ that depends only on $\xi$.

###### Proof.

Suppose that the value of $S_{\xi}(T^{\otimes n})$ is attained by a slice
decomposition of $T$ of size $(r_{1},r_{2},r_{3})$. By Lemma 28 there is an
$i\in[3]$ such that $r_{i}\geq\frac{m_{i}}{3}$. Therefore

$S_{\xi}(T)=r_{1}^{1/\xi_{1}}+r_{2}^{1/\xi_{2}}+r_{3}^{1/\xi_{3}}\geq
r_{i}^{1/\xi_{i}}\geq 3^{-1/\xi_{i}}m_{i}^{1/\xi_{i}}\geq
3^{-1/\min_{i}\xi_{i}}\min_{i}m_{i}^{1/\xi_{i}}.$

This proves the claim for $c_{\xi}=3^{-1/\min_{i}\xi_{i}}$. ∎

By applying Lemma 29 to powers of tensors we get the following sufficient
condition for the asymptotic weighted slice rank to be maximal.

######  Lemma 30.

Let $\xi\in\Xi$. For any tensor $T\in K^{m_{1}\times m_{2}\times m_{3}}$, if
for every $k\in\mathbb{N}$ the tensor $T^{\otimes k}$ is semistable, then
$G_{\xi}(T)=\min_{i}m_{i}^{1/\xi_{i}}$.

###### Proof.

By Lemma 29 we have $S_{\xi}(T^{\otimes k})\geq
c_{\xi}\min_{i}m_{i}^{k/\xi_{i}}$ and asymptotically
$G_{\xi}(T)\geq\min_{i}m_{i}^{1/\xi_{i}}$. ∎

Finally, we prove a necessary condition for the asymptotic weighed slice ranks
to be maximal. Combined with the above sufficient condition, this will prove
Theorem 26.

######  Lemma 31.

Let $T\in K^{m_{1}\times m_{2}\times m_{3}}$. If
$G_{\xi}(T)=\min_{i}m_{i}^{1/\xi_{i}}$ for every $\xi\in\Xi$, then for every
$k\in\mathbb{N}$ the tensor $T^{\otimes k}$ is semistable.

###### Proof.

Suppose at least one of the $m_{i}$ is strictly larger than 1. Assume that
some power $T^{\otimes q}$ is unstable. By [BCC+17a, Thm. 4.10] the powers
$T^{\otimes qk}$ have slice decompositions of size $(r_{1},r_{2},r_{3})$ with
$r_{i}\leq m_{i}^{qk}e^{-\alpha k}$ where $\alpha>0$ is the instability of
$T^{\otimes q}$ as defined in [BCC+17a]. Denote $m=\max_{i}m_{i}$ and take
$\xi_{i}=\log_{m}m_{i}$ so that $m_{i}^{1/\xi_{i}}=m$. Clearly,
$\xi=(\xi_{1},\xi_{2},\xi_{3})$ lies in $\Xi$. The corresponding weighted
slice rank can be bounded as

$S_{\xi}(T^{\otimes qk})\leq
r_{1}^{1/\xi_{1}}+r_{2}^{1/\xi_{2}}+r_{3}^{1/\xi_{3}}\leq
m^{qk}\left(e^{-\frac{\alpha}{\xi_{1}}k}+e^{-\frac{\alpha}{\xi_{2}}k}+e^{-\frac{\alpha}{\xi_{3}}k}\right)\leq
3\left(me^{-\frac{\alpha}{q\min_{i}\xi_{i}}}\right)^{qk}.$

It follows that the asymptotic $\xi$-weighted slice rank cannot have the
maximal possible value $m$.

Consider now the case when $m_{1}=1$. In this case, the tensor can be
presented as $T=e_{1}\otimes M$ for some matrix $M\in K^{m_{2}\times m_{3}}$.
Assume $M$ is nonzero and $\operatorname{rk}M=r$.

Every slice decomposition for $T$ of size $(0,r_{2},r_{3})$ can be transformed
into a rank decomposition for $M$ with $r_{2}+r_{3}$ summands by forgetting
the first tensor factor. Therefore $S_{(0,1,1)}(T)\geq\operatorname{rk}M$. On
the other hand, every rank decomposition for $M$ can be transformed into a
slice decomposition for $T$ of size $(0,r,0)$ or $(0,0,r)$ by tensoring one of
the factors with $e_{1}$, which gives an upper bound $S_{\xi}(T)\leq r$ if
$\xi_{1}=0$. Since $S_{(0,1,1)}(T)\leq S_{\xi}(T)$ if $\xi_{1}=0$, we have
$S_{\xi}(T)=r$ if $\xi_{1}=0$. If $\xi_{1}>0$, then $S_{\xi}(T)=1$ from the
decomposition $T=e_{1}\otimes M$.

Since matrix rank is multiplicative, we have the same values for the
asymptotic weighted slice ranks: $G_{\xi}(T)=1$ if $\xi_{1}>0$ and
$G_{\xi}(T)=r$ if $\xi_{1}=0$. The equality
$G_{\xi}(T)=\min_{i}m_{i}^{1/\xi_{i}}$ holds if and only if $r=m_{2}=m_{3}$.
If this is the case, the
$\mathrm{SL}_{1}\times\mathrm{SL}_{m_{2}}\times\mathrm{SL}_{m_{3}}$ orbit of
$T$ consists of tensors of the form $e_{1}\otimes M^{\prime}$ with $\det
M^{\prime}=\det M$, it is closed and does not contain $0$, so $T$ is
semistable. ∎

###### Proof of Theorem 26.

Combine the sufficient condition in Lemma 30 and the necessary condition in
Lemma 31. ∎

###  3.2 Arbitrary tensors over the complex numbers

In this section we characterize, for tensors over $\mathbb{C}$, the value of
the asymptotic weighted slice rank in terms of moment polytopes. Recall that
the moment polytope of a tensor $T$ is denoted by $\Pi(T)$, and that $H$
denotes the Shannon entropy of probability vectors. We gave a definition of
moment polytopes in terms of singular values in Section 1.2 and we will see
another of the standard descriptions soon. The characterization that we prove
is as follows.

######  Theorem 32.

Let $V_{1}$, $V_{2}$ and $V_{3}$ be vector spaces over $\mathbb{C}$ and let
$T\in V_{1}\otimes V_{2}\otimes V_{3}$ be a tensor. For every $\xi\in\Xi$ we
have

$G_{\xi}(T)=\max_{p\in\Pi(T)}\min_{i\in[3]}2^{H(p_{i})/\xi_{i}}.$

To carry out the proof of Theorem 32 we will use the following notation and
fundamental results. We write $\lambda\vdash_{m}k$ if $\lambda$ is a partition
of $k$ into $m$ parts. Such partitions are represented by non-decreasing
tuples in $\mathbb{N}^{m}$. We omit $k$ or $m$ if its value is not restricted.
We write $(\lambda_{1},\lambda_{2},\lambda_{3})\vdash_{(m_{1},m_{2},m_{3})}k$
if $\lambda_{i}\vdash_{m_{i}}k$ for every $i$. Again we may omit some of the
parameters when they are not restricted.

In characteristic $0$, the irreducible representations of the symmetric group
$S_{k}$ (Specht modules) correspond to partitions $\lambda\vdash k$. We denote
the representation corresponding to a partition $\lambda$ by $[\lambda]$. The
irreducible polynomial representations of $\mathrm{GL}_{n}$ (Schur modules)
correspond to partitions $\lambda\vdash_{n}$ and are denoted by
$\\{\lambda\\}$.

######  Theorem 33 (Schur–Weyl duality).

As a $\mathrm{GL}_{m}\times S_{k}$-representation, the space
$(\mathbb{C}^{m})^{\otimes k}$ decomposes into

$(\mathbb{C}^{m})^{\otimes
k}\cong\bigoplus_{\lambda\vdash_{d}n}\\{\lambda\\}\otimes[\lambda].$

We will need the following bounds on the dimensions of $\\{\lambda\\}$ and
$[\lambda]$.

######  Lemma 34 ([CM05]).

For any partition $\lambda\vdash_{m}k$ we have the following bounds on the
dimension of Schur and Specht modules:

$\displaystyle\dim\\{\lambda\\}\leq(k+1)^{\frac{m(m-1)}{2}}$
$\displaystyle\frac{n!}{\prod_{i}(\lambda_{i}+d-i)!}\leq\dim[\lambda]\leq\frac{n!}{\prod_{i}\lambda_{i}!}.$

In particular, it follows from Lemma 34 that for any partition
$\lambda=(\lambda_{1},\ldots,\lambda_{m})\vdash k$, if we consider the
stretched partition $n\lambda=(n\lambda_{1},\ldots,n\lambda_{m})\vdash nk$,
and we let $n$ go to infinity, then the rate of growth of the dimension of the
Specht module $[n\lambda]$ is precisely described in terms of the Shannon
entropy $H(\lambda/k)$ of the probability vector
$\lambda/k=(\lambda_{1}/k,\ldots,\lambda_{m}/k)$ as follows.

