aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USAbbinstitutetext: Martin Fisher School of Physics, Brandeis
University, Waltham, MA 02453, USA

# Entropy Cones and Entanglement Evolution for Dicke States

William Munizzi b Howard J. Schnitzer<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The $N$-qubit Dicke states ${|{D^{N}_{k}}\rangle}$, of Hamming-weight $k$, are
a class of entangled states which play an important role in quantum algorithm
optimization. We present a general calculation of entanglement entropy in
Dicke states, which we use to describe the ${|{D^{N}_{k}}\rangle}$ entropy
cone. We demonstrate that all ${|{D^{N}_{k}}\rangle}$ entropy vectors emerge
symmetrized, and use this to define a min-cut protocol on star graphs which
realizes ${|{D^{N}_{k}}\rangle}$ entropy vectors. We identify the stabilizer
group for all ${|{D^{N}_{k}}\rangle}$, under the action of the $N$-qubit Pauli
group and two-qubit Clifford group, which we use to construct
${|{D^{N}_{k}}\rangle}$ reachability graphs. We use these reachability graphs
to analyze and bound evolution of ${|{D^{N}_{k}}\rangle}$ entropy vectors in
Clifford circuits.

##  1 Introduction

There are numerous ways to classify sets of quantum states. For states in a
factorizable Hilbert space $\mathcal{H}=\bigotimes^{N}_{i}\mathcal{H}_{i}$,
one such classification is given by considering the entropy vector of each
state Bao2015 . The entropy vector of a pure state
${|{\psi}\rangle}\in\mathcal{H}$ is constructed as the ordered set of all
$2^{N}-1$ von Neumann entropies, computed by tracing out all tensor product
states ${|{\psi_{i}}\rangle}\in\mathcal{H}_{i}$. While every
${|{\psi}\rangle}\in\mathcal{H}$ can be assigned an entropy vector, specifying
an entropy vector does not uniquely determine a state. Instead, entropy
vectors describe an equivalence relation on states in $\mathcal{H}$, assigning
each state to a class based on its entanglement structure.

The space of allowed entropy vectors for a specific class of states defines
the entropy cone for that class Schnitzer:2022exe ; Linden2013 ; Bao2020 ;
Bao2020a . Since a particular entanglement structure often accompanies other
interesting state characteristics, identifying specific entropy cones which
contain the entropy vectors for different state classifications has received
significant research interest. One notable case is that of holographic states,
those with a smooth classical dual geometry via the AdS/CFT correspondence
Maldacena:1997re ; Witten:1998qj , which possess subsystem entanglement that
obeys the Ryu-Takayanagi formula Ryu:2006bv ; Hubeny:2007xt . The entropy
vectors of holographic states are likewise confined to a convex polyhedral
subspace of the ambient vector space, known as the holographic entropy cone.
In general, having an entropy vector that is contained within a particular
entropy cone is a necessary, though not sufficient, condition for a state to
belong to the class described by that cone.

The $N$-qubit Dicke states are a particular class of entangled quantum state
which have received notable recognition for numerous applications in quantum
algorithm development and precision measurement. Dicke states have found
significant use as the initial state for Quantum Approximate Optimization
Algorithm (QAOA) implementation, a quantum algorithm designed to approximate
combinatorial-optimization solutions farhi2014quantum . More recently,
techniques have been established to deterministically prepare $N$-qubit Dicke
states using circuits of depth $\mathcal{O}(n)$ Baertschi2019 . Much of the
NISQ-era utility of Dicke states for quantum computation is due to their
unique entanglement properties. Certain highly-entangled Dicke states may be
projected, via measurement, onto distinct lower-qubit states that cannot be
locally-transformed into each other.

An alternative, and useful, classification on $\mathcal{H}$ considers how
state sets transform under the action of a group. Consider a group $G\in
L(\mathcal{H})$, which transforms states ${|{\psi}\rangle}\in\mathcal{H}$.
Certain elements of $G$ may act trivially on some ${|{\psi}\rangle}$, mapping
the state to itself. Such elements define the stabilizer group for
${|{\psi}\rangle}$ under the action of $G$, which we denote
$\,\textnormal{Stab}\,_{G}({|{\psi}\rangle})$. The most generic quantum states
are stabilized by only the identity operator $\mathbb{1}$ in a given $G$.
Special sets of states, however, have larger stabilizer groups with respect to
certain sets of operators. A well-known example is the set of stabilizer
states, a class of classically-simulable quantum states, which are stabilized
by a maximal number, specifically $2^{N}$, of Pauli group elements
aaronson2004improved ; Gottesman1997 ; Gottesman1998 ; Bravyi2004 ;
knill2004faulttolerant ; Keeler2022 .

Acting on ${|{\psi}\rangle}$ with every $g\in G$ defines the orbit
$G\cdot{|{\psi}\rangle}$, which describes the trajectory of ${|{\psi}\rangle}$
through $\mathcal{H}$. For every finitely-generated $G$ the orbit is discrete,
lending itself to a natural graph-theoretic description. We construct the
graph which corresponds to $G\cdot{|{\psi}\rangle}$, known as a reachability
graph, by assigning vertices to states in the orbit, and edges to represent
generators of $G$. Viewed through the lens of quantum computation, each path
in a reachability graph defines a quantum circuit and the graph itself encodes
the state’s evolution under all possible circuits composed of the generating
gate set. States with isomorphic reachability graphs are congruent as they
share the same stabilizers under the action of the chosen group Keeler:2023xcx
.

We may also wish track the evolution of specific state properties under the
action of a group $G$. Analogous to the subgroups which stabilize a state,
there exist elements of $G$ which leave a particular state property invariant
Keeler2023b , e.g. all local gates preserve entanglement structure. If our
goal is to understand the dynamics of a chosen state parameter, we can
restrict consideration to the subset of $G$ which non-trivially evolves that
parameter. Represented in a reachability graph, this restriction corresponds
to eliminating vertices and edges from the graph, leaving only the graph
paths, i.e. the quantum circuits, which modify the parameter under study. This
modification on reachability graphs allows us to establish bounds on the
dynamics of a chosen state property under circuits composed of a certain gate
set.

In this paper we explore entanglement structure in Dicke states and the manner
in which that entanglement can evolve through quantum circuits. In Section 3,
we leverage the symmetric structure of Dicke states to explicitly compute all
entanglement entropies which arise in $N$-qubit Dicke systems. We use our
calculation of subsystem entanglement to generate all possible Dicke state
entropy vectors, which accordingly describe the $N$-qubit Dicke state entropy
cone. We demonstrate inclusion, and exclusion, of Dicke state entropy vectors
relative to other known entropy cones, and use our calculation to reproduce
the entropy vectors for $W$ states found in Schnitzer:2022exe . Additionally,
we propose a min-cut model on weighted star graphs which realizes the
symmetrized entropies of Dicke states.

In Section 4, we define the stabilizer group for all Dicke states under the
action of the Pauli and Clifford groups. We use this set of stabilizers to
construct reachability graphs Keeler2022 , which illustrate the orbit of Dicke
states under circuits composed of Pauli and Clifford gates. As we are
interested to understand the dynamics of Dicke state entropy vectors in
Clifford circuits, we restrict to a subset of Clifford gates consisting of
Hadamard and CNOT acting on two qubits. By analyzing reachability graphs built
of from these gates, we are able to ultimately bound the evolution of Dicke
state entropy vectors directly using properties of the reachability graph
Keeler2023b .

In forthcoming work, we apply the stabilizing operations identified in Section
4 to construct error-correcting codes for logical Dicke states. One reason for
using Dicke states in error-correcting codes is an increased resistance to
information loss upon single-qubit thermalization. Since tracing out a single
qubit from certain Dicke states recovers the same state, now defined on one
lower qubit-number, the redundancy of such states may yield comparative
advantages in codes. Relatedly, we also consider the potential of highly-
entangled Dicke states for magic distillation protocols. In this context, one
benefit of Dicke states is found in their ease of preparation and
consequential preference as initial states for computation. Another promising
feature of using Dicke states for magic distillation relies on the significant
amount of non-local magic which contained in these states. In both
applications, and perhaps others not considered in this paper, we believe this
exploration into Dicke state entanglement will prove useful.

##  2 Review: Dicke States, Entropy Cones, and the Stabilizer Formalism

We offer a short review of relevant background material used throughout this
work. Comprehensive discussions exist for the different topics covered here,
which we invite the curious reader to consult. Many significant papers discuss
the structure and properties of Dicke states, as well as their utility for
realizable quantum computation, of which we recommend PhysRev.93.99 ;
Baertschi2019 ; nepomechie2023qudit ; PhysRevA.80.052302 ; Stockton_2004 ;
PhysRevLett.103.020503 ; PhysRevA.95.013845 . For additional details on
entropy vectors and entropy cone construction, we suggest Bao2015 ; Hayden2013
; Schnitzer:2022exe ; Avis2023 ; Linden2013 ; Fadel2021 . Finally, the group-
theoretic constructs presented in this section are discussed extensively in
Keeler2022 ; aaronson2004improved ; Gottesman1997 ; Gottesman1998 ; Veitch2013
,and more formally in the text Alperin1995 .

###  2.1 Dicke States

The $N$-qubit Dicke states ${|{D^{N}_{k}}\rangle}$ compose an interesting
class of states which can be efficiently prepared using a polynomial number of
gates, despite having a larger-than-polynomial number $\binom{N}{k}$ of
excitations PhysRev.93.99 ; Baertschi2019 . This property affords significant
resource conservation compared to arbitrary state preparation, which relies on
an application of $\mathcal{O}(2^{N})$ gates. For this reason, Dicke states
often find preference as initial states for quantum optimization algorithms,
and have even been successfully implemented in experiment PhysRevA.95.013845 ;
PhysRevLett.103.020503 ; Stockton_2004 ; PhysRevA.80.052302 . Furthermore, the
highly-entangled structure of certain Dicke states can be used to project out
non-locally transformable states upon measurement, such as the $GHZ$ and $W$
states, with very little computational overhead.

