# Linear maps as sufficient criteria for entanglement depth and compatibility
in many-body systems

Maciej Lewenstein<EMAIL_ADDRESS>ICFO – Institut de Ciències
Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Spain  ICREA – Institució Catalana de Recerca i Estudis
Avançats, Lluis Companys 23, 08010 Barcelona, Spain  Guillem Müller-Rigat
ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Spain  Jordi Tura  Instituut-Lorentz,
Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands  Anna
Sanpera  Física Teórica: Informació i Fenòmens Quàntics. Departament de
Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain  ICREA –
Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010
Barcelona, Spain

###### Abstract

Physical transformations are described by linear maps that are completely
positive and trace preserving (CPTP). However, maps that are positive (P) but
not completely positive (CP) are instrumental to derive
separability/entanglement criteria. Moreover, the properties of such maps can
be linked to entanglement properties of the states they detect. Here, we
extend the results presented in [Phys. Rev A 93, 042335 (2016)], where
sufficient separability criteria for bipartite systems were derived. In
particular, we analyze the entanglement depth of an $N$-qubit system by
proposing linear maps that, when applied to any state, result in a bi-
separable state for the $1:(N-1)$ partitions, i.e., $(N-1)$-entanglement
depth. Furthermore, we derive criteria to detect arbitrary
$(N-n)$-entanglement depth tailored to states in close vicinity of the
completely depolarized state (the normalized identity matrix). We also provide
separability (or $1$\- entanglement depth) conditions in the symmetric sector,
including for diagonal states. Finally, we suggest how similar map techniques
can be used to derive sufficient conditions for a set of expectation values to
be compatible with separable states or local-hidden-variable theories. We
dedicate this paper to the memory of the late Andrzej Kossakowski, our
spiritual and intellectual mentor in the field of linear maps.

##  I Introduction

Linear maps are the fundamental tools to describe transformations in quantum
systems. In fact, any physical transformation corresponds to a completely
positive trace preserving operation (CPTP), that is, a linear map, $\Lambda$,
that is positive (P), $\Lambda(\rho)\geq 0$, on all quantum states ($\rho\geq
0$), and whose extension to a larger tensorial space leaves all states in the
composite space positive as well
$(\mathbbm{1}_{A}\otimes\Lambda_{B})\rho_{AB}\geq 0$. Such description applies
for closed and open quantum systems irrespectively from the fact that the
quantum system of interest is single or multipartite. However, maps that are
positive (P) but not completely positive (CP) are instrumental to derive
separability/entanglement criteria, that is, the absence/presence of shared
quantum correlations between the constituents of multipartite systems.
Moreover, the properties of such maps can be linked to entanglement properties
of the states they detect. Entanglement, the cornerstone feature of quantum
systems, has no classical analogue, and is a fundamental resource for quantum
information tasks.

Entanglement has been mostly studied and addressed and exploited in the
bipartite scenario. But even for this simplest case, checking if a given
composite system is separable (entangled) is an NP-hard
problemGurvits.JCSS.2003 . Despite the fact that in low dimensions one can use
semi-definite programming to numerically determine the presence of
entanglement Doherty+2.PRL.2002 , such method becomes unfeasible when the
dimensions of the composite Hilbert space grows, when the number of involved
parties increases, or when the aim, rather than checking the entanglement of a
specific quantum state, is to is characterize the entanglement properties of
families of states. For all these cases the common approaches include using
separability criteria or constructing entanglement witnesses.

To date, most of the known separability criteria are necessary but not
sufficient, starting with the prominent positive partial transpose criterion,
proposed by Peres Peres.PRL.1996 . In fact transposition $T$ is a paradigmatic
example of a positive map, i.e., a map that maps states onto states.
Transposition is not, however, completely positive, i.e., its extensions by
identity map to larger tensor spaces do not correspond to positive maps.
Therefore, this condition becomes a necessary criterion of separability. Such
map however, has proven to be sufficient condition of separability only for
two-qubits and one qubit-qutrit by Horodeckis 3Horodecki.PLA.1996 . For all
bipartite systems (i.e. Alice-Bob), it has also been proven in
(cf.4Horodeckis.RMP.2009 ), that any positive map acting exclusively on a
space of Alice’s (Bob’s) operators, transforms a separable state into a
positive definite state. Conversely, a state is separable if it remains
positive definite under the action of all positive maps on Alice’s (Bob’s)
side. Well known positive maps that have been proposed and applied to
determine separability are the celebrated reduction map M+PHorodecki.PRA.1999
; Cerf+2.PRA.1999 , the Breuer-Hall map Breuer.PRL.2006 ; Hall.JPA.2006 , or
the Choi map Choi.LAA.1975 .

Here, we follow the approach used to derive sufficient criteria of
separability for bipartite systems (necessary criteria of entanglement)
previously introduced by some of us in LAC16 . Now, our first aim is to derive
sufficient criteria to determine the entanglement depth of an ensemble of
$N$-qubit systems. The entanglement depth $n$ of an ensemble of $N$-parties
expresses the fact that the quantum state of the ensemble necessary involves
$n\leq N$ genuine multipartite entanglement. This characterization of
multipartite entanglement becomes extremely relevant for experiments involving
a large number of parties as it happens e.g., with cold gases. It is therefore
crucial for a meaningful comprehension of experimental results to have at our
disposal operational entanglement classification that can be used in
experiments.

The main point of our approach is to use families of (positive) maps,
$\Lambda_{\text{\boldmath$p$}}$, acting on global states of an $N$-qubit (or
qudit) multiparty system. These maps have the property that when applied to a
state (a state of a certain entanglement class) they produce a "weaker"
entanglement state (a state of another certain entanglement class; separable,
bi-separable, $n$-entanglement depth…). In particular, first we derive maps
whose action in the states of $(\mathbb{C}^{2})^{\otimes N}$ is to make them
bi-separable with respect the one partition, e.g., 1:(N-1). In other words,
maps all physical states onto the set of states with entanglement depth
$n=N-1$. The inverted map, $\Lambda^{-1}_{\text{\boldmath$p$}}$, (if it
exists) becomes then a sufficient criteria for bi-separability. Such a method
can be extended to derive maps that ensure (are sufficient) to reveal which
type of entanglement (entanglement depth) the multipartite system has. Further
we exploit such idea to derive sufficient conditions for compatibility with
separable states or local-hidden variable theories given a set of accessible
expectation values.

The paper is organized as follows. In Section II we present our notation,
definitions, and preliminary facts. In Section III we explain the general
theorem. Section IV we present our main results for $N$-qubits with several
maps leading to sufficient $N-1$ entanglement depth criteria and beyond. We
discuss as well the particular case symmetric subspaces. In Section V, we
focus on deriving sufficient conditions for compatibility with separable
states or local-hidden variable theories for a set of accessible expectation
values. We close the paper with the conclusions and outlook in Section VI.

##  II  Preliminaries

We consider $N$-qubit system (Alice, Bob, Charlie, … ) corresponding to a
composite Hilbert space ${\cal H}={\cal H}_{A}\otimes{\cal
H}_{B}\otimes\ldots=\mathbbm{C}^{2}\otimes\mathbbm{C}^{2}\otimes\ldots\mathbbm{C}^{2}=(\mathbbm{C}^{2})^{\otimes
N}$ of dimension $2^{N}$. The space of all bounded operators acting on it is
denoted by ${\cal B}({\cal H}_{A}\otimes{\cal H}_{B}\otimes\ldots)$, while the
set of states, that is, operators from ${\cal B}({\cal H}_{A}\otimes{\cal
H}_{B}\otimes\ldots)$ obeying $\rho\geq 0$, $\rho=\rho^{\dagger}$ and ${\rm
Tr}(\rho)=1$, as $R$. Note that $R$ is convex and compact. However, the
normalization condition of the states and, consequently, the compactness of
$R$ will only be assumed if necessary. Partially transposed operators are
denoted $\rho^{T}_{i}$, for $i=1,\cdots,N$. In the following, we will consider
also various classes of states such as:

  * •

Separable states $\Sigma$, also denoted as $\Sigma_{1}$, i.e., states that
admit the following decomposition

$\sigma=\sum_{k=1}^{K}p_{k}|\Psi_{k}\rangle_{1}\langle\Psi_{k}|_{1},$ (1)

where
$|\Psi_{k}\rangle_{1}=|e_{k_{1}}\rangle\otimes|e_{k_{2}}\rangle\otimes\ldots=\bigotimes_{i=1}^{N}|e_{k_{i}}\rangle$
are $N$ tensor products Peres.S.1995 .

  * •

States with entanglement depth $n\leq N$, $\Sigma_{n}$, i.e., the states that
admit the decomposition

$\sigma=\sum_{k=1}^{K}p_{k}|\Psi_{k}\rangle_{n}\langle\Psi_{k}|_{n},$ (2)

where $|\Psi_{k}\rangle_{n}=|e_{k_{i\cdots
j}}\rangle\otimes|e_{k_{i^{\prime},\cdots,j^{\prime}}}\rangle\otimes\ldots$
are product vectors that include genuinely entangled states of $i,\cdots,j\leq
n$ parties maximally, and at least one of the pure states must include an
$n$-partite genuinely entangled factor. TuraAloy1 ; TuraAloy2 ; TuraAloy3 .

  * •

PPT-operators $W_{PPT}$, are operators that have all possible partial
transposes positive, i.e., $W^{T_{i}}\geq$ for all $i=1,\cdots,N$ but also
$W^{T_{N/i,..j}}\geq 0$, etc. in particular, we will talk about PPT-states,
$R_{PPT}$, i.e., $\rho\in R$ and $\rho\in W_{PPT}$, i.e., $\rho^{T_{X}}\geq
0$, for any partition of the set of qubits into $X$ and the rest $N/X$.

Further, we consider various classes of linear maps

$\Lambda\colon{\cal B}({\cal H}_{A}\otimes{\cal H}_{B}\otimes\ldots)\to{\cal
B}({\cal H}_{A}\otimes{\cal H}_{B}\otimes\ldots)$

that preserve hermiticity. In particular, we pay particular attention to
family of maps parametrized by a multi-index $\bf p$ that map a certain subset
$S\subset{\cal B}({\cal H}_{A}\otimes{\cal H}_{B}\otimes\ldots)$ to another
certain subset of $S^{\prime}\subset{\cal B}({\cal H}_{A}\otimes{\cal
H}_{B}\otimes\ldots)$

$\Lambda_{{\bf p}}\colon S\to S^{\prime}$ (3)

such that for every $w\in S$, then $\Lambda_{\bf p}(w)\in S^{\prime}$. We
assume that those maps are invertible for almost all values of ${\bf p}$.

##  III  General theorem

First, we review the starting point of LAC16 , which is build upon the simple
observation:

######  Theorem 1.

Let $S,S^{\prime}$ be convex and compact subsets of ${\cal B}({\cal
H}_{A}\otimes{\cal H}_{B}\otimes\ldots)$, and let $\Lambda_{\bf p}\colon S\to
S^{\prime}$ be a family of maps, invertible for almost all $\bf p$. By ${\cal
P}_{SS^{\prime}}$ we denote the subset of the parameters set ${\bf p}$. The
maps have the property that for every $w\in S$, $\Lambda_{\bf p}(w)\in
S^{\prime}$ provided ${\bf p}\in{\cal P}_{SS^{\prime}}$. Then if
$\Lambda^{-1}_{\bf p}(\sigma)\in S$ $\Rightarrow$ $\sigma\in S^{\prime}$.

###### Proof.

Note that $\Lambda_{\bf p}\left[\Lambda^{-1}_{\bf p}(\sigma)\right]=\sigma$. ∎

In other words, the theorem provides sufficient criteria for $S^{\prime}$. In
practice, the choice of $\Lambda_{\mathbf{p}}$ is such that:

  * i)

we can easily check that $\Lambda_{\bf p}^{-1}(S^{\prime})\subset S$,

  * ii)

we can prove the assumption that $\Lambda_{\bf p}(S)\subset S^{\prime}$.

The difficulty on the derivation of such criteria is hidden in demonstrating
condition ii) and in technical difficulties of inverting $\Lambda_{\bf p}$.