######  Lemma 35.

For any $\lambda\vdash k$ we have $\dim[n\lambda]\leq
2^{H(\frac{\lambda}{k})kn}$ for any $n\in\mathbb{N}$ and
$2^{H(\frac{\lambda}{k})kn-o(n)}\leq\dim[n\lambda]$ as $n\to\infty$.

Finally, regarding dimensions, we will need the following inequality for the
dimension of the Specht module $[\lambda+\mu]$ where as usual $\lambda+\mu$ is
the partition $(\lambda_{1}+\mu_{1},\ldots,\lambda_{m}+\mu_{m})$.

######  Lemma 36.

For any partitions $\lambda$ and $\mu$ we have
$\dim[\lambda+\mu]\geq\dim[\lambda]\dim[\mu]$.

###### Proof.

For any partition $\nu\vdash k+\ell$ the $S_{k+\ell}$ representation $[\nu]$
decomposes as a $S_{k}\times S_{\ell}$ representation as

$[\nu]\\!\downarrow^{S_{k+\ell}}_{S_{k}\times
S_{\ell}}\cong\bigoplus_{\lambda\vdash k,\mu\vdash
l}c^{\nu}_{\lambda\mu}\,[\lambda]\otimes[\mu]$

where the $c^{\nu}_{\lambda\mu}$ are the so-called Littlewood–Richardson
coefficients [FH04, Ex. 4.43]. In particular, for $\nu=\lambda+\mu$ it is
known that $c^{\nu}_{\lambda\mu}=1$ [FH04, Ex. 15.24]. Therefore, supposing
$\lambda\vdash k$ and $\mu\vdash\ell$, we have that $[\lambda]\otimes[\mu]$ is
contained in $[\lambda+\mu]\\!\downarrow^{\smash{S_{k+\ell}}}_{S_{k}\times
S_{\ell}}$. It follows that $\dim[\lambda+\mu]\geq\dim[\lambda]\dim[\mu]$. ∎

The Schur–Weyl decompositions of the three representations
$(\mathbb{C}^{m_{1}})^{\otimes k}$, $(\mathbb{C}^{m_{2}})^{\otimes k}$ and
$(\mathbb{C}^{m_{3}})^{\otimes k}$ induce a decomposition of their tensor
product $(\mathbb{C}^{m_{1}})^{\otimes k}\otimes(\mathbb{C}^{m_{2}})^{\otimes
k}\otimes(\mathbb{C}^{m_{3}})^{\otimes k}$ as a direct sum of spaces
$(\\{\lambda_{1}\\}_{\mathrm{GL}_{m_{1}}}\otimes[\lambda_{1}])\otimes(\\{\lambda_{2}\\}_{\mathrm{GL}_{m_{2}}}\otimes[\lambda_{2}])\otimes(\\{\lambda_{3}\\}_{\mathrm{GL}_{m_{3}}}\otimes[\lambda_{3}])$.
For every tensor $T\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}}$ and every
partition triple $\lambda\vdash_{(m_{1},m_{2},m_{3})}k$ we denote by
$T^{\lambda}$ the projection of $T^{\otimes k}$ onto the direct summand
$(\\{\lambda_{1}\\}_{\mathrm{GL}_{m_{1}}}\otimes[\lambda_{1}])\otimes(\\{\lambda_{2}\\}_{\mathrm{GL}_{m_{2}}}\otimes[\lambda_{2}])\otimes(\\{\lambda_{3}\\}_{\mathrm{GL}_{m_{3}}}\otimes[\lambda_{3}])$.
The nonzero summands $T^{\lambda}$ satisfy the following semigroup property.

######  Theorem 37 ([WDGC13, §3c]).

Let $T\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}}$ be a tensor. The set of
all partitions $\lambda\vdash_{(m_{1},m_{2},m_{3})}$ such that
$T^{\lambda}\neq 0$ has the semigroup property: if $T^{\lambda}\neq 0$ and
$T^{\mu}\neq 0$, then $T^{\lambda+\mu}\neq 0$.

The semigroup property (Theorem 37) implies that
$\\{(\frac{\lambda_{1}}{k},\frac{\lambda_{2}}{k},\frac{\lambda_{3}}{k})\mid\lambda\vdash_{(m_{1},m_{2},m_{3})}k,T^{\lambda}\neq
0\\}$ is a convex subset of $\mathbb{Q}^{m_{1}+m_{2}+m_{3}}$. It is a
remarkable fact that this set coincides with the set of rational points in the
moment polytope $\Pi(T)$ defined in the introduction. This is the
representation-theoretic definition of the moment polytope. This was proven in
high generality in [Bri87] and the tensor case is explored in [WDGC13]. The
points of the moment polytope are tuples of $m_{1}+m_{2}+m_{3}$ real numbers
divided into three parts with $m_{i}$ numbers in each part. The three parts of
the tuple $p$ will denoted by $p_{1}$, $p_{2}$ and $p_{3}$. We denote by
$H(p)$ the tuple of entropies $(H(p_{1}),H(p_{2}),H(p_{3}))$.

We will now work towards the proof of the main theorem of this subsection,
Theorem 32. The proof of Theorem 32 is divided into several lemmas. Before we
start, let us establish some notation. Let $m_{i}=\dim V_{i}$ and
$m=(m_{1},m_{2},m_{3})$. Let $T\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}}$.
Recall that for a partition triple $\lambda\vdash_{m}k$ the tensor
$T^{\lambda}$ is the projection of $T^{\otimes k}\in V_{1}^{\otimes k}\otimes
V_{2}^{\otimes k}\otimes V_{3}^{\otimes k}$ onto the
$\mathrm{GL}(V_{1})\times\mathrm{GL}(V_{2})\times\mathrm{GL}(V_{3})\times
S_{k}$-subrepresentation
$(\\{\lambda_{1}\\}_{\mathrm{GL}(V_{1})}\otimes[\lambda_{1}])\otimes(\\{\lambda_{2}\\}_{\mathrm{GL}(V_{2})}\otimes[\lambda_{2}])\otimes(\\{\lambda_{3}\\}_{\mathrm{GL}(V_{3})}\otimes[\lambda_{3}])$.

For every $k\in\mathbb{N}$ we define the set
$R_{k}(T)=\\{\lambda\vdash_{m}k\mid T^{\lambda}\neq
0\\}\subset\mathbb{Z}^{m_{1}+m_{2}+m_{3}}$. The semigroup property (Theorem
37) implies that $R_{k+\ell}(T)\supset R_{k}(T)+R_{\ell}(T)$ and the
representation-theoretic characterization of the moment polytope gives that
$\Pi(T)$ is the closure of $\bigcup_{k}\frac{R_{k}(T)}{k}$. Define
$M_{\xi,k}(T)=\max_{\lambda\in
R_{k}(T)}\min_{i}(\dim[\lambda_{i}])^{1/\xi_{i}}$.

The next lemma gives a lower bound on the weighted slice rank of powers of
tensors, by using the semistability test (Lemma 23) and the lower bound on the
weighted slice rank of semistable tensors (Lemma 29).

######  Lemma 38.

For every tensor $T$ and $\xi\in\Xi$ we have $S_{\xi}(T^{\otimes k})\geq
c_{\xi}M_{\xi,k}(T)$ for some $c_{\xi}$ dependent only on $\xi$.

###### Proof.

Each component tensor $T^{\lambda}$ is obtained from the Kronecker power
$T^{\otimes k}$ by applying the projections $V_{i}^{\otimes
k}\to\\{\lambda_{i}\\}\otimes[\lambda_{i}]$ to the three tensor factors, so
$T^{\otimes k}\geq T^{\lambda}$.

Since $T^{\otimes k}$ is $S_{k}$-invariant and the projections are
equivariant, each $T^{\lambda}$ is $S_{k}$-invariant. Furthermore, because the
action of $S_{k}$ on $\\{\lambda_{i}\\}$ is trivial, for each linear form
$f_{i}\colon\\{\lambda_{i}\\}\to\mathbb{C}$ the restriction
$[(f_{1}\otimes\mathrm{id})\otimes(f_{2}\otimes\mathrm{id})\otimes(f_{3}\otimes\mathrm{id})]\,T^{\lambda}\in[\lambda_{1}]\otimes[\lambda_{2}]\otimes[\lambda_{3}]$
is $S_{k}$-invariant.

