# Total closure for permutation actions of
finite nonabelian simple groups

Saul D. Freedman  School of Mathematics and Statistics, University of St
Andrews  Michael Giudici  Centre for the Mathematics of Symmetry and
Computation, Department of Mathematics and Statistics, The University of
Western Australia  Cheryl E. Praeger  Centre for the Mathematics of Symmetry
and Computation, Department of Mathematics and Statistics, The University of
Western Australia

###### Abstract

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if
for each set $\Omega$ upon which $G$ acts faithfully, $G$ is the largest
subgroup of ${\rm Sym}(\Omega)$ that leaves invariant each of the $G$-orbits
in the induced action on $\Omega\times\cdots\times\Omega=\Omega^{k}$. Each
finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer
$k$ such that $G$ is totally $k$-closed. We address the question of
determining the closure number $k(G)$ for finite simple groups $G$. Prior to
our work it was known that $k(G)=2$ for cyclic groups of prime order and for
precisely six of the sporadic simple groups, and that $k(G)\geqslant 3$ for
all other finite simple groups. We determine the value for the alternating
groups, namely $k(A_{n})=n-1$. In addition, for all simple groups $G$, other
than alternating groups and classical groups, we show that $k(G)\leqslant 7$.
Finally, if $G$ is a finite simple classical group with natural module of
dimension $n$, we show that $k(G)\leqslant n+2$ if $n\geqslant 14$, and
$k(G)\leqslant\lfloor n/3+12\rfloor$ otherwise, with smaller bounds achieved
by certain families of groups. This is achieved by determining a uniform upper
bound (depending on $n$ and the type of $G$) on the base sizes of the
primitive actions of $G$, based on known bounds for specific actions. We pose
several open problems aimed at completing the determination of the closure
numbers for finite simple groups.

Keywords: k-closed permutation groups; primitive groups; base size; simple
groups

2020 Mathematics Subject Classification: 20B05, 20E32, 20B15

##  1 Introduction

The notion of $k$-closure of a permutation group $G$ on a set $\Omega$ was
introduced in 1969 by Wielandt [29, Definition 5.3]: namely the _$k$ -closure_
$G^{(k),\Omega}$ of $G$ is the set of all $g\in{\rm Sym}(\Omega)$
(permutations of $\Omega$) such that $g$ leaves invariant each $G$-orbit in
the induced $G$-action on ordered $k$-tuples from $\Omega$. The $k$-closure
$G^{(k),\Omega}$ is a subgroup of ${\rm Sym}(\Omega)$ containing $G$ [29,
Theorem 5.4], and a permutation group $G$ is said to be _$k$ -closed_ if
$G^{(k),\Omega}=G$. Determining when $G$ is $k$-closed can be important when
studying the _relational complexity_ of $G$. This is the smallest positive
integer $r$ such that, whenever $A$ and $B$ are ordered $n$-tuples from
$\Omega$ with $n\geqslant r$, and each $r$-subtuple of $A$ lies in the same
$G$-orbit as the corresponding $r$-subtuple of $B$, it is also the case that
$A\in B^{G}$. The relational complexity of $G$ is very difficult to calculate
in general, but it is bounded below by the smallest integer $k$ such that $G$
is $k$-closed, as shown in [15, Lemmas 1.6.5]; see that monograph for an
excellent introduction to relational complexity.

The $k$-closures for different faithful permutation representations of the
same group $G$ may be quite different, even for the symmetric group $S_{3}$
(see [12, p. 240]). We are interested in the stronger concept of _total
$k$-closure_, which is independent of the permutation representation of the
group. A group $G$ is said to be _totally $k$-closed_ if for each set $\Omega$
upon which $G$ acts faithfully we have $G=G^{(k),\Omega}$. The concept was
suggested in 2016 by Holt [30], and first studied by Abdollahi and Arezoomand
[1] when $k=2$.

For a permutation group $G\leqslant{\rm Sym}(\Omega)$, and $k\geqslant 2$,
Wielandt [29, Theorem 5.8] proved that

$G\leqslant G^{(k),\Omega}\leqslant G^{(k-1),\Omega},$ (1)

and in addition, see [29, Theorem 5.12], that if $G$ is finite then
$G=G^{(k),\Omega}$ for $k=|G|$. Thus if a finite group $G$ is totally
$(k-1)$-closed then it is automatically totally $k$-closed, and also $G$ is
totally $k$-closed for sufficiently large $k$. We define the _closure number_
$k(G)$ of a finite group $G$ as the least positive integer $k$ such that $G$
is totally $k$-closed.

In this paper we study the closure numbers for finite simple groups. In
previous work it has been shown that $k(G)=2$ if $G$ is a cyclic group of
prime order [12, Theorem 1.2], and for exactly six nonabelian simple groups
[2, Theorem 1.1], namely the Janko groups $J_{1},J_{3}$ and $J_{4}$, together
with $Ly,Th$ and the Monster $M$. By (1), $k(G)\geqslant 3$ for all other
finite simple groups. The problem of determining the closure number $k(G)$ for
the remaining simple groups was posed in [2, Question 3], and in particular
for determining all those with closure number $3$ (if any exist), see also
[12, Problem 3] and the Kourovka Notebook [18, New Problems 20.2 and 20.3].

A crucial tool for this work uses the notion of a _base_ for a permutation
group $G\leqslant{\rm Sym}(\Omega)$, that is to say, a set of points
$\alpha_{1},\dots,\alpha_{b}\in\Omega$ such that the only element of $G$
fixing each $\alpha_{i}$ is the identity. If $G\leqslant{\rm Sym}(\Omega)$ and
$G$ has a base of size $b$, then Wielandt [29, Theorem 5.12] proved that
$G=G^{(b+1),\Omega}$. To see how this is used, let us consider the finite
alternating groups $G=A_{n}$, $n\geqslant 5$. For the natural action of $G$ on
$\Omega=\\{1,\dots,n\\}$, and for $k\leqslant n-2$, the groups $G$ and ${\rm
Sym}(\Omega)$ have exactly the same orbits on $\Omega^{k}$, and hence
$G^{(k),\Omega}={\rm Sym}(\Omega)\neq G$, so by definition the closure number
satisfies $k(G)\geqslant n-1$. On the other hand any $(n-2)$-subset of
$\Omega$ is a base for $G$ and hence $G^{(n-1),\Omega}=G$ by [29, Theorem
5.12]. We prove in Lemma 3.8 that, in fact, in every faithful permutation
representation of $G$, the minimum size of a base is at most $n-2$, and hence
by [29, Theorem 5.12], the action is $(n-1)$-closed. Thus the closure number
is determined.

######  Theorem 1.1.

For $n\geqslant 5$, the closure number $k(A_{n})=n-1$.

On the other hand we exploit known upper bounds on the base sizes of primitive
permutation representations of the remaining simple groups in order to prove
the following upper bounds for their closure numbers.

######  Theorem 1.2.

Suppose that $G$ is a finite nonabelian simple group that is not isomorphic to
an alternating group. Then the following hold.

  1. (a)

If $G$ is a sporadic group, then $k(G)\leqslant 6$.

  2. (b)

If $G$ is an exceptional group of Lie type, then $k(G)\leqslant 7$.

  3. (c)

Suppose that $G$ is a classical group, with natural module of dimension $n$.
Then either $G$ and an upper bound for $k(G)$ are given in Table 1, or
$k(G)\leqslant\lfloor n/3+12\rfloor$. In particular, if $n\geqslant 14$ in any
case, then $k(G)\leqslant n+2$.

Table 1: Upper bounds for $k(G)$, for finite simple classical groups $G$ that satisfy the specified conditions, and are not isomorphic to alternating groups. Here, $\delta$ is the Kronecker delta. $G$ | Upper bound for $k(G)$ | Conditions
---|---|---
${\rm PSL}(n,q)$ | $n+2-\delta_{2q}$ | Always
${\rm PSU}(n,q)$ | $n+1$ |  $n\in\\{3,5\\}$ or $n\geqslant 16$
${\rm PSp}(n,q)$ | $n+1$ | $n\geqslant 14$
${\rm P}\Omega^{\varepsilon}(n,q)$ | $n$ |  $n\geqslant 16$, with $q$ even if $n=16$

We show in Lemma 5.4 that $k(M_{24})=6$. Thus Theorem 1.2(a) and [2, Theorem
1.1] leave 19 sporadic groups $G$ for which $3\leqslant k(G)\leqslant 6$ and
where we do not know the exact value.

######  Question 1.

Determine $k(G)$ for these remaining $19$ sporadic simple groups.

The bound $k(G)\leqslant 7$ for exceptional groups of Lie type is derived (see
Corollary 3.3) by using the fact that the minimum base size of primitive
actions of such groups is at most 6. By [6], equality only holds for certain
actions of $E_{6}(q)$ and $E_{7}(q)$, and so these are the only possibilities
to have $k(G)=7$. However, we do not know if this bound is met.

######  Question 2.

Determine a tight upper bound for $k(G)$ for an exceptional group $G$ of Lie
type.

In many of the low-dimensional classical cases not specified here, our work in
fact yields a smaller upper bound than the general bound of $\lfloor
n/3+12\rfloor$; see the comment following the proof of Theorem 1.2(c), before
Corollary 4.4. For simple $n$-dimensional classical groups we do not know
whether the closure numbers grow linearly with $n$ or are in fact
asymptotically smaller.

######  Question 3.

For a simple $n$-dimensional classical group $G$, is $k(G)=o(n)$?

In particular, deciding whether or not the closure numbers of finite simple
classical groups are bounded or not, as the dimension of their natural module
increases, would answer the open question [18, 20.3] in the Kourovka Notebook.

For a group $G$ we define $k_{\mathrm{trans}}(G)$ to be the smallest integer
$k$ such that $G^{(k),\Omega}=G$ for each set $\Omega$ upon which $G$ acts
faithfully and transitively. Similarly, we define $k_{\mathrm{prim}}(G)$ to be
the smallest integer $k$ such that $G^{(k),\Omega}=G$ for each set $\Omega$
upon which $G$ acts faithfully and primitively. Clearly,

$k(G)\geqslant k_{\mathrm{trans}}(G)\geqslant k_{\mathrm{prim}}(G).$

We note that we can use the results of [22] and [26] to give absolute upper
bounds on $k_{\mathrm{prim}}(G)$. In particular $k_{\mathrm{prim}}(G)\leqslant
6$ for all complete simple groups $G$.

### Acknowledgements

The research for this paper began at the 2022 Research Retreat of the Centre
for the Mathematics of Symmetry and Computation of the University of Western
Australia. The authors acknowledge discussions with their colleagues Vishnuram
Arumugam and Alice Devillers. The first author was supported by a St Leonard’s
International Doctoral Fees Scholarship and a School of Mathematics &
Statistics PhD Funding Scholarship at the University of St Andrews. The
research of the second and third authors was supported by ARC Discovery
Project Grants DP190101024 and DP190100450, respectively.

##  2 Preliminary results about $k$-closures

We begin by noting the following result of Wielandt which we will often use
without reference.

######  Lemma 2.1.

[29, Theorem 5.6] Let $G$ act on a set $\Omega$ and $k\geqslant 1$ be an
integer. Then $h\in G^{(k),\Omega}$ if and only if for all
$\alpha_{1},\ldots,\alpha_{k}\in\Omega$ there exists $g\in G$ such that
$(\alpha_{1},\ldots,\alpha_{k})^{g}=(\alpha_{1},\ldots,\alpha_{k})^{h}$.

Note that, because of our focus on the set of all faithful permutation
representations of a given group $G$, we never write $G^{(k)}$ for a
$k$-closure without giving also the set acted upon. So, for example, if $G$
acts on a set $\Omega$ and $\Delta$ is a $G$-invariant subset of $\Omega$,
then $(G^{(k)})^{\Delta}$ has no meaning in our work. We can write
$G^{(k),\Delta}$ for the $k$-closure of the action of $G$ on $\Delta$, and
$(G^{(k),\Omega})^{\Delta}$ for the permutation group induced on $\Delta$ by
the $k$-closure of the action of $G$ on $\Omega$.

We also note the following result.

######  Lemma 2.2.

Let $G$ act on a set $\Omega$ and let $\Delta$ be a $G$-invariant subset of
$\Omega$. Then $(G^{(k),\Omega})^{\Delta}\leqslant G^{(k),\Delta}$.

###### Proof.

[29, Exercise 5.25]. ∎

Next we prove a result about the $k$-closure of an intransitive action of a
nonabelian simple group.

######  Lemma 2.3.

Let $G$ be a nonabelian simple group acting on a set $\Omega$ with orbits
$\Delta_{1},\ldots,\Delta_{t}$ with $t\geqslant 2$. Suppose that for each
$i,j\in\\{1,\ldots,t\\}$ there is a point $\beta\in\Delta_{i}$ such that
$G_{\beta}$ is not transitive on $\Delta_{j}$, and also that
$G^{(k),\Delta_{i}}=G^{\Delta_{i}}\cong G$ for each $i=1,\ldots,t$ and some
$k\geqslant 2$. Then $G^{(k),\Omega}=G^{\Omega}\cong G$.