In this work, we generalize LAC16  to the multipartite case by initially
considering $S$ the set of all semidefinite-positive elements of
$\mathcal{B}((\mathbb{C}^{2})^{\otimes N})$. Consequently, the sufficient
condition will be written as $\Lambda^{-1}_{\mathbf{p}}(\sigma)\geq
0\Longrightarrow\sigma\in S^{\prime}$. The concrete sets $S^{\prime}$ that we
will examine are related to the entanglement properties of $N$-party quantum
states and are to be specified in the next section for each example. Our
approach bears similitudes with the works of Barnum and Gurvits
Gurvits+Barnum.PRA.2002  and Szarek et al. Szarek.PRA.2005 ;
Aubrun+Szarek.PRA.2006

##  IV  Main results

Here we present the main results on the construction of new sufficient
criteria for $N$-qubit systems from the inversion of linear maps. We will
consider families of such transformations inspired by the reduction and the
Breuer-Hall maps.

###  IV.1 Reduction-inspired maps

To start with, we consider the simple reduction-type family of maps defined as
$\Lambda_{\alpha}(\rho)=\mathrm{Tr}(\rho)\mathbbm{1}+\alpha\rho$.

####  IV.1.1 Sufficient criteria for $N-1$ \- entanglement depth

Now we fix $S^{\prime}\subset S$ the set of bi-separable states in the
partitions $1:(N-1)$, (one qubit subsystem versus the others $N-1$). In other
words, $S^{\prime}$ is the set of states with entanglement depth $(N-1)$, so
the set does not contain genuine $N$-qubit entanglement. The task is to find
the range of $\alpha$ in which for all states $\rho\geq 0$ we have that
$\Lambda_{\alpha}(\rho)$ is bi-separable for any partition $1:(N-1)$. Once
this is done, $\Lambda_{\alpha}^{-1}(\rho)\geq 0$ would be a sufficient
$(N-1)$-entanglement depth criterion. Bi-separability of
$\Lambda_{\alpha}(\rho)$ for $N$ qubits follows from the following known
result.

######  Theorem 2.

Compare Vidal.PRA.1999 ; Gurvits+Barnum.PRA.2002 . Let
$\Lambda_{\alpha}(\rho)={\rm Tr}(\rho)\mathbbm{1}+\alpha\rho$ be the family of
maps, and $-1\leq\alpha\leq 2$. Then $\rho\geq 0$ $\Rightarrow$
$\Lambda_{\alpha}(\rho)=:\sigma\in\Sigma_{N-1}$, i.e. $\sigma$ is bi-separable
in any $1:(N-1)$ partition (i.e. has entanglement depth $(N-1)$)

###### Proof.

The proof follows LAC16 . It is enough to prove the theorem for pure states,
$\rho=|\Psi\rangle\langle\Psi|$. For $N$ qubits, $|\Psi\rangle$ has maximally
Schmidt rank 2, in the partition Alice:All Others (any other $1:(N-1)$).
Without loosing generality, we can assume that
$|\Psi\rangle=\lambda_{0}|0\rangle\otimes|0\rangle_{N-1}+\lambda_{1}|1\rangle\otimes|1\rangle_{N-1}$.
It is then enough to check the positiveness and separability of
$\Lambda(|\Psi\rangle\langle\Psi|)$ on a $2\otimes 2$ space spanned by
$|0\rangle$, $|1\rangle$, $|0\rangle_{N-1}$, $|1\rangle_{N-1}$ in both Alice’s
and All Others’ spaces. PPT provides then necessary and sufficient condition
3Horodecki.PLA.1996 , and we easily get that indeed $-1\leq\alpha\leq 2$. The
condition $\alpha\geq-1$ actually follows already from a simpler requirement
of positiveness of $\Lambda_{\alpha}(\rho)$, as it appears in the definition
of the reduction map M+PHorodecki.PRA.1999 ; Cerf+2.PRA.1999 . ∎

The inverse of the reduction map leads to the following condition:

######  Corollary 3.

(Sufficient $N-1$-entanglement depth criterion 1) If
$\Lambda^{-1}_{\alpha}(\sigma):=[\sigma-{\rm
Tr}(\sigma)\mathbbm{1}/(2^{N}+\alpha)]/\alpha\geq 0$, then
$\sigma\in\Sigma_{N-1}$, i.e. $\sigma$ is bi-separable in any partition
$1:(N-1)$.

It is interesting to check whether simple criteria are strong enough to detect
some states outside the "separable ball" around identity for $N$
qubitsSzarek.PRA.2005 ; Aubrun+Szarek.PRA.2006 ; Gurvits+Barnum.PRA.2002  Of
course, we do not expect that all states from the separable ball are detected.
The reason for this weakness is that Barnum and Gurvits, and Szarek have
estimated the radius of the separable ball using a stronger result, namely: if
the operator norm of a Hermitian matrix $X$ is bounded by 1, then
$\mathbbm{1}+X$ is separable. In contrast, our simple criteria depend on the
spectrum, not on the norm, and are thereby independent. Nonetheless, we will
see in the next sections that with other choices of $\Lambda_{\bf p}$’s, we
could derive further stronger sufficient $(N-1)$-depth criteria.

####  IV.1.2 Sufficient criteria for arbitrary $N-n$-entanglement depth

The previous result can be generalized to detect arbitrary $N-n$-entanglement
depth around the maximally mixed state with the next Theorem, which is
analogous to Theorem 2:

######  Theorem 4.

Let $\Lambda_{\alpha}(\rho)={\rm Tr}(\rho)\mathbbm{1}+\alpha\rho$ be the
family of maps, and $-1\leq\alpha\leq 2$. Then $\rho\geq 0$ $\Rightarrow$
$\Lambda_{\alpha}(\rho)=\sigma\in\Sigma_{N-n}$, i.e. $\sigma$ has the
entanglement depth $N-n$.

Similarly, if $\sigma\geq 0$, and $\Lambda_{\alpha}^{-1}(\sigma)=\rho\geq 0$,
then $\sigma$ has the entanglement depth $N-n$.

###### Proof.

We discuss in detail the cases $n=2$; the generalizations to arbitrary $n$ are
straightforward. As before, it is enough to prove for pure states,
$\rho=|\Psi\rangle\langle\Psi|$. For $n=2$, $|\Psi\rangle$ has maximally
Schmidt rank 4, and, without loosing generality, we can assume that
$|\Psi\rangle=\lambda_{0}|0\rangle\otimes|0\rangle+\lambda_{1}|1\rangle\otimes|1\rangle+\lambda_{2}|2\rangle\otimes|2\rangle+\lambda_{3}|3\rangle\otimes|3\rangle$,
with $\lambda_{0}\geq\lambda_{1}\geq\lambda_{3}\geq\lambda_{4}$, and
$\sum_{i=0}^{3}\lambda_{i}^{2}=1$. It is enough to check then positiveness and
separability of $\Lambda(|\Psi\rangle\langle\Psi|)$ on a $4\otimes 4$ space
spanned by $|0\rangle$, $|1\rangle$, $|2\rangle$ and $|3\rangle$ in both
Alice’s and Bob’s spaces. This is achieved by observing first that if
$|\Psi\rangle$ is a normalized state, then obviously the positivity of
$\Lambda_{\alpha}(|\Psi\rangle\langle\Psi|)$ requires $\alpha\geq-1$. Direct
inspection shows that for $\alpha=-1$,
$\Lambda_{\alpha}(|\Psi\rangle\langle\Psi|)$ is a sum of four positive
matrices that effectively act on two-qubit Hilbert spaces and are given by

$\sigma_{ij}=\ket{\varphi_{ij}}\\!\bra{\varphi_{ij}}+\ket{ij}\\!\bra{ij}+\ket{ji}\\!\bra{ji},$
(4)

where $\ket{\varphi_{ij}}=\lambda_{j}\ket{ii}-\lambda_{i}\ket{jj}$ with
$i<j=0,1,2,3$.

These matrices are PPT and thus separable.

For $\alpha\geq 0$, we decompose
$\Lambda_{\alpha}(|\Psi\rangle\langle\Psi|)=\sigma^{T_{A}}+D$, where
$\sigma^{T_{A}}$ is separable by construction, and $D$ is positive diagonal in
the computational product basis, ergo separable by direct inspection. We
consider a family of product vectors
$|p(\phi,\psi)\rangle=A(1,ae^{i\phi},be^{i\psi},ce^{i\theta})^{\otimes 2}$ and
the separable state

$\sigma=\int d\phi/2\pi\int d\psi/2\pi\int
d\theta/2\pi|p(\phi,\psi)\rangle\langle p(\phi,\psi)|.$

The parameters can be chosen indeed in such a way that
$\Lambda_{\alpha}(|\Psi\rangle\langle\Psi|)-\sigma^{T_{A}}$ is diagonal. To
this aim we set $A^{2}=\alpha\lambda_{0}^{2}$,
$a^{2}=\lambda_{1}/\lambda_{0}$, $b^{2}=\lambda_{2}/\lambda_{0}$, and
$c^{2}=\lambda_{3}/\lambda_{0}$. Checking the explicit conditions that the
matrix $D$ is positive implies that
$1-A^{2}a^{2}=1-\alpha\lambda_{0}\lambda_{1}\geq 0$, i.e. $\alpha\leq 2$ since
$\lambda_{0}$ and $\lambda_{1}$ are the highest and the second-highest Schmidt
coefficients, respectively. The case $-1<\alpha<0$ follows from convexity.

∎

####  IV.1.3 Sufficient separability (or 1- entanglement depth) criteria for
symmetric states

In this subsection, we will start by considering maps acting on the bosonic
space of $N$ qubits, ${\cal
S}(\mathbbm{C}^{2}\otimes\ldots\otimes\mathbbm{C}^{2})$ by setting $S$ the set
of $N$-qubit symmetric states. The dimension of such subspace is $N-1$ and the
basis elements read:

$\left\\{\ket{S_{k}^{N}}={N\choose
k}^{-1/2}\sum_{\pi\in\mathfrak{S}_{N}}\pi(\ket{0}^{\otimes
N}\otimes\ket{1}^{\otimes{N-k}})\right\\}_{k=0}^{N}\ ,$ (5)

where the sum is over the distinct permutations. In the context of the
reduction map, we denote $\mathbbm{1}_{\mathcal{S}}$ as the identity in the
symmetric subspace (i.e. the subspace spanned by the vectors of Eq. 5).

In this section. we start by deriving sufficient separability criteria by
taking $S^{\prime}$ the subset of separable symmetric states of two or three
qubits.

For $N=2$ we obtain:

######  Theorem 5.

Let $\Lambda_{\alpha}(\rho_{\mathcal{S}})={\rm
Tr}(\rho_{\mathcal{S}})\mathbbm{1}_{\mathcal{S}}+\alpha\rho_{\mathcal{S}}$ be
the family of maps acting on the states of two qubits in the symmetric space,
and $-3/4\leq\alpha\leq 1$. Then $\rho_{\mathcal{S}}\geq 0$ $\Rightarrow$
$\Lambda_{\alpha}(\rho_{\mathcal{S}})=\sigma_{\mathcal{S}}\in\Sigma_{1}$, i.e.
$\sigma_{\mathcal{S}}$ is separable.

Similarly, if $\sigma_{\mathcal{S}}\geq 0$, and
$\Lambda_{\alpha}^{-1}(\sigma_{\mathcal{S}})=\rho_{\mathcal{S}}\geq 0$, then
$\sigma_{\mathcal{S}}$ has the entanglement depth $1$.

###### Proof.

The proof is straightforward. We consider a pure state,
$|\Psi\rangle=\lambda_{0}|0\rangle\otimes|0\rangle+\lambda_{1}|1\rangle\otimes|1\rangle$,
apply the map to it and check the positivness of the result and its
separability using the PPT criterion. ∎

Similar, but a little weaker result can be obtained for the maps acting on the
states of $N=3$ qubits in the symmetric space. In this subspace, PPT is
necessary but also sufficient for separability Eckert2002 .