If $\lambda\in R_{k}(T)$, that is, if $T^{\lambda}$ is nonzero, then there
exist $f_{i}$ such that this restriction is nonzero (this follows by a triple
application of a general fact that if $x\in U\otimes V$ is nonzero, then
$(f\otimes\mathrm{id})x\in V$ is nonzero for some $f$). Denote by $T^{\prime}$
a nonzero tensor of the form
$[(f_{1}\otimes\mathrm{id})\otimes(f_{2}\otimes\mathrm{id})\otimes(f_{3}\otimes\mathrm{id})]\,T^{\lambda}$.
By the semistability test (Lemma 23) the tensor $T^{\prime}$ is semistable.
Using Lemma 29 and the monotonicity of weighted slice ranks we get
$S_{\xi}(T^{\otimes k})\geq S_{\xi}(T^{\prime})\geq
c_{\xi}\min_{i}(\dim[\lambda_{i}])^{1/\xi_{i}}$. Maximizing over all
$\lambda\in R_{k}$ we get the required lower bound. ∎

######  Lemma 39.

For every tensor $T\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}}$ and
$\xi\in\Xi$ we have $S_{\xi}(T^{\otimes k})\leq(k+1)^{\gamma}M_{\xi,k}(T)$ for
some $\gamma$ not dependent on $k$.

###### Proof.

We prove the statement for the case when all three $\xi_{i}$ are nonzero. The
tensor $T^{\otimes k}$ is the sum of $T^{\lambda}$ for all $\lambda\in
R_{k}(T)$. By Lemma 18(iii) each summand $T^{\lambda}$ has slice
decompositions of sizes
$(\dim[\lambda_{1}]\dim\\{\lambda_{1}\\}_{\mathrm{GL}(V_{1})},0,0)$,
$(0,\dim[\lambda_{2}]\dim\\{\lambda_{2}\\}_{\mathrm{GL}(V_{2})},0)$ and
$(0,0,\dim[\lambda_{3}]\dim\\{\lambda_{3}\\}_{\mathrm{GL}(V_{3})})$. From the
definition of $M_{\xi,k}(T)$ it follows that for every $\lambda\in R_{k}(T)$
there is an index $i$ such that $\dim[\lambda_{i}]\leq M_{\xi,k}^{\xi_{i}}$.
By Lemma 34 the dimensions $\dim\\{\lambda_{i}\\}_{\mathrm{GL}(V_{i})}$ are
upper bounded by $(k+1)^{\alpha}$ for
$\alpha=\max_{i}\frac{m_{i}(m_{i}+1)}{2}$. Therefore, for every $\lambda\in
R_{k}(T)$ the tensor $T^{\lambda}$ has either a slice decomposition of size
$((k+1)^{\alpha}M_{\xi,k}^{\xi_{1}},0,0)$ or a slice decomposition of size
$(0,(k+1)^{\alpha}M_{\xi,k}^{\xi_{2}},0)$ or a slice decomposition of size
$(0,0,(k+1)^{\alpha}M_{\xi,k}^{\xi_{3}})$. Additionally, the number of
partition triples $\lambda\vdash_{m}k$ is upper bounded by
$(k+1)^{m_{1}+m_{2}+m_{3}}$. Summing the slice decompositions for all nonzero
$T^{\lambda}$, we get a slice decomposition for $T$ of size at most
$((k+1)^{\beta}M_{\xi,k}^{\xi_{1}},(k+1)^{\beta}M_{\xi,k}^{\xi_{2}},(k+1)^{\beta}M_{\xi,k}^{\xi_{3}})$
where $\beta=(m_{1}+m_{2}+m_{3})\alpha$. Therefore, the $\xi$-weighted slice
rank $S_{\xi}(T^{\otimes k})$ is at most $(k+1)^{\gamma}M_{\xi,k}$ where
$\gamma=\frac{\beta}{\min_{i}\xi_{i}}$.

The proof for the case when some $\xi_{i}=0$ is analogous, except that we do
not allow $i$-slices in our slice decompositions. ∎

######  Lemma 40.

The sequence $\sqrt[k]{M_{\xi,k}(T)}$ converges when $k$ goes to infinity, and
the limit is given by

$\lim_{k\to\infty}\sqrt[k]{M_{\xi,k}(T)}=\max_{p\in\Pi(T)}\min_{i}2^{H(p_{i})/\xi_{i}}.$

###### Proof.

Using $R_{k+\ell}(T)\supseteq R_{k}(T)+R_{\ell}(T)$ and Lemma 36 we can prove
the following supermultiplicativity property:

$\displaystyle M_{\xi,k+\ell}(T)$ $\displaystyle=\max_{\lambda\in
R_{k+\ell}(T)}\min_{i}(\dim[\lambda_{i}])^{\frac{1}{\xi_{i}}}$
$\displaystyle\geq\max_{\mu\in R_{k}(T),\nu\in
R_{\ell}(T)}\min_{i}(\dim[\mu_{i}+\nu_{i}])^{\frac{1}{\xi_{i}}}$
$\displaystyle\geq\max_{\mu\in R_{k}(T),\nu\in
R_{\ell}(T)}\min_{i}(\dim[\mu_{i}]\dim[\nu_{i}])^{\frac{1}{\xi_{i}}}$
$\displaystyle\geq\max_{\mu\in
R_{k}(T)}\min_{i}(\dim[\mu_{i}])^{\frac{1}{\xi_{i}}}\max_{\nu\in
R_{\ell}(T)}\min_{i}(\dim[\nu_{i}])^{\frac{1}{\xi_{i}}}$
$\displaystyle=M_{\xi,k}(T)M_{\xi,\ell}(T).$

From the upper bound on the dimension of $[\lambda_{i}]$ in terms of Shannon
entropy (Lemma 34) and the inclusion $\frac{R_{k}(T)}{k}\subseteq\Pi(T)$ we
obtain

$\sqrt[k]{M_{\xi,k}(T)}=\max_{\lambda\in
R_{k}}\min_{i}(\sqrt[k]{\dim[\lambda_{i}]})^{1/\xi_{i}}\leq\max_{\lambda\in
R_{k}}\min_{i}2^{H(\frac{\lambda_{i}}{k})/\xi_{i}}\leq\max_{p\in\Pi(T)}\min_{i}2^{H(p_{i})/\xi_{i}}.$

The convergence of $\sqrt[k]{M_{\xi,k}(T)}$ now follows from Fekete’s lemma
applied to the sequence

$\log\sqrt[k]{M_{\xi,k}(T)}=\frac{\log M_{\xi,k}(T)}{k},$

since it has a superadditive numerator and it is bounded.

On the other hand, using the lower bound $\dim[n\lambda_{i}]\geq
2^{H(\frac{\lambda_{i}}{k})kn-o(n)}$ (Lemma 34) we see that

$\sqrt[kn]{M_{\xi,kn}(T)}\geq\max_{\lambda\in
R_{k}}\min_{i}2^{H(\frac{\lambda_{i}}{k})/\xi_{i}-o(1)}$

and letting $n$ go to infinity we get

$\lim_{n\to\infty}\sqrt[n]{M_{\xi,n}(T)}=\lim_{n\to\infty}\sqrt[kn]{M_{\xi,kn}(T)}\geq\max_{\lambda\in
R_{k}}\min_{i}2^{H(\frac{\lambda_{i}}{k})/\xi_{i}}.$

Considering a sequence of rational points $\frac{\lambda^{(k)}}{k}\in\Pi(T)$
which converges to the point where the maximum
$\max_{p\in\Pi(T)}\min_{i}2^{H(p_{i})/\xi_{i}}$ is attained, we see that this
lower bound can be made arbitrarily close to the upper bound
$\max_{p\in\Pi(T)}\min_{i}2^{H(p_{i})/\xi_{i}}$. ∎

###### Proof of Theorem 32.

Lemmas 38 and 39 prove the bounds

$c_{\xi}M_{\xi,k}(T)\leq S_{\xi}(T^{\otimes
k})\leq(k+1)^{\gamma}M_{\xi,k}(T).$

Taking the $k$th root and letting $k$ go to infinity, and using Lemma 40, we
get the required

$G_{\xi}(T)=\lim_{k\to\infty}\sqrt[k]{M_{\xi,k}(T)}=\max_{p\in\Pi(T)}\min_{i}2^{H(p_{i})/\xi_{i}}.\qed$

######  Remark 41.

Note that not only have we obtained a characterization of $G_{\xi}(T)$ for
every tensor $T$ over $\mathbb{C}$, but from the proof it also follows that
the $\limsup$ in the definition of $G_{\xi}(T)$ can be replaced by a limit.

###  3.3 Tight tensors over arbitrary fields

In the previous section we proved an upper bound on the asymptotic weighted
slice rank for tensors over the complex numbers. Here we prove a similar upper
bound for tensors over arbitrary fields. Moreover, we show that this upper
bound is optimal for the subclass of _tight_ tensors. The idea is similar to
the complex case, but instead of the Schur–Weyl decomposition and moment
polytope, which are not available in positive characteristic, we use the
weight decomposition and the corresponding polytope.