We construct each $N$-qubit Dicke state ${|{D^{N}_{k}}\rangle}$ as the equal
superposition over all $N$-qubit states ${|{b}\rangle}$, where $b$ is a bit-
string of fixed Hamming-weight $h(b)=k$. Explicitly,

${|{D^{N}_{k}}\rangle}\equiv\binom{N}{k}^{-1/2}\sum_{b\in\\{0,1\\}^{n},\hskip
2.84544pth(b)=k}{|{b}\rangle}.$ (1)

Specific examples of Dicke states include,

$\begin{split}{|{D^{3}_{1}}\rangle}&=\frac{1}{\sqrt{3}}\left({|{100}\rangle}+{|{010}\rangle}+{|{001}\rangle}\right),\\\
{|{D^{4}_{2}}\rangle}&=\frac{1}{\sqrt{6}}\left({|{1100}\rangle}+{|{1010}\rangle}+{|{1001}\rangle}+{|{0110}\rangle}+{|{0101}\rangle}+{|{0011}\rangle}\right).\end{split}$
(2)

Dicke states of the form ${|{D^{N}_{1}}\rangle}$, those with Hamming-weight
$k=1$, are exactly the $N$-qubit $W$ states ${|{W_{N}}\rangle}$, defined

${|{W_{N}}\rangle}\equiv\frac{1}{\sqrt{N}}\left({|{100...00}\rangle}+{|{010...00}\rangle}+...+{|{000...01}\rangle}\right).$
(3)

Similarly, Dicke states of Hamming-weight $k=N$ are the $N$-qubit measurement
basis state

${|{D^{N}_{N}}\rangle}\equiv{|{000...01}\rangle}={|{1}\rangle}^{\otimes N}.$
(4)

###  2.2 Entropy Vectors and Entropy Cones

We compute the entanglement entropy of a state ${|{\psi}\rangle}$ according to
the von Neumann entropy

$S_{\psi}\equiv-\Tr\rho_{\psi}\ln\rho_{\psi},$ (5)

where $\rho_{\psi}\equiv{|{\psi}\rangle}\bra{\psi}$. Any pure state
${|{\psi}\rangle}$ has total entropy $S_{\psi}=0$, since
$\rho_{\psi}^{2}=\rho_{\psi}$. When information is measured in dits, as with a
state ${|{\psi}\rangle}\in\mathcal{H}^{d}$, the entropy in Eq. (5) is computed
using $\log_{d}$.

Even for an overall pure state, non-zero entanglement can exist when
considering complementary subsystems of ${|{\psi}\rangle}$. For a state
${|{\psi}\rangle}$, we can consider an $\ell$-party subsystem which we denote
$I$. The entanglement entropy between $I$ and its $(N-\ell)$-party complement
system $\bar{I}$ is then computed

$S_{I}=-\Tr\rho_{I}\ln\rho_{I}.$ (6)

The object $\rho_{I}$ is the reduced density matrix of subsystem $I$, computed
by tracing out its compliment $\bar{I}$.

For an $N$-party state ${|{\psi}\rangle}$, there are $2^{N}-1$ different
subsystems we can consider. Computing $S_{I}$ for each subsystem $I$, and
ordering the resulting set, defines the entropy vector $\vec{S}$ for
${|{\psi}\rangle}$. For example, the entropy vector for some $3$-party pure
state would have the form,

$\vec{S}=(S_{A},S_{B},S_{O};S_{AB},S_{AO},S_{BC};S_{ABO}).$ (7)

where we use a semicolon to distinguish entropies for regions of different
sizes $|I|$. The final party is often labeled with $O$, as it acts a purifier
for the remainder of the system.

If the overall state ${|{\psi}\rangle}$ is pure, we have the additional
constraint $S_{I}=S_{\bar{I}}$, which comes from the fact that $S_{\psi}=0$.
This condition allows the entropy vector for ${|{\psi}\rangle}$ to be
expressed using only $2^{N-1}-1$ entropies. Accordingly, the vector in Eq. (8)
can be described in the reduced form

$\vec{S}=(S_{A},S_{B};S_{AB}).$ (8)

We use this reduced entropy vector presentation throughout Section 4.

Subsystem entropies $S_{I}$ for multi-partite quantum states are required to
obey certain entropy inequalities Bao2015 ; Pippenger ; Hayden2013 , which can
also be used to classify that state. For example, all quantum states are
subadditive, meaning $S_{I}+S_{J}\geq S_{IJ}$ for all disjoint subsystems $I$
and $J$. Other entropy inequalities are more strict, and are not necessarily
satisfied by generic quantum states, e.g. the monogamy of mutual information
(MMI) Hayden2013  which states

$S_{IJ}+S_{IK}+S_{JK}\geq S_{I}+S_{J}+S_{K}+S_{IJK},$ (9)

for disjoint subsystems $I,\,J,$ and $K$. The MMI inequality is satisfied by
all holographic states, states with a smooth classical geometric dual through
the AdS/CFT correspondence.

Each entropy inequality defines a hyperplane in some $2^{N}-1$ dimensional
entropy-vector space, bisecting the space and placing entropy vectors which
satisfy the inequality on one side, and those which fail the inequality on the
other. Entropy vectors which saturate an inequality reside on the hyperplane
itself. The set of all inequalities satisfied by a class of quantum states,
bounds a convex polyhedral cone in the entropy vector space, known as an
entropy cone Bao2015 . Entropy vectors which correspond to a certain class of
quantum states, e.g. holographic states or stabilizer states, must have an
entropy vector which lies in the convex hull111We again highlight that this
condition is necessary, but not sufficient, for identifying states
corresponding to a particular class. of the corresponding entropy cone.
Alternatively, entropy cones can be specified by identifying all extremal
rays, the entropy vectors which saturate two inequalities and lie at the
intersection of two hyperplanes, or by directly identifying all possible
entropy vectors for the class of states, as performed in Section 3.

Entropy cones for various classes of states are well-understood for low party
number. However as system size increases, so too does the number of necessary
inequalities, and complexity of each inequality, needed to characterize each
entropy cone. To navigate this complexity increase, much effort has turned
towards studying more fundamental properties of entropy cones. The symmetrized
entropy cone prescription Czech2021 ; Fadel2021  focuses the extremal
properties of a cone’s structure under a symmetry projection. Symmetrized
entropies are defined as in Eq. (8), with the addition of a normalization
factor based on the cardinality of the subsystem. For subsystems $I$, we have

$\tilde{S}_{k}\equiv\left[\binom{N+1}{k}\right]^{-1}\sum_{I\in\\{I\\}_{k}}S_{I},$
(10)

where the sum is computed over all subsystems $I$, of and $N$-party states,
with fixed cardinality $k=|I|$. As an example, computing the symmetrized
entropy of all single-party subsystems for a $4$-party state, we have

$\tilde{S}_{1}=\frac{1}{4}\left(S_{A}+S_{B}+S_{C}+S_{O}\right).$ (11)

Mathematical graphs also offer a useful description of entanglement in multi-
partite quantum systems, particularly with regards to holographic systems
Bao2015 . The entropy vectors of holographic states can be realized as a min-
cut protocol on weighted undirected graphs, where edge cuts in the graph
correspond to traversing minimal-length geodesics in the dual geometry.
Broader classes of states require more generic graph descriptions, including
hypergraphs Bao2020a ; Bao2020  or topological links Bao2022a . Symmetrized
entropy vectors can likewise be realized using a min-cut prescription on
weighted star graphs Czech2021 ; Fadel2021 . In Section 3.2, we extend this
star graph proposal to describe the structure of Dicke state entropy vectors.

###  2.3 Stabilizer Formalism and Reachability Graphs

An essential set of gates in quantum computing is the set of Pauli gates,
defined in a unitary matrix representation as

$\mathbb{1}\equiv\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\,\,\sigma_{X}\equiv\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\,\,\sigma_{Y}\equiv\begin{bmatrix}0&-i\\\
i&0\end{bmatrix},\,\,\sigma_{Z}\equiv\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$
(12)

In a fixed measurement basis $\\{{|{0}\rangle},\,{|{1}\rangle}\\}$, the
matrices in Eq. (12) act as operators on a Hilbert space $\mathbb{C}^{2}$. The
set $\\{\sigma_{X},\,\sigma_{Y},\,\sigma_{Z}\\}$ generates the $16$-element
Pauli group under multiplication, denoted $\Pi_{1}$.

We can extend the matrix representation of Pauli gates to arbitrary qubit
number by composing sets of Pauli strings. Each Pauli string describes a set
of local actions performed on specified qubits in an $N$-qubit system. Every
$N$-qubit Pauli string can be defined222This Pauli string representation as
the $N$-fold tensor product of $2\times 2$ matrices requires two conditions:
first, we assume the Hilbert space factorizes into a product of $N$ qubits
($N$ copies of $\mathcal{H}^{2}$), and secondly, we ascribe an ordering to the
set of qubits which will serve as an indexing system. as a tensor product over
$2\times 2$ matrices. For example, the action of $\sigma_{X}$ on the $k^{th}$
qubit of an $N$-qubit system can be written

$\sigma^{k}_{X}\equiv\mathbb{1}^{1}\otimes\ldots\otimes\mathbb{1}^{k-1}\otimes\sigma_{X}\otimes\mathbb{1}^{k+1}\otimes\ldots\otimes\mathbb{1}^{N}.$
(13)

Eq. (15) is an example of a weight-$1$ Pauli string, where the weight of a
string denotes the number of non-identity operations in the tensor product.
The $N$-qubit Pauli group $\Pi_{N}$ is generated by the set of all weight-$1$
Pauli strings.