######  Theorem 6.

Let $\Lambda_{\alpha}(\rho_{\mathcal{S}})={\rm
Tr}(\rho_{\mathcal{S}})\mathbbm{1}_{\mathcal{S}}+\alpha\rho_{\mathcal{S}}$ be
the family of maps acting on the states of three qubits in the symmetric
space, and $-1/3=-\lambda_{\rm min}\leq\alpha\leq 2\lambda_{\rm min}=2/3$,
where $\lambda_{\rm min}>0$ is the minimal eigenvalue of $(\mathbbm{1}_{\cal
S})^{T_{A}}$. Then $\rho_{\mathcal{S}}\geq 0$ $\Rightarrow$
$\Lambda_{\alpha}(\rho_{\mathcal{S}})=\sigma_{\mathcal{S}}\in\Sigma_{1}$, i.e.
$\sigma_{\mathcal{S}}$ is separable.

Similarly, if $\sigma_{\mathcal{S}}\geq 0$, and
$\Lambda_{\alpha}^{-1}(\sigma_{\mathcal{S}})=\rho_{\mathcal{S}}\geq 0$, then
$\sigma_{\mathcal{S}}$ has the entanglement depth $1$.

###### Proof.

Obviously, the positiveness of the map requires $\alpha\geq-1$. Now we observe
that $(\mathbbm{1}_{\cal S})^{T_{A}}$ is a positive definite operator acting
on $\mathbbm{C}^{2}\otimes{\cal S}(\mathbbm{C}^{2}\otimes\mathbbm{C}^{2})$,
with minimal eigenvalue $\lambda_{\rm min}$. The partial transpose of any
projector on a pure state has eigenvalues between $[-1/2,1]$Swapan_neg , so
for positive $\alpha$, we must have $\lambda_{\rm min}-\alpha/2\geq 0$, and
for negative $\alpha$, $\lambda_{\rm min}-|\alpha|\geq 0$. The eigenvalues of
$(\mathbbm{1}_{\cal S})^{T_{A}}$ can be obtained analytically and are:
$(1/3,1/3,1/3,1/3,4/3,4/3)$, so that $\lambda_{\rm min}=1/3$. ∎

The latter result can be generalized to $N$ qubits, where $\lambda_{\rm
min}=1/N$. There, we cannot conclude PPT states are separable: starting from
$N=4$, there exist PPT entangled symmetric states Tura-sym .

Diagonal states.– However, for diagonal qubit symmetric states for arbitrary
$N$,

$\rho_{DS}=\sum_{k=0}^{N}p_{k}\ket{S_{k}^{N}}\bra{S_{k}^{N}}\ ,$ (6)

PPT for the largest bipartitions
$\left\lfloor{\frac{N}{2}}\right\rfloor:N-\left\lfloor{\frac{N}{2}}\right\rfloor$
is necessary and sufficient for separability Ruben_Sanpera . The next result
gives an efficient way to compute the partial transpose for such states:

######  Lemma 7.

Yu ; Ruben_Sanpera  The state $\rho_{DS}$ has positive partial transpose with
respect the bipartition
$\left\lfloor{\frac{N}{2}}\right\rfloor:N-\left\lfloor{\frac{N}{2}}\right\rfloor$
iff

$M_{l}(\\{p_{k}\\}_{k=0}^{N})=\left[\begin{matrix}p_{0+l}/{N\choose
0+l}&p_{1+l}/{N\choose 1+l}&\dots&p_{n_{l}+l}/{N\choose n_{l}+l}\\\
p_{1+l}/{N\choose 1+l}&p_{2+l}/{N\choose 2+l}&\dots&p_{n_{l}+1+l}/{N\choose
n_{l}+1+l}\\\ \vdots&\vdots&&\vdots\\\ p_{n_{l}+l}/{N\choose
n_{l}+l}&p_{n_{l}+1+l}/{N\choose n_{l}+1+l}&\cdots&p_{2n_{l}+l}/{N\choose
2n_{l}+l}\end{matrix}\right]\geq 0,\\\ $ (7)

where $l\in\\{0,1\\}$ and
$n_{l=0}=\left\lfloor{\frac{N}{2}}\right\rfloor,n_{l=1}=\left\lfloor{\frac{N-1}{2}}\right\rfloor$.

This allows us to formulate the basic result on diagonal $N$-qubit symmetric
states:

######  Theorem 8.

Let $\Lambda_{\alpha}(\rho_{DS})=\sigma_{DS}={\rm
Tr}(\rho)\mathbbm{1}_{\mathcal{S}}+\alpha\rho_{DS}$ be the family of maps
acting on diagonal qubit symmetric states. Let $\alpha$ be such that the two
associated matrices $M^{\prime}_{0}$ and $M^{\prime}_{1}$ are positive
semidefinite, where $M^{\prime}_{0}=M_{0}(\\{1+\alpha p_{k}\\}_{k=0}^{N})$,
$M^{\prime}_{1}=M_{1}(\\{1+\alpha p_{k}\\}_{k=0}^{N})$.

Similarly, if $\sigma_{DS}\geq 0$, and
$\Lambda_{\alpha}^{-1}(\sigma_{DS})=\rho_{DS}\geq 0$, then $\sigma_{DS}$ is
separable.

###### Proof.

The map transforms $p_{k}\to p_{k}^{\prime}=1+\alpha p_{k}$. Note that
$p_{k}^{\prime}$ are not normalized, but this does not affect positivity
condition. Thus, the relevant Hankel matrices for $\sigma_{DS}$ have indeed
the form presented in Lemma 7. ∎

Via a convex roof argument, it is sufficient to consider pure states. Hence,
to compute the bounds of $\alpha$ for diagonal symmetric states it is
sufficient to check $\Lambda_{\alpha}(\ket{D^{N}_{k}}\bra{D^{N}_{k}})$ for
each basis element $k=0,1,..,N$ and take the intersection of ranges
$\alpha_{{\rm min,max}_{k}}$. In order to compute the bound of $\alpha$ we
know that for $\alpha=0$ the state is PPT so we just need to solve $\det
M^{\prime}_{0}=0,\det M^{\prime}_{1}=0$ with $M_{0},M_{1}$ corresponding to
each basis element to find at least one vector in the kernel. By finding the
closest roots of $\det M^{\prime}_{0}=0,\det M^{\prime}_{1}=0$ with
$M_{0},M_{1}$ from both sides, we obtain

$N=2$ | $N=3$ | $N=4$ | $N=5$
---|---|---|---
$(-0.750,1.000)$ | $(-0.667,0.732)$ | $(-0.326,0.440)$ | $(-0.231,0.268)$
Table 1: Bounds on $\alpha$, $(\alpha_{\rm{min}},\alpha_{\rm{max})}$ for
different number of parties $N$. The bounds $(\alpha_{\min},\alpha_{\max})$
are algebraic numbers and they take the following closed expressions: for
$N=2:[-3/4,1]$, for $N=3:[-2/3,\sqrt{3}-1]$, for
$N=4:[(3\sqrt{6}-8)/2,5(\sqrt{21}-3)/18]$, and for $N=5$ the lower bound is
given by the smallest root of the polynomial
$18\alpha^{3}+54\alpha^{2}-378\alpha-90$ and the upper bound is
$(\sqrt{114}-8)/10.$

This result can be extended to qudits. In such case, the symmetric basis is
generalized as:

$\left\\{\ket{S_{\mathbf{k}}^{N}}={N\choose\mathbf{k}}^{-1/2}\sum_{\pi\in\mathfrak{S}_{N}}\pi\left(\bigotimes_{a=0}^{d-1}\ket{a}^{\otimes
k_{a}}\right)\right\\}_{\mathbf{k}\vdash N}\ ,$ (8)

where now $\mathbf{k}=(k_{0},k_{1},..,k_{d-1})$ is a partition of $N$.
Presently we consider states of the form introduced in Ref. Rut_Mar :

$\rho_{DD}=\sum_{k=0}^{N(d-1)}p_{k}\ket{D_{k}^{N}}\bra{D_{k}^{N}}\ ,$ (9)

where $\ket{D_{k}^{N}}$ are the natural generalization of Dicke states for
higher local dimension and read as coherent superposition of product vectors
with the same number of excitations. In the symmetric basis,

$\left\\{\ket{D_{k}^{N}}=\mathcal{N}_{k}^{-1/2}\sum_{\mathbf{d}\cdot\mathbf{k}=k}{N\choose\mathbf{k}}^{1/2}\ket{S_{\mathbf{k}}^{N}}\right\\}_{k=0}^{N(d-1)}\
,$ (10)

where $\mathcal{N}_{k}^{1/2}$ is the normalization factor (without simple
formula) and $\mathbf{d}=(0,1,..,d-1)$. We denote by
$\mathbbm{1}_{\mathcal{D}}$ the identity (projector) on the basis of Eq. (10).

Similarly, for $N$ even PPT with respect the largest bipartition
$\left\lfloor{\frac{N}{2}}\right\rfloor:\left(N-\left\lfloor{\frac{N}{2}}\right\rfloor\right)$
is a necessary and sufficient separability criteria for states of the form
$\rho_{DD}$ Rut_Mar . In complete analogy with the qubit case, we can pose the
following theorems:

######  Lemma 9.

Rut_Mar  The state $\rho_{DD}$ has positive partial transpose with respect the
bipartition
$\left\lfloor{\frac{N}{2}}\right\rfloor:\left(N-\left\lfloor{\frac{N}{2}}\right\rfloor\right)$
if, and only if,

$M_{l}(\\{p_{k}\\}_{k=0}^{N(d-1)})=\left[\begin{matrix}p_{0+l}/\mathcal{N}_{0+l}&p_{1+l}/\mathcal{N}_{1+l}&\dots&p_{n_{l}+l}/\mathcal{N}_{n_{l}+l}\\\
p_{1+l}/\mathcal{N}_{1+l}&p_{2+l}/\mathcal{N}_{2+l}&\dots&p_{n_{l}+1+l}/\mathcal{N}_{n_{l}+l+l}\\\
\vdots&\vdots&&\vdots\\\
p_{n_{l}+l}/\mathcal{N}_{n_{l}+l}&p_{n_{l}+1+l}/\mathcal{N}_{n_{l}+l+1}&\cdots&p_{2n_{l}+l}/\mathcal{N}_{2n_{l}+l}\end{matrix}\right]\geq
0,\\\ $ (11)

where $l\in\\{0,1\\}$ and
$n_{l=0}=\left\lfloor{\frac{N(d-1)}{2}}\right\rfloor,n_{l=1}=\left\lfloor{\frac{N(d-1)-1}{2}}\right\rfloor$.

This allows us to formulate the basic result regarding the qudit diagonal
states of Eq.(9) :

######  Theorem 10.

Let $\Lambda_{\alpha}(\rho_{DD})=\sigma_{DD}={\rm
Tr}(\rho)\mathbbm{1}_{\mathcal{D}}+\alpha\rho_{DD}$ be the family of maps
acting on diagonal qudit Dicke states. Let $\alpha$ be such that the two
matrices $M^{\prime}_{0}$ and $M^{\prime}_{1}$ be positive semidefinite, where
$M^{\prime}_{0}=M_{0}(\\{1+\alpha p_{k}\\}_{k=0}^{N(d-1)})$,
$M^{\prime}_{1}=M_{1}(\\{1+\alpha p_{k}\\}_{k=0}^{N(d-1)})$.

Similarly, if $\sigma_{DD}\geq 0$, and
$\Lambda_{\alpha}^{-1}(\sigma_{DD})=\rho_{DD}\geq 0$, then $\sigma_{DD}$ is
separable.

###### Proof.

The map transforms $p_{k}\to p_{k}^{\prime}=1+\alpha p_{k}$. Note that
$p_{k}^{\prime}$ are not normalized, but this does not affect positivity
condition. Thus, the relevant Hankel matrices for $\sigma_{DD}$ have indeed
the form presented in Lemma 9. ∎

Note that for qubits ($d=2$), $\rho_{DD}$ and $\rho_{DS}$ are equivalent, thus
yielding Theorem 8 for $N$ even as a particular case.