Let $e_{1},\dots,e_{m}$ be the standard basis of the vector space $K^{m}$. The
tensor products $e_{j_{1}}\otimes e_{j_{2}}\otimes e_{j_{3}}$ form a basis for
the tensor product space $K^{m_{1}\times m_{2}\times m_{3}}$. Every tensor
$T\in K^{m_{1}\times m_{2}\times m_{3}}$ can be written as
$T=\sum_{j_{1}=1}^{m_{1}}\sum_{j_{2}=1}^{m_{2}}\sum_{j_{3}=1}^{m_{3}}t_{j_{1}j_{2}j_{3}}\,e_{j_{1}}\otimes
e_{j_{2}}\otimes e_{j_{3}}.$ The _support_ of $T$, denoted by
$\operatorname{supp}(T)$, is the set of all triples $(j_{1},j_{2},j_{3})$ for
which $t_{j_{1}j_{2}j_{3}}\neq 0$.

For any $k$-tuple $s\in[m]^{k}$ we define the vector $e_{s}\coloneqq
e_{s_{1}}\otimes e_{s_{2}}\otimes\dots\otimes e_{s_{k}}$. These vectors form a
basis of $(K^{m})^{\otimes k}$. We define the _type_ of $s$ as the $m$-tuple
$w\in\mathbb{N}^{m}$ with coefficients $w_{j}=\\#\\{i\mid s_{i}=j\\}$ for
$j\in[m]$. That is, for every $j\in[m]$ the value of $w_{j}$ is the number of
appearances of the element $j$ in the tuple $s$. The tuple $s$ contains $k$
elements in total, so $\sum_{i=1}^{m}w_{i}=k$. The set of possible types of
elements in $[m]^{k}$ is the set of all ordered partitions of $k$ into $m$
parts. We may think of these types as rational probability distributions
$\frac{w}{k}$ on $[m]$. Let $[[w]]\subseteq K^{m}$ be the subspace spanned by
the basis elements $e_{s}$ such that $s\in[m]^{k}$ is of type $w$. The
dimension of $[[w]]$ equals the multinomial coefficient $\binom{k}{w}\coloneqq
k!/(w_{1}!\cdots w_{m}!)$. The space $(K^{m})^{\otimes k}$ decomposes as the
direct sum of the subspaces $[[w]]$ over all possible types $w$.

We write $(w_{1},w_{2},w_{3})\vDash_{(m_{1},m_{2},m_{3})}k$ if $w_{i}$ is the
type of an element of $[m_{i}]^{k}$ for $i=1,2,3$. For every tensor $T\in
K^{m_{1}\times m_{2}\times m_{3}}$ the Kronecker power $T^{\otimes k}$
decomposes into a direct sum of tensors
$T^{w}\in[[w_{1}]]\otimes[[w_{2}]]\otimes[[w_{3}]]$ going over all types
$w\vDash_{m}k$. From the decomposition

$T^{\otimes
k}=\sum_{s_{1}\in[m_{1}]^{k}}\sum_{s_{2}\in[m_{2}]^{k}}\sum_{s_{3}\in[m_{3}]^{k}}\prod_{\ell=1}^{k}t_{s_{1\ell}s_{2\ell}s_{3\ell}}e_{s_{1}}\otimes
e_{s_{2}}\otimes e_{s_{3}},$

it follows that the component $T^{w}$ is nonzero if and only if there is a
sequence of triples
$(s_{1\ell},s_{2\ell},s_{3\ell})\in(\operatorname{supp}(T))^{k}$ such that the
sequences $(s_{i1},\dots,s_{ik})$ have types $w_{i}$. Or, in terms of
probability distributions, if and only if there is a rational probability
distribution on $\operatorname{supp}(T)$ with common denominator $k$ such that
the marginal distributions are $\frac{w_{i}}{k}$. Let $W(T)$ be the polytope
consisting of all triples of marginal distributions for all probability
distributions on $\operatorname{supp}(T)$.

######  Theorem 42.

Let $T\in K^{m_{1}\times m_{2}\times m_{3}}$ be a tensor. For every
$\xi\in\Xi$

$G_{\xi}(T)\leq\max_{p\in W(T)}\min_{i\in[3]}2^{H(p_{i})/\xi_{i}}.$

###### Proof.

Define

$M_{\xi,k}(T)=\max_{\begin{subarray}{c}w\vDash_{m}k\\\ T^{w}\neq
0\end{subarray}}\min_{i}(\dim[[w_{i}]])^{1/\xi_{i}}.$

The number of possible types $w$ is at most $(k+1)^{m_{1}+m_{2}+m_{3}}$, and
by the same reasoning as in Lemma 39 we have $S_{\xi}(T^{\otimes
k})\leq(k+1)^{\gamma}M_{\xi,k}(T)$ for some $\gamma$. Using a well-known upper
bound on the multinomial coefficients in terms of entropy,

$M_{\xi,k}(T)\leq\max_{\begin{subarray}{c}w\vDash_{m}k\\\ T^{w}\neq
0\end{subarray}}\min_{i}2^{kH(\frac{w_{i}}{k})/\xi_{i}}\leq\max_{p\in
W(T)}\min_{i}2^{kH(p_{i})/\xi_{i}}.$

This gives the required upper bound. ∎

A tensor $T\in K^{m_{1}\times m_{2}\times m_{3}}$ is called _tight_ (in the
standard basis) if there exist three injective maps
$u_{i}\colon[m_{i}]\to\mathbb{Z}$ such that for every
$(j_{1},j_{2},j_{3})\in\operatorname{supp}(T)$ it holds that
$u_{1}(j_{1})+u_{2}(j_{2})+u_{3}(j_{3})=0$ [Str91].

######  Lemma 43.

If a tensor $T\in K^{m_{1}\times m_{2}\times m_{3}}$ is tight, then all
nonzero components $T^{w}$ are semistable.

###### Proof.

We may assume that $K$ is algebraically closed. Since $T$ is tight there is by
definition for every $i\in[3]$ an injective map
$u_{i}\colon\\{1,\dots,m_{i}\\}\to\mathbb{Z}$ such that for every
$(j_{1},j_{2},j_{3})\in\operatorname{supp}(T)$ it holds that
$u_{1}(j_{1})+u_{2}(j_{2})+u_{3}(j_{3})=0$.

Equip the spaces $V_{i}=K^{m_{i}}$ with actions of $K^{\times}$ by letting
$t\in K^{\times}$ act as $t\cdot e_{j}=t^{u_{i}(j)}e_{j}$. Taking $k$ copies
of these representations, we obtain actions of the algebraic torus
$(K^{\times})^{k}$ on $V_{i}^{\otimes k}$. Additionally, the symmetric group
$S_{k}$ acts on $V_{i}^{\otimes k}$ by permuting the tensor factors. These two
actions combine to give actions of the semidirect product
$(K^{\times})^{k}\rtimes S_{k}$ on the three spaces $V_{i}^{\otimes k}$.

Note that the type spaces $[[w]]\subset V_{i}^{\otimes k}$ are irreducible
$(K^{\times})^{k}\rtimes S_{k}$-subrepresentations of $V_{i}^{\otimes k}$.
Indeed, as a $(K^{\times})^{k}$-representation $V_{i}^{\otimes k}$ decomposes
into a direct sum of one-dimensional representations spanned by the basis
vectors $e_{j}$, which all have different weights
$(u_{i}(j_{1}),\dots,u_{i}(j_{k}))$. Therefore, every $(K^{\times})^{k}\rtimes
S_{k}$-subrepresentation is spanned by some subset of these basis vectors
which is closed under permutations. Since every two sequences of the same type
can be transformed into each other by a permutation, every such
subrepresentation is a sum of type spaces.

Actions of $K^{\times}$ on $V_{i}$ define the action on the tensor product
$V_{1}\otimes V_{2}\otimes V_{3}$. An element $t\in K^{\times}$ acts by
$t\cdot e_{j_{1}}\otimes e_{j_{2}}\otimes
e_{j_{3}}=t^{u_{1}(j_{1})+u_{2}(j_{2})+u_{3}(j_{3})}e_{j_{1}}\otimes
e_{j_{2}}\otimes e_{j_{3}}$. Since $u_{1}(j_{1})+u_{2}(j_{2})+u_{3}(j_{3})=0$
for all triples $(j_{1},j_{2},j_{3})\in\operatorname{supp}(T)$, the tensor
$T\in V_{1}\otimes V_{2}\otimes V_{3}$ is invariant under this action of
$K^{\times}$. It follows that the Kronecker power $T^{\otimes k}$ is invariant
under $(K^{\times})^{k}$. Since $T^{\otimes k}$ is also invariant under
permutation of the $k$ factors, it is invariant under the action of
$(K^{\times})^{k}\rtimes S_{k}$ on $V_{1}^{\otimes k}\otimes V_{2}^{\otimes
k}\otimes V_{3}^{\otimes k}$.