Having constructed $\Pi_{N}$, we could further consider operations which map
$\Pi_{N}$ to itself. The $N$-qubit Clifford group $\mathcal{C}_{N}$ is the set
of unitaries which normalizes the Pauli group, i.e. $\mathcal{C}_{N}$ maps
elements of $\Pi_{N}$ to elements of $\Pi_{N}$ via conjugation. In the single-
qubit case, we can define $\mathcal{C}_{1}$ as the group generated by the
Hadamard sylvester1867lx ; hadamard1893resolution  and phase gates, defined as
the matrices

$H\equiv\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix},\quad
P\equiv\begin{bmatrix}1&0\\\ 0&i\end{bmatrix}.$ (14)

Just as with Pauli gates we can generalize to an $N$-qubit description by
composing strings of Clifford operators, where we use a subscript to indicate
the qubit being acted on, e.g.

$H_{k}\equiv\mathbb{1}^{1}\otimes\ldots\otimes\mathbb{1}^{k-1}\otimes
H\otimes\mathbb{1}^{k+1}\otimes\ldots\otimes\mathbb{1}^{N}.$ (15)

Unlike the Pauli group, however, the group $\mathcal{C}_{N}$ is not generated
by only weight-$1$ Clifford strings. For $N>1$, constructing $\mathcal{C}_{N}$
requires the addition of the bi-local $CNOT$ gate, defined

$C_{i,j}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{bmatrix}.$ (16)

The $CNOT$ gate acts on two qubits in an $N$-qubit system by first evaluating
the state of the $i^{th}$ qubit, the control bit, then performing a $NOT$
operation on the $j^{th}$ qubit, the target bit, if the $i^{th}$ qubit is
found in the state ${|{1}\rangle}$. It is important to note that $C_{i,j}\neq
C_{j,i}$. We may now define the group $\mathcal{C}_{N}$ as

$\mathcal{C}_{N}\equiv\langle
H_{1},\,...,\,H_{N},\,P_{1},\,...,\,P_{N},\,C_{1,2},\,C_{2,1},\,...,\,C_{N-1,N},\,C_{N,N-1}\rangle.$
(17)

Given a Hilbert space $\mathcal{H}$ and group $G\subset L(\mathcal{H})$, an
element of $G$ is said to stabilize ${|{\psi}\rangle}\in\mathcal{H}$ if it
acts trivially on ${|{\psi}\rangle}$. The set of all $g\in G$ that stabilize
${|{\psi}\rangle}$, defined

$\,\textnormal{Stab}\,_{G}({|{\psi}\rangle})\equiv\\{g\in
G\>|\>g{|{\psi}\rangle}={|{\psi}\rangle}\\},$ (18)

makes up the stabilizer subgroup of ${|{\psi}\rangle}$ under the action of
$G$. Otherwise stated, ${|{\psi}\rangle}$ is a $+1$ eigenvector of each
$g\in\,\textnormal{Stab}\,_{G}({|{\psi}\rangle})$.

For $G$ a finite group, Lagrange’s theorem Alperin1995  ensures a partition of
$|G|$ for any subgroup $H\leq G$, explicitly

$|G|=\left[G\mathrel{\mathop{\mathchar 58\relax}}H\right]\cdot|H|,$ (19)

with $[G\mathrel{\mathop{\mathchar 58\relax}}H]$ the index of $H$ in $G$.
Furthermore when $G$ acts on a set $X$, the Orbit-Stabilizer theorem
Alperin1995  gives the orbit of $x\in X$ under the action of $G$ as

$|G\cdot x|=\left[G\mathrel{\mathop{\mathchar
58\relax}}\,\textnormal{Stab}\,_{G}(x)\right]=\frac{|G|}{|\,\textnormal{Stab}\,_{G}(x)|}.$
(20)

When $G$ acts on a Hilbert space $\mathcal{H}$, when can use Eqs. (19) and
(20) to construct orbits of ${|{\psi}\rangle}\in\mathcal{H}$ under the group
action Keeler:2023xcx .

When considering the action of $\Pi_{N}$ on $\mathcal{H}$, states which are
stabilized by a $2^{N}$-element subgroup of $\Pi_{N}$ are known as stabilizer
states aaronson2004improved ; Gottesman1997 ; garcia2017geometry ; Keeler2022
. Stabilizer states play a critical role in near-term realization of quantum
computing, as they represent the set of quantum systems which can be
efficiently classically simulated Gottesman1998 . One way to construct the set
of $N$-qubit stabilizer states is by acting on a state in the measurement
basis $\\{{|{0}\rangle},{|{1}\rangle}\\}^{N}$ with the Clifford group
$\mathcal{C}_{N}$. The set of $N$-qubit stabilizer states $S_{N}$ is exactly
the orbit of each state in $\\{{|{0}\rangle},{|{1}\rangle}\\}^{N}$, under the
action of $\mathcal{C}_{N}$. The number of $N$-qubit stabilizer states
generated by the orbit $S_{N}$ is derived in doi:10.1063/1.4818950 , and has
order

$\left|S_{n}\right|=2^{n}\prod_{k=0}^{n-1}(2^{n-k}+1).$ (21)

The process of constructing state orbits under group action naturally admits a
graph-theoretic description aaronson2004improved ; Keeler:2023xcx . For some
$G$ acting on ${|{\psi}\rangle}\in\mathcal{H}$, we can assign a vertex to each
state in the orbit $[G\cdot{|{\psi}\rangle}]$, and an edge to each generator
of $G$. This graph is known as the reachability graph for ${|{\psi}\rangle}$,
and maps the evolution of ${|{\psi}\rangle}$ through $\mathcal{H}$ under the
action of $G$. Figure 1 depicts the reachability graph for ${|{0}\rangle}$
under the single-qubit Clifford group $\mathcal{C}_{1}$, with vertices
representing the $6$ single-qubit stabilizer states.

Figure 1: Orbit of ${|{0}\rangle}$ under the action of $\mathcal{C}_{1}$,
depicted as a reachability graph. Graph vertices represent the $6$ single-
qubit stabilizer states, and edges correspond to $\mathcal{C}_{1}$ generators.
For generators which are self-inverse, e.g. the Hadamard gate, we use
undirected edges.

It is often useful to consider only the action of a subgroup $H\leq G$ on
$\mathcal{H}$. Focusing on the action of $H$ highlights specific features of a
state’s orbit, and can better-exhibit the evolution of certain state
properties through the orbit. When composing the orbit of a state
${|{\psi}\rangle}$ under the action of some $H\leq G$, the term restricted
graph is sometimes used to discuss the emergent reachability graph Keeler2022
.

##  3 The Entropy Cone for ${|{D^{N}_{k}}\rangle}$

In this section, we describe the entropy cone for $N$-qubit Dicke states by
explicitly building all ${|{D^{N}_{k}}\rangle}$ entropy vectors for qubit
number $N$ and Hamming-weight $k$. We highlight the symmetric properties of
${|{D^{N}_{k}}\rangle}$ entropy vectors, and note the relative containment of
the ${|{D^{N}_{k}}\rangle}$ entropy cone in other known entropy cones. We
demonstrate that our construction for Dicke states reproduces the $W$ state
entropy cone found in Schnitzer:2022exe . Additionally, we give a realization
of ${|{D^{N}_{k}}\rangle}$ entropy vectors as a min-cut prescription on
weighted star graphs. In later sections we analyze the evolution of the
${|{D^{N}_{k}}\rangle}$ entropy vectors defined here, under the action of
Clifford circuits.

###  3.1 Dicke State Entropy Vectors

The symmetric structure of Dicke states ${|{D^{N}_{k}}\rangle}$ enables a
direct calculation of subsystem entanglement entropy from the non-zero
diagonal elements of the density matrix Witten:2018zva . For an $N$-party pure
state ${|{D^{N}_{k}}\rangle}$ of Hamming-weight $k$, the entanglement entropy
of an $\ell$-party subsystem is computed

$S_{\ell}\left({|{D^{N}_{k}}\rangle}\right)\equiv-\binom{N}{k}^{-1}\sum_{i=0}^{min(\ell,k)}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{N}{k}^{-1}\binom{\ell}{i}\binom{N-\ell}{k-i}\right].$
(22)

We can directly verify that $S_{\ell}=S_{N-\ell}$ and $S_{N}=0$, from Eq.
(22). Furthermore, we highlight the property that
$S_{\ell}\left({|{D^{N}_{k}}\rangle}\right)$ depends only on the cardinality
of a chosen subsystem.

The calculation of $S_{\ell}$ admits simplifications for specific values of
$\ell$ and $k$. For states ${|{D^{N}_{k}}\rangle}$ with $\ell\geq k$, Eq. (22)
becomes

$S_{\ell}=\ln\left[\binom{N}{k}\right]-\binom{N}{k}^{-1}\sum_{i=0}^{k}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{\ell}{i}\binom{N-\ell}{k-i}\right],$
(23)

where we note the $\ell$-independence of the first term. A derivation of (23)
is given in Appendix A. A similar decoupling exists for the case $\ell<k$, and
is shown in Eq. (52).

For $k=1$, states ${|{D^{N}_{k}}\rangle}$ are the subset of $N$-qubit $W$
states, ${|{W_{N}}\rangle}\equiv{|{D^{N}_{1}}\rangle}$. For an $\ell$-party
subsystem of ${|{D^{N}_{1}}\rangle}$, the expression for $S_{\ell}$ in Eq.
(22) gives

$S_{\ell}\left({|{D^{N}_{1}}\rangle}\right)=\frac{\ell}{N}\ln\left[\frac{N}{\ell}\right]+\frac{(N-\ell)}{N}\ln\left[\frac{N}{N-\ell}\right],$
(24)

in agreement with the calculations given in Schnitzer:2022exe .