For qutrits ($d=3$) and, more generally, qudits (unrestricted $d$) we can find
the critical values of $\alpha$ by following these steps: We begin by
considering $\Pi_{DD}:=\sum_{k=0}^{N(d-1)}\ket{D_{k}^{N}}\bra{D_{k}^{N}}$ and,
for every $k\in\\{0,\ldots,N(d-1)\\}$ we compute the characteristic polynomial
$Q_{k}(x;\alpha)$ of
$(\Pi_{DD}+\alpha\ket{D_{k}^{N}}\bra{D_{k}^{N}})^{\Gamma}$, where $\Gamma$
denotes partial transposition with respect to some subsystems. By virtue of a
convex roof argument, it suffices to consider projectors onto pure basis
states $\ket{D_{k}^{N}}\bra{D_{k}^{N}}$. To analytically find a range of
$\alpha$ we proceed by computing the polynomial greatest common divisor
between $Q_{k}(x;\alpha)$ and $Q_{k}(x;0)$. Since $\alpha$ is treated as a
parameter, the polynomial Euclidian algorithm using long division can be used,
as $Q_{k}$ is a univariate polynomial. Computing
$\gcd(Q_{k}(x;\alpha),Q_{k}(x;0))$ allows us to ignore all the eigenvalues of
the partial transposition that do not depend on alpha without even having to
compute them. All the eigenvalues that do not depend on $\alpha$ are non-
negative if and only if $(\Pi_{DD})^{\Gamma}\geq 0$ on all possible partial
transpositions. If this holds, we can therefore define
$R_{k}(x;\alpha):=Q_{k}(x;\alpha)/\gcd(Q_{k}(x;\alpha),Q_{k}(x;0))$, which is
a polynomial by construction. If all the roots of $R_{k}(x;0)$ are strictly
positive, then by continuity of $\alpha$, we will have
${\alpha_{\max}}-{\alpha_{\min}}>0$. To find the candidates for
${\alpha_{\max}}$ and ${\alpha_{\min}}$ we can find all the roots of
$R_{k}(0;\alpha)$ which can be expressed in terms of radicals for small $N$
and $d$. Note, however, that the projector $\Pi_{DD}$ violates the PPT
criterion for $N=3$ and $d\geq 4$ and it is therefore entangled. In this case
the range for $\alpha$ is $\emptyset$.

For qutrits and $N=3$, $R_{k}(x;\alpha)$ splits in cubic factors in $x$.
Therefore, we can write their roots as radicals in function of $\alpha$, which
we can denote $\lambda_{i}(\alpha)$. By solving for $\lambda_{i}(\alpha)=0$
and keeping all the real solutions we can iterate on $k$ to find
$\alpha_{\min}$ and $\alpha_{\max}$. In the case fof $N=d=3$, $\alpha_{\max}$
is given by $(7\sqrt{10}-22)/36\approx 0.00377621$ whereas $\alpha_{\min}$ is
the intermediate root of the polynomial
$49\alpha^{3}+147\alpha^{2}-315\alpha-1$, approximately $-0.00316992$.

###  IV.2 Breuer-Hall- inspired maps

Let us start the subsection by presenting generalizations of Theorem 2 derived
in a similar spirit as the generalization of the reduction map by the Breuer-
Hall map Breuer.PRL.2006 ; Hall.JPA.2006 . Note that Breuer-Hall’s
construction does not work in the two qubit case—Breuer-Hall’s map identically
vanishes for qubits. In our case when we consider maps acting on the composite
space of Alice and Bob and Charlie, etc. this restriction will not apply. In
the two-dimensional space there exist (up to a phase) a single unitary,
$\sigma_{2}$, with the property that for every $|e\rangle$,
$\sigma_{2}|e^{*}\rangle=|e^{\perp}\rangle$, where $*$ indicates complex
conjugation and $\perp$ perpendicular. This operator can be applied to all of
the $N$ qubits. It is thus easy to construct analogous Breuer-Hall unitary
operators $V$, such that
$V|f_{1}^{*}\rangle\otimes\ldots\otimes|f_{k}\rangle=|f_{1}^{\perp}\rangle\otimes\ldots\otimes|f_{k}^{\perp}\rangle$,
being simple tensor products of $\sigma_{2}$.

Below we will denote, for brevity:

$\tilde{\rho}_{A}=\sigma^{A}_{2}\rho^{T_{A}}\sigma^{A}_{2},$
$\tilde{\rho}_{AB}=\sigma^{A}_{2}\sigma^{B}_{2}\rho^{T_{AB}}\sigma^{B}_{2}\sigma^{A}_{2},$

etc.

We can now prove the following result about a generalized $\Lambda_{p}$.

######  Theorem 11.

Let $\Lambda^{A}_{\alpha,\beta}(\rho)={\rm
Tr}(\rho)\mathbbm{1}+\alpha\rho+\beta\tilde{\rho}_{A}$ be a family of maps.
Let $\alpha\geq{\rm max}[-1,\beta/2-1]$ and $\beta\geq{\rm
max}[-1,\alpha/2-1]$. Then $\rho\geq 0$ $\Rightarrow$
$\Lambda_{\alpha,\beta}(\rho)=:\sigma\in\Sigma_{N-1}$, i.e., $\sigma$ is bi-
separable in the partition $1^{A}:(N-1)^{BC\ldots}$

###### Proof.

The proof is very similar to that of Theorem 2 as above, and it follows LAC16
. It is enough to prove it for pure states, $\rho=|\Psi\rangle\langle\Psi|$.
Without any loss of generality, take
$|\Psi\rangle=\lambda_{0}|0\rangle_{A}\otimes|0\rangle_{A.O.}+\lambda_{1}|1\rangle_{A}\otimes|1\rangle_{A.O.}$.
It is enough to check then positiveness and separability of
$\Lambda(|\Psi\rangle\langle\Psi|)$ on a $2\otimes 2$ space spanned by
$|0\rangle_{A,A.O.}$, $|1\rangle_{A,A.O.}$ in both Alice’s and All Others’
(A.O.) spaces, where PPT provides then necessary and sufficient condition. ∎

The above theorem has an obvious corollary:

######  Corollary 12.

Let $\Lambda^{i}_{\alpha_{i},\beta_{i}}(\rho)={\rm
Tr}(\rho)\mathbbm{1}+\alpha_{i}\rho+\beta_{i}\tilde{\rho}_{i}$ be a family of
maps. Let
$\Lambda(\rho)=\sum_{i=1}^{N}p_{i}\Lambda^{i}_{\alpha_{i},\beta_{i}}(\rho)$,
with $p_{i}\geq 0$, and $\sum_{i=1}^{N}p_{i}=1$. Let $\alpha_{i}\geq{\rm
max}[-1,\beta_{i}/2-1]$ and $\beta_{i}\geq{\rm max}[-1,\alpha_{i}/2-1]$. Then
$\rho\geq 0$ $\Rightarrow$ $\Lambda(\rho)=:\sigma\in\Sigma_{N-1}$, i.e.,
$\sigma$ has entanglement depth $N-1$ (is a mixture of states that are bi-
separable in the partition $1^{i}:(N-1)^{A.O.}$

###### Proof.

The proof is obvious: a mixture of bi-separable state in the partitions
$1^{i}:(N-1)^{A.O.}$ has $(N-1)$-depth. The problem of using the above result
for entanglement depth criteria lies in the difficulty of inverting the map
$\Lambda(\rho)$. ∎

The general Theorem 1 then immediately implies the following $(N-1)$-depth
result.

######  Theorem 13.

(Sufficient $(N-1)$-depth criterion 2) Let
$\rho=\Lambda_{\alpha,\beta}^{-1}(\sigma)\geq 0$, i.e.,

$\frac{1}{\alpha^{2}-\beta^{2}}\left[\alpha\sigma-\beta\tilde{\sigma}_{A}-\frac{(\alpha-\beta)}{2^{N}+\alpha+\beta}{\rm
Tr}(\sigma)\mathbbm{1}\right]\geq 0.$ (12)

Then $\sigma$ is bi-separable in the partition $1_{A}:(N-1)_{A.O.}$

Noticeably, the contributions from $\alpha$ and $\beta$ compensate each other,
diminishing in effect off-diagonal parts of $\sigma$ in the Alice space. The
criterion derived above is particularly strong at the boundary of the
parameter region, i.e., for $\beta=\alpha/2-1$ or $\alpha=\beta/2-1$.

######  Corollary 14.

(Sufficient $(N-1)$-depth criterion 3) Let

$\left[\alpha\sigma-(\alpha/2-1)\tilde{\sigma}_{A}-\frac{(1+\alpha/2)}{2N+3\alpha/2-1}{\rm
Tr}(\sigma)\mathbbm{1}\right]\geq 0$ (13)

or

$\left[\beta\tilde{\sigma}_{A}-(\beta/2-1)\sigma-\frac{(1+\beta/2)}{2N+3\beta/2-1}{\rm
Tr}(\sigma)\mathbbm{1}\right]\geq 0.$ (14)

Then $\sigma$ is bi-separable in the partition $1_{A}:(N-1)_{A.O.}$

The above corollary has a particularly interesting limit, when both $\alpha$
and $\beta$ tend to $\infty$.

######  Corollary 15.

(Sufficient $(N-1)$-depth criterion 4) Let

$\sigma-\tilde{\sigma}_{A}/2>0$ (15)

or

$\tilde{\sigma}_{A}-\sigma/2>0.$ (16)

Then $\sigma$ is bi-separable in the partition $1_{A}:(N-1)_{A.O.}$

Note that the latter inequalities are valid only in the asymptotic sense and
require strict positiveness. One should also note that in the above results we
used maps that explicitly involved partially transposed matrices. For this
reason, it is useful to remind the reader of some results of Ref.
Kraus+3.PRA.2000 .

######  Theorem 16.

Kraus+3.PRA.2000  If $\sigma=\sigma^{T_{A}}\geq 0$ then $\sigma$ is bi-
separable in the partition $1_{A}:(N-1)_{A.O.}$

######  Corollary 17.

Kraus+3.PRA.2000  Let $\sigma\geq 0$ be a state. If $\sigma+\sigma^{T_{A}}$ is
of full rank and $\|(\sigma+\sigma^{T_{A}})^{-1}\|\
\|\sigma-\sigma^{T_{A}}\|\leq 1$, then $\sigma$ is bi-separable in the
partition $1_{A}:(N-1)_{A.O.}$. This corollary implies that if $\sigma$ is of
full rank and is very close to $\sigma^{T_{A}}$ then $\sigma$ is separable.
Here we use the operator norm
$\|A\|:=\max_{\|\Psi\rangle\|=1}\|A|\Psi\rangle\|$.

Note that the present results are clearly independent since they involve
matrices $\tilde{\sigma}_{A}$ and related ones. Note that the corollary 17
leads also to an obvious $(N-1)$-depth result, in analogy to the theorem (19),
see below.

######  Theorem 18.

(Sufficient $(N-1)$-depth criterion 3) Let
$\rho=\Lambda_{\alpha,\beta}^{-1}(\sigma)\geq 0$, i.e.,

$\frac{1}{\alpha^{2}-\beta^{2}}\left[\alpha\sigma-\beta\tilde{\sigma}_{A}-\frac{(\alpha-\beta)}{2^{N}+\alpha+\beta}{\rm
Tr}(\sigma)\mathbbm{1}\right]\geq 0.$ (17)

Then $\sigma$ has depth $(N-1)$.

Let us now present the strongest theorem of this section, which involves
Breuer-Hall unitary operators on the both sides of Alice and Bob:

######  Theorem 19.

Let ${\bf p}=\\{\alpha,\beta,\gamma,\delta\\}$ and $\Lambda_{\bf p}(\rho)={\rm
Tr}(\rho)\mathbbm{1}+\alpha\rho+\beta\tilde{\rho}_{A}+\gamma\tilde{\rho}_{A.O.}+\delta\tilde{\rho}$
be the family of maps; let there exist $0\leq a,b\leq 1$, $a+b\leq 1$ such
that the parameters fulfill the four conditions:

$\displaystyle\alpha$ $\displaystyle\geq\beta/2-a,$ (18a) $\displaystyle\beta$
$\displaystyle\geq\alpha/2-a,$ (18b) $\displaystyle\gamma$
$\displaystyle\geq\delta/2-b,$ (18c) $\displaystyle\delta$
$\displaystyle\geq\gamma/2-b,$ (18d)

and $\alpha\geq-1$, $\beta\geq-1$, $\gamma\geq-1$, and $\delta\geq-1$. Then
$\rho\geq 0$ $\Rightarrow$ $\sigma=\Lambda_{\bf p}(\rho)\in\Sigma$, i.e.
$\sigma$ is separable.