Since $T^{\otimes k}$ is invariant under the action of
$(K^{\times})^{k}\rtimes S_{k}$ and the projections $V_{i}^{\otimes
k}\to[[w_{i}]]$ are equivariant, all components $T^{w}$ are also invariant.
Using the semistability test (Lemma 23) we conclude that if $T^{w}$ is
nonzero, then it is semistable. ∎

######  Lemma 44.

Let $T\in K^{m_{1}\times m_{2}\times m_{3}}$ be a tight tensor. For every
$\xi\in\Xi$ we have $S_{\xi}(T^{\otimes k})\geq c_{\xi}M_{\xi,k}(T)$ for some
$c_{\xi}$ dependent only on $\xi$.

###### Proof.

The proof of this statement is along the same lines as the proof of Lemma 38.
For every $w\vDash_{(m_{1},m_{2},m_{3})}k$ the tensor $T^{w}$ is a restriction
of $T^{\otimes k}$. By Lemma 43, $T^{w}$ is semistable whenever it is nonzero,
and using monotonicity of weighted slice ranks and Lemma 29 we obtain

$S_{\xi}(T^{\otimes k})\geq S_{\xi}(T^{w})\geq
c_{\xi}\min_{i}(\dim[[w_{i}]])^{1/\xi_{i}}.$

Maximizing over all $w$ such that $T^{w}$ is nonzero, we get the required
bound. ∎

######  Theorem 45.

Let $T\in K^{m_{1}\times m_{2}\times m_{3}}$ be a tight tensor. For every
$\xi\in\Xi$ we have

$G_{\xi}(T)=\max_{p\in W(T)}\min_{i\in[3]}2^{H(p_{i})/\xi_{i}}.$

###### Proof.

The upper bound is given by Theorem 42. The proof of the lower bound follows
the same reasoning as the proof of Lemma 40 and Theorem 32. The set $R_{k}(T)$
in this case is defined as $\\{w\vDash_{m}k\mid T^{w}\neq 0\\}$. Instead of
Lemma 36 we use the basic inequality
$\binom{k+\ell}{w+v}\geq\binom{k}{w}\binom{\ell}{v}$. ∎

##  4 Minimax correspondence

In this section we introduce the notion of a _dual pair_ , a variation on the
well-known Legendre–Fenchel transformation for convex functions which is also
known as the convex conjugate. For functions that form a dual pair, we use the
von Neumann minimax theorem to prove what we call a _minimax correspondence_ :
each function can be computed as an optimization problem in terms of the
other. The result in this section is general. We will apply it to the quantum
functionals and the asymptotic weighted slice ranks, and variations, in
Section 5.

###  4.1 Definition of a dual pair

######  Definition 46 (Dual pair).

Let $\Pi\subseteq\mathbb{R}^{n}$ be a compact convex set. Let
$h_{1},\dots,h_{k}$ be functions $\Pi\to\mathbb{R}_{\geq 0}$ that are
continuous and concave. We will write $h(x)=(h_{1}(x),\ldots,h_{k}(x))$.
Define the parameter spaces

$\displaystyle\Theta$
$\displaystyle=\\{(\theta_{1},\dots,\theta_{k})\mid\theta_{i}\geq
0,\,\sum_{i=1}^{k}\theta_{i}=1\\},$ $\displaystyle\Xi$
$\displaystyle=\\{(\xi_{1},\dots,\xi_{k})\mid\xi_{i}\geq
0,\,\max\\{\xi_{1},\dots,\xi_{k}\\}=1\\}.$

Let $\langle v,w\rangle\coloneqq\sum_{i=1}v_{i}w_{i}$ denote the standard dot
product on real vectors. We define the functions $f\colon\Theta\to\mathbb{R}$
and $g\colon\Xi\to\mathbb{R}$ by

$\displaystyle f(\theta)$
$\displaystyle=\max_{x\in\Pi}\,\langle\theta,h(x)\rangle,$ $\displaystyle
g(\xi)$ $\displaystyle=\max_{x\in\Pi}\min_{i\in[k]}\frac{h_{i}(x)}{\xi_{i}}.$

We call $(f,g)$ a _dual pair_.

Clearly the template and motivation for the above definition are pairs formed
by (the logarithms of) the quantum functionals and the asymptotic weighted
slice rank. Namely, for any fixed tensor $T$ we have that the pair
$(f_{\theta}(T),g_{\xi}(T))$ is a dual pair where $f_{\theta}(T)$ is the
logarithm of the quantum functional at $T$ and $g_{\xi}(T)$ is the logarithm
of the asymptotic $\xi$-weighted slice rank of $T$.

######  Remark 47.

The notion of a dual pair is not symmetric. That is, if $(f,g)$ is a dual
pair, then $(g,f)$ is not necessarily a dual pair.

###  4.2 Von Neumann minimax theorem

To prove the minimax correspondence for dual pairs we use the von Neumann
minimax theorem for quasiconcave-quasiconvex functions.

Let $X$ be a convex set and let $f\colon X\to\mathbb{R}$ be a function. We
define $f$ to be _quasiconvex_ if $f(\lambda
x+(1-\lambda)y)\leq\max\\{f(x),f(y)\\}$ for every $\lambda\in[0,1]$. We define
$f$ to be _quasiconcave_ if $f(\lambda x+(1-\lambda)y)\geq\max\\{f(x),f(y)\\}$
for every $\lambda\in[0,1]$. Equivalently, $f$ is quasiconvex if the sublevel
set $\\{x\mid f(x)\leq c\\}$ is convex for every $c$, and $f$ is quasiconcave
if the superlevel set $\\{x\mid f(x)\geq c\\}$ is convex for every $c$. Convex
functions are quasi-convex and concave functions are quasi-concave.

We define $f$ to be _quasilinear_ if it is both quasiconvex and quasiconcave.
Every fractional linear function $f(x)=\langle u,x\rangle/\langle v,x\rangle$
is a quasilinear function on any convex set on which $\langle v,x\rangle$ does
not change its sign. To see this, note that $\langle u,x\rangle/\langle
v,x\rangle\leq c$ if and only if $\langle u-cv,x\rangle\leq 0$ in case when
$\left<v,x\right>$ is positive and similarly when $\leq$ is replaced by
$\geq$. Thus the sublevel and superlevel sets are halfspaces, and in
particular they are convex. However, generally a fractional linear function is
neither convex nor concave.

Von Neumann proved the following minimax theorem for quasiconcave-quasiconvex
functions in 1937, extending the earlier version from 1928 which was for
bilinear functions. See, for instance, [Sim95, Theorem 2, page 3].

######  Theorem 48.

Let $X\subset\mathbb{R}^{m}$ and $Y\subset\mathbb{R}^{n}$ be nonempty compact
convex sets and let $f\colon X\times Y\to\mathbb{R}$ be a continuous function
such that $f(x,\cdot)$ is quasiconvex for every $x\in X$ and $f(\cdot,y)$ is
quasiconcave for every $y\in Y$. Then

$\max_{x\in X}\min_{y\in Y}f(x,y)=\min_{y\in Y}\max_{x\in X}f(x,y).$

We will also use the following property of quasiconvex functions.

######  Lemma 49.

Let $\Pi\subseteq\mathbb{R}^{m}$ be a convex polytope and
$f\colon\Pi\to\mathbb{R}$ be a quasiconcave function. Then the minimal value
of $f$ is attained at some vertex of $\Pi$.

###### Proof.

Let $c$ be minimal among the values of $f$ on all vertices of $\Pi$. Since the
set $\\{x\mid f(x)\geq c\\}$ is convex and contains all the vertices of $\Pi$,
it contains its convex hull $\Pi$. Thus, $f$ does not attain values that are
strictly less than $c$. ∎

In particular, it follows from Lemma 49 that every fractional linear function
$\langle u,x\rangle/\langle v,x\rangle$ defined on a convex polytope attains
its maxima and minima at vertices of the convex polytope.

###  4.3 Minimax correspondence

We will now show using the minimax theorem that there is a correspondence
between functions that form a dual pair.

For $\xi\in\Xi$ define the restricted parameter space

$\Theta(\xi)=\\{\theta\in\Theta\mid\xi_{i}=0\Rightarrow\theta_{i}=0\\}.$

Note that if $\theta\in\Theta(\xi)$, then $\left<\theta,\xi\right>$ is
nonzero.

######  Theorem 50 (Minimax correspondence).

Let $(f,g)$ be a dual pair. Then $f$ and $g$ are continuous and $f$ is convex.
Moreover, $f$ and $g$ are expressed in terms of each other via

$\displaystyle g(\xi)$
$\displaystyle=\min_{\theta\in\Theta(\xi)}\frac{f(\theta)}{\langle\theta,\xi\rangle},$
$\displaystyle f(\theta)$
$\displaystyle=\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle.$

###### Proof.

First of all, the function $f(\theta)$ is continuous and convex since it is
the pointwise maximum of the continuous and convex functions
$\theta\mapsto\langle\theta,h(x)\rangle$.