###### Proof.

For each $i=1,\ldots,t$, $G$ acts faithfully on $\Delta_{i}$ since
$G^{(k),\Delta_{i}}\cong G$. Also, by Lemma 2.2,
$G^{(k),\Omega}\leqslant\prod_{i=1}^{t}(G^{(k),\Omega})^{\Delta_{i}}\leqslant\prod_{i=1}^{t}G^{(k),\Delta_{i}}\cong
G^{t}$. It follows that $G^{(k),\Omega}$ is a subdirect subgroup of
$G^{(k),\Delta_{1}}\times\cdots\times G^{(k),\Delta_{t}}\cong G^{t}$, and so,
by a lemma of Scott ([28, p. 328] or see [27, Theorem 4.16]), the group
$G^{(k),\Omega}$ is equal to a direct product of $\ell$ diagonal subgroups
$H_{j}\cong G$ for $j=1,\ldots,\ell$ (where $\ell\geqslant 1$), with pairwise
disjoint supports, that is to say, there is a partition
$\\{\mathcal{O}_{1},\ldots,\mathcal{O}_{\ell}\\}$ of
$\\{\Delta_{1},\ldots,\Delta_{t}\\}$ such that, for each $j$,
$(H_{j})^{\Delta}\cong G$ for all $\Delta\in\mathcal{O}_{j}$ and
$(H_{j})^{\Delta}=1$ for all $\Delta\notin\mathcal{O}_{j}$. In particular,
$G^{(k),\Omega}\cong G^{\ell}$. Suppose that $\ell>1$. Choose
$\Delta_{i}\in\mathcal{O}_{1}$ and $\Delta_{j}\in\mathcal{O}_{2}$. Then the
kernel $(G^{(k),\Omega})_{(\Delta_{i})}$ of the action on $\Delta_{i}$
contains the diagonal subgroup $H_{2}$, which induces $G$ on $\Delta_{j}$. By
assumption, there exists $\beta\in\Delta_{i}$ such that $G_{\beta}$ is not
transitive on $\Delta_{j}$. Choose $\alpha\in\Delta_{j}$ and consider the
$k$-tuple $(\alpha,\beta,\beta,\ldots,\beta)$ of points. By definition of the
$k$-closure, for each $h\in H_{2}<G^{(k),\Omega}$, there exists $g\in G$ such
that
$(\alpha,\beta,\beta,\ldots,\beta)^{g}=(\alpha,\beta,\ldots,\beta)^{h}=(\alpha^{h},\beta,\beta,\ldots,\beta)$.
In particular $\alpha^{g}=\alpha^{h}$ and $g\in G_{\beta}$. Since $H_{2}$ is
transitive on $\Delta_{j}$ it follows that $G_{\beta}$ is transitive on
$\Delta_{j}$, which is a contradiction. Thus $\ell=1$, and so
$G^{(k),\Omega}=G^{\Omega}\cong G$. ∎

######  Corollary 2.4.

Let $G$ be a nonabelian simple group acting on a set $\Omega$ with orbits
$\Delta_{1},\ldots,\Delta_{t}$ with $t\geqslant 2$ such that the actions of
$G$ on each orbit are pairwise equivalent. If, for some $k$,
$G^{(k),\Delta_{i}}=G^{\Delta_{i}}\cong G$, then $k\geqslant 2$ and
$G^{(k),\Omega}=G^{\Omega}\cong G$.

###### Proof.

Since the actions of $G$ are pairwise equivalent, if $\beta\in\Delta_{i}$ then
$G_{\beta}$ fixes a point in each orbit and so is not transitive on any
$\Delta_{j}$. Suppose that $G^{(k),\Delta_{i}}=G^{\Delta_{i}}\cong G$, for
some $k$. Then $k\geqslant 2$, since $G^{(1),\Delta_{i}}={\rm
Sym}(\Delta_{i})>G^{\Delta_{i}}$ as $G^{\Delta_{i}}\cong G$ is a nonabelian
simple group. The assertion now follows from Lemma 2.3. ∎

Next we need some results about the $k$-closure of an imprimitive group.

######  Lemma 2.5.

Let $G$ be a group with a transitive action on a set $\Omega$, and suppose
that $\mathcal{B}$ is a nontrivial $G$-invariant partition of $\Omega$. Let
$k$ be an integer such that $2\leqslant k\leqslant|\mathcal{B}|$ and let
$B\in\mathcal{B}$. Then the following hold:

  1. (a)

$\mathcal{B}$ is a $G^{(k),\Omega}$-invariant partition.

  2. (b)

$(G^{(k),\Omega})^{\mathcal{B}}\leqslant G^{(k),\mathcal{B}}$.

  3. (c)

$(G^{(k),\Omega})_{B}^{B}\leqslant(G_{B})^{(k),B}$.

  4. (d)

If there exist $B_{1},\ldots,B_{k}\in\mathcal{B}$ such that
$G_{B_{1},\ldots,B_{k}}=1$, then $G^{(k),\Omega}$ acts faithfully on
$\mathcal{B}$.

###### Proof.

Let $g\in G^{(k),\Omega}$. Let $B\in\mathcal{B}$ and choose $\alpha,\beta\in
B$ with $\alpha\neq\beta$, and consider the $k$-tuple
$(\alpha,\beta,\beta,\ldots,\beta)$. Since $g\in G^{(k),\Omega}$, there exists
$h\in G$ such that
$(\alpha,\beta,\beta,\ldots,\beta)^{h}=(\alpha,\beta,\beta,\ldots,\beta)^{g}$.
Then since $\mathcal{B}$ is a $G$-invariant partition of $\Omega$, it follows
that $\alpha^{g}$ and $\beta^{g}$ lie in the same part of $\mathcal{B}$. Hence
$\mathcal{B}$ is also invariant under $G^{(k),\Omega}$ and part (a) holds

Now consider $g^{\mathcal{B}}$. Let $B_{1},\ldots,B_{k}\in\mathcal{B}$ and
choose $\beta_{i}\in B_{i}$, for $i=1,\dots,k$. Then there exists $h\in G$
such that $(\beta_{1},\ldots,\beta_{k})^{g}=(\beta_{1},\ldots,\beta_{k})^{h}$.
Thus
$(B_{1},\ldots,B_{k})^{g^{\mathcal{B}}}=(B_{1},\ldots,B_{k})^{h^{\mathcal{B}}}$
and so $g^{\mathcal{B}}\in G^{(k),\mathcal{B}}$, as required for part (b).

Next, let $h\in(G^{(k),\Omega})_{B}$, so that
$h^{B}\in(G^{(k),\Omega})_{B}^{B}$. To show that $h^{B}\in(G_{B})^{(k),B}$ we
prove the criteria of Lemma 2.1, namely for all $k$-tuples
$(\alpha_{1},\dots,\alpha_{k})\in B^{k}$, we find an element $x\in(G_{B})^{B}$
such that
$(\alpha_{1},\dots,\alpha_{k})^{h^{B}}=(\alpha_{1},\dots,\alpha_{k})^{x}$.
Since $h\in(G^{(k),\Omega})_{B}$, there exists $y\in G$ such that
$(\alpha_{1},\dots,\alpha_{k})^{h}=(\alpha_{1},\dots,\alpha_{k})^{y}$.
Further, since $h$ fixes $B$ setwise it follows that $y\in G_{B}$. Then taking
$x:=y^{B}$ we have the required equality for (c).

Finally, suppose that there exist $B_{1},\ldots,B_{k}\in\mathcal{B}$ such that
$G_{B_{1},\ldots,B_{k}}=1$. Let $h$ lie in the kernel of the action of
$G^{(k),\Omega}$ on $\mathcal{B}$, and choose $\alpha\in\Omega$. Since $G$
acts transitively on $\Omega$ there exists $g\in G$ such that
$\alpha^{g^{-1}}\in B_{1}$. For each $i=2,3,\ldots,k$ choose $\beta_{i}\in
B_{i}^{g}$. Since $h\in G^{(k),\Omega}$, there exists $x\in G$ such that
$(\alpha,\beta_{2},\ldots,\beta_{k})^{h}=(\alpha,\beta_{2},\ldots,\beta_{k})^{x}$.
By its definition, $h$ fixes each $B_{i}^{g}$ setwise, and so
$\beta_{i}^{x}\in B_{i}^{g}$ for all $i$ and $\alpha^{x}\in B_{1}^{g}$. Thus
$x\in
G_{B_{1}^{g},B_{2}^{g},\ldots,B_{k}^{g}}=(G_{B_{1},\ldots,B_{k}})^{g}=1$.
Hence $\alpha^{h}=\alpha^{x}=\alpha$. Since $\alpha$ was arbitrary and
$G^{(k),\Omega}\leqslant{\rm Sym}(\Omega)$, it follows that $h=1$ and so
$G^{(k),\Omega}$ acts faithfully on $\mathcal{B}$ as stated in part (d). ∎

##  3 Links between bases and closures

For a finite group $G$ acting on a set $\Omega$, let $b(G^{\Omega})$ denote
the minimum size of a base for the permutation group $G^{\Omega}\leqslant{\rm
Sym}(\Omega)$ induced by $G$ on $\Omega$; $b(G^{\Omega})$ is called the _base
size_ of $G^{\Omega}$. When $G$ is transitive, a _maximal block system_ for
$G$ on $\Omega$ is a $G$-invariant partition $\Sigma$ of $\Omega$ such that
$|\Sigma|>1$ and the induced action $G^{\Sigma}$ is primitive. In other words,
the only $G$-invariant partition strictly coarser than $\Sigma$ (that is, with
each part a union of more than one part of $\Sigma$) is the universal
partition $\\{\Omega\\}$ with just one part. Note that $\Sigma$ is allowed to
be the partition into singletons.

######  Lemma 3.1.

Let $G$ be a finite nonabelian simple group, and suppose that $G$ has a
faithful permutation representation on a set $\Omega$. Let $\Omega_{0}$ be a
$G$-orbit on $\Omega$ with $|\Omega_{0}|>1$, and let $\Sigma$ be a maximal
block system for $G$ on $\Omega_{0}$. Then the base size $b(G^{\Omega})$
satisfies $b(G^{\Omega})\leqslant b(G^{\Sigma})$.

###### Proof.

Since $|\Sigma|>1$ and $G$ is simple it follows that $G$ acts faithfully on
$\Sigma$. Let $b=b(G^{\Sigma})$. Then $G$ has a base
$\\{\sigma_{1},\dots,\sigma_{b}\\}$ in $\Sigma$, that is
$G_{\sigma_{1},\dots,\sigma_{b}}=1$. Let $\alpha_{i}\in\sigma_{i}$ for each
$i$. Then $G_{\alpha_{1},\dots,\alpha_{b}}$ fixes each of the $\sigma_{i}$
setwise, and hence $G_{\alpha_{1},\dots,\alpha_{b}}\leqslant
G_{\sigma_{1},\dots,\sigma_{b}}=1$. It follows that $b(G^{\Omega})\leqslant
b$. ∎

We may restate Wielandt’s result [29, Theorem 5.12] as follows.

######  Lemma 3.2.

If $G$ is a finite group, and $G$ has a permutation representation on a set
$\Omega$, such that $b(G^{\Omega})=b$, then $G^{(b+1),\Omega}=G$.

This has the following useful consequence.

######  Corollary 3.3.

Let $G$ be a finite nonabelian simple group, and let $b_{maxprim}(G)$ denote
the maximum of $b(G^{\Omega})$ over each faithful primitive permutation
representation of $G$ on a set $\Omega$. Then the closure number $k(G)$ is at
most $b_{maxprim}(G)+1$.

###### Proof.

Let $G$ be faithfully represented on a set $\Omega$. Then, by Lemma 3.1,
$b(G^{\Omega})$ is at most the base size of some primitive representation of
$G$, and hence $b(G^{\Omega})\leqslant b_{maxprim}(G)=b$, say. It follows from
Lemma 3.2 that $G^{(b+1),\Omega}=G$. Hence, by definition of the closure
number, $k(G)\leqslant b+1$. ∎

In the light of Corollary 3.3, we next summarise some recent results on base
sizes of finite primitive permutation groups that are nonabelian simple
groups. For several families of such actions the base size is much larger than
all others; these are called standard actions and are defined as follows.

######  Definition 3.4.

Let $G$ be a finite nonabelian simple group. A primitive action of $G$ on a
set $\Omega$ is _standard_ if, up to equivalence of actions, one of (a) or (b)
holds for $G^{\Omega}$, and the action is non-standard otherwise.

  1. (a)

$G=A_{n}$ for some $n\geqslant 5$, and $\Omega$ is an orbit of subsets or
partitions of $\\{1,\dots,n\\}$; or

  2. (b)

$G$ is a classical simple group with natural module $V$ over a field of
characteristic $p$, so $G={\rm PSL}(V),{\rm PSp}(V),{\rm PSU}(V)$ or ${\rm
P}\Omega^{\varepsilon}(V)$, and $G$ has a _subspace action_ on $\Omega$, that
is to say, one of the following holds:

    1. (i)

for some positive integer $k\leqslant\dim(V)/2$, $\Omega$ is a $G$-orbit of
$k$-dimensional totally singular, nondegenerate, or (if G is orthogonal, $p=2$
and $k=1$) nonsingular subspaces of $V$; or

    2. (ii)

$G={\rm PSp}(V)$, $p=2$, and $\Omega$ is the set of nondegenerate quadratic
forms on $V$ of fixed type $+$ or $-$ that polarise to the symplectic form
preserved by $G$.

If $G={\rm PSL}(V)$ in case (b)(i) above, then $\Omega$ consists of all
$k$-dimensional subspaces of $V$. Note also that, as the group $G$ is simple,
the definition above is slightly simpler than that for a standard action of an
almost simple group as given by Cameron and Kantor111In some cases (e.g., [5,
Definition 2.1]), subspace actions are defined even more generally, allowing
for certain additional actions of (non-simple) almost simple groups with socle
${\rm PSp}(4,q)^{\prime}$ ($q$ even) or ${\rm P}\Omega^{+}(8,q)$; see the
proof of [24, Proposition 8].in [11], where they conjectured that for each
non-standard primitive action of an almost simple group $G$ on a finite set
$\Omega$, the base size $b(G^{\Omega})$ is bounded above by an absolute
constant $c$. In [23, Theorem 1.3], Liebeck and Shalev proved this conjecture,
but did not specify the constant $c$. In the case of a non-standard primitive
group $G$ with socle $A_{n}$, a sketch of a proof was given in [11,
Proposition 2.3] that $c=2$ should usually suffice. A detailed proof in [8,
Theorem 1.1 and Corollary 1.2] showed that this is indeed true for all
$n\geqslant 13$, and all possibilities for non-standard actions of $A_{n}$ and
$S_{n}$ with base-size at least $3$ were given in [8, Table 1] (although the
group $A_{6}$ appears in this table, the corresponding action is standard).
Burness and others [5, 9, 10] determined the best possible value for $c$ such
that $b(G^{\Omega})\leqslant c$ for all other non-standard primitive actions
of almost simple groups $G$. In particular, upper bounds for $b(G^{\Omega})$
were specified in [5, Theorem 1.1, Proposition 4.1] in the case where
$G^{\Omega}$ is a non-standard action of an almost simple classical group.
Thus we obtain the following.

######  Lemma 3.5.

Let $G$ be a finite nonabelian simple group, and suppose that $G$ has a non-
standard primitive action on a finite set $\Omega$ with $|\Omega|>1$. Then the
following statements hold.

  1. (i)

$b(G^{\Omega})\leqslant 7$, with equality if and only if $G=M_{24}$ in its
$5$-transitive action of degree $24$.