###### Proof.

The proof is similar to the proof of Theorem 11, but more complex and
technical. Again it is enough to prove for pure states,
$\rho=|\Psi\rangle\langle\Psi|$ and, without loosing generality, take
$|\Psi\rangle=\lambda_{0}|0\rangle\otimes|0\rangle+\lambda_{1}|1\rangle\otimes|1\rangle$.
But, now we have to consider three cases: i) when the Breuer-Hall unitary $V$
acts as $\sigma_{2}$ on Bob’s subspace spanned by $|0\rangle$ and $|1\rangle$;
ii) when $V$ transforms $|0\rangle$ and $|1\rangle$ to $|2\rangle$ and
$|3\rangle$; iii) when $V$ transforms $|0\rangle$ and $|1\rangle$ to
orthogonal vectors in a subspace spanned by $|0\rangle$ and $|2\rangle$, and
$|1\rangle$ and $|3\rangle$, respectively.

The conditions $\alpha\geq-1$, $\beta\geq-1$, $\gamma\geq-1$, and
$\delta\geq-1$ follow again from the analysis of the case when $\rho$ is a
projector on a product state, say $|0\rangle\otimes|0\rangle$, so that
$\tilde{\rho}_{A}$ is a projector on $|1\rangle\otimes|0\rangle$,
$\tilde{\rho}_{B}$ on $|0\rangle\otimes|\tilde{1}\rangle$, and $\tilde{\rho}$
on $|1\rangle\otimes|\tilde{1}\rangle$, where $|\tilde{1}\rangle$ is a vector
orthogonal to $|0\rangle$. To derive the conditions in Eq. (18), we split
$\mathbbm{1}=a\mathbbm{1}+b\mathbbm{1}+(1-a-b)\mathbbm{1}$ and apply the
result of Theorem 11 to $a\mathbbm{1}+\alpha\rho+\beta\tilde{\rho}_{A}$, and
$(1-a)\mathbbm{1}+\gamma\tilde{\rho}_{A.O.}+\delta\tilde{\rho}$. Clearly, this
estimate of the region of parameters, where $\Lambda_{\bf p}(\rho)$ is
separable, is very conservative, and probably can be improved significantly. ∎

The conditions above are sufficient for separability and correspond to the
case iii), which probably is not the most demanding, but we were not able to
find weaker sufficient conditions, i.e., the largest allowed regions of the
parameters. Case i) leads to the less demanding restrictions that follow from
the above conditions, when we set $a+b=1$. Finally, case ii) is the least
restrictive – indeed it allows to exceed the restriction $a+b\leq 2$ and reach
$a=b=1$. Note that the proof of the Theorem 11 is just the same, when we set
$\gamma=\delta=0$, while the proof of Theorem 19, as discussed above, is more
involved.

Of course, there exists a generalization of the above theorems to the case of
mixtures:

######  Corollary 20.

Let $\Lambda^{i}_{\alpha_{i},\beta_{i},\gamma_{i},\delta_{i}}(\rho)={\rm
Tr}(\rho)\mathbbm{1}+\alpha_{i}\rho+\beta_{i}\tilde{\rho}_{i}+\gamma_{i}\tilde{\rho}_{A.O.\
than\ i}+\delta_{i}\tilde{\rho}$ be a family of maps. Let
$\Lambda(\rho)=\sum_{i=1}^{N}p_{i}\Lambda^{i}_{\alpha_{i},\beta_{i}}(\rho)$,
with $p_{i}\geq 0$, and $\sum_{i=1}^{N}p_{i}=1$. Let $\alpha_{i}$,
$\beta_{i}$, $\gamma_{i}$ and $\delta_{i}$ fulfill the conditions of the
theorem (19). Then $\rho\geq 0$ $\Rightarrow$
$\Lambda(\rho)=:\sigma\in\Sigma_{N-1}$, i.e., $\sigma$ has entanglement depth
$N-1$ (is a mixture of states that are bi-separable in the partition
$1^{i}:(N-1)^{A.O.}$).

###### Proof.

The proof is obvious: a mixture of bi-separable state in the partitions
$1^{i}:(N-1)^{A.O.}$ has $(N-1)$-depth. The problem of using the above result
for entanglement depth criteria lies in the difficulty of inverting the map
$\Lambda(\rho)$. Below we present explicit calculations for the case of 3
qubits. ∎

For simplicity, we consider a map

######  Corollary 21.

Let

$\displaystyle\Lambda(\rho)$ $\displaystyle=$ $\displaystyle\sigma={\rm
Tr}(\rho)\mathbbm{1}$ $\displaystyle+$ $\displaystyle
p_{A}[\alpha_{A}\rho+\beta_{A}\tilde{\rho}_{A}+\gamma_{A}\tilde{\rho}_{BC}+\delta_{A}\tilde{\rho}]$
$\displaystyle+$ $\displaystyle
p_{B}[\alpha_{B}\rho+\beta_{B}\tilde{\rho}_{B}+\gamma_{B}\tilde{\rho}_{AC}+\delta_{B}\tilde{\rho}]$
$\displaystyle+$ $\displaystyle
p_{C}[\alpha_{C}\rho+\beta_{C}\tilde{\rho}_{C}+\gamma_{C}\tilde{\rho}_{AB}+\delta_{C}\tilde{\rho}],$

with $p_{i}\geq 0$, and $\sum_{i=A,B,C}p_{i}=1$. Applying different
transpositions and $\sigma_{2}$s, we get:

$\displaystyle\tilde{\sigma}_{A}={\rm Tr}(\rho)\mathbbm{1}$ $\displaystyle+$
$\displaystyle
p_{A}[\alpha_{A}\tilde{\rho}_{A}+\beta_{A}\rho+\gamma_{A}\tilde{\rho}+\delta_{A}\tilde{\rho}_{BC}]$
$\displaystyle+$ $\displaystyle
p_{B}[\alpha_{B}\tilde{\rho}_{A}+\beta_{B}\tilde{\rho}_{AB}+\gamma_{B}\tilde{\rho}_{C}+\delta_{B}\tilde{\rho}_{BC}]$
$\displaystyle+$ $\displaystyle
p_{C}[\alpha_{C}\tilde{\rho}_{A}+\beta_{C}\tilde{\rho}_{AC}+\gamma_{C}\tilde{\rho}_{B}+\delta_{C}\tilde{\rho}_{BC}],$

and so on. In this way we obtain 8 linear inhomogeneous equations for $\rho$,
$\tilde{\rho}_{A}$, $\ldots$, which can be easily solved to find
$\rho=\Lambda^{-1}(\sigma)$. We use also the relation ${\rm
Tr}(\sigma)=[8+\sum_{i=A,B,C}p_{i}(\alpha_{i}+\beta_{i}+\gamma_{i}+\delta_{i})]{\rm
Tr}(\rho)$. Obviously, we have then: If $\sigma\geq 0$ and
$\rho=\Lambda^{-1}(\sigma)\geq 0$, then $\sigma$ does not exhibit any genuine
3-qubit entanglement.

Let $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$ and $\delta_{i}$ fulfill the
conditions of the theorem (19). Then $\rho\geq 0$ $\Rightarrow$
$\Lambda(\rho)=:\sigma\in\Sigma_{N-1}$, i.e., $\sigma$ has entanglement depth
$N-1$ (is a mixture of states that are bi-separable in the partition
$1^{i}:(N-1)^{A.O.}$).

##  V  Sufficient criteria for compatibility

In this section, we demonstrate how similar linear map methods can be used in
the space of correlations to construct sufficient criteria for compatibility
of a given set of correlations with certain models. In the first subsection,
we consider $S^{\prime}$ to be the set of statistics compatible with a global
separable state. We finish by deriving sufficient criteria for compatibility
with local-hidden variable models.

###  V.1 Sufficient criteria for compatibility with a global entangled state

Consider a system of $N$-qubits from which the first and second moments of the
collective spin, $\mathbf{J}=\sum_{i=1}^{N}\bm{\sigma}^{(i)}/2$, are being
inferred. Here we denote $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ the
vector of Pauli matrices. Concretely, we focus on the unpolarized case
$\langle\mathbf{J}\rangle=\mathbf{0}$ and assume knowledge of the second
moments $\\{\langle J_{a}^{2}\rangle\\}_{a\in\\{x,y,z\\}}$ for three
orthogonal directions $\\{x,y,z\\}$. Based on these expectation values, we
want to verify compatibility with a global $N$-qubit separable state.
Remarkably, there is a closed-form answer to this question, which we remind
the reader in the following Lemma:

######  Lemma 22.

Toth  If the expectation values $\\{\langle J_{a}\rangle,\langle
J_{a}^{2}\rangle\\}_{a\in\\{x,y,z\\}}$ fulfill the inequalities

$\begin{array}[]{crl}\sum_{a\in\\{x,y,z\\}}\langle J_{a}^{2}\rangle\leq
N(N+2)/4\\\ \sum_{a\in\\{x,y,z\\}}(\Delta J_{a})^{2}\geq N/2\\\ \langle
J_{a}^{2}\rangle+\langle J_{b}^{2}\rangle-(N-1)(\Delta J_{c})^{2}\leq N/2\\\
(N-1)((\Delta J_{a})^{2}+(\Delta J_{b})^{2})-\langle J_{c}^{2}\rangle\geq
N(N-2)/4,\end{array}$ (21)

where $(\Delta J_{a})^{2}=\langle J_{a}^{2}\rangle-\langle J_{a}\rangle^{2}$
is the variance and in the last two inequalities $\\{a,b,c\\}$ vary over all
the permutations of $\\{x,y,z\\}$. Then, in the thermodynamic limit
$N\rightarrow\infty$ there exists a global separable state $\rho_{\rm
SEP}\in\mathcal{B}((\mathbb{C}^{2})^{\otimes N})$ exhibiting $\\{\langle
J_{a}\rangle_{{\rm SEP}},\langle J_{a}^{2}\rangle_{{\rm
SEP}}\\}_{a\in\\{x,y,z\\}}$ arbitrary close to $\\{\langle
J_{a}\rangle,\langle J_{a}^{2}\rangle\\}_{a\in\\{x,y,z\\}}$.

Now, we apply the reduction map to the underlying state
$\Lambda_{\alpha}(\rho)=\mathrm{Tr}(\rho)\mathbbm{1}+\alpha\rho$. Instead of
transforming the second moment $\langle J_{a}^{2}\rangle$, we take
equivalently $\langle J_{a}^{2}\rangle-N/4=\langle\mathcal{C}_{a}\rangle$,
where $\langle\mathcal{C}_{a}\rangle=\sum_{i\neq
j=1}^{N}\langle\sigma^{(i)}_{a}\sigma^{(j)}_{a}\rangle/4$ is the correlation
function. The action of the map on the state rescales the correlation function
$\langle\mathcal{C}_{a}\rangle\mapsto\alpha\langle\mathcal{C}_{a}\rangle:=\langle\mathcal{C}_{a}\rangle_{\alpha}$
equally for any orientation $a\in\\{x,y,z\\}$.