The ordered set of all $2^{N}-1$ subsystem entropies for a state
${|{D^{N}_{k}}\rangle}$, computed according to Eq. (22), compose the entropy
vector $\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$. Since each $S_{\ell}$
depends only on $|\ell|$, all Dicke state entropy vectors share the form

$\vec{S}\left({|{D^{N}_{k}}\rangle}\right)\equiv\Bigl{(}\underbrace{S_{1},\,...,\,S_{1}}_{\binom{N}{1}};\,\underbrace{S_{2},\,...,\,S_{2}}_{\binom{N}{2}};\,...;\,\underbrace{S_{N-1},\,...,\,S_{N-1}}_{\binom{N}{N-1}};\,0\Bigl{)}.$
(25)

The entropy vectors in Eq. (25) are manifestly symmetrized Czech2021 , leaving
$\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$ invariant up to exchange of
subsystems of equal size $|\ell|$.

All $N$-qubit Dicke state entropy vectors can be calculated using Eqs. (22)
and (25). Collectively, these equations describe the $N$-qubit Dicke state
entropy cone, defined for each $N$ as the convex hull of all
$\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$. Since each
$\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$ is symmetrized, all Dicke state
entropy vectors automatically satisfy the symmetrized quantum entropy cone
(SQEC) inequalities Fadel2021 ,

$-S_{\ell-1}+2S_{\ell}-S_{\ell+1}\geq 0,\qquad\forall\,1\leq\ell\leq\lceil
N/2\rceil,$ (26)

which verify symmetric instances of subadditivity and strong-subadditivity.
The $N$-qubit Dicke state entropy cone is therefore contained within the SQEC
for all $N$.

The Monogamy of Mutual Information (MMI) inequality, given in Eq. (9), defines
a subset of facet which bound the holographic entropy cone. For
${|{D^{N}_{k}}\rangle}$, MMI is saturated333MMI is trivially saturated by
entropy vectors of the unentangled Dicke states
${|{D^{N}_{N}}\rangle}={|{1}\rangle}^{\otimes N}$. when $N=3$, and violated
otherwise. Similarly for the symmetrized holographic entropy cone (SHEC), the
${|{D^{N}_{k}}\rangle}$ entropy vectors violate requisite inequalities for
$N>3$, namely

$-\ell(\ell+1)S_{\ell-1}+2(\ell-1)(\ell+1)S_{\ell}-\ell(\ell-1)S_{\ell+1}\geq
0,\qquad\forall\,\ell\in[2,n/2].$ (27)

Consequently, the $N$-qubit Dicke state entropy cone is not contained within
the holographic or symmetrized holographic entropy cones.

The entropy cone of stabilizer states is completely characterized up through
$N=4$ parties ($5$ qubits including the purifier). For $N\leq 5$ qubits, we
find that all Dicke state entropy vectors are contained within the stabilizer
entropy cone despite ${|{D^{N}_{k}}\rangle}$ not being a stabilizer state for
all444We again note the exception for all ${|{D^{N}_{N}}\rangle}$, and all
$D^{N}_{k}$ with $N=2$, as these are stabilizer states. $N\geq 3$.
Furthermore, applying Clifford circuits to ${|{D^{N}_{k}}\rangle}$ generates
additional entropy vectors beyond those in Eq. (25), a process described in
Section 4, which we also find reside within the stabilizer entropy cone. We
conjecture, but do not prove, that the Dicke state entropy cone is completely
contained within the convex hull of the stabilizer entropy cone.

We have given an explicit calculation of all $\ell$-party entanglement
entropies in Dicke states ${|{D^{N}_{k}}\rangle}$, for arbitrary system size
$N$ and Hamming-weight $k$. We used this result to construct all Dicke state
entropy vectors $\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$, and showed that
our results reproduce previous entropy vector calculations for $W$ states for
$k=1$. We present the set of all $\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$,
for a fixed qubit number $N$, as the $N$-qubit Dicke state entropy cone. We
have highlighted that, since ${|{D^{N}_{k}}\rangle}$ entropy vectors emerge
symmetrized, the ${|{D^{N}_{k}}\rangle}$ entropy cone is contained within the
SQEC. At $N\geq 4$ we observed that
$\vec{S}\left({|{D^{N}_{k}}\rangle}\right)$ violates holographic inequalities,
e.g. MMI, and lies outside both the holographic and symmetrized holographic
entropy cones. In the next section, we use our ${|{D^{N}_{k}}\rangle}$ entropy
vector construction to define a min-cut protocol which realizes Dicke state
entropy vectors using weighted star graphs.

###  3.2 A Graph Model for ${|{D^{N}_{k}}\rangle}$ Entropies

We now outline a protocol which compute Dicke state entropies, as given in Eq.
(22), as a sum over minimum-weight edge cuts on star graphs. Initial
descriptions using star graphs to represent average entropies were presented
in Czech:2021rxe ; Fadel2021 , and later extended to include the possibility
of negative edge weights in Harper:2022sky ; Schnitzer:2022exe . We
demonstrate an explicit example of this star graph construction for
${|{D^{N}_{1}}\rangle}$ entropy vectors, and describe how to recursively
generalize the model for $k>1$.

To construct our representation of entanglement entropy, we consider a graph
$G=(V,E)$, with vertex set $V$ partitioned into subsets of internal vertices
$V_{Int.}\subseteq V$, and external vertices $V_{Ext.}\subseteq V$. For an
$N$-party ${|{\psi}\rangle}$ with purifier, each disjoint subsystem $\ell$ is
assigned a vertex $v_{\ell}\in V_{Ext.}$, where $|V_{Ext.}|=N+1$. The entropy
$S_{\ell}$ is then computed as the total weight of a min-cut on $G$ which
separates $v_{\ell}$ from its complement subsystem $v_{\ell^{C}}\subset
V_{Ext.}$.

For Dicke states ${|{D^{N}_{k}}\rangle}$ we represent each $S_{\ell}$ using a
star graph with $N$ edges of unit weight, and one edge of weight $w\leq 0$.
One novel feature of these graphs is that $w$ may take on select negative
values, subject to the required inequalities555We often consider a tuple of
non-negative entropies which lives in a totally non-negative sector of some
$2^{N}-1$ vector space. If instead one considers a perfect tensor
decomposition, negative entropies are permitted as long as they sum to a
positive value in the entropy basis and satisfy the required inequalities of a
chosen entropy cone. For further detail we recommend Harper:2022sky . of a
particular entropy cone, which ultimately sum to non-negative entropies. Since
$\ell$-party ${|{D^{N}_{k}}\rangle}$ entropies depend only on the cardinality
$|\ell|$, we compute $S_{\ell}$ as the min-cut

$S_{\ell}=\min\\{|\ell|,N-1-|\ell|+w\\},$ (28)

following Schnitzer:2022exe ; Fadel2021 . Figure 2 gives an example of a star
graph which realizes $S_{\ell}$ for a state ${|{D^{3}_{k}}\rangle}$.

\begin{overpic}[width=170.71652pt]{BasicStarGraph.pdf} \put(54.0,95.0){A}
\put(-6.0,50.0){B} \put(54.0,0.0){C} \put(100.0,50.0){O} \end{overpic} Figure
2: Example of a $4$-legged star graph, with $3$ legs of unit weight and one
leg of weight $w\leq 0$, which realizes the entropies of
${|{D^{3}_{k}}\rangle}$. The weight $w$ is negative in this graph, and defined
as a function of $k$.

The value of $w$ is defined in terms of $k$, as shown in Eqs. (29)–(37).

Since ${|{D^{N}_{k}}\rangle}$ entropies in Eq. (22) obey the symmetry
$S_{\ell}=S_{N-\ell}$, we can define $\tilde{S}_{\ell}$ to be the symmetrized
variable over all $\ell$-party entanglement entropy

$\tilde{S}_{\ell}=\binom{N}{\ell}^{-1}\left[\binom{N-1}{\ell}S_{\ell}+\binom{N-1}{N-\ell}S_{N-\ell}\right].$
(29)

As shown in Eq. (23), each $S_{\ell}$ is computed as a sum over $\ell+1$ terms
when $\ell<k$, or $k+1$ terms when $\ell\geq k$. Accordingly, each
$\tilde{S}_{\ell}$ in Eq. (29) is realized as a sum over $\ell+1$ (or $k+1$)
star graphs, for $1\leq\ell\leq\lceil N/2\rceil$. This sum over graphs for
each $S_{\ell}$ generalizes the previous constructions in Schnitzer:2022exe
to all ${|{D^{N}_{k}}\rangle}$.

To demonstrate this min-cut model, we construct an explicit representation of
$S_{\ell}$ for states ${|{D^{N}_{1}}\rangle}$. Applying Eq. (28) to Eq. (29)
we have

$\tilde{S}_{\ell}=\frac{1}{N}\left[(N-\ell)\min\\{\ell,N-1-\ell+w_{1}\\}+\ell\min\\{N-\ell,w_{2}+\ell-1\\}\right].$
(30)

Figure 3 shows an example pair of graphs, for the state
${|{D^{4}_{1}}\rangle}$, whose sum over min-cuts realizes Eq. (30). In both
graphs, the negatively-weighted edge connects to the external vertex for the
purifier $O$.

Figure 3: Pair of star graphs whose min-cut sum calculates $\tilde{S}_{\ell}$,
as in Eq. (30), for ${|{D^{5}_{1}}\rangle}$. The values $w_{1}$, $w_{2}$ are
both negative and set by inserting Eq. (22) into Eq. (28).