  2. (ii)

If $G=A_{n}$ for some $n\geqslant 5$, then either $b(G^{\Omega})<n-2$ or
$(n,|\Omega|,b(G^{\Omega}))=(5,6,3)$.

  3. (iii)

If $G$ is a classical group, then $b(G^{\Omega})\leqslant 4$.

  4. (iv)

If $b(G^{\Omega})=4$ and $G$ is equal to ${\rm PSL}(n,q)$, ${\rm PSU}(n,q)$ or
${\rm PSp}(n,q)$ with $n\leqslant 5$, then $G={\rm PSU}(4,3)$.

We are now in a position to prove part of Theorem 1.2.

_Proof of Theorem 1.2(b)._ Let $G$ be an exceptional group of Lie type. Then
all primitive actions of $G$ are non-standard and so by Lemma 3.5,
$b_{maxprim}(G)\leqslant 6$. Hence Corollary 3.3 implies that $k(G)\leqslant
7$, as required. ∎

###  3.1 Bases for alternating groups

For the actions of $A_{n}$ or $S_{n}$ on $k$-subsets, the best general bounds
have been obtained by Halasi in [16]. To extract the information we need from
his work it is simplest to use his monotonicity result from [16, Section 2].

######  Lemma 3.6.

Let $G=A_{n}$ or $S_{n}$, where $n\geqslant 5$ and $n=3m+\delta$ with
$\delta\in\\{0,1,2\\}$. Consider the action of $G$ on the set $\Omega$ of
$k$-subsets of $X=\\{1,2,\dots,n\\}$, where $2\leqslant k\leqslant n/2$. Then
$b(G^{\Omega})\leqslant 2m$ if $\delta\in\\{0,1\\}$ and
$b(G^{\Omega})\leqslant 2m+1$ if $\delta=2$; so in particular
$b(G^{\Omega})\leqslant 2n/3$. Moreover, $b(G^{\Omega})\leqslant n-2$, with
equality if and only if $(n,k,G)=(6,2,S_{6})$ or $(5,2,S_{5})$.

###### Proof.

Since $b(A_{n}^{\Omega})\leqslant b(S_{n}^{\Omega})$, we may assume that
$G=S_{n}$ for proving the main assertion. As in [16], let $f(n,k)$ denote the
(minimum) base size of the action of $S_{n}$ on the set of $k$-subsets of $X$.
By [16, Corollary 2.2], $f(n,k)\leqslant f(n,2)$, so we may assume further
that $k=2$. Write $n=3m+\delta$ as in the statement, and consider the
following subset of $\Omega$:

$B=\\{\\{3j+1,3j+2\\},\\{3j+2,3j+3\\}\mid 0\leqslant j\leqslant m-1\\}.$

Let $H$ be the pointwise stabiliser of $B$ in $G$. Then $H$ fixes each point
$i$ of $X$ such that $1\leqslant i\leqslant 3m$. Suppose first that $\delta=0$
or $1$ (so that $n\geqslant 6$). Then $H=1$ and hence
$b(G^{\Omega})\leqslant|B|=2m\leqslant 2n/3\leqslant n-2$. Further if
$b(G^{\Omega})=n-2$ then $2n/3=n-2$, and hence $n=6$ and $b(G^{\Omega})=4$. By
[16, Theorem 3.1], $f(6,3)=\lceil\log_{2}6\rceil=3$, which is less than
$b(G^{\Omega})=4$, and hence $k=2$. Also, by [16, Theorem 3.2],
$f(6,2)\geqslant\lceil(2n-2)/(k+1)\rceil=4$, and it follows that
$b(S_{6}^{\Omega})=4$. However $b(A_{6}^{\Omega})=3$ since the pairs
$\\{1,2\\},\\{1,3\\},\\{1,4\\}$ form a base. Thus the result holds if
$\delta\leqslant 1$.

Suppose now that $\delta=2$. Then $H=\langle(n-1,n)\rangle$, and the pointwise
stabiliser of $B\cup\\{\\{1,n\\}\\}$ is trivial. Hence
$b(G^{\Omega})\leqslant|B|+1=2m+1$ and we have both $2m+1<2n/3$ and
$2m+1\leqslant 3m=n-2$. In this case, if $b(G^{\Omega})=n-2$, then $2m+1=3m$,
so $n=5$. The proof is complete on noting that $b(S_{5}^{\Omega})=3$ while
$b(A_{5}^{\Omega})=2$. ∎

Similarly for the base sizes of the actions of $A_{n}$ or $S_{n}$ on
partitions, precise results have been obtained by Burness, Garonzi and
Lucchini in [7] for small part sizes, building on the work of Benbenishty,
Cohen and Niemeyer [3], and by Morris and Spiga in [25] for large part sizes.
Again, the information we require is most easily obtained for most cases from
earlier work, this time [7] and an argument in Liebeck’s 1984 paper [20].

######  Lemma 3.7.

Let $G=A_{n}$ or $S_{n}$, where $n\geqslant 5$ and $n=ab$ with $a\geqslant
2,b\geqslant 2$. Consider the action of $G$ on the set $\Omega$ of partitions
of $X=\\{1,2,\dots,n\\}$ with $b$ parts of size $a$. Then
$b(G^{\Omega})\leqslant n-2$ with equality if and only if
$(a,b,G)=(2,3,S_{6})$ or $(a,b,G)=(3,2,S_{6})$.

###### Proof.

Since $b(A_{n}^{\Omega})\leqslant b(S_{n}^{\Omega})$, we may assume that
$G=S_{n}$ for proving the main assertion. If $a\geqslant 3$ then in [20, Case
(i) on pp. 13–14], Liebeck proved that $b(G^{\Omega})\leqslant(a-1)(b-1)+2=ab-
a-b+3\leqslant n-2$, where the second equality is strict unless $(a,b)=(3,2)$.
In this case we have that [25, Theorems 1.1 and 1.2] $b(S_{n}^{\Omega})=4=n-2$
while $b(A_{n}^{\Omega})=3<n-2$. Thus we may assume that $a=2$, so $b\geqslant
3$. In this case, by [7, Theorem 2], the base size satisfies
$b(S_{n}^{\Omega})=3<n-2$ if $b\geqslant 4$, and $b(S_{n}^{\Omega})=4=n-2$ if
$b=3$. Further, $b(A_{n}^{\Omega})=3<n-2$ if $b=3$, see [7, Remark 2.8]. ∎

Finally, we prove the following lemma, which completes the proof of Theorem
1.1.

######  Lemma 3.8.

Let $G=A_{n}$ where $n\geqslant 5$, and suppose that $G$ has a faithful
primitive permutation representation on a finite set $\Omega$. Then
$b(G^{\Omega})\leqslant n-2$, with equality if and only if either $|\Omega|=n$
or $(n,|\Omega|,b(G^{\Omega}))=(5,6,3)$. Moreover the assertions of Theorem
1.1 are valid.

###### Proof.

If $|\Omega|=n$ then the $G$-action on $\Omega$ is permutationally isomorphic
to its natural action and $b(G^{\Omega})=n-2$. Suppose this is not the case.
If the primitive permutation group $G^{\Omega}$ is standard, as in Definition
3.4(a), then by Lemmas 3.6 and 3.7, $b(G^{\Omega})<n-2$. On the other hand if
$G^{\Omega}$ is non-standard, then by Lemma 3.5, either $b(G^{\Omega})<n-2$ or
$(n,|\Omega|,b(G^{\Omega}))=(5,6,3)$. This proves the first assertion.

Now fix $n\geqslant 5$, and let $\Omega$ be such that $G=A_{n}$ has a faithful
primitive action on $\Omega$ and $b(G^{\Omega})$ is maximal over all such
actions, that is, $b(G^{\Omega})=b_{maxprim}(G)$. By Corollary 3.3, the
closure number $k(G)$ satisfies $k(G)\leqslant b(G^{\Omega})+1$, and we have
just proved that $b(G^{\Omega})\leqslant n-2$. Thus $k(G)\leqslant n-1$. On
the other hand, as we noted in the introduction, for the natural action of $G$
on $\Omega=\\{1,\dots,n\\}$, $G$ and $S_{n}$ have the same orbits on
$\Omega^{n-2}$, and hence $G^{(n-2),\Omega}=S_{n}\neq G$. Thus $k(G)>n-2$, and
hence $k(G)=n-1$. This proves Theorem 1.1. ∎

##  4 Bases for simple classical groups

In this subsection, we consider in detail base sizes of finite simple
classical groups $G$ that are not isomorphic to alternating groups. We are
concerned with

$b_{maxprim}(G):=\max\\{b(G^{\Omega})\mid G\text{ acts faithfully and
primitively on the set }\Omega\\}$. (2)

In particular, we prove the following theorem, which gives lower and upper
bounds for $b_{maxprim}(G)$. Together with Corollary 3.3, these bounds allow
us to prove Theorem 1.2(c). The bounds depend on certain quantities which we
define as follows.

$\mathcal{O}:=\\{\mathrm{P}\Omega^{\varepsilon}(n,q)\mid\mbox{$n$ even,
$\varepsilon=\pm$, $q\leqslant 3$, and if $q=3$, then $n\equiv
1-\varepsilon.1\pmod{4}\\}$}$ (3)

$\alpha:=\left\\{\begin{array}[]{ll}1&\mbox{if $(q+1)\mid n$}\\\
0&\mbox{otherwise,}\end{array}\right.\
\beta:=\left\\{\begin{array}[]{ll}1&\mbox{if $G\in\mathcal{O}$}\\\
0&\mbox{otherwise,}\end{array}\right.\
\gamma:=\left\\{\begin{array}[]{ll}1&\mbox{if $q$ even,
$n\in\\{10,12,14,16\\}$}\\\ 0&\mbox{otherwise.}\end{array}\right.$ (4)

Table 2: Lower and upper bounds for $b_{maxprim}(G)$, for finite simple classical groups $G$ not isomorphic to alternating groups, with $\alpha,\beta,\gamma$ as in (4), and $\delta$ the Kronecker delta. $G$ | Lower bound | Upper bound
---|---|---
${\rm PSL}(n,q)$ | $n+1-\delta_{2q}$ | $n+1-\delta_{2q}$
${\rm PSU}(4,q)$ | $4-\alpha$ | $5$
${\rm PSU}(6,q)$ | $6-\alpha$ | $12$
${\rm PSU}(n,q)$, $7\leqslant n\leqslant 15$ | $n-\alpha$ | $\lfloor n/3+11\rfloor$
${\rm PSU}(n,q)$, $n\in\\{3,5\\}$ or $n\geqslant 16$ | $n-\alpha$ | $n$
${\rm PSp}(4,q)$ | $4$ | $5$
${\rm PSp}(n,q)$, $6\leqslant n\leqslant 12$ | $n$ | $\lfloor n/3+10\rfloor$
${\rm PSp}(n,q)$, $n\geqslant 14$ | $n$ | $n$
${\rm P}\Omega^{\varepsilon}(n,q)$, $7\leqslant n\leqslant 16$ | $n-1-\beta$ | $\lfloor n/3+11-\gamma\rfloor$
${\rm P}\Omega^{\varepsilon}(n,q)$, $n\geqslant 17$ | $n-1-\beta$ | $n-1$

######  Theorem 4.1.

Let $G$ be a finite simple classical group that is not isomorphic to an
alternating group. Then Table 2 gives a lower and upper bound for
$b_{maxprim}(G)$.

In order to prove this theorem, it will be necessary to consider primitive
subspace actions $G^{\Omega}$, as in Definition 3.4. The next result, Lemma
4.2, summarises results from [17]. Note that if $\Omega$ is a $G$-orbit of
$k$-dimensional subspaces of the natural module $V$ for $G$, with
$n:=\dim(V)$, then (up to equivalence of actions) $1\leqslant k\leqslant n/2$.
In fact, if $G={\rm P}\Omega^{-}(n,q)$, then $n/2-1$ is the maximum dimension
of a totally singular subspace of $V$. On the other hand, if $G\neq{\rm
P}\Omega^{-}(n,q)$ or if $4\nmid n$, then $G$ does not act primitively on any
set of $n/2$-dimensional nondegenerate subspaces of $V$ (see [4, Table 2.2]).

######  Lemma 4.2.

Let $G$ be a finite simple classical group with natural module $V$ of
dimension $n$. Additionally, let $\Omega$ be a set of $k$-dimensional
nondegenerate or totally singular subspaces of $V$, such that $G^{\Omega}$ is
a primitive subspace action. Then $b(G^{\Omega})\leqslant n/k+11$.
Furthermore, $b(G^{\Omega})\leqslant n/k+10$ if $G={\rm PSp}(n,q)$ and $k$ is
odd; if $G={\rm P}\Omega^{\pm}(n,q)$, $k$ is odd and $q$ is even; or if
$k=n/2$ and either $G\neq{\rm P}\Omega^{-}(n,q)$ or $4\nmid n$.

###### Proof.

Theorem 3.3 of [17] shows that $b(G^{\Omega})\leqslant n/k+11$. If $k=n/2$ and
either $G\neq{\rm P}\Omega^{-}(n,q)$ or $4\nmid n$, then $\Omega$ is a set of
totally singular subspaces of $V$ (since $G^{\Omega}$ is primitive). This is
also true if $k$ is odd and either $G={\rm PSp}(n,q)$, or $G={\rm
P}\Omega^{\pm}(n,q)$ with $q$ even, as here no nondegenerate subspace of $V$
has odd dimension. Thus in any of these special cases, $b(G^{\Omega})\leqslant
n/k+10$ by [17, p. 28]. ∎

It would be possible to use Lemmas 3.5 and 4.2 directly (along with some
additional arguments) to obtain upper bounds for $b_{maxprim}(G)$, at least in
the case where $G$ has no primitive action on (degenerate) nonsingular
$1$-dimensional subspaces of $V$. However, in most cases, we can deduce
significantly improved bounds by considering in detail the known bounds on
base sizes of primitive actions of $G$ on subspaces of dimension at most $2$.

By case (b) of Definition 3.4, any subspace action of $G$ is defined with
respect to the natural module for $G$. Thus $G$ may have standard actions that
are not subspace actions of $G$, but are equivalent to subspace actions of
isomorphic classical groups. Therefore, in the lemma below, it is not
sufficient to consider $G$ only up to isomorphism. For example, the orthogonal
groups $G={\rm P}\Omega^{\varepsilon}(n,q)$ (with $\varepsilon=-$ if $n=4$ so
that $G$ is simple) occur as classical simple groups for any integer
$n\geqslant 3$, even though these groups are often considered as linear,
unitary or symplectic groups of lower dimension when $n\leqslant 6$.