In order to derive a sufficient condition for compatibility with a separable
state, we need to find the range of $\alpha$ such that the modified
correlations $\\{\langle\mathcal{C}_{a}\rangle_{\alpha}\\}_{a\in\\{x,y,z\\}}$
do not violate any inequality (21). Provided that the state is unpolarized
$\langle\mathbf{J}\rangle=0$, $\\{(\Delta J_{a})^{2}=\langle
J_{a}^{2}\rangle=\langle\mathcal{C}_{a}\rangle+N/4\\}_{a\in\\{x,y,z\\}}$, the
parameter $\alpha$ appears linearly in the modified inequalities (21).
Consequently, for fixed correlations
$\\{\langle\mathcal{C}_{a}\rangle\\}_{a\in\\{x,y,z\\}}$ the extreme value of
$\alpha$ may be found via the linear program (LP):

$\begin{array}[]{crl}\max_{\alpha}&\alpha&\\\ \mbox{s.t.}&-N\leq
4\alpha\sum_{a\in\\{x,y,z\\}}\langle\mathcal{C}_{a}\rangle&\leq N(N-1)\\\
&4\alpha(\langle\mathcal{C}_{a}\rangle+\langle\mathcal{C}_{b}\rangle-(N-1)\langle\mathcal{C}_{c}\rangle)&\leq
N(N-1)\\\
&4\alpha((N-1)(\langle\mathcal{C}_{a}\rangle+\langle\mathcal{C}_{b}\rangle)-\langle\mathcal{C}_{c}\rangle)&\geq
N(1-N)\\\ \end{array}$ (22)

Note that in the same LP (22) one can consider minimization instead of
maximization to find the extreme value in the opposite direction. The range of
$\alpha$ is all we need to formulate the sufficient condition:

######  Theorem 23.

(Compatibility with a global separable state) Let
$\rho\in\mathcal{B}((\mathbb{C}^{2})^{\otimes N})$ be a quantum state and
consider the set of expectation values $\\{\langle J_{a}^{2}\rangle,\langle
J_{a}\rangle=0\\}_{a\in\\{x,y,z\\}}$ descending from it. Let $\alpha^{*}$ be
the solution of LP (22). If $\alpha^{*}>1$, then
$\sigma:=\Lambda_{\beta}^{-1}(\rho)/\mathrm{Tr}[\Lambda_{\beta}^{-1}(\rho)]=((2^{N}+\beta)\rho-\mathbbm{1})/\beta$
admits a separable description in the thermodynamic limit $N\rightarrow\infty$
from the same set of observables for $\beta\in[2^{N}/(\alpha^{*}-1),\infty)$

###### Proof.

Optimality of $\alpha$ implies that
$\\{\langle\mathcal{C}_{a}\rangle_{\alpha^{*}}=\langle
J_{a}^{2}\rangle_{\alpha^{*}}-N/4\\}_{a\in\\{x,y,z\\}}$ do not violate any
inequality (21) for all $\alpha\in[0,\alpha^{*}]$. Consequently, according to
Lemma 21, in the thermodynamic limit, it exists a separable state explaining
the expectation values. Since the observables are fixed and
$\\{\langle\mathcal{C}_{a}\rangle\\}_{a\in\\{x,y,z\\}}$ is generated from
$\rho$, we just have to find the relation between $\alpha$ and $\beta$. For a
given $\beta$, we have that
$\\{\langle\mathcal{C}_{a}\rangle_{\alpha}=\alpha\langle\mathcal{C}_{a}\rangle=\frac{2^{N}+\beta}{\beta}\langle\mathcal{C}_{a}\rangle\\}_{a=\\{x,y,z\\}}$.
Hence, $\beta=2^{N}/(\alpha-1)$ yields the range
$\beta\in[2^{N}/(\alpha^{*}-1),\infty)$. ∎

###  V.2 Sufficient criteria for compatibility with a local-hidden-variable
theory

The last set $S^{\prime}$ we want to consider is that of the correlations that
are compatible with a local hidden variable model (LHVM) Bell64 . The set of
LHVM correlations forms a polytope. Let us denote this polytope $\mathbbm{P}$.
Membership of a set of correlators
$\text{\boldmath$P$}(\text{\boldmath$a$}|\text{\boldmath$x$})$ in
$\mathbbm{P}$ can be determined by the feasibility of the following LP:

$\begin{array}[]{crl}\min_{\text{\boldmath$\lambda$}}&0&\\\
\mbox{s.t.}&\sum_{i}\lambda_{i}\text{\boldmath$v$}_{i}&=\text{\boldmath$P$}(\text{\boldmath$a$}|\text{\boldmath$x$})\\\
&\sum_{i}\lambda_{i}&=1\\\ &\lambda_{i}&\geq 0,\\\ \end{array}$ (23)

where $\text{\boldmath$v$}_{i}$ enumerate all possible deterministic local
strategies. If we take the no-signalling principle into consideration, in a
Bell scenario with $n$ parties, $m$ measurement choices each with $d$ possible
outcomes, a vertex $\text{\boldmath$v$}_{i}$ is given by $(1+m(d-1))^{n}-1$
coordinates, and $i$ ranges from $0$ to $d^{mn}-1$. The local deterministic
strategies encoded in $i$ can be decoded by expressing the integer $i$ with
digits $i_{1},\ldots,i_{n}$ in base $d^{m}$, and then each $i_{j}$ as
${i_{j}}_{1},\ldots,{i_{j}}_{n}$ as digits in base $d$. Then, ${i_{j}}_{k}$
indicates that party $j$ outputs the ${i_{j}}_{k}$-th outcome upon measuring
the $k$-th observable, thus defining a deterministic local strategy.

Note that quantum correlations take the form
$\text{\boldmath$P$}(\text{\boldmath$a$}|\text{\boldmath$x$})=\mathrm{Tr}[\rho\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]$,
where $\rho$ is some quantum state and
$\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}$ a POVM element. The image of
the reduction map $\Lambda_{\alpha}$ on any $\rho$ shrinks the correlation
vector $P$ by a factor $\alpha$ and adds a small bias:
$\mathrm{Tr}[\Lambda_{\alpha}(\rho)\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]=\mathrm{Tr}[(\mathrm{Tr}(\rho)\mathbbm{1}+\alpha\rho)\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]=\mathrm{Tr}[\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]+\alpha\mathrm{Tr}[\rho\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]$.
In the case of von Neumann, rank one projective measurements,
$\mathrm{Tr}[\Pi_{\text{\boldmath$a$}|\text{\boldmath$x$}}]=1$. If we take
$\rho$ to be an $N-$partite $d$-level quantum state, then a proper
normalization of $\Lambda_{\alpha}(\rho)$ maps $P$ to
$(\text{\boldmath$1$}+\alpha\text{\boldmath$P$})/(d^{N}+\alpha)$, where $1$ is
a vector of ones.

It is customary to take into consideration correlation functions instead of
probabilities, to get rid of these inconvenient biases: In a Bell scenario
with dichotomic measurements, one would normally label the measurement
outcomes $\pm 1$ (and in more general scenarios, as $d$-th roots of unity
ArnaultJPA ). For instance, for a bipartite scenario with dichotomic
measurements labelled $\pm 1$, one would have $\langle
A_{x}B_{y}\rangle:=\sum_{a,b}(-1)^{a+b}P(ab|xy)$; i.e., the discrete Fourier
transform of the correlators AugusiakNJP ; SATWAP ; MaxMaxMax . In this case,
the vector of correlators
$\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle$
maps to
$\alpha\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle$
since
$\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\mathbbm{1}}=0$
(we omit the correction for normalization of $\Lambda_{\alpha}(\rho)$ without
loss of generality). Note that for binary outcomes the superindex
$(\text{\boldmath$k$})$ is not necessary, and that whenever $k_{i}=0$ this
corresponds to a marginal correlator that does not involve the $i$-th party.

Hence, the condition for a set of correlators
$\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle$
to admit a LHVM interpretation reads

$\begin{array}[]{crl}\max_{\text{\boldmath$\lambda$},\alpha}&\alpha&\\\
\mbox{s.t.}&\sum_{i}\lambda_{i}\text{\boldmath$w_{i}$}&=\alpha\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle\\\
&\sum_{i}\lambda_{i}&=1\\\ &\lambda_{i}&\geq 0,\\\ \end{array}$ (24)

where $\text{\boldmath$w$}_{i}$ are the same $\text{\boldmath$v$}_{i}$ that
have undergone the affine transformation given by the discrete Fourier
transform. Naturally, in LP (24) one may as well consider
$\min_{\text{\boldmath$\lambda$},\alpha}\alpha$ instead of the maximization to
obtain the scaling factor in the opposite direction.

Along the lines of Theorem 23, we can similarly pose a sufficient condition
for compatibility with a LHVM:

######  Theorem 24.

(Compatibility with a local hidden variable model) Let $\rho$ be a quantum
state and consider a set of traceless local observables $A_{x}^{{[i]}^{(k)}}$,
yielding a vector of correlations
$\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\rho}$
on $\rho$. Let $\alpha^{*}$ be the solution of LP (24). If $\alpha^{*}>1$,
then
$\sigma:=\Lambda_{\beta}^{-1}(\rho)/\mathrm{Tr}[\Lambda_{\beta}^{-1}(\rho)]=((d^{N}+\beta)\rho-\mathbbm{1})/\beta$
admits an LHVM for the same set of observables for
$\beta\in[d^{N}/(\alpha^{*}-1),\infty)$.

###### Proof.

Optimality of $\alpha^{*}$ implies that
$\alpha\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\rho}\in\mathbbm{P}$
for all $\alpha\in[0,\alpha^{*}]$. Since the observables are fixed and
$\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\rho}$
is generated from $\rho$, we just have to find the relation between $\alpha$
and $\beta$. For a given $\beta$, we have that
$\alpha\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\rho}=\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\sigma}=\frac{d^{N}+\beta}{\beta}\langle\text{\boldmath$A$}_{\text{\boldmath$x$}}^{(\text{\boldmath$k$})}\rangle_{\rho}$.
Hence, $\beta=d^{N}/(\alpha-1)$ yields the range
$\beta\in[d^{N}/(\alpha^{*}-1),\infty)$. ∎

Since the characterization of $S^{\prime}$ quickly becomes intractable in the
multipartite case ($S^{\prime}$ is a polytope living in an
$O(\exp(N))$-dimensional space determined by $O(\exp(N))$ vertices), here we
may consider a relaxed version of the problem, by focusing on the projection
of $S^{\prime}$ onto the space of permutationally invariant one- and two-body
correlators TuraScience ; TuraAnnPhys . In the latter case, $S^{\prime}$ is
embedded in a five-dimensional affine space and its vertices can be enumerated
by the partitions of $N$, thus making the linear program in Eq. (24) efficient
in practice.

In that case, if we consider $N$-partite permutationally invariant Bell
inequalities with one- and two-body correlators for the two-input two-output
scenario, one only needs to consider symmetric correlators ${\cal
S}_{x}:=\sum_{i}\langle A_{x}^{[i]}\rangle$ and ${\cal S}_{x,y}:=\sum_{i\neq
j}\langle A_{x,y}^{[i,j]}\rangle$, with $0\leq x\leq y\leq 1$. The vertices
$\text{\boldmath$v$}_{i}$ are such that ${\cal S}_{x}=p^{(1)}_{x}(c_{i})$,
where $p^{(1)}_{x}$ is a linear polynomial and $c_{i}\vdash N$ is a partition
of $N$ in $d^{m}=4$ elements, and ${\cal S}_{x,y}=p^{(2)}_{x,y}(c_{i})$. The
analogous of the LP (24) for this case is

$\begin{array}[]{crl}\max_{\text{\boldmath$\lambda$},\alpha}&\alpha&\\\
\mbox{s.t.}&\sum_{i}\lambda_{i}\text{\boldmath$w_{i}$}&=\alpha[{\cal
S}_{0},{\cal S}_{1},{\cal S}_{00},{\cal S}_{01},{\cal S}_{11}]\\\
&\sum_{i}\lambda_{i}&=1\\\ &\lambda_{i}&\geq 0.\\\ \end{array}$ (25)

In this case it follows that:

######  Corollary 25.

(Local Hidden Variable Model explainable through symmetric correlators) For
the same conditions as in Theorem 24, let $\tilde{\alpha}$ be the optimal
solution of LP (25). Then $\sigma$ admits a LHVM describable through
symmetrized local response functions expressable as one- and two-body
marginals, for $\beta\in[d^{N}/(\tilde{\alpha}-1),\infty)$.

For very complex Bell scenarios, even the polynomial scaling of LP (25) may be
too computationally intensive in practice. In this case, one can further relax
the membership problem in the set $S^{\prime}$ to that of a superset
$S^{\prime\prime}$ via a symmetrized moment method based on outer
approximations of convex hulls of semialgebraic sets, which admits a much more
efficient description as a semidefinite program that does not depend on $N$
FadelTuraPRL :

$\begin{array}[]{crl}\max_{\text{\boldmath$y$},\alpha}&\alpha&\\\
\mbox{s.t.}&y_{i}&=\alpha{(\text{\boldmath$\cal S$})}_{i}\\\ &y_{0}&=1\\\
&\sum_{i}y_{i}\Gamma_{i}&\geq 0,\\\ \end{array}$ (26)

where $\Gamma_{i}$ are constant matrices defined by $p_{x}^{(1)}$ and
$p_{xy}^{(2)}$ FadelTuraPRL .