Beginning with the first term in Eq. (30), which we denote
$(\tilde{S}_{\ell})_{1}$, we have

$(\tilde{S}_{\ell})_{1}=\frac{1}{N}(N-\ell)\min\\{\ell,N-1-\ell+w_{1}\\}.$
(31)

From Eq. (22) we know

$\min\\{\ell,N-1-\ell+w_{1}\\}=\ln\left[\frac{N}{N-\ell}\right],$ (32)

which we solve for $w_{1}$ to give the bound

$w_{1}=\ell+\ln\left[\frac{N}{N-\ell}\right]-(N-1).$ (33)

The weight $w_{1}$ in Eq. (33) takes on negative values for

$\ell<(N-1)-\ln\left[\frac{N}{N-\ell}\right].$ (34)

Evaluating the second term $(\tilde{S}_{\ell})_{2}$ in Eq. (29), we have

$(\tilde{S}_{\ell})_{2}=\frac{\ell}{N}\min\\{N-\ell,w_{2}+\ell-1\\}.$ (35)

We solve Eq. (35) to find the weight

$w_{2}=\ln\left[\frac{N}{\ell}\right]-\ell+1,$ (36)

which is negative while

$\ell>1+\ln\left[\frac{N}{\ell}\right].$ (37)

The procedure in Eqs. 30–37 can be applied for all ${|{D^{N}_{1}}\rangle}$,
with the resulting symmetrized entropies $\tilde{S}_{\ell}$ described as a
min-cut protocol on a pair of weighted star graphs analogous to those in
Figure 3. Each graph possesses a single edge of negative weight, and the
values of each weight can be determined as in Eqs. (34) and (37). We now
describe how to generalize this model to arbitrary $k$, by inserting sequences
of star graphs to evaluate each $S_{\ell}$.

We can naturally extend the protocol described in Eqs. 30–37 to all
${|{D^{N}_{k}}\rangle}$ with $k\geq 1$. For any $k>1$, each of the terms
$(\tilde{S}_{\ell})_{1}$ and $(\tilde{S}_{\ell})_{2}$ in Eq. (29) are computed
as a sum over min-cuts on $\ell+1$ star graphs for $\ell\geq k$, or $k+1$ star
graphs for $\ell<k$. For example, consider the symmetrized entropies of
${|{D^{5}_{2}}\rangle}$,

$\begin{split}\tilde{S}_{1}&=(\tilde{S}_{1})_{1}+(\tilde{S}_{1})_{2}=2\binom{5}{1}^{-1}\left(\frac{3}{5}\ln\left[\frac{5}{3}\right]+\frac{2}{5}\ln\left[\frac{5}{2}\right]\right)=\tilde{S}_{4},\\\
\tilde{S}_{2}&=(\tilde{S}_{2})_{1}+(\tilde{S}_{2})_{2}=2\binom{5}{2}^{-1}\left(\frac{3}{5}\ln\left[\frac{5}{3}\right]+\frac{3}{10}\ln\left[\frac{10}{3}\right]+\frac{1}{10}\ln\left[\frac{10}{1}\right]\right)=\tilde{S}_{3},\\\
\tilde{S}_{5}&=0\\\ \end{split}$ (38)

The quantity $\tilde{S}_{1}$ in Eq. (38) admits a star graph representation
exactly as described in Eqs. (29)–(37). Meanwhile, $\tilde{S}_{2}$ is given by
a sum over three star graphs, each having a single edge of negative weight.

We have given a min-cut protocol on weighted star graphs which realizes the
symmetrized entropies $\tilde{S}_{\ell}$ of all Dicke states
${|{D^{N}_{k}}\rangle}$. We gave a direct example showing graph realizations
of $S_{\ell}$ for states ${|{D^{N}_{1}}\rangle}$, in agreement with results
demonstrated in Schnitzer:2022exe . We generalized this technique to $k>1$ by
computing each term in $\tilde{S}_{\ell}$ as a sum over $\ell+1$ star graphs,
each having a single edge of negative weight. In the next section we explore
group stabilizers for Dicke states under action of the Pauli and Clifford
groups. We likewise analyze the orbits of Dicke states under these groups, as
well as the dynamics of ${|{D^{N}_{k}}\rangle}$ entropy vectors under Clifford
circuits. We illustrate ${|{D^{N}_{k}}\rangle}$ orbits as reachability graphs,
using the methods given in Keeler2022 ; Keeler:2023xcx .

##  4 Stabilizers and Orbits of ${|{D^{N}_{k}}\rangle}$

In this section we construct the stabilizer subgroups for all Dicke states
${|{D^{N}_{k}}\rangle}$ under action of the Pauli and Clifford groups. We use
each subgroup to construct the reachability graph for all
${|{D^{N}_{k}}\rangle}$, under the action of Pauli and Clifford group elements
Keeler:2023xcx . We highlight differences between the reachability graph
structures observed for Dicke states, and those seen among the $N$-qubit
stabilizer states. We later remark on the utility of the
${|{D^{N}_{k}}\rangle}$ stabilizer groups identified for error-correcting
codes. We analyze ${|{D^{N}_{k}}\rangle}$ reachability graphs, with vertices
colored to indicate the entropy vector, and determine the evolution of
${|{D^{N}_{k}}\rangle}$ entropy vectors under a restricted subgroup of
Clifford operators. Further, we establish bounds on how much each
${|{D^{N}_{k}}\rangle}$ entropy vector can change under the select set of
gates. The Mathematica data and packages used to generate all graphs is
publicly available github .

###  4.1 Pauli Group Orbits

We first consider the action of the Pauli group666In the case of $\Pi_{N}$, as
well as with $\mathcal{C}_{N}$, we first mod out each group by elements which
act as a global phase on the group. For $\mathcal{C}_{N}$ this global phase
element is $\omega\equiv(H_{i}P_{i})^{3}$, which has the property
$\omega^{8}=\mathbb{1}$. Likewise for $\Pi_{N}$, this global phase is
$\omega^{2}$. For details, see Section 5.1 of Keeler:2023xcx . $\Pi_{N}$ on
the set of $N$-qubit Dicke states. While all quantum states are trivially
stabilized by $\mathbb{1}\in\Pi_{N}$, Dicke states admit larger stabilizer
subgroups in $\Pi_{N}$, which make them useful for stabilizer code
construction. Every ${|{D^{N}_{k}}\rangle}$ is stabilized by, at least, $2$
elements of $\Pi_{N}$, but some are stabilized by more.

In addition to $\mathbb{1}$, all Dicke states ${|{D^{N}_{k}}\rangle}$ are
stabilized by the $\Pi_{N}$ element

$\begin{split}&\bigotimes_{i=1}^{N}\sigma_{Z}^{i},\quad\textnormal{for k
even},\\\ -&\bigotimes_{i=1}^{N}\sigma_{Z}^{i},\quad\textnormal{for k odd}.\\\
\end{split}$ (39)

The operator in Eq. (39) acts as a $\sigma_{Z}$ on every qubit of an $N$-qubit
system, with an additional $-1$ phase for $k$ odd. For example, states
${|{D^{3}_{1}}\rangle}$ and ${|{D^{5}_{2}}\rangle}$ have respective stabilizer
subgroups

$\begin{split}\,\textnormal{Stab}\,_{\Pi_{3}}({|{D^{3}_{1}}\rangle})&=\\{\mathbb{1},-\sigma_{Z}^{1}\sigma_{Z}^{2}\sigma_{Z}^{3}\\},\\\
\,\textnormal{Stab}\,_{\Pi_{5}}({|{D^{5}_{2}}\rangle})&=\\{\mathbb{1},\sigma_{Z}^{1}\sigma_{Z}^{2}\sigma_{Z}^{3}\sigma_{Z}^{4}\sigma_{Z}^{5}\\}.\\\
\end{split}$ (40)

The stabilizer subgroup containing $\mathbb{1}$ and Eq. (39) quotients
$\Pi_{N}$ into a group of order $2^{2n-1}$. Figure 4 illustrates the
reachability graph for ${|{D^{3}_{1}}\rangle}$ under the action of $\Pi_{3}$.

Figure 4: Orbit of ${|{D^{3}_{1}}\rangle}$ under the $3$-qubit Pauli group
$\Pi_{3}$, which contains $32$ vertices. In general, states stabilized by only
$\mathbb{1}$ and Eq. (39) will have a Pauli orbit of length $2^{2n-1}$.

The Dicke state ${|{D^{N}_{N}}\rangle}$ is a stabilizer state, specifically
${|{D^{N}_{N}}\rangle}={|{1}\rangle}^{\otimes N}$, as are all
${|{D^{N}_{k}}\rangle}$ for $N\leq 2$. Accordingly, ${|{D^{N}_{N}}\rangle}$ is
stabilized by a $2^{N}$-element subgroup of $\Pi_{N}$. In addition to
$\mathbb{1}$ and the operator in Eq. (39), ${|{D^{N}_{N}}\rangle}$ is
stabilized by the action of $-\sigma_{Z}$ on any single qubit, as well as
$\sigma_{Z}$ on any qubit pair for $N\geq 2$. Written explicitly, the set

$\,\textnormal{Stab}\,_{\Pi_{N}}({|{D^{N}_{N}}\rangle})\supseteq\\{-\sigma_{Z}^{i},\,\sigma_{Z}^{i}\sigma_{Z}^{j}\\},\quad\forall\,i,j\in\\{1,N\\}.$
(41)

As an example, the stabilizer subgroup of ${|{D^{3}_{3}}\rangle}$ consists of
the $6$ operations

$\,\textnormal{Stab}\,_{\Pi_{3}}({|{D^{3}_{3}}\rangle})=\\{\mathbb{1},\,-\sigma_{Z}^{1},\,-\sigma_{Z}^{2},\,-\sigma_{Z}^{3},\,\sigma_{Z}^{1}\sigma_{Z}^{2},\,\sigma_{Z}^{1}\sigma_{Z}^{3},\,\sigma_{Z}^{2}\sigma_{Z}^{3},\,-\sigma_{Z}^{1}\sigma_{Z}^{2}\sigma_{Z}^{3}\\}.$
(42)

The corresponding Pauli orbit for ${|{D^{3}_{3}}\rangle}$ is shown in Figure
5, where we note that all edges of the graph simultaneously represent the
actions of $\sigma_{X}^{i}$ and $\sigma_{Y}^{i}$, as they act identically on
${|{D^{3}_{3}}\rangle}$ up to global phase.