######  Lemma 4.3.

Let $G$ be a finite simple classical group that is not isomorphic to an
alternating group, and let $V$ be the natural module for $G$. Additionally,
let $\Omega$ be a set of $k$-dimensional subspaces of $V$, such that
$k\leqslant 2$ and $G^{\Omega}$ is a primitive subspace action. Then Table 3
gives an upper bound for $b(G^{\Omega})$.

Table 3: Upper bounds for the base sizes of primitive subspace actions in Lemma 4.3. Here, $\alpha:=1$ if $\varepsilon=-$, and $\alpha:=0$ otherwise. Additionally, $\delta$ is the Kronecker delta. $G$ | $k$ | Upper bound for $b(G^{\Omega})$
---|---|---
${\rm PSL}(n,q)$ | $1$ | $n+1-\delta_{2q}$
${\rm PSL}(n,q)$, $n\geqslant 4$ | $2$ | $\lceil n/2\rceil+2+\delta_{4n}$
${\rm PSU}(n,q)$ | $1$ | $n+\delta_{2n}$
${\rm PSU}(n,q)$, $n\geqslant 4$ | $2$ | $\lceil n/2\rceil+3\delta_{4n}+\delta_{5n}+\delta_{6n}$
${\rm PSp}(n,q)$ | $1$ | $n+\delta_{2n}$
${\rm PSp}(n,q)$, $n\geqslant 4$ | $2$ | $\lceil n/2\rceil+2\delta_{4n}+\delta_{6n}$
${\rm P}\Omega^{\varepsilon}(n,q)$, $n\geqslant 3$ | $1$ | $n-1+\delta_{3n}+\delta_{5n}$
${\rm P}\Omega^{\varepsilon}(n,q)$, $n\geqslant 4$ | $2$ | $\lceil n/2\rceil+\delta_{4n}+\delta_{5n}+\alpha\delta_{6n}$

###### Proof.

Let $b(G,k)$ be the value in the final column of Table 3 corresponding to the
pair $(G,k)$. We first consider the case where $G={\rm PSL}(n,q)$ and $k=1$.
Let $\\{e_{1},\ldots,e_{n}\\}$ be a basis for $V$. It is well known (and easy
to verify) that $B:=\\{\langle e_{1}\rangle,\ldots,\langle e_{n}\rangle\\}$ is
a base of minimal size for $G^{\Omega}$ if $q=2$, and that $B\cup\\{\langle
e_{1}+\cdots+e_{n}\rangle\\}$ is a base of minimal size if $q>2$. Hence
$b(G^{\Omega})=b(G,k)$.

Next, if $G\in\\{{\rm PSU}(2,q),{\rm PSp}(2,q)\\}$ with $k=1$, then the
subspaces in $\Omega$ are $1$-dimensional and totally singular. Every
$1$-dimensional subspace of ${\rm PSp}(2,q)$ is totally singular, and so the
action in this case is precisely the unique primitive subspace action of ${\rm
PSL}(2,q)$ (which is equal to ${\rm PSp}(2,q)$). This action is in fact also
equivalent to the unique subspace action of ${\rm PSU}(2,q)$ (see [4, Tables
2.3 & 8.1]). As $G$ is simple, $q\neq 2$ for these groups, and we deduce that
$b(G^{\Omega})=3=b(G,k)$.

Thus we may assume that $n\geqslant 3$, and that if $k=1$ then $G$ is not
linear. Additionally, if $G$ is orthogonal, we assume here that $n\geqslant
4$, that $k=1$ if $n\leqslant 6$, and that $\Omega$ is a set of nondegenerate
or nonsingular subspaces if $n=4$ (we deal with the other orthogonal cases in
the next paragraph). The results in [24, §2.1–2.2] show that $b(G,k)$ is an
upper bound for the base size of the action on $\Omega$ of the projective
isometry group ${\rm PGL}(n,q),{\rm PGU}(n,q),{\rm PSp}(n,q)$ or ${\rm
PGO}^{\varepsilon}(n,q)$ associated with and containing $G$. As
$b(G^{\Omega})\leqslant b(R^{\Omega})$ for any overgroup $R$ of $G$, it
follows that $b(G^{\Omega})\leqslant b(G,k)$.

Finally we deal with the actions of small dimensional orthogonal groups not
covered by the previous paragraph, namely the groups $G={\rm
P}\Omega^{\varepsilon}(n,q)\cong H$, where $(n,\varepsilon,H)$ is one of the
following:

$(3,\circ,{\rm PSL}(2,q)),(4,-,{\rm PSL}(2,q^{2})),(5,\circ,{\rm
PSp}(4,q)),(6,+,{\rm PSL}(4,q)),(6,-,{\rm PSU}(4,q)).$

We can use the tables in [4, §2.2, §8.2] to determine the maximal subgroup
type of a point stabiliser of $G^{\Omega}$, considered as a subgroup of $H$.
Using this information, we deduce from [5, Tables 2–3] that if $G^{\Omega}$ is
not equivalent to a subspace action of $H$, then $b(G^{\Omega})\leqslant 3$,
and in each case this is at most the relevant bound $b(G,k)$. We also see that
if $G^{\Omega}$ is equivalent to a subspace action of $H$ in the case where
$n=6$ and $k=2$, then $\varepsilon=-$ and the action of $H$ is on totally
singular $1$-dimensional subspaces. In each case, the bounds on base sizes of
subspace actions of $H$ from above show that $b(G^{\Omega})\leqslant b(G,k)$.
∎

We can now prove Theorem 4.1. Since $b_{maxprim}(G)$ is a property of $G$ as
an abstract group, we may now consider $G$ up to isomorphism.

Proof of Theorem 4.1 Let $V$ be the natural module for $G$ and $n:=\dim(V)$.
We begin by determining a lower bound for $b_{maxprim}(G)$. For each positive
integer $k\leqslant n/2$, [14] gives a lower bound for the base size of a
primitive action of $G$ on a set of $k$-dimensional subspaces of $V$
(depending only on $G$ and $k$). Additionally, the tables in [19, §3.5] and
[4, §8.2] show that $G$ has a primitive action on a set of $1$-dimensional
subspaces of $V$. In each case, the lower bound for $b_{maxprim}(G)$ given in
Table 2 is the lower bound from [14] in the case $k=1$ (when $G$ is linear,
this bound is well known; see the proof of Lemma 4.3).

Now, let $b_{maxsub}(G)$ and $b_{maxstand}(G)$ denote the maximum of the base
sizes of all primitive subspace actions of $G$, and of all primitive standard
actions of $G$, respectively. The latter accounts for all actions of $G$ that
are equivalent to subspace actions of isomorphic classical groups.
Additionally, let $b_{upper}(G)$ be the upper bound for $b_{maxprim}(G)$ from
Table 2. We will divide the remainder of the proof into several cases,
corresponding to the type of $G$. In each case, we utilise the upper bounds
from Lemma 4.3 for base sizes of primitive subspace actions of $G$ on
subspaces of $V$ of dimension at most two.

Case (a): $G\cong{\rm PSL}(n,q)$. Any primitive action of $G$ on
$k$-dimensional subspaces of $V$, with $3\leqslant k\leqslant n/2$, has base
size at most $n/k+3$ if $k\mid n$, or at most $n/k+5$ otherwise [17, p. 24].
As $\lceil n/2\rceil+2$ and $\lfloor n/3+5\rfloor$ are less than or equal to
$n$ for all $n\geqslant 7$, it follows easily from Lemma 4.3 (noting that
$G\not\cong{\rm PSL}(4,2)\cong A_{8}$) that $b_{maxsub}(G)\leqslant
n+1-\delta_{2q}=b_{upper}(G)$.

We now consider standard actions of $G$ that are not subspace actions. The
relevant isomorphisms here are ${\rm PSL}(2,7)\cong{\rm PSL}(3,2)$, ${\rm
PSL}(2,q)={\rm PSp}(2,q)\cong{\rm PSU}(2,q)\cong\Omega(3,q)$, ${\rm
PSL}(2,q)\cong{\rm P}\Omega^{-}(4,q^{1/2})$ (when $q$ is a square) and ${\rm
PSL}(4,q)\cong{\rm P}\Omega^{+}(6,q)$. The tables in [4, §2.2, §8.2] show that
the action of ${\rm P}\Omega^{+}(6,q)$ on $3$-dimensional totally singular
subspaces is equivalent to a subspace action of ${\rm PSL}(4,q)$. It follows
from Lemma 4.3 that each subspace action of ${\rm P}\Omega^{+}(6,q)$ or of
${\rm PSL}(4,q)$ has a base size of at most $5$, while the remaining actions
on subspaces here all have a base size of at most $3$. When $q$ is even, the
action of ${\rm PSp}(2,q)$ corresponding to case (b)(ii) of Definition 3.4
also has base size at most $3$ [17, p. 30]. Thus $b_{maxstand}(G)\leqslant
b_{upper}(G)$ in each case.

It remains to consider the cases where our upper bound $b_{upper}(G)$ for
$b_{maxstand}(G)$ may be strictly less than $b_{maxprim}(G)$. By Lemma 3.5,
this can occur only if $n+1-\delta_{2q}<4$, i.e., only if $n=2$ or $G={\rm
PSL}(3,2)$. This same lemma now shows that the base size of a non-standard
action of $G$ is at most $3=n+1-\delta_{2q}$. Therefore, in each linear case,
$b_{maxprim}(G)\leqslant b_{upper}(G)$.

Case (b): $G\cong{\rm PSU}(n,q)$, with $n\geqslant 3$. As $\lfloor
n/3+11\rfloor\leqslant n$ for all $n\geqslant 16$, we see from Lemmas 4.2 and
4.3 that $b_{maxsub}(G)\leqslant b_{upper}(G)$.

The remaining standard actions correspond to the isomorphisms ${\rm
PSU}(4,q)\cong{\rm P}\Omega^{-}(6,q)$ and ${\rm PSU}(4,2)\cong{\rm
PSp}(4,3)\cong{\rm P}\Omega(5,3)$. Lemma 4.3 shows that
$b_{maxstand}(G)\leqslant b_{upper}(G)$ (since $4\nmid 6$, the group ${\rm
P}\Omega^{-}(6,q)$ has no primitive subspace action on three-dimensional
subspaces of its natural module). Moreover, if $b_{upper}(G)<4$, then
$b_{maxsub}(G)\leqslant 3=n$. Hence Lemma 3.5 yields $b_{maxprim}(G)\leqslant
b_{upper}(G)$ in each case.

Case (c): $G\cong{\rm PSp}(n,q)$, with $n\geqslant 4$. When $q$ is even, the
subspace action of $G$ from case (b)(ii) of Definition 3.4 has base size $n$
if $n\geqslant 6$ [24, Proposition 1], or base size at most $5$ if $n=4$ [17,
p. 30]. Since $\lfloor n/3+10\rfloor$ and $\lfloor n/4+11\rfloor$ are less
than or equal to $n$ for all $n\geqslant 14$, we obtain
$b_{maxsub}(G)\leqslant b_{upper}(G)$ from Lemmas 4.2 and 4.3.

We now consider the isomorphisms ${\rm PSp}(4,3)\cong{\rm PSU}(4,2)\cong{\rm
P}\Omega^{-}(6,2)$ and ${\rm PSp}(4,q)\cong{\rm P}\Omega(5,q)$. In each case,
Lemma 4.3 yields $b_{maxstand}(G)\leqslant b_{upper}(G)$. As no symplectic $G$
satisfies $b_{upper}(G)<4$, it follows from Lemma 3.5 that
$b_{maxprim}(G)\leqslant b_{upper}(G)$.

Case (d): $G\cong{\rm P}\Omega^{\varepsilon}(n,q)$, with $n\geqslant 7$.
Noting that $\lfloor n/3+11\rfloor\leqslant n-1$ for all $n\geqslant 17$ and
$\lfloor n/4+11\rfloor\leqslant\lfloor n/3+10\rfloor$ for all $n\geqslant 9$,
we see from Lemmas 4.2 and 4.3 that $b_{maxsub}(G)\leqslant b_{upper}(G)$.

As $n\geqslant 7$, there are no nontrivial isomorphisms of classical groups to
consider. Moreover, $b_{upper}(G)>3$, and so $b_{maxprim}(G)\leqslant
b_{upper}(G)$ by Lemma 3.5. ∎

We can now prove Theorem 1.2 for classical groups.

_Proof of Theorem 1.2(c)._ Let $G$ be a classical group that is not isomorphic
to an alternating group. Then Theorem 1.2(c) follows from Corollary 3.3 and
the upper bound for $b_{maxprim}(G)$ given in Theorem 4.1. ∎

Of course, if $G$ is a low-dimensional unitary, symplectic or orthogonal group
that is not explicitly mentioned in Theorem 1.2(c), then the general upper
bound of $\lfloor n/3+12\rfloor$ for $k(G)$ from this theorem is often higher
than the actual upper bound that follows from Theorem 4.1. Notice also that
the lower and upper bounds for $b_{maxprim}(G)$ from Table 2 often coincide.
The following corollary of Theorem 4.1 summarises when this occurs. Here,
$\delta$ again denotes the Kronecker delta, and $\mathcal{O}$ is as in (3).

######  Corollary 4.4.

Let $G$ be a finite simple classical group that is not isomorphic to an
alternating group.

  1. (i)

If $G={\rm PSL}(n,q)$, then $b_{maxprim}(G)=n+1-\delta_{2,q}$.

  2. (ii)

If $G={\rm PSU}(n,q)$ with $(q+1)\nmid n$, and with $n\in\\{3,5\\}$ or
$n\geqslant 16$, then $b_{maxprim}(G)=n$.

  3. (iii)

If $G={\rm PSp}(n,q)$ with $n\geqslant 14$, then $b_{maxprim}(G)=n$.

  4. (iv)

If $G={\rm P}\Omega^{\varepsilon}(n,q)$ with $n\geqslant 16$, with $q$ even if
$n=16$, and with $G\notin\mathcal{O}$, then $b_{maxprim}(G)=n-1$.