######  Corollary 26.

(Relaxation of Corollary 25) Let $\cal S$ be the vector of symmetrized
correlators generated by $\rho$ and the same choice of observables at every
site. Let us denote $\tilde{\alpha}$ the optimal solution to LP (25) and
$\hat{\alpha}$ that of SDP (26). Let $\Delta$ be a uniform (independent of
$\cal S$) upper bound to $\hat{\alpha}-\tilde{\alpha}$. If
$\hat{\alpha}-\Delta>1$ then $\sigma$ fulfills the conditions of Corollary 25
for $\beta\in[d^{N}/(\hat{\alpha}-\Delta-1),\infty)$.

##  VI  Conclusions and outlook

We have presented in this paper several families of linear maps, that have a
property that when applied to $N$-qubit states, result in states separable
with respect a bipartition. When convertible, such maps allow deriving
sufficient criteria for entanglement depth in $N$ qubit systems. In
particular, in the present work we derive sufficient criteria for arbitrary
entanglement depth tailored to states close to the normalized identity matrix
and sufficient entanglement criteria for different classes of symmetric and
symmetric-diagonal states. Finally, we propose how the application of linear
maps can be used to detect states leading to expectation values compatible
with separable states and local-hidden-variable theories. Such criteria have
rather obvious possible applications and implications in quantum information
science and quantum many body physics: they can namely be applied to determine
whether and how much entanglement is needed for various specific protocols for
quantum computation or other quantum tasks, similarly as it was discussed in
Ref. Braunstein+5.PRL.1999 .

###### Acknowledgements.

M.L. and G.M.R. acknowledges support from: ERC AdG NOQIA; Ministerio de
Ciencia y Innovation Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I "Retos Colaboración" QUSPIN
RTC2019-007196-7); Fundació Cellex; Fundació Mir-Puig; Generalitat de
Catalunya (European Social Fund FEDER and CERCA program (AGAUR Grant No. 2017
SGR 134, QuantumCAT U16-011424, co-funded by ERDF Operational Program of
Catalonia 2014-2020); Barcelona Supercomputing Center MareNostrum
(FI-2022-1-0042); EU Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU
Horizon Europe Program (Grant Agreement 101080086 - NeQST), National Science
Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314); European Union’s
Horizon 2020 research and innovation program under the Marie-Skłodowska-Curie
grant agreement No 101029393 (STREDCH) and No 847648 ("La Caixa" Junior
Leaders fellowships ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031,
LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). Views and opinions expressed in
this work are, however, those of the author(s) only and do not necessarily
reflect those of the European Union, European Climate, Infrastructure and
Environment Executive Agency (CINEA), nor any other granting authority.
Neither the European Union nor any granting authority can be held responsible
for them. M.L., G.M.R. and A.S. acknowledge support from MCIN Recovery,
Transformation and Resilience Plan with funding from European Union
NextGenerationEU (PRTR C17.I1). J.T. has received support from the European
Union’s Horizon Europe program through the ERC StG FINE-TEA-SQUAD (Grant No.
101040729). A.S. acknowledges financial support from the Spanish Agencia
Estatal de Investigación, Grant No. PID2019-107609GB-I00, the European
Commission QuantERA grant ExTRaQT (Spanish MICINN project PCI2022-132965), and
Catalan Government for the project QuantumCAT 001-P-001644, co-financed by the
European Regional Development Fund (FEDER).

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11institutetext:  Instituto de Física Fundamental, CSIC, Calle Serrano 123,
E-28006 Madrid, Spain
11email<EMAIL_ADDRESS>22institutetext: Observatorio Astronómico
Nacional, IGN, Calle Alfonso XII 3, E-28014 Madrid, Spain 33institutetext:
Observatorio de Yebes, IGN, Cerro de la Palera s/n, E-19141 Yebes,
Guadalajara, Spain

# Aromatic cycles are widespread in cold clouds††thanks: Based on observations
carried out with the Yebes 40m telescope (projects 20A016, 21A006, and
22B023). The 40m radio telescope at Yebes Observatory is operated by the
Spanish Geographic Institute (IGN; Ministerio de Transportes, Movilidad y
Agenda Urbana).

M. Agúndez 11 N. Marcelino 2233 B. Tercero 2233 J. Cernicharo 11

(Received; accepted)

We report the detection of large hydrocarbon cycles toward several cold dense
clouds. We observed four sources (L1495B, Lupus-1A, L483, and L1527) in the Q
band (31-50 GHz) using the Yebes 40m radiotelescope. Using the line stack
technique, we find statistically significant evidence of benzonitrile (C6H5CN)
in L1495B, Lupus-1A, and L483 at levels of 31.8 $\sigma$, 15.0 $\sigma$, and
17.2 $\sigma$, respectively, while there is no hint of C6H5CN in the fourth
source, L1527. The column densities derived are in the range (1.7-3.8)
$\times$ 1011 cm-2, which is somewhat below the value derived toward the cold
dense cloud TMC-1. When we simultaneously analyze all the benzonitrile
abundances derived toward cold clouds in this study and in the literature, a
clear trend emerges in that the higher the abundance of HC7N, the more
abundant C6H5CN is. This indicates that aromatic cycles are especially favored
in those interstellar clouds where long carbon chains are abundant, which
suggests that the chemical processes that are responsible for the formation of
linear carbon chains are also behind the synthesis of aromatic rings. We also
searched for cycles other than benzonitrile, and found evidence of indene
(C9H8), cyclopentadiene (C5H6), and 1-cyano cyclopentadiene (1-C5H5CN) at
levels of 9.3 $\sigma$, 7.5 $\sigma$, and 8.4 $\sigma$, respectively, toward
L1495B, which shows the strongest signal from C6H5CN. The relative abundances
between the various cycles detected in L1495B are consistent –within a factor
of three– with those previously found in TMC-1. It is therefore likely that
not only C6H5CN but also other large aromatic cycles are abundant in clouds
rich in carbon chains.

###### Key Words.:

astrochemistry – line: identification – ISM: molecules – radio lines: ISM

##  1 Introduction

Cold dense clouds with temperatures of around 10 K and volume densities of a
few 104 molecules of H2 per cm3 (Bergin & Tafalla, 2007) are known to contain
a rich variety of molecules, most of which are organic in nature. The chemical
composition of this kind of environment has long been thought to be dominated
by linear unsaturated carbon chains with a backbone of alternating triple and
single bonds, such as polyynes and cyanopolyynes. Chemical models have been
developed to explain the formation of molecules in these clouds, and they are
reasonably successful in accounting for the abundances of the carbon chains
observed in TMC-1, which is the paradigm of a cold dense cloud rich in carbon
chains (Agúndez & Wakelam, 2013). The quest for chemical complexity in these
clouds has for years been focused on the search for increasingly long linear
chains, the largest one found to date being HC11N (Loomis et al., 2021).

However, in recent years we have witnessed a paradigm shift concerning the
chemistry of cold dense clouds. Large hydrocarbon cycles, some of them
aromatic in character, have been discovered in TMC-1 by means of
radioastronomical techniques. The first such molecule detected was
benzonitrile (C6H5CN; McGuire et al. 2018), which is the CN derivative of
benzene (C6H6) and is thought to be a proxy for the radioastronomically
invisible aromatic molecule C6H6 (Cooke et al., 2020). The CCH derivative of
benzene (C6H5CCH) has also been identified toward TMC-1 (Cernicharo et al.,
2021c; Loru et al., 2023), and the six-membered ring benzyne (C6H4) has also
been observed (Cernicharo et al., 2021b).

The five-membered ring cyclopentadiene (C5H6), which is nonaromatic but
moderately polar, has been directly observed toward TMC-1 (Cernicharo et al.,
2021a). In addition, the two CN isomeric derivatives (McCarthy et al., 2021;
Lee et al., 2021) and the two CCH substituted isomers (Cernicharo et al.,
2021c) have been detected. Moreover, fulvenallene (C5H4CCH2), which is also a
derivative of cyclopentadiene, has been observed as well (Cernicharo et al.,
2022).

In addition to single cycles, hydrocarbons composed of fused rings have also
been detected toward TMC-1. The two simplest members of the family of
polycyclic aromatic hydrocarbons, indene (C9H8) and naphthalene (C10H8), have
been detected, either directly or indirectly. The nonzero, albeit small dipole
moment of indene permitted its detection (Cernicharo et al., 2021a; Burkhardt
et al., 2021a), while one of the CN-substituted isomers of indene was also
detected (Sita et al., 2022). Naphthalene is nonpolar and therefore cannot be
observed through radioastronomical techniques. Nevertheless, the two CN-
substituted isomers have been observed (McGuire et al., 2021) and, similarly
to the case of benzene, can be considered as proxies for naphthalene itself.

The observation of large cycles in cold dense clouds is challenging and has
for this reason been restricted to TMC-1. In a recent study, Burkhardt et al.
(2021b) reported the detection of C6H5CN toward four additional clouds. There
are still many open questions regarding the presence of large hydrocarbon
cycles in cold dense clouds. For example, the circumstances under which their
presence is favored and how they are synthesized remain unknown. Here we
report the detection of large cycles toward new sources, which allows us to
build a larger statistical basis from which to understand the occurrence of
large hydrocarbon cycles in cold interstellar clouds.

##  2 Astronomical observations

Figure 1: Two individual lines of C6H5CN (left and middle panels) and the
stacked spectrum of C6H5CN (right panels) for L1495B (top row), Lupus-1A
(middle row), and L483 (bottom row), using the 130,13-120,12 line at 32833.813
MHz as a reference. The number of stacked lines and the significance level of
the line stack signal (in units of $\sigma$) are indicated in the right
panels. The systemic velocity of each source is indicated by a vertical dotted
line.

We carried out observations of four cold dense clouds (L1495B, Lupus-1A, L483,
and L1527) with the Yebes 40m telescope in the frame of a line survey in the Q
band and a search for negative ions (details and source coordinates are given
in Agúndez et al. 2023). The observations, which used the 7 mm receiver
described in Tercero et al. (2021), were taken between July 2020 and February
2023. The data consist of a full spectrum across the Q band (31.08-49.52 GHz)
in horizontal and vertical polarizations with a spectral resolution of 38 kHz.

The observations were carried out using the frequency-switching technique. The
frequency throw adopted, which changed between different observing periods as
a result of tests done at the 40m telescope, was 10.52 MHz for L1495B and
Lupus-1A, 10 MHz and 10.52 MHz for L483, and 8 MHz for L1527. We used two
spectral setups with the frequency of the local oscillator separated by 5 MHz
to identify spurious signals. Total on-source telescope times in each
polarization were 45 h for L1495B, 120 h for Lupus-1A, 103 h for L483, and 40
h for L1527. The intensity scale at the 40m telescope is antenna temperature,
$T_{A}^{*}$, for which we estimate a calibration error of 10 %. The
$T_{A}^{*}$ noise levels after averaging horizontal and vertical polarizations
are in the range 0.8-2.6 mK for L1495B, 0.7-2.8 mK for Lupus-1A, 0.4-1.0 mK
for L483, and 0.7-2.7 mK for L1527. We convert $T_{A}^{*}$ to $T_{mb}$ (main
beam brightness temperature) by dividing by $B_{\rm eff}$/$F_{\rm eff}$, where
for the 40m telescope in the Q band $B_{\rm eff}$ = 0.797
$\exp$[$-$($\nu$(GHz)/71.1)2] and $F_{\rm eff}$ = 0.97. The half power beam
width (HPBW) can be fitted as a function of frequency as HPBW(′′) =
1763/$\nu$(GHz).

##  3 Detection by line stack

Figure 2: Stacked spectra of L1495B for C9H8 (left panel), C5H6 (middle
panel), and 1-C5H5CN (right panel), using the 140,14-130,13 line at 32528.329
MHz, the 31,2-22,1 line at 37453.992 MHz, and the 100,10-90,9 line at
33050.464 MHz as references, respectively. The number of stacked lines and the
significance level of the line stack signal (in units of $\sigma$) is
indicated in each panel. The systemic velocity of L1495B is indicated by a
vertical dotted line.