Next, we will prove that
$g(\xi)=\min_{\theta\in\Theta(\xi)}f(\theta)/\langle\theta,\xi\rangle$. By
definition we have

$g(\xi)=\max_{x\in\Pi}\min_{i\in[k]}\frac{h_{i}(x)}{\xi_{i}}=\max_{x\in\Pi}\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}.$ (1)

From applying Lemma 49 to the quasiconcave function
$\theta\mapsto\langle\theta,h(x)\rangle/\langle\theta,\xi\rangle$ on the
simplex $\Theta(\xi)$ we get the equation

$\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}=\min_{\theta\in\Theta(\xi)}\frac{\langle\theta,h(x)\rangle}{\langle\theta,\xi\rangle}$

and so

$g(\xi)=\max_{x\in\Pi}\min_{\theta\in\Theta(\xi)}\frac{\langle\theta,h(x)\rangle}{\langle\theta,\xi\rangle}.$

Then applying the minimax theorem (Theorem 48) gives

$g(\xi)=\min_{\theta\in\Theta(\xi)}\max_{x\in\Pi}\frac{\langle\theta,h(x)\rangle}{\langle\theta,\xi\rangle}=\min_{\theta\in\Theta(\xi)}\frac{f(\theta)}{\langle\theta,\xi\rangle}.$

This proves the claim. It also shows that $g(\xi)$ is continuous, since it is
the pointwise minimum of the continuous functions $\theta\mapsto
f(\theta)/\langle\theta,\xi\rangle$.

We will now prove
$f(\theta)=\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle$.

Plugging (1) into $\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle$ we get

$\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle=\max_{\xi\in\Xi}\max_{x\in\Pi}\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\langle\theta,\xi\rangle.$

Exchanging the maximizations gives

$\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle=\max_{x\in\Pi}\max_{\xi\in\Xi}\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\langle\theta,\xi\rangle.$

Note that for each $j$ such that $\xi_{j}\neq 0$ we have

$\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\xi_{j}\leq\frac{h_{j}(x)}{\xi_{j}}\xi_{j}=h_{j}(x),$

and if $\xi_{j}=0$, then the same inequality $\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\xi_{j}\leq h_{j}(x)$ holds because $h_{j}(x)$ is
nonnegative. Therefore

$\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\langle\theta,\xi\rangle=\sum_{j=1}^{k}\left(\theta_{j}\cdot\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\xi_{j}\right)\leq\sum_{j=1}^{k}\theta_{j}h_{j}(x)=\left<\theta,h(x)\right>.$

This upper bound is attained if $\xi$ is proportional to $h(x)$, that is,
$\xi=\frac{h(x)}{\max_{i\in[k]}h_{i}(x)}$, or arbitrary $\xi$ if $h(x)=0$.
Thus

$\max_{\xi\in\Xi}\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\langle\theta,\xi\rangle=\left<\theta,h(x)\right>$

and

$\max_{\xi\in\Xi}\,g(\xi)\langle\theta,\xi\rangle=\max_{x\in\Pi}\max_{\xi\in\Xi}\min_{i\in[k],\xi_{i}\neq
0}\frac{h_{i}(x)}{\xi_{i}}\langle\theta,\xi\rangle=\max_{x\in\Pi}\langle\theta,h(x)\rangle=f(\theta).$

We conclude that the claim holds. ∎

###  4.4 Preordered semirings and the minimax correspondence

We extend the ideas of the previous section to families of dual pairs that are
parametrized by elements of a preordered semiring. This is mostly a notational
extension. In our application, preordered semiring will be the set of tensors
multiplication given by tensor product, addition given by direct sum and
preoder given by restriction. We will then discuss how the parametrization
behaves under the minimax correspondence.

######  Definition 51.

Let $T\mapsto\Pi(T)\subseteq\mathbb{R}^{n}$ be a map that assigns a compact
convex set to every element $T$ from some index set that has defined on it a
product $\otimes$, a sum $\oplus$ and a preorder $\leq$.131313In our
applications this index set will be the set of $k$-tensors, with product given
by the tensor product, sum given by the direct sum and preorder given by
restriction. Let $h_{1},\ldots,h_{k}$ be functions
$\cup_{T}\Pi(T)\to\mathbb{R}_{\geq 0}$ that are continuous and quasiconcave,
as before. For every $\theta\in\Theta$ and $\xi\in\Xi$ we define the functions

$\displaystyle f_{\theta}(T)$
$\displaystyle=\max_{x\in\Pi(T)}\langle\theta,h(x)\rangle,$ $\displaystyle
g_{\xi}(T)$
$\displaystyle=\max_{x\in\Pi(T)}\min_{i\in[k]}\frac{h_{i}(x)}{\xi_{i}}.$

Note that compared to the notation for dual pairs, in our new notation the
parameters $\theta$ and $\xi$ have been moved into the subscript. For every
$T$ the functions $\xi\mapsto g_{\xi}(T)$ and $\theta\mapsto f_{\theta}(T)$
form a dual pair. We say that $(f_{\theta},g_{\xi})$ is a _dual pair family_.

Again our main example are the quantum functionals and the asymptotic weighted
slice ranks: they form a dual pair family.

If $(f_{\theta},g_{\xi})$ is a dual pair family, then by the minimax
correspondence (Theorem 50) we know that $f_{\theta}(T)$ is continuous in
$\theta$ and $g_{\xi}(T)$ is continuous in $\xi$ and $f_{\theta}(T)$ is convex
in $\theta$. Moreover, $f_{\theta}(T)$ and $g_{\xi}(T)$ are related via

$\displaystyle g_{\xi}(T)$
$\displaystyle=\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(T)}{\langle\theta,\xi\rangle},$
$\displaystyle f_{\theta}(T)$
$\displaystyle=\max_{\xi\in\Xi}\,g_{\xi}(T)\langle\theta,\xi\rangle.$

We define $F_{\theta}(T)=2^{f_{\theta}(T)}$ and $G_{\xi}(T)=2^{g_{\xi}(T)}$.
The minimax correspondence between $g_{\xi}(T)$ and $f_{\theta}(T)$ allows us
to transfer properties:

######  Theorem 52.

Let $S$ and $T$ be fixed elements from the index set.

  1. (i)

If $F_{\theta}(S\otimes T)\geq F_{\theta}(S)F_{\theta}(T)$ for all $\theta$,
then $G_{\xi}(S\otimes T)\geq G_{\xi}(S)G_{\xi}(T)$ for all $\xi$.

  2. (ii)

If $F_{\theta}(S\oplus T)\geq F_{\theta}(S)+F_{\theta}(T)$ for all $\theta$,
then $G_{\xi}(S\oplus T)\geq G_{\xi}(S)+G_{\xi}(T)$ for all $\xi$.

  3. (iii)

If $G_{\xi}(S\otimes T)\leq G_{\xi}(S)G_{\xi}(T)$ for all $\xi$, then
$F_{\theta}(S\otimes T)\leq F_{\theta}(S)F_{\theta}(T)$ for all $\theta$.

  4. (iv)

If $G_{\xi}(S\oplus T)\leq G_{\xi}(S)+G_{\xi}(T)$ for all $\xi$, then
$F_{\theta}(S\oplus T)\leq F_{\theta}(S)+F_{\theta}(T)$ for all $\theta$.

  5. (v)

$F_{\theta}(S)\geq F_{\theta}(T)$ for all $\theta$ if and only if
$G_{\xi}(S)\geq G_{\xi}(T)$ for all $\xi$.

  6. (vi)

$F_{\theta}(T)=\alpha$ for all $\theta$ if and only if $G_{\xi}(T)=\alpha$ for
all $\xi$.

###### Proof.

(i) We will show $G_{\xi}(S\otimes T)\geq G_{\xi}(S)G_{\xi}(T)$ by showing
that $g_{\xi}(T_{1}\otimes T_{2})\geq g_{\xi}(T_{1})+g_{\xi}(T_{2})$. We apply
the minimax correspondence and then the assumption $f_{\theta}(S\otimes T)\geq
f_{\theta}(S)+f_{\theta}(T)$ to get

$g_{\xi}(S\otimes T)=\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(S\otimes
T)}{\langle\theta,\xi\rangle}\geq\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(S)+f_{\theta}(T)}{\langle\theta,\xi\rangle}$

Separating the minimization over $\theta$ into two parts and then applying the
minimax correspondence again, we get

$\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(S)+f_{\theta}(T)}{\langle\theta,\xi\rangle}=\min_{\theta\in\Theta(\xi)}\left[\frac{f_{\theta}(S)}{\langle\theta,\xi\rangle}+\frac{f_{\theta}(T)}{\langle\theta,\xi\rangle}\right]\\\
\geq\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(S)}{\langle\theta,\xi\rangle}+\min_{\theta\in\Theta(\xi)}\frac{f_{\theta}(T)}{\langle\theta,\xi\rangle}=g_{\xi}(S)+g_{\xi}(T).$

This proves the claim.