######  Remark 4.5.

Observe that if $G$ is one of the groups in Corollary 4.4 for which
$b_{maxprim}(G)$ is known exactly, then the upper bound for the closure number
$k(G)$ from Theorem 1.2 is equal to $b_{maxprim}(G)+1$. Hence this theorem
gives the best upper bound for $k(G)$ that can be obtained solely using
Corollary 3.3. On the other hand, for many low-dimensional unitary, symplectic
and orthogonal groups $G$, improved upper bounds on the base sizes of
primitive actions of $G$ on subspaces of dimension greater than two would
likely result in improved upper bounds for $b_{maxprim}(G)$ and hence for
$k(G)$.

##  5 $k$-closures of sporadic groups

We start with the following lemma on certain simple groups.

######  Lemma 5.1.

Let $G$ be a finite nonabelian simple group acting primitively on a set
$\Omega$ such that $\operatorname{Out}(G)=1$ and $G$ is maximal in ${\rm
Alt}(\Omega)$. If $G$ is $k$-transitive on $\Omega$ but not $(k+1)$-transitive
for some integer $k$ such that $1\leqslant k<|\Omega|-2$, then
$G^{(k+1),\Omega}=G$ but $G^{(k),\Omega}\neq G$.

###### Proof.

Note that since $G$ is $k$-transitive we have that $G^{(k),\Omega}={\rm
Sym}(\Omega)\neq G$. Since $G$ is maximal in ${\rm Alt}(\Omega)$ it follows
that $G^{(k+1),\Omega}\cap{\rm Alt}(\Omega)=G$ or ${\rm Alt}(\Omega)$.
However, ${\rm Alt}(\Omega)$ is $(k+1)$-transitive on $\Omega$ while $G$ is
not, so ${\rm Alt}(\Omega)\not\leqslant G^{(k+1),\Omega}$. Hence
$G^{(k+1),\Omega}\cap{\rm Alt}(\Omega)=G$. Thus $|G^{(k+1),\Omega}:G|\leqslant
2$. Since $G$ acts primitively on $\Omega$ (and $|G|$ is not prime) we have
that $C_{{\rm Sym}(\Omega)}(G)=1$. Then as $\operatorname{Out}(G)=1$ it
follows that $G^{(k+1),\Omega}=G$. ∎

######  Corollary 5.2.

  1. (a)

Let $G=M_{23}\leqslant{\rm Sym}(\Omega)$ with $|\Omega|=23$. Then
$G^{(5),\Omega}=G$.

  2. (b)

Let $G=Co_{3}\leqslant{\rm Sym}(\Omega)$. with $|\Omega|=276$. Then
$G^{(3),\Omega}=G$.

  3. (c)

Let $G=Co_{2}\leqslant{\rm Sym}(\Omega)$ with $|\Omega|=2300$. Then
$G^{(2),\Omega}=G$.

###### Proof.

By [21], $G$ is maximal in ${\rm Alt}(\Omega)$. Additionally,
$\operatorname{Out}(G)=1$, and so the result follows from Lemma 5.1. ∎

When $G=M_{23}$, $Co_{2}$ or $Co_{3}$ we have from [10, Table 1] that
$b_{maxprim}(G)=6$ so our next result is a slight improvement on what can be
deduced from Corollary 3.3. We suspect that it is not best possible.

######  Lemma 5.3.

Let $G=M_{23}$, $Co_{2}$ or $Co_{3}$. Then $k(G)\leqslant 6$.

###### Proof.

Let $n=23$ if $G=M_{23}$, $n=276$ if $G=Co_{3}$ and $n=2300$ if $G=Co_{2}$,
and let $G$ act faithfully on a set $\Omega$. We will prove that
$G^{(6),\Omega}=G^{\Omega}\cong G$, and from this it will follow that
$k(G)\leqslant 6$. If $G$ has an orbit with a maximal block system
$\mathcal{B}$ such that $|\mathcal{B}|\neq n$, then
$b(G^{\mathcal{B}})\leqslant 5$, by [10, Theorem 1], and so
$b(G^{\Omega})\leqslant 5$, by Lemma 3.1. Hence by Lemma 3.2,
$G^{(6),\Omega}=G^{\Omega}\cong G$. Thus it remains to consider the case where
on each orbit $\Delta$, each maximal block system has size $n$. Let $\Delta$
be an orbit of $G$ and let $\mathcal{B}$ be a maximal block system on
$\Delta$. By Lemma 2.5 and Corollary 5.2 we have that $G^{(6),\Delta}$
preserves $\mathcal{B}$ and
$(G^{(6),\Delta})^{\mathcal{B}}=G^{(6),\mathcal{B}}=G^{\mathcal{B}}\cong G$.
Moreover, by [10, Theorem 1] there exist $B_{1},\ldots,B_{6}\in\mathcal{B}$
such that $G_{B_{1},\ldots,B_{6}}=1$ (since $G$ is faithful on $\mathcal{B}$),
and it follows from Lemma 2.5 that $G^{(6),\Delta}$ acts faithfully on
$\mathcal{B}$. Thus $G^{(6),\Delta}\cong G$ so $G^{(6),\Delta}=G^{\Delta}$.

If $\Omega=\Delta$ then we are done so we may assume that $G$ has more than
one orbit. Recall that, on each $G$-orbit, each maximal block system has size
$n$. Let $\Delta_{1}$ be a $G$-orbit and let $\mathcal{B}_{1}$ be a maximal
block system for $G$ on $\Delta_{1}$. Let $\beta\in B\in\mathcal{B}_{1}$. Then
$G_{\beta}\leqslant G_{B}$. When $G=M_{23}$ we have that $G_{B}=M_{22}$, when
$G=Co_{3}$ we have $G_{B}=McL.2$ and when $G=Co_{3}$ we have $G_{B}={\rm
PSU}_{6}(2).2$. In all cases $|G_{B}|$ is not divisible by $23$ but $n$ is,
and so $G_{B}$ does not have a transitive action of degree $n$. Thus $G_{B}$,
and hence also $G_{\beta}$ is not transitive on any orbit of $G$. Since
$G^{(6),\Delta}=G^{\Delta}$ for each $G$-orbit $\Delta$, it follows from Lemma
2.3 that $G^{(6),\Omega}=G^{\Omega}\cong G$ and the result follows. ∎

We use Lemma 5.3 for $M_{23}$, as well as the fundamental Lemma 2.1, to
determine the closure number of $M_{24}.$

######  Lemma 5.4.

$k(M_{24})=6$.

###### Proof.

Let $G=M_{24}$ act faithfully on a set $\Omega$. We divide the orbits $\Delta$
of $G$ into three types:

  1. (a)

$|\Delta|=24$.

  2. (b)

$G$ has a maximal block system on $\Delta$ of size other than 24.

  3. (c)

Each maximal block system of $G$ on $\Delta$ has size 24 and at least one of
these maximal block systems is not the partition into singletons.

Recall that we allow a maximal block system to be the partition into
singletons.

Let $\Delta$ be an orbit of $G$ and suppose first that $\Delta$ is of type
(a). Then $G$ acts 5-transitively but not 6-transitively on $\Delta$. By [21]
$G$ is a maximal subgroup of $A_{24}$ and so by Lemma 5.1, $k(G)\geqslant 6$
and $G^{(6),\Delta}=G^{\Delta}\cong G$.

Next suppose that $\Delta$ has type (b) and let $\mathcal{B}$ be a maximal
block system on $\Delta$ with $|\mathcal{B}|\neq 24$. Then by [10, Theorem 1]
the base size $b(G^{\mathcal{B}})\leqslant 4$. Hence by Lemma 3.1, also
$b(G^{\Delta})\leqslant 4$ and so by Lemma 3.2,
$G^{(5),\Delta}=G^{\Delta}\cong G$. This implies that
$G^{(6),\Delta}=G^{\Delta}\cong G$.

Finally suppose that $\Delta$ is of type (c) and let $\mathcal{B}$ be a
maximal block system of size 24 that is not the partition into singletons. Let
$B\in\mathcal{B}$. By Lemma 2.5 and our result on orbits of type (a), we have
$M_{24}\cong G^{\mathcal{B}}\leqslant(G^{(6),\Delta})^{\mathcal{B}}\leqslant
G^{(6),\mathcal{B}}\cong M_{24}$, and so $(G^{(6),\Delta})^{\mathcal{B}}\cong
G$. Moreover, by Lemma 2.5,
$M_{23}\cong(G^{\Delta})_{B}^{B}\leqslant(G^{(6),\Delta})_{B}^{B}\leqslant(G_{B})^{(6),B}=M_{23}^{(6),B}$,
which is isomorphic to $M_{23}$ by Lemma 5.3. Thus $G^{(6),\Delta}\leqslant
M_{23}\wr M_{24}$ acting imprimitively. Furthermore, since the minimal degree
of a transitive representation of $M_{23}$ is 23 (see for example [13, p71])
we have $|B|\geqslant 23$. Let $K$ be the kernel of the action of
$G^{(6),\Delta}$ on $\mathcal{B}$ and suppose that $K\neq 1$. Then by a lemma
of Scott ([28, p. 328] or see [27, Theorem 4.16]) the group $K$ is a direct
product of $\ell$ diagonal subgroups $H_{j}\cong M_{23}$ for $j=1,\ldots,\ell$
(where $\ell\geqslant 1$), with pairwise disjoint supports, that is to say,
there is a partition $\\{\mathcal{O}_{1},\ldots,\mathcal{O}_{\ell}\\}$ of
$\mathcal{B}$ such that, for each $j$, $(H_{j})^{B}\cong M_{23}$ for all
$B\in\mathcal{O}_{j}$ and $(H_{j})^{B}=1$ for all $B\notin\mathcal{O}_{j}$.
Suppose that $\ell\geqslant 2$. Then there exist distinct
$B_{j},B_{k}\in\mathcal{B}$ such that for all $\alpha_{1},\ldots,\alpha_{5}\in
B_{j}$ and all $\beta_{1},\beta_{2}\in B_{k}$, there exists $h\in K$ such that
$(\alpha_{1},\ldots,\alpha_{5})^{h}=(\alpha_{1},\ldots,\alpha_{5})$ and
$\beta_{1}^{h}=\beta_{2}$. Since $K\leqslant G^{(6),\Delta}$, there exists
$g\in G$ such that
$(\alpha_{1},\ldots,\alpha_{5},\beta_{1})^{g}=(\alpha_{1},\ldots,\alpha_{5},\beta_{1})^{h}=(\alpha_{1},\ldots,\alpha_{5},\beta_{2})$.
As $\beta_{1},\beta_{2}$ were arbitrary points of $B_{k}$, it follows that
$G_{\alpha_{1},\ldots,\alpha_{5}}$ acts transitively on $B_{k}$. Note that
$G_{B_{j}}=M_{23}$, so $G_{B_{j}}$ acts faithfully on $B_{j}$. Now
$G_{\alpha_{1},\ldots,\alpha_{5}}=(G_{B_{j}})_{\alpha_{1},\ldots,\alpha_{5}}=(M_{23})_{\alpha_{1},\ldots,\alpha_{5}}$
while $|B_{k}|\geqslant 23$. Suppose first that $G_{B_{j}}$ has a maximal
block system of size greater than 23 on $B_{j}$. Then by [10, Theorem 1] and
Lemma 3.1, we have $b(G_{B_{j}}^{B_{j}})\leqslant 4$. Hence we could have
chosen $\alpha_{1},\ldots,\alpha_{5}$ such that
$(M_{23})_{\alpha_{1},\ldots,\alpha_{5}}=1$, contradicting
$G_{\alpha_{1},\ldots,\alpha_{5}}$ acting transitively on $B_{k}$. Hence all
maximal block systems $\Sigma$ of $G_{B_{j}}$ on $B_{j}$ have size 23.
However, in this case we can choose $\alpha_{1},\ldots,\alpha_{5}\in B_{j}$
from pairwise distinct blocks of $\Sigma$ such that
$|(M_{23})_{\alpha_{1},\ldots,\alpha_{5}}|<23$, again contradicting
$G_{\alpha_{1},\ldots,\alpha_{5}}$ acting transitively on $B_{k}$.

Thus $\ell=1$ so $K\cong M_{23}$, and $G^{(6),\Delta}=M_{23}\times M_{24}$.
Hence $G^{(6),\Delta}$ has an intransitive normal subgroup $N$ isomorphic to
$M_{24}$ such that the normal subgroup $L\cong M_{23}$ acts transitively on
the set $\Sigma$ of $N$-orbits. Now $\Sigma$ is a $G$-invariant partition of
$\Delta$, and a maximal block system for the $G$-action on $\Sigma$
corresponds to a maximal block system, say $\mathcal{B}^{\prime}$, for
$G^{\Delta}$. By assumption all maximal block systems for $G$ on $\Delta$ have
size 24 and so $|\mathcal{B}^{\prime}|=24$. Hence the $G$-action on $\Sigma$
has a block system of size $24$. Thus the subgroup $L$ of $G$ is transitive on
$\Sigma$ preserving a block system of size $24$. This implies that $L\cong
M_{23}$ has a subgroup of index $24$, which is a contradiction (see for
example [13, p71]). Therefore $K=1$ and $G^{(6),\Delta}=M_{24}$.

Thus we have shown that $G^{(6),\Delta}=G^{\Delta}\cong G$ for each orbit
$\Delta$. In particular, if $G$ is transitive on $\Omega$ then we have shown
that $G^{(6),\Omega}=G^{\Omega}\cong G$. Suppose now that $G$ is intransitive
on $\Omega$. If there is some orbit $\Delta$ of type (b) then we saw above
that $b(G^{\Delta})\leqslant 4$. Since $G$ acts faithfully on $\Delta$ we have
$b(G^{\Omega})\leqslant 4$ and so by Lemma 3.2,
$G^{(5),\Omega}=G^{\Omega}\cong G$. If all orbits are of type (a) then all
actions of $G$ on orbits are pairwise equivalent and so by Corollary 2.4 we
have $G^{(6),\Omega}=G^{\Omega}\cong G$.