Figure 5: Reachability graph of state ${|{D^{3}_{3}}\rangle}$ under the action
of $\Pi_{3}$. This graph contains $8$ vertices, with gates $\sigma_{X}^{i}$
and $\sigma_{Y}^{i}$ acting the same on ${|{D^{3}_{3}}\rangle}$. States of the
form ${|{D^{N}_{N}}\rangle}$ are stabilizer states and are stabilized by
$2^{N}$ elements of $\Pi_{N}$.

Finally, we consider Dicke states ${|{D^{N}_{k}}\rangle}$ where $N=2k$. States
${|{D^{2k}_{k}}\rangle}$ are stabilized by the simultaneous action of
$\sigma_{X}$ and $\sigma_{Y}$ on every qubit. For states
${|{D^{2k}_{k}}\rangle}$, we have

$\,\textnormal{Stab}\,_{\Pi_{N}}({|{D^{2k}_{k}}\rangle})\supset\left\\{\bigotimes_{i=1}^{2k}\sigma_{X}^{i},\,\bigotimes_{i=1}^{2k}\sigma_{Y}^{i}\right\\},$
(43)

as well as $\mathbb{1}$ and Eq. (39). The two additional stabilizers in Eq.
(43) result in $4$-element stabilizer subgroup for ${|{D^{2k}_{k}}\rangle}$
under the action of $\Pi_{N}$.

For the exemplar state ${|{D^{4}_{2}}\rangle}$, its stabilizer subgroup under
$\Pi_{4}$ can be written

$\,\textnormal{Stab}\,_{\Pi_{4}}({|{D^{4}_{2}}\rangle})=\\{\mathbb{1},\,\sigma_{X}^{1}\sigma_{X}^{2}\sigma_{X}^{3}\sigma_{X}^{4},\,\sigma_{Y}^{1}\sigma_{Y}^{2}\sigma_{Y}^{3}\sigma_{Y}^{4},\,\sigma_{Z}^{1}\sigma_{Z}^{2}\sigma_{Z}^{3}\sigma_{Z}^{4}\\}.$
(44)

The orbit of ${|{D^{4}_{2}}\rangle}$ under $\Pi_{4}$ action contains $64$
states, and is depicted in Figure 6.

Figure 6: Orbit of ${|{D^{4}_{2}}\rangle}$ under the action of $\Pi_{4}$. This
reachability graph contains $64$ vertices as ${|{D^{4}_{2}}\rangle}$ is only
stabilized by $4$ elements of $\Pi_{4}$.

We have given the stabilizer subgroup for all ${|{D^{N}_{k}}\rangle}$ under
action of the $N$-qubit Pauli group. We used the stabilizer subgroup to
generate a reachability graph for ${|{D^{N}_{k}}\rangle}$, which represents
each state’s orbit under $\Pi_{N}$. In the following section we extend our
analysis to consider action of the $N$-qubit Clifford group $\mathcal{C}_{N}$,
as well as $\mathcal{C}_{N}$ subgroups. We use the reachability graphs of
${|{D^{N}_{k}}\rangle}$ to analyze entanglement structures observed in Dicke
state orbits.

###  4.2 Clifford Group Orbits and Entanglement Evolution

Dicke states ${|{D^{N}_{k}}\rangle}$ are not stabilizer states for $N\geq 3$
and $N\neq k$. However, interestingly, states ${|{D^{N}_{k}}\rangle}$ are
stabilized by more Clifford group elements than just $\mathbb{1}$. In this
section, we extend our study of ${|{D^{N}_{k}}\rangle}$ orbits by considering
the action of the two-qubit Clifford group $\mathcal{C}_{2}$. Since
entanglement modification via Clifford gates occurs through bi-local action,
this restriction to $\mathcal{C}_{2}$ is sufficient for exploring the
evolution of Dicke state entropy vectors under Clifford circuits. We construct
the stabilizer subgroup for each ${|{D^{N}_{k}}\rangle}$ under the action of
$\mathcal{C}_{2}$, and compute the size of each orbit.

We also present reachability graphs for ${|{D^{N}_{k}}\rangle}$ under the
action of the $\mathcal{C}_{2}$ subgroup $(HC)_{1,2}\equiv\langle
H_{1},\,H_{2},\,C_{1,2},\,C_{2,1}\rangle$, with vertices colored by entropy
vector as in Keeler2022 ; Keeler:2023xcx . Since the $P_{1}$ and $P_{2}$ gates
cannot modify a state’s entropy vector, the subgroup $(HC)_{1,2}$ contains all
non-trivial entropy vector dynamics. Furthermore, graph representations of
$(HC)_{1,2}$ orbits are easier to parse than $\mathcal{C}_{2}$ orbits, as they
contain a factor of $10$ less vertices. We use $(HC)_{1,2}$ orbits of Dicke
states to give a bound on the number of times ${|{D^{N}_{k}}\rangle}$ entropy
vectors can change under this gate set. A more general bound on entropy vector
dynamics using quotient graphs is derived in Keeler2023b .

All one and two-qubit Dicke states, ${|{D^{1}_{1}}\rangle}$ and
${|{D^{2}_{k}}\rangle}$, are also stabilizer states. Likewise, every
${|{D^{N}_{N}}\rangle}$ is a stabilizer state as well. Accordingly, states
${|{D^{2}_{k}}\rangle}$ and ${|{D^{N}_{N}}\rangle}$ are stabilized by a
$192$-element subgroup of $\mathcal{C}_{2}$, and their reachability graph is
exactly the two-qubit stabilizer state graph shown in Figure 10 of Appendix B.

When we restrict to the action of $(HC)_{1,2}=\langle
H_{1},\,H_{2}\,C_{1,2}\,C_{2,1}\rangle$, the orbit of ${|{D^{2}_{k}}\rangle}$
and ${|{D^{N}_{N}}\rangle}$ contains $24$ states. Figure 7 illustrates this
orbit, showing the reachability graph of ${|{D^{N}_{N}}\rangle}$ under the
action of $(HC)_{1,2}$. While the state ${|{D^{N}_{N}}\rangle}$ is
unentangled, elements of the $(HC)_{1,2}$ subgroup are capable of generating
instances of $GHZ$-type entanglement throughout the orbit.

\begin{overpic}[width=369.88582pt]{HCGraphD21.pdf}
\put(61.5,23.0){\footnotesize{$\swarrow{|{D^{N}_{N}}\rangle}$}}
\put(13.5,16.0){\footnotesize{$\uparrow{|{GHZ}\rangle}_{N}$}} \end{overpic}
Figure 7: Orbit of ${|{D^{N}_{N}}\rangle}$ under $\langle
H_{1},\,H_{2},\,C_{1},\,C_{2}\rangle$ subgroup. This reachability graph has
$24$ vertices and $2$ entanglement possibilities, unentangled and maximally-
entangled. Since ${|{D^{N}_{N}}\rangle}={|{1}\rangle}^{\otimes N}$, this
reachability graph is shared by a subset of the $N$-qubit stabilizer states.

Dicke states of the form ${|{D^{N}_{1}}\rangle}$, which define the set of
$N$-qubit $W$ states, as well as states ${|{D^{N}_{N-1}}\rangle}$, are
stabilized by $4$ elements of $\mathcal{C}_{2}$. Specifically, the states
${|{D^{N}_{1}}\rangle}$ and ${|{D^{N}_{N-1}}\rangle}$ have stabilizer subgroup

$\begin{split}\,\textnormal{Stab}\,_{\mathcal{C}_{2}}({|{D^{N}_{1}}\rangle})&=\\{\mathbb{1},\,H_{2}C_{1,2}H_{2},\,H_{1}C_{1,2}H_{2}C_{2,1},\,H_{2}C_{1,2}C_{2,1}C_{1,2}H_{1}\\}\\\
&=\,\textnormal{Stab}\,_{\mathcal{C}_{2}}({|{D^{N}_{N-1}}\rangle}).\end{split}$
(45)

The stabilizer group in Eq. (45) yields an orbit of $2880$ states for
${|{D^{N}_{1}}\rangle}$ and ${|{D^{N}_{N-1}}\rangle}$, under the action of
$\mathcal{C}_{2}$.

Restricting group action to $(HC)_{1,2}$, the orbits of all
${|{D^{N}_{1}}\rangle}$ and ${|{D^{N}_{N-1}}\rangle}$, for $N\geq 3$, consist
of $288$ states. Figure 8 depicts the ${|{D^{N}_{1}}\rangle}$ reachability
graph under $(HC)_{1,2}$, shown for the example state ${|{D^{3}_{1}}\rangle}$.
While this reachability graph has $288$ vertices, it is not isomorphic to the
$288$-vertex graph observed for stabilizer states under the action of
$(HC)_{1,2}$, presented in Keeler2022 .

\begin{overpic}[width=369.88582pt]{HCGraphD31.pdf}
\put(53.0,67.9){\footnotesize{$\leftarrow{|{D^{3}_{1}}\rangle}$}}
\end{overpic} Figure 8: Orbit of ${|{D^{3}_{1}}\rangle}$ under $\langle
H_{1},\,H_{2},\,C_{1},\,C_{2}\rangle$ action. The graph has $288$ vertices,
and contains $5$ different entropy vectors. We especially note the topological
distinction of this graph, compared to the $288$-vertex stabilizer state
graph. Numerical approximations for entanglement entropies are shown in the
figure, with exact values in Table 1.