(ii) For this proof we use the general fact that
$x+y=\max_{0\leq\varepsilon\leq
1}x^{\varepsilon}y^{1-\varepsilon}2^{h(\varepsilon)}$ for any nonnegative
reals $x,y$ [Str91, Eq. 2.13], where $h$ denotes the binary entropy as usual.
We know that $F_{\theta}(T_{1}\oplus T_{2})\geq
F_{\theta}(T_{1})+F_{\theta}(T_{2})$ and so

$\displaystyle G_{\xi}(T_{1}\oplus T_{2})$
$\displaystyle=\min_{\theta\in\Theta(\xi)}F_{\theta}(T_{1}\oplus
T_{2})^{\tfrac{1}{\langle\theta,\xi\rangle}}\geq\min_{\theta\in\Theta(\xi)}(F_{\theta}(T_{1})+F_{\theta}(T_{2}))^{\tfrac{1}{\langle\theta,\xi\rangle}}.$

From the aforementioned general fact it follows that

$\min_{\theta\in\Theta(\xi)}(F_{\theta}(T_{1})+F_{\theta}(T_{2}))^{\tfrac{1}{\langle\theta,\xi\rangle}}=\min_{\theta\in\Theta(\xi)}\max_{0\leq\varepsilon\leq
1}(F_{\theta}(T_{1})^{\varepsilon}F_{\theta}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}.$

Now we apply the von Neumann minimax theorem to get that

$\displaystyle\min_{\theta\in\Theta(\xi)}\max_{0\leq\varepsilon\leq
1}(F_{\theta}(T_{1})^{\varepsilon}F_{\theta}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}$
$\displaystyle=\max_{0\leq\varepsilon\leq
1}\min_{\theta\in\Theta(\xi)}(F_{\theta}(T_{1})^{\varepsilon}F_{\theta}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}.$

We pull the minimization over $\theta$ inside the product and then apply the
minimax correspondence to get

$\displaystyle\max_{0\leq\varepsilon\leq
1}\min_{\theta}(F_{\theta}(T_{1})^{\varepsilon}F_{\theta}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}$
$\displaystyle\geq\max_{0\leq\varepsilon\leq
1}\min_{\theta}(F_{\theta}(T_{1})^{\varepsilon})^{\tfrac{1}{\langle\theta,\xi\rangle}}\min_{\theta}(F_{\theta}(T_{2})^{1-\varepsilon})^{\tfrac{1}{\langle\theta,\xi\rangle}}\min_{\theta}(2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}$
$\displaystyle=\max_{0\leq\varepsilon\leq
1}G_{\xi}(T_{1})^{\varepsilon}G_{\xi}(T_{2})^{1-\varepsilon}\min_{\theta}(2^{h(\varepsilon)})^{\tfrac{1}{\langle\theta,\xi\rangle}}$
$\displaystyle\geq\max_{0\leq\varepsilon\leq
1}G_{\xi}(T_{1})^{\varepsilon}G_{\xi}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)}.$

Another application of $x+y=\max_{0\leq\varepsilon\leq
1}x^{\varepsilon}y^{1-\varepsilon}2^{h(\varepsilon)}$ gives

$\displaystyle\max_{0\leq\varepsilon\leq
1}G_{\xi}(T_{1})^{\varepsilon}G_{\xi}(T_{2})^{1-\varepsilon}2^{h(\varepsilon)}$
$\displaystyle=G_{\xi}(T_{1})+G_{\xi}(T_{2}).$

This proves the claim.

The proofs for the other claims of the theorem go similarly. ∎

##  5 Quantum functionals over arbitrary fields

The quantum functionals are a family of functions on complex tensors. We
discussed their special properties already in the introduction and will recall
them now. For $\theta\in\Theta$ and any complex tensor $T$ the quantum
functionals are defined by

$F_{\theta}(T)\coloneqq\max_{p\in\Pi(T)}2^{\langle\theta,H(p)\rangle}.$

Their key properties [CVZ18] are

  * •

Monotonicity under restriction: $S\leq T\Rightarrow F_{\theta}(S)\leq
F_{\theta}(T)$,

  * •

Additivity under direct sum: $F_{\theta}(S\oplus
T)=F_{\theta}(S)+F_{\theta}(T)$,

  * •

Multiplicativity under tensor product: $F_{\theta}(S\otimes
T)=F_{\theta}(S)\cdot F_{\theta}(T)$,

  * •

Normalization on unit tensors: $F_{\theta}(I_{n})=n$ for any $n\in\mathbb{N}$
where $I_{n}=\sum_{i=1}^{n}e_{i}\otimes e_{i}\otimes e_{i}$.

Functions with the above properties are called _universal spectral points_
[Str88]. Such universal spectral points are powerful tools for the study of,
for example, barriers for fast matrix multiplication [CGLZ20], and other
question about large tensor powers of tensors.

The above construction of universal spectral points does not allow immediate
extension to tensors over other fields than the complex numbers. The purpose
of this section is to construct a family of functionals on tensors over
arbitrary fields which we conjecture to be universal spectral points, and give
evidence for the validity of this conjecture.

The construction is as follows. Fix some field $K$ and let $T$ be a tensor
over $K$. Let $G_{\xi}(T)$ denote the asymptotic $\xi$-weighted slice rank of
$T$. We define the _quantum functionals over $K$_ by

$F^{K}_{\theta}(T)\coloneqq\max_{\xi\in\Xi}G_{\xi}(T)^{\langle\theta,\xi\rangle}.$

######  Proposition 53.

The $F^{K}_{\theta}$ are monotone and normalized.

###### Proof.

We know that the weighted slice rank $S_{\xi}$ is monotone and normalized
(Section 2.2). Therefore, the same holds for the asymptotic weighted slice
rank $G_{\xi}$, and in turn the same holds for the function $F_{\theta}^{K}$.
∎

######  Conjecture 54.

The $F^{K}_{\theta}$ are additive and multiplicative.

Proposition 53 and 54 (if true) combined say that the $F_{\theta}^{K}$ are
universal spectral points for tensors over $K$.

We discuss two pieces of evidence for 54 in the following subsections. The
first is that the conjecture is true over the complex numbers, since then
$F^{\mathbb{C}}_{\theta}$ coincides with the quantum functionals $F_{\theta}$
from [CVZ18]. The second is that the conjecture is true for the subclass of
tight tensors over arbitrary fields, since then $F^{K}_{\theta}$ coincides
with a family of tensor parameters called the support functionals [Str91].
Both proofs will use the minimax correspondence of Section 4.

Finally, in Section 5.3, as another application of the minimax correspondence,
we introduce and discuss the min-entropy quantum functionals.

###  5.1 Arbitrary tensors over the complex numbers

######  Theorem 55.

The quantum functionals and asymptotic slice ranks are related as follows.

$G_{\xi}(T)=\min_{\theta\in\Theta(\xi)}\,F_{\theta}(T)^{1/\langle\theta,\xi\rangle},\quad
F_{\theta}(T)=\max_{\xi\in\Xi}\,G_{\xi}(T)^{\langle\theta,\xi\rangle}.$

###### Proof.

We will use Theorem 50. Let $\Pi$ be the moment polytope of $T$. For $p\in\Pi$
let $h_{i}(p)=H(p_{i})$ be the Shannon entropy of the $i$th component of $p$.
Then $h_{i}(p)$ is a continuous and concave function. We see that $f(\theta)$
is the logarithmic quantum functional and by Theorem 32 we see that $g(\xi)$
is the asymptotic $\xi$-weighted slice rank. Theorem 50 proves the claim. ∎

In particular, Theorem 55 implies that 54 holds for $K=\mathbb{C}$.

Moreover, from Theorem 55 and the general Theorem 52 it follows that:

######  Corollary 56.

The asymptotic $\xi$-weighted slice rank $G_{\xi}$ on complex tensors is
monotone, normalized, super-multiplicative, and super-additive.

###  5.2 Tight tensors over arbitrary fields

Let $K$ be an arbitrary field. Recall that a tensor $T\in K^{m_{1}\times
m_{2}\times m_{3}}$ is called _tight_ in the standard bases if there exist
injective maps $u_{i}\colon[m_{i}]\to\mathbb{Z}$ such that the sum
$u_{1}(j_{1})+u_{2}(j_{2})+u_{3}(j_{3})$ is constant when we vary over all
$(j_{1},j_{2},j_{3})\in\operatorname{supp}(T)$. For tight tensors over an
arbitrary field the support functionals of [Str91] are defined as

$\zeta_{\theta}(T)=\max_{p\in W(T)}2^{\langle\theta,H(p)\rangle},$

where $W(T)$ is the polytope defined in Section 3.3. For tight tensors these
functionals are spectral points, that is, they are monotone, multiplicative,
additive and normalised [Str91]141414In fact, this is true for the slightly
larger class of _oblique_ tensors. We refer to [Str91] and [CVZ18] for
details..