It remains to consider the case where there is at least one orbit $\Delta$ of
type (c) and all the remaining orbits are of type (a) or (c). Recall that this
implies that $G^{\Delta}$ preserves a block system $\mathcal{B}$ of size $24$
on $\Delta$, and moreover in each $G$-orbit $\Delta^{\prime}$, each maximal
block system for $G$ on $\Delta^{\prime}$ has size $24$. Let $\beta\in
B\in\mathcal{B}$. Then $G_{\beta}\leqslant G_{B}=M_{23}$. Since $M_{23}$ does
not have any transitive actions of degree 24, it follows that $G_{B}$, and
hence also $G_{\beta}$, is not transitive on any orbit of $G$. Thus by Lemma
2.3, and the fact that $G^{(6),\Delta^{\prime}}=M_{24}$ for all $G$-orbits
$\Delta^{\prime}$, we deduce that $G^{(6),\Omega}\cong G$. Thus we have shown
that $k(G)\leqslant 6$. Moreover, equality holds since if $|\Omega|=24$ then
$G$ is 5-transitive on $\Omega$ and hence $G^{(5),\Omega}={\rm Sym}(\Omega)$.
∎

We can now complete the proof of Theorem 1.2.

_Proof of Theorem 1.2(a)._ Let $G$ be a sporadic simple group. If $G\cong
M_{24}$ then by Lemma 5.4 we have $k(G)=6$, so suppose further that
$G\not\cong M_{24}$. By Corollary 3.3, the closure number $k(G)$ satisfies
$k(G)\leqslant b_{maxprim}(G)+1$, where $b_{maxprim}(G)$ is as in (2). Let
$\Omega$ be a finite set on which $G$ acts faithfully and primitively such
that $b(G^{\Omega})=b_{maxprim}(G)$, so $k(G)\leqslant b(G^{\Omega})+1$. By
[10, Theorem 1] we in fact have that either $b(G^{\Omega})\leqslant 5$ and so
$k(G)\leqslant 6$, or $G=M_{23}$, $Co_{2}$ or $Co_{3}$. Thus Theorem 1.2(a)
follows from Lemma 5.3. ∎

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Moreover, by studying the first derivative of $f(x)$ we find the following
critical points (where $f^{\prime}(x)=0$):

$\displaystyle x_{1}$ $\displaystyle=\frac{1}{4}\left(1-\sqrt{1+8s}\right)$
(B.104) $\displaystyle x^{\prime}_{1}$
$\displaystyle=\frac{1}{4}\left(1+\sqrt{1+8s}\right)$ (B.105) $\displaystyle
x^{\prime\prime}_{1}$ $\displaystyle=1.$ (B.106)

We can easily deduce that $x_{1}<0$ and that $\sqrt{s}<x^{\prime}_{1}<1$. By
combining this with the fact that $f^{\prime\prime}(1)=2(1-s)>0$, we conclude
that $f(x)\geq 0$ for $x\in[\sqrt{s},\sqrt{2}]$ and that it presents a local
minimum in $x=1$.

From the above analysis, we deduce that $f(x)<0$ for $x\in[0,\sqrt{s})$ when
$0<s<1$. However, as anticipated, the condition $0\leq x<\sqrt{s}$ is not
compatible with the definitions of $s$ and $c$. Indeed, by using the
definitions (B.100) and (B.101) we show that $x\geq\sqrt{s}$ holds:

$\displaystyle x\geq\sqrt{s}\,\,$
$\displaystyle\Leftrightarrow\,\,\sqrt{1-\sin a_{1}\cos b_{-}\cos
c_{-}}\geq\sqrt{\sin b_{-}\sin c_{-}}$ $\displaystyle\Leftrightarrow\,\,1-\sin
a_{1}\cos b_{-}\cos c_{-}\geq\sin b_{-}\sin c_{-}$ (B.107)

which can be proved via the sufficient condition (B.87). This implies that
$f(\sqrt{1-\sin a_{1}\cos b_{-}\cos c_{-}})\geq 0$ for every $a_{1}$, $b_{-}$
and $c_{-}$, which proves (B.102).

We thus proved (B.97a), which completes the proof of (B.83a), which in turn
completes the proof of the validity of (B.64a). This proves the lower bound
(B.62), which employed in (B.61) provides us with the bound (B.6) on the
conditional entropy of a fixed state $\rho_{\alpha}$ in the mixture:

$H(A_{0}|E)_{\rho_{\alpha}}\geq
1-h\left[\frac{1}{4}\left(\beta^{\alpha}_{\mathrm{H}}+1+\sqrt{(\beta^{\alpha}_{\mathrm{H}})^{2}+2\beta^{\alpha}_{\mathrm{H}}-3}\right)\right]$
(B.108)

Finally, by the convexity of (B.108), we extend the derived bound to the whole
mixed state (B.3) as shown in (B.7), thus obtaining the entropy bound (B.1).
This concludes the proof of Theorem 1. ∎

###  B.3 Tightness of the bound

In order to demonstrate that the entropy bound (B.1) is tight we need to show
that, for every Bell value $\beta_{\mathrm{H}}$, there exists a quantum state
and a set of measurements performed by the parties such that the Bell value is
exactly given by $\beta_{\mathrm{H}}$ and such that the conditional entropy of
Alice’s outcome $A_{0}$ is equal to the rhs of (B.1).

The states that satisfy the above conditions (for every $\beta_{\mathrm{H}}$)
belong to the following family of states diagonal in the GHZ basis:

$\tau(\nu)=\nu|\hskip 0.7pt\psi_{0,0,0}\rangle\langle\psi_{0,0,0}\hskip
0.7pt|+(1-\nu)|\hskip 0.7pt\psi_{1,0,0}\rangle\langle\psi_{1,0,0}\hskip
0.7pt|,$ (B.109)

where $\nu\in[1/2,1]$. In order to see this, we first assign to Eve maximum
knowledge by letting her hold the purifying system $E$ of $\tau(\nu)$:

$\ket{\Psi_{ABCE}}=\sqrt{\nu}\ket{\psi_{0,0,0}}\otimes\ket{e_{0}}_{E}+\sqrt{1-\nu}\ket{\psi_{1,0,0}}\otimes\ket{e_{1}}_{E}.$
(B.110)

We now fix Alice’s observable $A_{0}$ to be $Z$ and compute the classical-
quantum state of Alice’s $Z$ outcome and Eve’s quantum system:

$\displaystyle\tau_{ZE}(\tau)$ $\displaystyle=\sum_{z=0}^{1}|\hskip
0.7ptz\rangle\langle z\hskip
0.7pt|_{Z}\otimes\operatorname{Tr}_{BC}[\braket{z}{\Psi_{ABCE}}\braket{\Psi_{ABCE}}{z}]$
$\displaystyle=\sum_{z=0}^{1}\frac{1}{2}|\hskip 0.7ptz\rangle\langle z\hskip
0.7pt|_{Z}\otimes\rho^{z}_{E}.$ (B.111)

In the above expression, $\rho^{z}_{E}$ is the conditional state of Eve given
that Alice obtained outcome $Z=z$ and can be easily computed as:

$\displaystyle\rho^{z}_{E}=\nu|\hskip 0.7pte_{0}\rangle\langle e_{0}\hskip
0.7pt|+(-1)^{z}\sqrt{\nu(1-\nu)}(|\hskip 0.7pte_{0}\rangle\langle e_{1}\hskip
0.7pt|+\mathrm{h.c.})+(1-\nu)|\hskip 0.7pte_{1}\rangle\langle e_{1}\hskip
0.7pt|,$ (B.112)

with eigenvalues $\\{0,1\\}$. Then, the conditional entropy of $\tau(\nu)$ can
be computed in terms of the parameter $\nu$ as follows:

$\displaystyle H(Z|E)_{\tau(\nu)}$
$\displaystyle=H(E|Z)_{\tau(\nu)}+H(Z)_{\tau(\nu)}-H(E)_{\tau(\nu)}$
$\displaystyle=1-h(\nu)$ (B.113)

where $h(x)$ is the binary entropy.

The second step is to derive the maximal Bell value achievable by the state
$\tau(\nu)$. We parametrize the parties’ observables and partially fix the
measurement angles888The measurement angle $a_{0}$ of Alice’s observable
$A_{0}$ is already fixed by the fact that we chose to study the conditional
entropy of Alice’s $Z$ outcome when the parties share the state $\tau(\nu)$.
$a_{0},b_{+}$ and $c_{+}$ as in Subsec. B.1.1. Then, by computing the Bell
value (B.8) for the state $\tau(\nu)$, we get:

$\displaystyle\beta^{\tau(\nu)}_{\mathrm{H}}=(2\nu-1)\sin a_{1}\cos b_{-}\cos
c_{-}+\sin b_{-}+\sin c_{-}-\sin b_{-}\sin c_{-}$ (B.114)

The above expression can be maximized over the remaining measurement
directions $a_{1}$, $b_{-}$ and $c_{-}$ thus yielding the following maximal
Bell value:

$\displaystyle\beta^{\tau(\nu)}_{\mathrm{H}}=2\nu+\frac{1}{2\nu}-1,$ (B.115)

with corresponding optimal angles:

$\displaystyle a_{1}=\frac{\pi}{2}\quad,\quad
b_{-}=\arctan\frac{1}{\sqrt{4\nu^{2}-1}}\quad,\quad
c_{-}=\arcsin\frac{1}{2\nu}.$ (B.116)

By reverting (B.115) and by inserting the result in (B.113), we express the
conditional entropy of $\tau(\nu)$ in terms of its achievable Bell value
$\beta^{\tau(\nu)}_{\mathrm{H}}$:

$H(Z|E)_{\tau(\nu)}=1-h\left[\frac{1}{4}\left(\beta^{\tau(\nu)}_{\mathrm{H}}+1+\sqrt{(\beta^{\tau(\nu)}_{\mathrm{H}})^{2}+2\beta^{\tau(\nu)}_{\mathrm{H}}-3}\right)\right],$
(B.117)

which coincides with the lower bound (B.1) on the conditional entropy of
Alice’s $A_{0}$ outcome. This proves that the entropy bound in (B.1) is tight,
since there exists an honest implementation that attains it.

Interestingly, we notice that the optimal measurement angles (B.116) of Bob
and Charlie for the state (B.109) are given by $b_{-}=c_{-}=\pi/2$ when
$\nu\to 1/2$. In other words, the optimal observables of Bob and Charlie that
maximize the Bell value tend to be compatible ($B_{0}=-B_{1}$ and
$C_{0}=-C_{1}$) when $\tau(\nu)$ tends to a separable state. This fact has
been recently observed in [59] for the CHSH inequality [34], where less
incompatible observables yield higher Bell values while demanding less
entanglement from the state.

##  Appendix C Proof of one-outcome entropy bound for MABK inequality

The proof technique based on the uncertainty relation, used to derive the
entropy bound of Theorem 1, can be easily adapted to obtain further entropy
bounds.

In this Appendix we rederive the conditional entropy bound on Alice’s outcome
$A_{0}$ when Alice, Bob and Charlie test the MABK inequality [42, 43, 44].
This bound was first obtained in [46] via a correspondence between the MABK
inequality and its bipartite counterpart, the CHSH inequality [34]. The bound
is also obtained in [47] by direct minimization of the conditional entropy for
a fixed violation $\beta_{\mathrm{M}}$. We report the bound for clarity.

######  Theorem 2.

Let Alice, Bob and Charlie test the MABK inequality [42, 43, 44] and obtain a
Bell value of $\beta_{\mathrm{M}}$. Then, the von Neumann entropy of Alice’s
outcome $A_{0}$ conditioned on Eve’s information $E$ satisfies

$H(A_{0}|E)\geq
1-h\left(\frac{1}{2}+\frac{1}{2}\sqrt{\frac{\beta_{\mathrm{M}}^{2}}{8}-1}\right),$
(C.1)

where $h(x)=-x\log_{2}x+(1-x)\log_{2}(1-x)$ is the binary entropy.

We point out that the bound in (C.1) also holds for Alice’s observable $A_{1}$
due to the symmetry of the MABK inequality.

###### Proof.

From [47], we know that we can reduce the state shared by the parties to a
three-qubit state and their measurements to rank-one projective measurements
on their respective qubits.

We start by deriving an upper bound on the three-party MABK value. In order to
do so, we consider its expression as obtained from the recursive definition
[47]:

$\displaystyle\beta_{M}=\frac{1}{2}\braket{\left[A_{0}(B_{0}+B_{1})+A_{1}(B_{0}-B_{1})\right](C_{0}+C_{1})+\left[A_{1}(B_{0}+B_{1})-A_{0}(B_{0}-B_{1})\right](C_{0}-C_{1})},$
(C.2)

and we exploit the degrees of freedom in the choice of the local reference
frames to impose that every party’s observable lies in the $(x,y)$ plane of
the Bloch sphere:

$\displaystyle A_{i}$ $\displaystyle=X\cos a_{i}+Y\sin a_{i}$ (C.3)
$\displaystyle B_{i}$ $\displaystyle=X\cos b_{i}+Y\sin b_{i}$ (C.4)
$\displaystyle C_{i}$ $\displaystyle=X\cos c_{i}+Y\sin c_{i}.$ (C.5)

Moreover, we rotate Alice’s reference frame along the $z$ axis such that:

$\displaystyle A_{0}$ $\displaystyle=X$ (C.6) $\displaystyle A_{1}$
$\displaystyle=X\cos a+Y\sin a.$ (C.7)

We define new operators $B_{\pm}:=(B_{0}\pm B_{1})/2$ and $C_{\pm}:=(C_{0}\pm
C_{1})/2$ for Bob and Charlie, such that they can be recast as follows:

$\displaystyle B_{+}$ $\displaystyle=\cos b_{-}(X\cos b_{+}+Y\sin b_{+})$
(C.8) $\displaystyle B_{-}$ $\displaystyle=-\sin b_{-}(X\sin b_{+}-Y\cos
b_{+})$ (C.9) $\displaystyle C_{+}$ $\displaystyle=\cos c_{-}(X\cos
c_{+}+Y\sin c_{+})$ (C.10) $\displaystyle C_{-}$ $\displaystyle=-\sin
c_{-}(X\sin c_{+}-Y\cos c_{+}),$ (C.11)

where $b_{\pm}:=(b_{0}\pm b_{1})/2$ and $c_{\pm}:=(c_{0}\pm c_{1})/2$. We
rotate Bob’s and Charlie’s reference frames such that $b_{+}=c_{+}=0$. By
inserting everything in (C.2) we obtain the following simplified MABK value:

$\displaystyle\beta_{M}$ $\displaystyle=2\cos b_{-}\cos
c_{-}\braket{XXX}-2\sin b_{-}\sin c_{-}\braket{XYY}$
$\displaystyle+2\braket{(X\cos a+Y\sin a)(\sin b_{-}\cos c_{-}YX+\cos
b_{-}\sin c_{-}XY)}$ $\displaystyle=:2\,\vec{V}\cdot\vec{W},$ (C.12)

where in the last line we defined the vectors:

$\displaystyle\vec{V}$
$\displaystyle=\left(\braket{XXX},\braket{XYY},\braket{XYX},\braket{XXY},\braket{YYX},\braket{YXY}\right)$
(C.13) $\displaystyle\vec{W}$ $\displaystyle=\left(\cos b_{-}\cos c_{-},-\sin
b_{-}\sin c_{-},\cos a\sin b_{-}\cos c_{-},\cos a\cos b_{-}\sin c_{-},\sin
a\sin b_{-}\cos c_{-},\right.$ $\displaystyle\left.\sin a\cos b_{-}\sin
c_{-}\right).$ (C.14)

By the Cauchy-Schwarz inequality and by observing that $\left\|W\right\|=1$,
we obtain:

$\displaystyle\beta_{M}\leq
2\sqrt{\braket{YXY}^{2}+\braket{YYX}^{2}+(\braket{XXX}^{2}+\braket{XXY}^{2})+(\braket{XYY}^{2}+\braket{XYX}^{2})}.$
(C.15)

We now prove that for a generic three-qubit state the following inequalities
hold:

$\displaystyle\braket{XXX}^{2}+\braket{XXY}^{2}$ $\displaystyle\leq 1$ (C.16)
$\displaystyle\braket{XYY}^{2}+\braket{XYX}^{2}$ $\displaystyle\leq 1.$ (C.17)

To show this, we write the generic three-qubit state in the GHZ basis:

$\rho=\sum_{i,j,k=0}^{1}\lambda_{ijk}|\hskip
0.7pt\psi_{i,j,k}\rangle\langle\psi_{i,j,k}\hskip
0.7pt|+\sum_{j,k=0}^{1}\left(c_{jk}|\hskip
0.7pt\psi_{0,j,k}\rangle\langle\psi_{1,j,k}\hskip
0.7pt|+\mathrm{h.c.}\right)+\dots,$ (C.18)

where $\lambda_{ijk}$ are the real diagonal elements, $c_{jk}$ are the complex
coherences between the states $\ket{\psi_{0,j,k}}$ and $\ket{\psi_{1,j,k}}$
and $\mathrm{h.c.}$ stands for the Hermitian conjugate of the term preceding
it. In the dots “$\dots$” we include every other coherence of the state
$\rho$, since they do not play a role in the expectation values appearing in
(C.16) and (C.17). This can be readily seen by considering the action of the
operators of (C.16) and (C.17) on the states of the GHZ basis:

$\displaystyle XXX\ket{\psi_{i,j,k}}$
$\displaystyle=(-1)^{i}\ket{\psi_{i,j,k}}$ (C.19) $\displaystyle
XXY\ket{\psi_{i,j,k}}$
$\displaystyle=-\mathbbm{i}(-1)^{i+k}\ket{\psi_{\bar{i},j,k}}$ (C.20)
$\displaystyle XYY\ket{\psi_{i,j,k}}$
$\displaystyle=(-1)^{i+j+k+1}\ket{\psi_{i,j,k}}$ (C.21) $\displaystyle
XYX\ket{\psi_{i,j,k}}$
$\displaystyle=-\mathbbm{i}(-1)^{i+j}\ket{\psi_{\bar{i},j,k}},$ (C.22)

which yield the following expectation values on $\rho$:

$\displaystyle\braket{XXX}$
$\displaystyle=\sum_{j,k=0}^{1}\lambda_{0jk}-\lambda_{1jk}$ (C.23)
$\displaystyle\braket{XXY}$
$\displaystyle=\sum_{j,k=0}^{1}(-1)^{k}2\mathrm{Im}(c_{jk})$ (C.24)
$\displaystyle\braket{XYY}$
$\displaystyle=-\sum_{j,k=0}^{1}(-1)^{j+k}(\lambda_{0jk}-\lambda_{1jk})$
(C.25) $\displaystyle\braket{XYX}$
$\displaystyle=\sum_{j,k=0}^{1}(-1)^{j}2\mathrm{Im}(c_{jk}).$ (C.26)

Before proving (C.16) and (C.17), we need to derive a couple of conditions
satisfied by the parameters describing the state $\rho$. Since $\rho\geq 0$,
then also its restriction to every $2$-dimensional subspace spanned by
$\\{\ket{\psi_{0,j,k}},\ket{\psi_{1,j,k}}\\}$ (for $j,k\in\\{0,1\\}$) must be
positive-semidefinite. Indeed, let $P_{jk}:=|\hskip
0.7pt\psi_{0,j,k}\rangle\langle\psi_{0,j,k}\hskip 0.7pt|+|\hskip
0.7pt\psi_{1,j,k}\rangle\langle\psi_{1,j,k}\hskip 0.7pt|$ be the projector on
such a subspace. Then,

$\displaystyle P_{jk}\rho P_{jk}\geq 0\quad\iff\quad\lambda_{0jk}|\hskip
0.7pt\psi_{0,j,k}\rangle\langle\psi_{0,j,k}\hskip 0.7pt|+\lambda_{1jk}|\hskip
0.7pt\psi_{1,j,k}\rangle\langle\psi_{1,j,k}\hskip 0.7pt|+c_{jk}|\hskip
0.7pt\psi_{0,j,k}\rangle\langle\psi_{1,j,k}\hskip 0.7pt|+\mathrm{h.c.}\geq 0,$
(C.27)

for all $j$ and $k$. A necessary condition for (C.27) is that its determinant
is non-negative (the eigenvalues of positive-semidefinite operators are all
non-negative):

$\lambda_{0jk}\lambda_{1jk}\geq\left|c_{jk}\right|^{2}\geq\mathrm{Im}^{2}(c_{jk}).$
(C.28)

A second condition on the state’s parameters can be found starting from the
following chain of equivalences (note that $\lambda_{ijk}\geq 0$ follows from
$\rho\geq 0$):

$\displaystyle
2\left(\sqrt{\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}}-\sqrt{\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}}\right)^{2}\geq
0$
$\displaystyle\iff\,\,2\left(\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}+\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}-2\sqrt{\lambda_{0jk}\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\lambda_{1j^{\prime}k^{\prime}}}\right)\geq
0$
$\displaystyle\iff\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}+\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\geq
4\sqrt{\lambda_{0jk}\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\lambda_{1j^{\prime}k^{\prime}}}-\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}-\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}.$
(C.29)

By using (C.28) twice we can lower bound the square-root on the rhs as
follows:

$\displaystyle
4\sqrt{\lambda_{0jk}\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\lambda_{1j^{\prime}k^{\prime}}}\geq
4\left|\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}})\right|\geq
4(-1)^{k+k^{\prime}}\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}}).$
(C.30)

By employing the last expression in (C.29), we obtain the second condition
needed to prove inequalities (C.16) and (C.17):

$\displaystyle\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}+\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\geq
4(-1)^{k+k^{\prime}}\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}})-\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}-\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}.$
(C.31)

We now proceed on proving (C.16). The lhs of (C.16) can be expressed using
(C.23) and (C.24) as follows:

$\displaystyle\braket{XXX}^{2}+\braket{XXY}^{2}$
$\displaystyle=\left[\sum_{j,k=0}^{1}(\lambda_{0jk}-\lambda_{1jk})\right]^{2}+4\left[\sum_{j,k=0}^{1}(-1)^{k}\mathrm{Im}(c_{jk})\right]^{2}$
$\displaystyle=\sum_{j,k=0}^{1}\left[(\lambda_{0jk}-\lambda_{1jk})^{2}+4\mathrm{Im}^{2}(c_{jk})\right]$
$\displaystyle+2\sum_{(j,k)\neq(j^{\prime},k^{\prime})}\left[(\lambda_{0jk}-\lambda_{1jk})(\lambda_{0j^{\prime}k^{\prime}}-\lambda_{1j^{\prime}k^{\prime}})+4(-1)^{k+k^{\prime}}\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}})\right]$
$\displaystyle=\sum_{j,k=0}^{1}\left[(\lambda_{0jk}-\lambda_{1jk})^{2}+4\mathrm{Im}^{2}(c_{jk})\right]$
$\displaystyle+2\sum_{(j,k)\neq(j^{\prime},k^{\prime})}\left[\lambda_{0jk}\lambda_{0j^{\prime}k^{\prime}}+\lambda_{1jk}\lambda_{1j^{\prime}k^{\prime}}+4(-1)^{k+k^{\prime}}\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}})-\lambda_{0jk}\lambda_{1j^{\prime}k^{\prime}}-\lambda_{1jk}\lambda_{0j^{\prime}k^{\prime}}\right].$
(C.32)

We now upper bound the above expression by using (C.28) in the first sum and
(C.31) in the second sum. We obtain:

$\displaystyle\braket{XXX}^{2}+\braket{XXY}^{2}$
$\displaystyle\leq\sum_{j,k=0}^{1}(\lambda_{0jk}+\lambda_{1jk})^{2}+2\sum_{(j,k)\neq(j^{\prime},k^{\prime})}(\lambda_{0jk}+\lambda_{1jk})(\lambda_{0j^{\prime}k^{\prime}}+\lambda_{1j^{\prime}k^{\prime}})$
$\displaystyle=\left[\sum_{j,k=0}^{1}\lambda_{0jk}+\lambda_{1jk}\right]^{2}=1,$
(C.33)

where in the last equality we used the fact that $\operatorname{Tr}\rho=1$.
This proves (C.16).

Similarly, the lhs of (C.17) can be expressed through (C.25) and (C.26) and
upper bounded using (C.28) and (C.31) as follows:

$\displaystyle\braket{XYY}^{2}+\braket{XYX}^{2}$
$\displaystyle=\sum_{j,k=0}^{1}\left[(\lambda_{0jk}-\lambda_{1jk})^{2}+4\mathrm{Im}^{2}(c_{jk})\right]$
$\displaystyle+2\sum_{(j,k)\neq(j^{\prime},k^{\prime})}(-1)^{j+j^{\prime}+k+k^{\prime}}\left[(\lambda_{0jk}-\lambda_{1jk})(\lambda_{0j^{\prime}k^{\prime}}-\lambda_{1j^{\prime}k^{\prime}})+4(-1)^{k+k^{\prime}}\mathrm{Im}(c_{jk})\mathrm{Im}(c_{j^{\prime}k^{\prime}})\right]$
$\displaystyle\leq\sum_{j,k=0}^{1}(\lambda_{0jk}+\lambda_{1jk})^{2}+2\sum_{(j,k)\neq(j^{\prime},k^{\prime})}(-1)^{j+j^{\prime}+k+k^{\prime}}(\lambda_{0jk}+\lambda_{1jk})(\lambda_{0j^{\prime}k^{\prime}}+\lambda_{1j^{\prime}k^{\prime}})$
$\displaystyle=\left[\sum_{j,k=0}^{1}(-1)^{j+k}(\lambda_{0jk}+\lambda_{1jk})\right]^{2}\leq
1,$ (C.34)

which proves (C.17).

Finally, by employing (C.16) and (C.17) in (C.15) we derive the following
upper bound on the MABK value:

$\displaystyle\beta_{M}\leq
2\sqrt{2+2\max\\{\left|\braket{YXY}\right|,\left|\braket{YYX}\right|\\}^{2}}.$
(C.35)

Now, we can turn to the conditional entropy of Alice’s outcome when she
measures $A_{0}=X$ and lower bound it with the uncertainty relation [49]:

$\displaystyle H(X|E)\geq 1-H(Y|BC)\geq 1-h(Q_{Y,O_{B}O_{C}}),$ (C.36)

where in the second inequality we followed the same steps that lead to (B.58)
and (B.59), and where $Q_{Y,O_{B}O_{C}}$ is defined as the probability that
the $Y$ outcome of Alice differs from the product of the outcomes of Bob
($O_{B}$) and Charlie ($O_{C}$), that is:

$Q_{Y,O_{B}O_{C}}:=\Pr[YO_{B}O_{C}=-1]=\frac{1-\braket{YO_{B}O_{C}}}{2}.$
(C.37)

Moreover, by using the properties of the binary entropy $h(x)$ as in (B.60),
we obtain:

$\displaystyle H(X|E)\geq
1-h\left(\frac{1+\left|\braket{YO_{B}O_{C}}\right|}{2}\right).$ (C.38)

We emphasize that we can employ the uncertainty relation and derive a bound
like the one in (C.38) independently of the measurement settings $A_{i}$,
$B_{i}$ and $C_{i}$ of the parties in the DI scenario. Of course, in order to
make the derived inequality useful in our case, we set one of Alice’s
measurements to $X$ –which is also one of Alice’s settings in the DI scenario,
see (C.6)– so that we obtain an inequality for the conditional entropy
$H(X|E)$. For this argument, we can arbitrarily choose Bob’s and Charlie’s
measurements in (C.38) to be either $O_{B}=X$ and $O_{C}=Y$ or $O_{B}=Y$ and
$O_{C}=X$. Both cases lead to valid lower bounds on the conditional entropy of
Alice’s $X$ outcome. We can then lower bound the conditional entropy $H(X|E)$
by:

$\displaystyle H(X|E)\geq
1-h\left(\frac{1+\max\\{\left|\braket{YXY}\right|,\left|\braket{YYX}\right|\\}}{2}\right).$
(C.39)

By reverting the upper bound on the MABK value (C.35), we obtain:

$\max\\{\left|\braket{YXY}\right|,\left|\braket{YYX}\right|\\}\geq\sqrt{\frac{\beta_{\mathrm{M}}^{2}}{8}-1}.$
(C.40)

Finally, by employing (C.40) in (C.39) we recover the entropy bound (C.1).

We point out that, although the proof derives a lower bound on the conditional
entropy of Alice’s $X$-basis outcome, the derived bound is general and holds
for any measurement Alice implements. This is because at the beginning of the
proof, we purposely set Alice’s local reference frame such that her qubit
observable $A_{0}$ coincides with the Pauli operator $X$. This has no effect
on the description of the state (C.18) shared by Alice, Bob and Charlie since
it is a completely generic three-qubit state for any choice of local reference
frames. ∎

##  Appendix D Numerical computation of two-outcome entropy bounds

In this Appendix we describe the steps that allow us to numerically compute
lower bounds on the two-outcome entropy $H(A_{0}B_{0}|E)$ for the Holz
inequality, the Parity-CHSH inequality and the CHSH inequality. The bounds are
plotted in Fig. 2 and are used in Sec. 5 to compare the performance of DIRE
protocols based on different Bell inequalities.