Every ${|{D^{N}_{1}}\rangle}$ orbit, for $N\geq 3$, under $(HC)_{1,2}$
contains $5$ different entropy vectors. The number $5$ maximizes the number of
entropy changes which can occur under all circuits realized as paths Figure 8,
which can only modify entanglement $5$ times Keeler2023b . The specific
entropy vectors in Figure 8, those arising in the orbit of
${|{D^{3}_{1}}\rangle}$, are composed of $4$ distinct entanglement entropies.
While the number of entropy vectors found on graphs like Figure 8 is fixed,
the different entanglement entropies comprising those entropy vectors does
continue to grow with increasing $N$. Beginning with $N\geq 3$, we conjecture
the number of distinct entanglement entropies which makes up all entropy
vectors in the $(HC)_{1,2}$ orbit of ${|{D^{N}_{1}}\rangle}$ to be

$|s|_{N}=\lfloor\frac{5N-7}{2}\rfloor.$ (46)

All remaining Dicke states, i.e. all ${|{D^{N}_{k}}\rangle}$ with $1<k<N-1$,
are stabilized by $2$ elements of $\mathcal{C}_{2}$. The stabilizer subgroup
for these ${|{D^{N}_{k}}\rangle}$ is given in Eq. (47), and the orbit of such
states under $\mathcal{C}_{2}$ has size $5760$.

$\,\textnormal{Stab}\,_{\mathcal{C}_{2}}\left({|{D^{N}_{k}}\rangle}\right)=\\{\mathbb{1},\,H_{1}C_{1,2}C_{2,1}C_{1,2}H_{2}\\},\quad\forall\,1<k<N-1,$
(47)

Considering the action of $(HC)_{1,2}$ on ${|{D^{N}_{k}}\rangle}$, with
$1<k<N-1$, gives an orbit of $576$ states. The orbit of these
${|{D^{N}_{k}}\rangle}$ states under $(HC)_{1,2}$ is particularly interesting,
as it differs in size from any stabilizer state orbit under the same
$(HC)_{1,2}$ action Keeler2022 ; Keeler:2023xcx . Stated another way, the
stabilizer subgroup in Eq. (47) is not shared by any stabilizer state.
Consequently, reachability graphs with $576$ vertices, like the example shown
in Figure 9 for ${|{D^{4}_{2}}\rangle}$, are never witnessed for stabilizer
states.

\begin{overpic}[width=426.79134pt]{HCGraphD42.pdf}
\put(19.6,55.5){\footnotesize{$\uparrow{|{D^{4}_{2}}\rangle}$}} \end{overpic}
Figure 9: Reachability graph showing the orbit of ${|{D^{4}_{2}}\rangle}$
under $\langle H_{1},\,H_{2},\,C_{1},\,C_{2}\rangle$. The graph has $576$
vertices, an orbit size never observed among stabilizer states, and contains
$6$ different entanglement possibilities. We again provide numerical
approximations for entanglement entropies in the figure, with exact values
given in Table 2.

Reachability graphs like that in Figure 9 admit $6$ unique entropy vectors
throughout the orbit. The bounds proposed in Keeler2023b  limit graphs
isomorphic to Figure 9 to having, at most, $9$ different entropy vectors.
However, since entanglement dynamics additionally depends on the state being
evolved through the quantum circuit, the symmetries of Dicke states constrain
the number of entropy vectors in these graphs to $6$. As with the orbits of
${|{D^{N}_{1}}\rangle}$, while the overall number of entropy vectors in the
reachability graph is fixed, for all $N$ and $k$, the number of distinct
entanglement entropies which make up those vectors continues to increase for
larger and larger $N$, and varies for different values of $k$.

We have identified the stabilizer subgroups for all Dicke states
${|{D^{N}_{k}}\rangle}$, under the action of the $N$-qubit Pauli group
$\Pi_{N}$, as well as the two-qubit Clifford group $\mathcal{C}_{2}$. We
demonstrated that there exist three distinct stabilizer subgroups for states
${|{D^{N}_{k}}\rangle}$, depending on the values of $N$ and $k$. States
${|{D^{N}_{N}}\rangle}={|{1}\rangle}^{\otimes N}$ belong to the set of
$N$-qubit stabilizer states, and share the corresponding stabilizer groups
Keeler2022 ; Keeler:2023xcx . States ${|{D^{N}_{1}}\rangle}$ and
${|{D^{N}_{N-1}}\rangle}$ share a stabilizer subgroup in $\Pi_{N}$ and
$\mathcal{C}_{2}$, as do all ${|{D^{N}_{k}}\rangle}$ with $1<k<N-1$. We
illustrated the orbit of ${|{D^{N}_{k}}\rangle}$, under the action of
$\Pi_{N}$ using reachability graphs.

In order to understand the evolution of Dicke state entropy vectors, we
likewise constructed reachability graphs for all ${|{D^{N}_{k}}\rangle}$ under
the action of the $\mathcal{C}_{2}$ subgroup $(HC)_{1,2}=\langle
H_{1},\,H_{2},\,C_{1},\,C_{2}\rangle$. Since entanglement modification in
Clifford circuits occurs through bi-local action, restricting to this subgroup
enabled us to place constraints on the dynamics of ${|{D^{N}_{k}}\rangle}$
entropy vectors under Clifford gates. We found the number of entropy vectors
in each $(HC)_{1,2}$ orbit to be constant, with $5$ entropy vectors possible
on graphs of $288$ vertices, and $6$ entropy found on graphs with $576$
vertices. While the number of entropy vectors on these graphs is fixed, the
number of distinct entropies continued to increase.

##  5 Discussion

In this work we constructed the entropy cone for $N$-qubit Dicke states
${|{D^{N}_{k}}\rangle}$ by calculating all entropy vectors for arbitrary
values of $N$ and $k$. We first defined a function to compute the entanglement
entropy $S_{\ell}$ of any $\ell$-party subsystem of ${|{D^{N}_{k}}\rangle}$.
We demonstrated that ${|{D^{N}_{k}}\rangle}$ entropy vectors are manifestly
symmetric, with $S_{\ell}$ only dependent on the size of subsystem $\ell$, and
therefore lie within the convex hull of the SQEC. We likewise find that
${|{D^{N}_{k}}\rangle}$ entropy vectors are contained within the Stabilizer
entropy cone as far as it is characterized, up to $N=5$. Dicke state entropy
vectors do not, however, satisfy the necessary HEC or SHEC conditions for all
$N\geq 3$, where $k\neq N$. We verified that our calculation accurately
reproduces all vectors of the $N$-qubit $W$ state entropy cone
Schnitzer:2022exe , since ${|{W_{N}}\rangle}={|{D^{N}_{1}}\rangle}$.

We additionally define a prescription which realizes average entropies
$\tilde{S}_{\ell}$, for ${|{D^{N}_{k}}\rangle}$, as a min-cut protocol on
weighted star graphs. Entropies $S_{\ell}$, as in Eq. 22, are computed as the
minimum-weight edge cut on $\ell+1$ star graphs of $N+1$ legs each. Every star
graph has a single edge of weight $w\leq 0$, with the precise value of $w$
constrained by the values of $N$ and $\ell$. The sum of two set of star
graphs, one representing $S_{\ell}$ and one $S_{N-\ell}$, defines
$\tilde{S}_{\ell}$ for all ${|{D^{N}_{k}}\rangle}$. This graph representation
of ${|{D^{N}_{k}}\rangle}$ entropies builds upon other min-cut protocols for
symmetrized entropies Czech:2021rxe ; Fadel2021 ; Schnitzer:2022exe , and is
an interesting direction for future study.

We studied the orbits of ${|{D^{N}_{k}}\rangle}$ under action of $N$-qubit
Pauli group $\Pi_{N}$ and the $2$-qubit Clifford group $\mathcal{C}_{2}$.
Interestingly, Dicke states form a set of non-stabilizer states which are
stabilized by more Clifford elements than just $\mathbb{1}$. We identified the
stabilizer subgroup for every ${|{D^{N}_{k}}\rangle}$, under the action of
both groups, which we used to generate ${|{D^{N}_{k}}\rangle}$ reachability
graphs Keeler:2023xcx . Each ${|{D^{N}_{k}}\rangle}$ reachability graph
depicts the state’s orbit under the action of a chosen group, and generates
all states which can be reached through circuits built of the generating
gates. Since ${|{D^{N}_{k}}\rangle}$ is often initialized as the starting
state for many quantum algorithms, the reachability graphs in Section 4
provide a map through the Hilbert space for algorithms that begin with
${|{D^{N}_{k}}\rangle}$. Furthermore, since construction of reachability
graphs is not limited to the Clifford group Munizzi_2022 , it would be
interesting to explore Dicke state orbits under a universal set of gates.

Reachability graphs can also be used to bound entanglement evolution under a
chosen set of gates, by examining how many times the entropy can be changed by
circuits in the graph Keeler2023b . Motivated to explore the dynamics of
${|{D^{N}_{k}}\rangle}$ entropy vectors under $\mathcal{C}_{2}$, we focused on
the subgroup $(HC)_{1,2}=\langle H_{1},\,H_{2},\,C_{1,2},\,C_{2,1}\rangle$
since entanglement in Clifford circuits occurs, at most, through the bi-local
CNOT gate. We found that the number of entropy vectors on each
${|{D^{N}_{k}}\rangle}$ reachability graph is constant, with $5$ entropy
vectors on the $288$-vertex graphs, like that in Figure 8, and $6$ entropy
vectors on the $576$-vertex graphs, like Figure 9. While the number of entropy
vectors is fixed, the number of distinct entanglement structures which compose
each vector continues to increase for larger and larger systems.

We expect our analysis of the $N$-qubit Dicke state entropy cone and
${|{D^{N}_{k}}\rangle}$ orbits to generalize for qudit Dicke states. Circuits
for deterministically preparing arbitrary qudit Dicke states are known, and
many recursive generalizations Dicke state properties have also been
demonstrated. We expect the ${|{D^{N}_{k}}\rangle}$ entropy vectors presented
in this paper to generalize similarly, with preliminary efforts towards $W$
state entropy cone generalization given in Schnitzer:2022exe . The Pauli and
Clifford groups can likewise be extended to arbitrary Hilbert space dimension
Jagannathan:2010sb ; Hostens_2005 , which would allow us to extend our orbit
model and consider the evolution of entanglement evolution for higher-
dimensional Dicke systems.