######  Theorem 57.

Let $T$ be a tight tensor. The support functionals and asymptotic slice ranks
of $T$ are related as follows:

$G_{\xi}(T)=\min_{\theta\in\Theta(\xi)}\,\zeta_{\theta}(T)^{\langle\theta,\xi\rangle},\quad\zeta_{\theta}(T)=\max_{\xi\in\Xi}\,G_{\xi}(T)^{\langle\theta,\xi\rangle}.$

###### Proof.

The characterization of the asymptotic slice ranks of tight tensors proven in
Theorem 45 shows that $(\log\zeta_{\theta},g_{\xi})$ form a dual pair family.
The claim follows from the minimax correspondence (Theorem 50). ∎

In particular, Theorem 57 implies that 54 holds for all tight tensors over an
arbitrary field $K$.

Moreover, from Theorem 57 and the general Theorem 52 it follows that:

######  Corollary 58.

The asymptotic $\xi$-weighted slice rank $G_{\xi}$ on tight tensors over
arbitrary fields is monotone, normalized, super-multiplicative, and super-
additive.

###  5.3 Min-entropy quantum functionals

Let $F_{\theta,\infty}$ be the min-entropy quantum functionals as defined in
Section 1.5.

######  Theorem 59.

The min-entropy quantum functionals are monotone, normalized and super-
multiplicative.

###### Proof.

Monotonicity follows from the fact that if $S\leq T$, then
$\Pi(S)\subseteq\Pi(T)$. For the normalization, for every $r\in\mathbb{N}$ it
is clear that $F_{\theta,\infty}(I_{r})\leq F_{\theta}(I_{r})=r$. On the other
hand, it follows directly from the spectral definition of the moment polytope
that the flattenings of $I_{r}$ in the standard basis have uniform singular
values and so $F_{\theta,\infty}(I_{r})\geq r$. Super-multiplicativity follows
from the characterization of the min-entropy quantum functionals as

$F_{\theta,\infty}(T)=\sup_{S\in G\cdot
T}\prod_{i=1}^{3}\Biggl{(}\frac{\mathinner{\lVert
S_{(i)}\rVert}_{F}^{2}}{\mathinner{\lVert
S_{(i)}\rVert}_{2}^{2}}\Biggr{)}^{\\!\\!\theta_{i}},$

and the fact that the Frobenius norm and spectral norm are multiplicative
under the Kronecker product. ∎

Since the min-entropy $H_{\infty}$ is continuous and concave, we find via the
minimax correspondence (Theorem 50) that:

######  Theorem 60.

For every $\xi\in\Xi$ and every tensor $T$ we have

$G_{\xi,\infty}(T)=\min_{\theta\in\Theta(\xi)}{F_{\theta,\infty}(T)}^{1/\langle\theta,\xi\rangle}.$

For every $\theta\in\Theta$ and every tensor $T$ we have

$F_{\theta,\infty}(T)=\max_{\xi\in\Xi}G_{\xi,\infty}(T)^{\langle\theta,\xi\rangle}.$

Again, via the general Theorem 52 we find that:

######  Corollary 61.

For every $\xi\in\Xi$ the function $G_{\xi,\infty}$ is monotone, normalized
and super-multiplicative.

The function $G_{\xi,\infty}(T)$ is similar to the G-stable rank that was
introduced in [Der20]. However, our weighting is slightly different from the
weighting using in the definition of G-stable rank.

##  Appendix A Invariance, polystability and semistability

We will prove the semistability test given in Lemma 23. This lemma is
essentially an adaptation of a result proven by Kempf [Kem78] to the tensor
action of the group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$.

The result of Kempf actually deals with a notion that is stronger than
semistability, called polystability. Given a representation $V$ of a reductive
group $G$, any nonzero element $v\in V$ is called _polystable_ if the orbit
$G\cdot v$ is Zariski closed. Clearly, polystable implies semistable. When
working in projective space, we call an element $[v]\in\mathbb{P}V$ polystable
if and only if $v\in V$ is polystable.

Kempf proved the following result:

######  Theorem 62 (Kempf’s polystability test [Kem78, Cor. 5.1]).

Let $V$ be a representation of a reductive group $G$. Then $G$ acts on the
projective space $\mathbb{P}V$ of lines in $V$. Assume that $G$ has no
nontrivial central $1$-parameter subgroup. If the stabilizer of an element
$y\in\mathbb{P}V$ in $G$ is not contained in any proper parabolic subgroup of
$G$, then $y$ is polystable under $G$.

We now apply Kempf’s theorem to tensors. Let $V=V_{1}\otimes V_{2}\otimes
V_{3}$ and let the (reductive) group
$G=\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ act in
the usual way on $V$. We call a nonzero tensor $T\in V_{1}\otimes V_{2}\otimes
V_{3}$ _polystable_ if the orbit $G\cdot T$ is closed for the group
$G=\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$. Since
polystability implies semistability, the following lemma implies Lemma 23.

######  Lemma 63.

Let $\Gamma$ be a group and let $V_{1},V_{2},V_{3}$ be irreducible
representations of $\Gamma$. Let $\Gamma$ act diagonally on the tensor product
$V_{1}\otimes V_{2}\otimes V_{3}$. Every nonzero tensor $T\in V_{1}\otimes
V_{2}\otimes V_{3}$ invariant under $\Gamma$ is polystable.

###### Proof.

The center of $\mathrm{SL}_{n}$ is the finite subgroup
$\\{\alpha\,\mathrm{id}\mid\alpha\in K,\,\alpha^{n}=1\\}$. Therefore, the
center of
$G=\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ is also
a finite group and thus does not contain a nontrivial $1$-parameter subgroup.

For any vector space $V$, the parabolic subgroups of $\mathrm{SL}(V)$
correspond to flags in $V$, in the following way. If $\\{0\\}=U_{0}\subset
U_{1}\subset\dots\subset U_{k}=V$ is a flag, then the corresponding parabolic
subgroup consists of all maps in $\mathrm{SL}(V)$ that preserve the flag (that
is, leave each space $U_{i}$ invariant). Each proper parabolic subgroup is
therefore contained in the subgroup $P_{U}$ which leaves some nonzero proper
subspace $U$ invariant (these are maximal proper parabolic subgroups). The
parabolic subgroups of the product group
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ are
precisely the products of parabolic subgroups of the $\mathrm{SL}(V_{i})$.
Therefore, each proper parabolic subgroup of
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$ leaves
invariant a nonzero proper subspace $U_{i}\subset V_{i}$ in one of the three
factors.

Each element $g\in\Gamma$ acts on $V_{i}$ via some linear map
$\rho_{i}(g)\in\mathrm{GL}(V_{i})$. We can scale these linear maps to obtain
elements $\hat{\rho}_{i}(g)\coloneqq(\det\rho_{i}(g))^{-1/{\dim
V_{i}}}\,\rho_{i}(g)\in\mathrm{SL}(V_{i})$. Note that for every group element
$g\in\Gamma$ the corresponding triple
$(\hat{\rho}_{1}(g),\hat{\rho}_{2}(g),\hat{\rho}_{3}(g))\in\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$
leaves the line $[T]\in\mathbb{P}(V_{1}\otimes V_{2}\otimes V_{3})$ invariant,
because $T$ is invariant under $\Gamma$.

Since the representations $V_{1}$, $V_{2}$ and $V_{3}$ are irreducible, the
only subspaces $U_{i}\subseteq V_{i}$ that are invariant under $\rho_{i}(g)$
for all $g\in\Gamma$ are $V_{i}$ itself and the zero subspace. This means that
there is no proper parabolic subgroup of
$\mathrm{SL}(V_{1})\times\mathrm{SL}(V_{2})\times\mathrm{SL}(V_{3})$
containing $\\{(\hat{\rho}_{1}(g),\hat{\rho}_{2}(g),\hat{\rho}_{3}(g))\mid
g\in\Gamma\\}$. Using Kempf’s polystability test (Theorem 62) we conclude that
$T$ is polystable. ∎

#### Acknowledgements

MC and VL acknowledge financial support from the European Research Council
(ERC Grant Agreement No. 818761) and VILLUM FONDEN via the QMATH Centre of
Excellence (Grant No. 10059). JZ thanks Visu Makam, Michael Walter, Yinan Li
and Harold Nieuwboer for helpful discussions. The authors thank anonymous
referees for helpful comments.

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