##### Holz inequality

We compute a numerical lower bound on $H(A_{0}B_{0}|E)$ as a function of the
violation $\beta_{\mathrm{H}}$ of the Holz inequality for three parties. The
bound is obtained by direct optimization of the entropy once the violation is
fixed. Based on the numerical bound, we also conjecture the correspondent
analytical expression (Conjecture 1).

In order to make the optimization numerically feasible, we arbitrarily fix the
local reference frames of Alice, Bob and Charlie and parametrize the state
they share as shown in Subsec. B.1. Then, the measurement angle $b_{1}$ is
fixed by $b_{0}$ through (B.17) as follows: $b_{1}=\pi-b_{0}$, which implies
that $b_{-}=b_{0}-\pi/2$. By substituting in the Bell value (B.8) of the Holz
inequality, we obtain:

$v_{\mathrm{H}}=\left(\cos a_{1}\braket{ZXX}+\sin a_{1}\braket{XXX}\right)\sin
b_{0}\cos c_{-}-\cos b_{0}\braket{ZZ\mathbbm{1}}+\sin
c_{-}\braket{Z\mathbbm{1}Z}+\cos b_{0}\sin c_{-}\braket{\mathbbm{1}ZZ},$ (D.1)

which is now written in terms of the free measurement angles $a_{1}$ and
$c_{-}$ and the angle $b_{0}$ that instead appears in the conditional entropy
expression. Note that, thanks to the parametrization of the state in Subsec.
B.1.2, the expectation values in (D.1) are easily written in terms of the
state parameters $\\{\rho_{ijk},t_{jk}\\}$ as follows:

$\displaystyle\braket{XXX}$
$\displaystyle=\sum_{j,k=0}^{1}(\rho_{0jk}-\rho_{1jk})\cos(2t_{jk})$ (D.2)
$\displaystyle\braket{ZXX}$
$\displaystyle=\sum_{j,k=0}^{1}(\rho_{0jk}-\rho_{1jk})\sin(2t_{jk})$ (D.3)
$\displaystyle\braket{ZZ\mathbbm{1}}$
$\displaystyle=\sum_{j,k=0}^{1}(-1)^{j}(\rho_{0jk}-\rho_{1jk})\cos(2t_{jk})$
(D.4) $\displaystyle\braket{Z\mathbbm{1}Z}$
$\displaystyle=\sum_{j,k=0}^{1}(-1)^{k}(\rho_{0jk}-\rho_{1jk})\cos(2t_{jk})$
(D.5) $\displaystyle\braket{\mathbbm{1}ZZ}$
$\displaystyle=\sum_{j,k=0}^{1}(-1)^{j+k}(\rho_{0jk}+\rho_{1jk}).$ (D.6)

By also expressing the entropy $H(A_{0}B_{0}|E)$ in terms of the state
parameters $\\{\rho_{ijk},t_{jk}\\}$ and the angle $b_{0}$ (recall that
Alice’s reference frame is chosen such that $a_{0}=0$), we numerically solve
the following optimization problem:

$\displaystyle\min_{\\{\rho_{ijk},t_{jk},b_{0},a_{1},c_{-}\\}}H(A_{0}B_{0}|E)(\rho_{ijk},t_{jk},b_{0})$
$\displaystyle\quad\quad\mbox{sub. to}\quad
v_{\mathrm{H}}(\rho_{ijk},t_{jk},b_{0},a_{1},c_{-})=\beta_{\mathrm{H}}\,;\,{\textstyle\sum_{ijk}}\,\rho_{ijk}=1\,;\,\rho_{ijk}\geq
0,$ (D.7)

when varying $\beta_{\mathrm{H}}$ in the interval $(1,3/2]$. In reality, we
solve the equivalent –in the sense that leads to the same entropy for every
$\beta_{\mathrm{H}}$– but simpler optimization problem:

$\displaystyle\min_{\\{\rho_{ijk},t_{jk},b_{0}\\}}H(A_{0}B_{0}|E)(\rho_{ijk},t_{jk},b_{0})$
$\displaystyle\quad\mbox{sub.
to}\quad\bar{v}_{\mathrm{H}}(\rho_{ijk},t_{jk},b_{0})\geq\beta_{\mathrm{H}}\,;\,{\textstyle\sum_{ijk}}\,\rho_{ijk}=1\,;\,\rho_{ijk}\geq
0,$ (D.8)

where $\bar{v}_{\mathrm{H}}$ is the maximum of (D.1) over the free angles
$a_{1}$ and $c_{-}$:

$\displaystyle\bar{v}_{\mathrm{H}}=\sqrt{\sin^{2}b_{0}(\braket{ZXX}^{2}+\braket{XXX}^{2})+\left(\braket{Z\mathbbm{1}Z}+\cos
b_{0}\braket{\mathbbm{1}ZZ}\right)^{2}}-\cos b_{0}\braket{ZZ\mathbbm{1}}.$
(D.9)

The numerical plot points obtained by solving (D.8) with the built-in
functions of Wolfram Mathematica [60] are reported in Fig. 3, together with
our conjectured bound on $H(A_{0}B_{0}|E)$. Our conjecture on the analytical
expression of the entropy bound is given in Conjecture 1.

##### Parity-CHSH inequality

The bound on $H(A_{0}B_{0}|E)$ when three parties test the Parity-CHSH
inequality is also obtained by direct numerical optimization, similarly to the
bound for the Holz inequality.

As a matter of fact, note that the Parity-CHSH inequality (3) is a particular
case (upon relabeling the observables) of the Holz inequality (2) for
$c_{0}=c_{1}$, i.e., when Charlie’s two measurements coincide. Thus, the
optimization problem yielding the entropy bound for the Parity-CHSH inequality
is equal to (D.7), where we set $c_{-}=0$.

##### CHSH inequality

The bound on $H(A_{0}B_{0}|E)$ when two parties test the CHSH inequality is
again obtained by direct numerical optimization. In order to simplify the
optimization, we apply the results of [47] to the CHSH scenario and
parametrize the state shared by Alice and Bob as a Bell-diagonal state:

$\rho=\sum_{i,j=0}^{1}\lambda_{ij}|\hskip
0.7pt\psi_{i,j}\rangle\langle\psi_{i,j}\hskip 0.7pt|,$ (D.10)

where $\ket{\psi_{i,j}}=(\ket{0j}+(-1)^{i}\ket{1\bar{j}})/\sqrt{2}$ are the
states of the Bell basis. We also assume without loss of generality that the
parties’ observables are rank-one projective measurements in the
$(x,y)$-plane, defined by the eigenstates:

$\displaystyle\ket{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}+(-1)^{a}e^{\mathbbm{i}\varphi_{A_{k}}}\ket{1}\right)$
(D.11) $\displaystyle\ket{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{0}+(-1)^{b}e^{\mathbbm{i}\varphi_{B_{k}}}\ket{1}\right)$
(D.12)

where $a,b\in\\{0,1\\}$ are the outcomes of Alice’s and Bob’s observables
$A_{k}$ and $B_{k}$ and $\varphi_{A_{k}},\varphi_{B_{k}}$ are the
corresponding measurement directions, respectively. Then, one can compute the
joint probability of obtaining outcomes $a$ and $b$ when Alice and Bob
measured $A_{k}$ and $B_{l}$. We obtain:

$\displaystyle
p(a,b|k,l)=\frac{1}{4}\left[1+(-1)^{a+b}\cos(\varphi_{A_{k}}+\varphi_{B_{l}})(\lambda_{00}-\lambda_{10})+(-1)^{a+b}\cos(\varphi_{A_{k}}-\varphi_{B_{l}})(\lambda_{01}-\lambda_{11})\right],$
(D.13)

and observe that $p(0,0|k,l)=p(1,1|k,l)$ and
$p(0,1|k,l)=p(1,0|k,l)=1/2-p(0,0|k,l)$. We can thus express all the
probabilities appearing in the conditional entropy $H(A_{0}B_{0}|E)$ in terms
of $p:=p(0,0|0,0)$.

We can now derive a simple expression for the conditional entropy of interest:

$\displaystyle H(A_{0}B_{0}|E)$
$\displaystyle=H(A_{0}B_{0})+H(E|A_{0}B_{0})-H(E)$
$\displaystyle=-2p\log_{2}p-2(1/2-p)\log_{2}(1/2-p)-H(\\{\lambda_{ij}\\})$
$\displaystyle=1+h(2p)-H(\\{\lambda_{ij}\\}),$ (D.14)

where in the second equality we used the fact that the state shared by Alice,
Bob and Eve is pure (hence $H(E)=H(\rho)$) and the conditional state
$\rho^{a,b}_{E}$ of Eve, given that Alice and Bob obtained outcomes $a$ and
$b$, is still pure (thus $H(E|A_{0}B_{0})=0$).

The CHSH Bell value, for the parametrization described above, reduces to:

$\displaystyle v_{\mathrm{C}}$
$\displaystyle=\braket{A_{0}B_{0}}+\braket{A_{0}B_{1}}+\braket{A_{1}B_{0}}-\braket{A_{1}B_{1}}$
$\displaystyle=(\lambda_{00}-\lambda_{10})(\cos(\varphi_{A_{0}}+\varphi_{B_{0}})+\cos(\varphi_{A_{0}}+\varphi_{B_{1}})+\cos(\varphi_{A_{1}}+\varphi_{B_{0}})-\cos(\varphi_{A_{1}}+\varphi_{B_{1}}))$
$\displaystyle+(\lambda_{01}-\lambda_{11})(\cos(\varphi_{A_{0}}-\varphi_{B_{0}})+\cos(\varphi_{A_{0}}-\varphi_{B_{1}})+\cos(\varphi_{A_{1}}-\varphi_{B_{0}})-\cos(\varphi_{A_{1}}-\varphi_{B_{1}})).$
(D.15)

We then numerically solved the following optimization problem with the built-
in functions of Wolfram Mathematica [60]:

$\displaystyle\min_{\\{\lambda_{ij},\varphi_{A_{0}},\varphi_{B_{0}},\varphi_{A_{1}},\varphi_{B_{1}}\\}}1+h(2p)-H(\\{\lambda_{ij}\\})$
$\displaystyle\quad\quad\mbox{sub. to}\quad
v_{\mathrm{C}}(\lambda_{ij},\varphi_{A_{0}},\varphi_{B_{0}},\varphi_{A_{1}},\varphi_{B_{1}})=\beta_{\mathrm{C}}\,;\,{\textstyle\sum_{ij}}\,\lambda_{ij}=1\,;\,\lambda_{ij}\geq
0,$ (D.16)

where $p$ is given by:

$\displaystyle
p=\frac{1}{4}\left[1+\cos(\varphi_{A_{0}}+\varphi_{B_{0}})(\lambda_{00}-\lambda_{10})+\cos(\varphi_{A_{0}}-\varphi_{B_{0}})(\lambda_{01}-\lambda_{11})\right].$
(D.17)

The numerical solution of (D.16) is the entropy bound reported in Fig. 2. We
remark that the same bound has been independently computed in [38] by
combining an analytical simplification similar to the one reported here with
numerical techniques.

##  Appendix E Tightness of one-outcome entropy bound for Parity-CHSH
inequality

In this Appendix we demonstrate that the lower bound on the entropy of Alice’s
outcome $A_{0}$ when three parties test the Parity-CHSH inequality, reported
in (A.14), is tight.

######  Lemma 3.

Let Alice, Bob and Charlie test the Parity-CHSH inequality [20] and obtain a
Bell value of $\beta_{\mathrm{pC}}$. Then, the following lower bound on the
von Neumann entropy of Alice’s outcome $A_{0}$, conditioned on Eve’s
information $E$,

$H(A_{0}|E)\geq
1-h\left(\frac{1}{2}+\frac{1}{2}\sqrt{\left(\beta_{\mathrm{pC}}\right)^{2}-1}\right),$
(E.1)

is tight. Namely, that there exists a quantum state and a set of measurements
yielding a Bell value of $\beta_{\mathrm{pC}}$ with conditional entropy of
Alice’s outcome $A_{0}$ given by the rhs of (E.1).

###### Proof.

Consider the same family of states used to prove the tightness of the bound in
(11), that is:

$\tau(\nu)=\nu|\hskip 0.7pt\psi_{0,0,0}\rangle\langle\psi_{0,0,0}\hskip
0.7pt|+(1-\nu)|\hskip 0.7pt\psi_{1,0,0}\rangle\langle\psi_{1,0,0}\hskip
0.7pt|,$ (E.2)

where $\nu\in[1/2,1]$. Then, the conditional entropy of Alice’s outcome
$A_{0}=Z$ can be computed in terms of the parameter $\nu$ and reads:

$\displaystyle H(A_{0}|E)_{\tau(\nu)}=1-h(\nu).$ (E.3)

Now, we compute the maximal violation of the Parity-CHSH inequality (3)
achieved by the state $\tau(\nu)$. To do this, we first parametrize the
parties’ observables as in Subsec. B.1.1 and orient the reference frames such
that $C=X$, $A_{0}=Z$ and $b_{+}=0$ (thus $B_{+}=\cos b_{0}Z$ and $B_{-}=\sin
b_{0}X$). Then, the Bell value of the Parity-CHSH inequality reads:

$\displaystyle\beta_{\rm pC}=\sin b_{0}(\cos a_{1}\braket{ZXX}+\sin
a_{1}\braket{XXX})+\cos b_{0}\braket{ZZ\mathbbm{1}}.$ (E.4)

By computing the expectation values on the state $\tau(\nu)$, we obtain:

$\displaystyle\beta^{\tau(\nu)}_{\mathrm{pC}}=\sin b_{0}\sin
a_{1}(2\nu-1)+\cos b_{0}.$ (E.5)

The above expression can be maximized over the remaining measurement
directions $a_{1}$ and $b_{0}$ yielding the following maximal Bell value:

$\displaystyle\beta^{\tau(\nu)}_{\mathrm{pC}}=\sqrt{(2\nu-1)^{2}+1}.$ (E.6)

By reverting the last expression we obtain:

$\displaystyle\nu=\frac{1}{2}+\frac{1}{2}\sqrt{(\beta^{\tau(\nu)}_{\mathrm{pC}})^{2}-1}$
(E.7)

which substituted in (E.3) returns exactly the lower bound in (E.1). Hence we
proved that the bound is tight. ∎