The Dicke state stabilizers presented in this work find immediate application
in stabilizer code construction. Given a scheme for encoding logical Dicke
states and a suitable choice of measurement, we can construct an error-
correcting channel directly using the stabilizers in Section 4. For specific
Dicke states, such as ${|{D^{N}_{k}}\rangle}$, the entanglement structure
renders the state robust to single-qubit loss, particularly at large $N$. We
expect this characteristic to offer significant error-correction advantages
when using Dicke state encoding for noisy processing. In future work, we
explore this proposal and construct a class of Dicke stabilizer codes,
evaluating their performance when compared to existing schemes.

Finally, the entanglement structure of certain Dicke states makes
${|{D^{N}_{k}}\rangle}$ an interesting candidate for magic distillation
protocols Bravyi2004 . Coupled with the ease of preparing
${|{D^{N}_{k}}\rangle}$, Dicke states can enable improved protocols for
distilling magic with minimal overhead. The states ${|{D^{5}_{1}}\rangle}$,
${|{D^{5}_{2}}\rangle}$, and ${|{D^{5}_{4}}\rangle}$ specifically possess a
significant amount non-local magic Bao2022a , though ultimately experience
error rates slightly above the fault-tolerant Bravyi-Kitaev threshold. These
error rates can be improved however, by passing ${|{D^{5}_{k}}\rangle}$
through a short sequence of gates. This further motivates an understanding
${|{D^{N}_{k}}\rangle}$ orbits under universal gate sets, which can provide
circuits to improve the utility of Dicke states in distillation schemes.

## Acknowledgements

HJS wishes to thank Matt Headrick for alerting him to the possible interest
overlaps of the Arizona State University group with his. WM wishes to thank
Adam Burchardt, ChunJun Cao, Jonathan Harper, Cynthia Keeler and Jason Pollack
for helpful discussions. WM is supported by the U.S. Department of Energy
under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.
Dicke states were brought to our attention by Rafael Nepomechie in
nepomechie2023qudit , and in private communication to HJS.

##  Appendix A Simplification for fixed $\ell$ and $k$

In this Appendix we give a few reductions for the ${|{D^{N}_{k}}\rangle}$
entropies in Eq. (22). Expanding $\ln$ in Eq. (22) gives,

$\begin{split}S_{\ell}=-\binom{N}{k}^{-1}\sum_{i=0}^{min(\ell,k)}&\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{N}{k}^{-1}\right]\\\
&-\binom{N}{k}^{-1}\sum_{i=0}^{min(\ell,k)}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{\ell}{i}\binom{N-\ell}{k-i}\right].\\\
\end{split}$ (48)

We invoke the Chu-Vandermonde identity, which reads

$\sum_{i=0}^{n}\binom{r}{i}\binom{s}{n-i}=\binom{r+s}{n},$ (49)

to simplify the first sum as

$\begin{split}S_{\ell}=\binom{N}{k}^{-1}&\binom{N}{\min(\ell,k)}\ln\left[\binom{N}{k}\right]\\\
&-\binom{N}{k}^{-1}\sum_{i=0}^{min(\ell,k)}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{\ell}{i}\binom{N-\ell}{k-i}\right].\\\
\end{split}$ (50)

For the case when $\ell\geq k$, Eq. (50) further simplifies to,

$S_{\ell}=\ln\left[\binom{N}{k}\right]-\binom{N}{k}^{-1}\sum_{i=0}^{k}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{\ell}{i}\binom{N-\ell}{k-i}\right],$
(51)

where we note the $\ell$-independence of the first term.

Likewise for $\ell<k$ we have the reduction

$S_{\ell}=\frac{k!(N-k)!}{\ell!(N-\ell)!}\ln\left[\binom{N}{k}\right]-\binom{N}{k}^{-1}\sum_{i=0}^{\ell}\binom{\ell}{i}\binom{N-\ell}{k-i}\ln\left[\binom{\ell}{i}\binom{N-\ell}{k-i}\right].$
(52)

##  Appendix B Additional Reachability Graphs

###  B.1 Clifford Orbits of ${|{D^{N}_{N}}\rangle}$

The reachability graph for ${|{D^{N}_{N}}\rangle}$, and accordingly for all
stabilizer states, under the action of $\mathcal{C}_{2}$ is shown in Figure
10.

Figure 10: Orbit of all ${|{D^{N}_{N}}\rangle}$ under the action of the
$2$-qubit Clifford group $\mathcal{C}_{2}$. The state ${|{D^{N}_{N}}\rangle}$
is a stabilizer state, and therefore its reachability graph is isomorphic to
that of all $2$-qubit stabilizer states.

Figure 11 gives the reachability graph for ${|{D^{N}_{N}}\rangle}$ under the
action of $\mathcal{C}_{3}$.

Figure 11: Orbit of ${|{D^{N}_{N}}\rangle}$ under the action of the $3$-qubit
Clifford group $\mathcal{C}_{3}$. Since ${|{D^{N}_{N}}\rangle}$ is a
stabilizer state, this reachability graph is isomorphic to the orbit shared by
all $3$-qubit stabilizer states.

###  B.2 $(HC)_{1,2}$ Orbits of Higher-Qubit ${|{D^{N}_{1}}\rangle}$

Below we include additional examples of ${|{D^{N}_{1}}\rangle}$ orbits under
the action of $(HC)_{1,2}\equiv\langle
H_{1},\,H_{2}\,C_{1,2}\,C_{2,1}\rangle$. Figure 12 shows the orbit for
${|{D^{4}_{1}}\rangle}$ under $(HC)_{1,2}$, which contains $288$ states and
$4$ different entropy vectors. This set of $5$ entropy vectors is built of $6$
different entanglement entropies.

Figure 12: Orbit of ${|{D^{4}_{1}}\rangle}$, and stabilizer state, under the
action of $(HC)_{1,2}$.

Figure 13 gives the orbit of ${|{D^{5}_{1}}\rangle}$ under $(HC)_{1,2}$. This
orbit likewise has $5$ different entropy vectors, which are composed of $9$
different entanglement entropies.

Figure 13: Orbit of ${|{D^{5}_{1}}\rangle}$, and stabilizer state, under the
action of $(HC)_{1,2}$.

##  Appendix C Entropy Vector Tables

In this Appendix we give exact entropy vectors seen in Figures 8 and 9. Table
1 gives each entropy vector from Figure 8. There are $4$ entanglement
entropies observed in the orbit of ${|{D^{3}_{1}}\rangle}$ under the action of
$\langle H_{1},\,H_{2}\,C_{1,2}\,C_{2,1}\rangle$, which we define as variables
in Eq. (53) for presentation clarity.

$\begin{split}s_{0}&\equiv 1,\\\
s_{1}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[3\right],\\\
s_{2}&\equiv\frac{5}{6}\log_{2}\left[\frac{6}{5}\right]+\frac{1}{6}\log_{2}\left[6\right],\\\
s_{3}&\equiv\frac{3-\sqrt{5}}{6}\log_{2}\left[\frac{6}{3-\sqrt{5}}\right]+\frac{3+\sqrt{5}}{6}\log_{2}\left[\frac{6}{3+\sqrt{5}}\right],\end{split}$
(53)

The four entanglement entropies in Eq. (53) compose the entropy vectors in
Table 1.

Label | Entropy Vector
---|---
| $(s_{1},\,s_{1},\,s_{1})$
| $(s_{3},\,s_{1},\,s_{1})$
| $(s_{1},\,s_{3},\,s_{1})$
| $(s_{0},\,s_{0},\,s_{1})$
| $(s_{2},\,s_{2},\,s_{1})$
Table 1: Table showing the $5$ entropy vectors found in the orbit of
${|{D^{3}_{1}}\rangle}$, shown in Figure 8, under the action of $(HC)_{1,2}$.
For a clearer presentation, we introduce the variables in Eq. (53) to present
these entropy vectors.

Similarly for the orbit of ${|{D^{4}_{2}}\rangle}$ under $\langle
H_{1},\,H_{2}\,C_{1,2}\,C_{2,1}\rangle$ action, there are $5$ entanglement
entropies observe. We likewise define the variables,

$\begin{split}s_{0}&\equiv\frac{5}{6}\log_{2}\left[\frac{12}{5}\right]+\frac{1}{6}\log_{2}\left[12\right],\\\
s_{1}&\equiv\frac{3-\sqrt{5}}{6}\log_{2}\left[\frac{12}{3-\sqrt{5}}\right]+\frac{3+\sqrt{5}}{6}\log_{2}\left[\frac{12}{3+\sqrt{5}}\right],\\\
s_{2}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[6\right],\\\
s_{3}&\equiv\frac{3-2\sqrt{2}}{6}\log_{2}\left[\frac{12}{3-2\sqrt{2}}\right]+\frac{3+2\sqrt{2}}{6}\log_{2}\left[\frac{12}{3+2\sqrt{2}}\right],\\\
s_{4}&\equiv 1,\\\
s_{5}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[3\right],\\\
s_{6}&\equiv\frac{5}{6}\log_{2}\left[\frac{6}{5}\right]+\frac{1}{6}\log_{2}\left[6\right].\end{split}$
(54)

Label | Entropy Vector
---|---
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{2},\,s_{2}\right)$
| $\left(s_{6},\,s_{5},\,s_{4},\,s_{4},\,s_{2},\,s_{1},\,s_{1}\right)$
| $\left(s_{5},\,s_{6},\,s_{4},\,s_{4},\,s_{2},\,s_{1},\,s_{1}\right)$
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{0},\,s_{0}\right)$
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{2},\,s_{2}\right)$
| $\left(s_{6},\,s_{6},\,s_{4},\,s_{4},\,s_{2},\,s_{3},\,s_{3}\right)$
Table 2: The $6$ entropy vectors contained in the orbit of
${|{D^{4}_{2}}\rangle}$ under the action of $(HC)_{1,2}$, illustrated in
Figure 9. We introduce variables in Eq. (54) to display the entropy vectors in
the table.

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