The Yebes 40m data of the L1495B, Lupus-1A, L483, and L1527 were analyzed
using the program CLASS of the GILDAS software (Pety,
2005)111https://www.iram.fr/IRAMFR/GILDAS/. While inspecting the data, we
noticed the presence of several individual lines of C6H5CN in the spectrum of
L1495B. The lines are weak, with $T_{mb}$ $\sim$ 2-5 mK (see left panels in
the top row in Fig. 1). We also searched for C6H5CN in the spectra of Lupus-1A
and L483, although we only find a few lines, which are barely visible above
the noise level (see left panels in the middle and bottom rows in Fig. 1),
while many others are hidden within the noise. In the case of L1527, we see no
obvious line of C6H5CN. Given that benzonitrile has many lines with similar
intensities within the Q band, as seen in the very sensitive spectrum of TMC-1
(Cernicharo et al., 2021c), we decided to use the line stack technique to
evaluate whether or not there is spectral evidence of this molecule in the
four aforementioned sources.

In order to carry out the line stack, we restricted to the most intense lines
of C6H5CN in the Q band, which are those with quantum number $K_{a}$ = 0, 1, 2
in Figs. C.1-C.4 of Cernicharo et al. (2021c) (35 lines in total). The
hyperfine splitting of these lines is small (10-30 kHz). Lines with $K_{a}$
$>$ 2 are weaker and have significant hyperfine splitting, which further
dilutes the intensity among the different hyperfine components (see spectrum
of TMC-1 in Cernicharo et al. 2021c). The line frequencies and strengths of
benzonitrile were computed from the rotational constants and dipole moments
reported by Wohlfart et al. (2008). We selected a spectral region of $\pm$ 30
km s-1 around each line and cleaned the channels belonging to spectral
features other than the C6H5CN target line; that is, lines from other
molecules, negative frequency-switching artifacts, or spurious signals. In
order to take into account the different expected intensities of the lines
under consideration, we computed the spectrum under local thermodynamic
equilibrium (LTE) at a temperature of 10 K. We note that the only
determination of the rotational temperature of C6H5CN reported in the
literature, which corresponds to TMC-1 (Cernicharo et al., 2021c), yields a
value of 9 K, which is equal to the gas kinetic temperature in this source
(Agúndez et al., 2023). Benzonitrile is therefore likely to be thermalized in
cold dense clouds. The gas kinetic temperature is around 10 K in the four
studied sources (Agúndez et al., 2023), and therefore we adopt this round
value here for simplicity. We arbitrarily selected the 130,13-120,12 line at
32833.813 MHz as a reference and used the calculated brightness temperatures
of the LTE spectrum to compute relative intensities with respect to the
reference line. We then scaled the spectrum of each line by the inverse of the
theoretical relative intensity.

Once we had each spectrum cleaned from signals other than the target C6H5CN
line and conveniently scaled according to the expected relative intensity, we
fitted a baseline to a polynomial of degree 10 (the frequency-switching
technique produces baseline ripples) and subtracted it. We then stacked all
spectra aligned in the velocity scale using the inverse square of the rms
noise level as weight. The line-stacking procedure used here, which is similar
to that employed to map C6H5CN in TMC-1 (Cernicharo et al., 2023), has two
benefits. On the one hand, it controls the absolute scale of the intensity,
because all lines are stacked with the same intensity as that of the reference
line. Therefore, the intensity of the signal after the line stack can be
assigned to the reference line, which permits a correct evaluation of the
column density. On the other hand, lines contribute to the signal according to
their expected relative intensities. That is, the weaker the line, the higher
the factor by which the spectrum is multiplied, the higher the noise level,
and the lower the weight. This way, we avoid dilution of the line stack signal
due to the weak lines.

The results from the line stack of C6H5CN are shown in the right panels of
Fig. 1 for L1495B, Lupus-1A, and L483. In the three cases, there is a clear
signal well above the noise level and precisely centered at the systemic
velocity of each source: 7.67 km s-1 for L1495B (Cordiner et al., 2013;
Agúndez et al., 2023), 5.00 km s-1 for Lupus-1A (Sakai et al., 2010), and 5.30
km s-1 for L483 (Agúndez et al., 2019). We note that in the case of L1495B, we
removed one line from the analysis because it overlaps with a spurious signal.
The signal-to-noise ratio (S/N) is evaluated as $\int T_{A}^{*}dv$ / [rms
$\times$ $\sqrt{\Delta v\times\delta v}$], where rms is the measured noise
level of the stacked spectrum, $\delta v$ is the spectral resolution in
velocity of the stacked spectrum, and $\int T_{A}^{*}dv$ and $\Delta v$ are
the velocity-integrated intensity and full width at half maximum (FWHM),
respectively, given by a Gaussian fit to the line profile. Given the
limitations associated to the line-stacking procedure, we consider that a
detection is statistically significant when the S/N is above 5 $\sigma$. We
carried out the same line-stack analysis to search for evidence of C6H5CN in
L1527, although no statistically significant signal was detected. The
nondetection of C6H5CN in L1527 allows us to impose a stringent upper limit to
the column density of C6H5CN. This is interesting because L1527 is known to be
rich in carbon chains (Sakai et al., 2008).

Motivated by the detection of benzonitrile in the three clouds L1495B,
Lupus-1A, and L483, we searched for evidence of other large cycles in these
sources. Guided by previous observations toward TMC-1 (Cernicharo et al.,
2021a, c; McGuire et al., 2021), we searched for cyclopentadiene and its CN
and CCH derivatives, for indene, and for the CN derivatives of naphthalene.
For all these molecules, we computed the LTE spectrum at 10 K, arbitrarily
selecting one intense line for reference (see caption of Fig. 2), and
evaluated the relative line intensities from the calculated brightness
temperatures. We only included those lines that are calculated to have at
least half the intensity of the reference line. The line frequencies and
strengths were computed from the rotational constants and dipole moments
reported in the literature: C5H6 (Bogey et al., 1988; Laurie, 1956), 1- and
2-C5H5CN (Cernicharo et al., 2021c; Sakaizumi et al., 1987), 1- and 2-C5H5CCH
(Cernicharo et al., 2021c), C9H8 (Cernicharo et al., 2021a; Caminati, 1993),
and 1- and 2-C10H7CN (McNaughton et al., 2018). We then carried out the line-
stack analysis in exactly the same manner as for C6H5CN. We found a
statistically significant line stack signal for indene, cyclopentadiene, and
1-C5H5CN in the L1495B cloud. The stacked spectra are shown in Fig. 2, where
we also give the number of lines included in the line stack analysis and the
S/N reached. The significance levels are well above the 5 $\sigma$ threshold.
In any case, we plan to perform new observations of L1495B to provide a more
robust identification of these three cycles and to search for other cycles
that have not yet been detected. The detection of these three particular
cycles in L1495B is consistent with (1) the fact that this is the cloud with
the highest signal for C6H5CN among the sources studied here, and (2) the
relative intensities of the different cycles observed in TMC-1. In this
regard, it makes sense that the only derivative of cyclopentadiene detected in
L1495B is 1-C5H5CN because this is the one showing the higher line intensities
in TMC-1 (Cernicharo et al., 2021c). We also note that in the case of the CN
derivatives of naphthalene, the search would be better carried out at
frequencies below 30 GHz because of the small rotational constants.

##  4 Discussion

Table 1: Observed column densities.

Molecule | Source |  $T_{\rm rot}$ a | $N$
---|---|---|---
|  | (K) |  (cm-2)
C6H5CN | L1495B |  |  3.8 $\times$ 1011
C6H5CN | Lupus-1A |  |  2.1 $\times$ 1011
C6H5CN | L483 |  |  1.7 $\times$ 1011
C6H5CN | L1527 |  |  $<$6.5 $\times$ 1010 b
C9H8 | L1495B |  |  1.1 $\times$ 1013
C5H6 | L1495B |  |  7.8 $\times$ 1012
1-C5H5CN | L1495B |  |  6.6 $\times$ 1010
HC7N | L1495B |  9.3 $\pm$ 0.4 |  (2.7 $\pm$ 0.4) $\times$ 1012
HC7N | Lupus-1A |  9.7 $\pm$ 0.4 |  (6.3 $\pm$ 0.9) $\times$ 1012
HC7N | L483 |  11.1 $\pm$ 0.4 |  (2.2 $\pm$ 0.3) $\times$ 1012
HC7N | L1527 |  17.0 $\pm$ 1.2 |  (7.4 $\pm$ 1.2) $\times$ 1011

a The rotational temperature for the cycles detected through line stack is
assumed to be 10 K in all sources.
b Upper limit calculated at the 5 $\sigma$ level.

To compute column densities for the cycles detected through line stack, we
assumed that the velocity-integrated intensity of the stacked line corresponds
to the reference line and adopted a round value of 10 K for the rotational
temperature. The column densities derived are given in Table 1. In the case of
L1527, we derived a 5 $\sigma$ upper limit to the column density of C6H5CN.

Benzonitrile is the aromatic cycle that has been most widely observed in cold
dense clouds. In addition to the detection in TMC-1 (McGuire et al., 2018;
Cernicharo et al., 2021c), it has also been observed in four other cold clouds
(Burkhardt et al., 2021b), three clumps in the Serpens South star-forming
region (named 1a, 1b, and 2; Friesen et al. 2013), and in the L1521F core,
located in the Taurus Molecular Cloud region (Crapsi et al., 2004). In their
study, Burkhardt et al. (2021b) find that the abundance of benzonitrile
relative to HC7N varies considerably between the Taurus and Serpens sources.
More specifically, the abundance ratio C6H5CN/HC7N is about four times higher
in the Taurus clouds TMC-1 and L1521F compared to the three Serpens clouds.
Here we report the detection of benzonitrile in three additional cold clouds
and the nondetection in a fourth one. Therefore, we now have a sizable sample
of nine sources where benzonitrile has been detected, or at least searched
for, which can be used to examine how benzonitrile behaves among different
cold clouds. The nine clouds are similar in that they are rich in carbon
chains. It is therefore interesting to evaluate whether the abundance of
benzonitrile is linked to that of carbon chains. To this purpose, we
calculated column densities for HC7N in the four clouds studied here using our
Yebes 40m data and the rotation diagram method. The calculated column
densities, together with the rotational temperatures derived, are given in
Table 1.

Figure 3: Column density of C6H5CN versus column density of HC7N for all the
clouds where C6H5CN has been detected or searched for. The colors indicate
different locations in the sky. Values were taken from Cabezas et al. (2022)
and Cernicharo et al. (2021c) for TMC-1, from Friesen et al. (2013) and
Burkhardt et al. (2021b) for the three clouds in Serpens and L1521F, and from
the present work elsewhere.

In Fig. 3 we represent the column densities of C6H5CN with respect to those of
HC7N. We can appreciate a very suggestive trend according to which the higher
the column density of HC7N, the more abundant C6H5CN is. The L1527 data point,
although it only provides an upper limit to the column density of
benzonitrile, brings valuable information supporting the positive correlation
between aromatic cycles and linear chains. We can also see that the abundance
of benzonitrile varies depending on the region where the cloud is located. For
example, in Serpens-s1a, Serpens-s1b, L1521F, and Lupus-1A the column
densities of HC7N are similar, although the same is not true for C6H5CN, which
is significantly more abundant in the two Serpens clouds than in Lupus-1A.
Regarding the evolutionary status of the clouds, most sources in Fig. 3 do not
contain a protostar, while L1521F hosts a very low-luminosity object (Bourke
et al., 2006), and L483 and L1527 are known to contain a protostar (Agúndez et
al., 2019; Sakai et al., 2008). It is interesting to note that the two
protostellar sources are the ones where benzonitrile is less abundant,
although it is also true that these two clouds have lower HC7N column
densities compared to the other sources. It therefore appears that aromatic
cycles are favored in those clouds where carbon chains are abundant, and these
clouds tend to be associated with early evolutionary stages along the process
of star formation. The relationship between aromatic cycles and carbon chains
is strengthened by the fact that the spatial distribution of C6H5CN in TMC-1
coincides with that of carbon chains, such as HC3N, HC5N, HC7N, and C6H
(Cernicharo et al., 2023).

Given that we detect several large cycles in L1495B, we are curious to know
whether or not the relative abundances between the various cycles are similar
to those found in TMC-1. Indene is the only species composed of fused rings
that has been detected in L1495B. According to the values given in Table 1,
indene has an abundance 29 times higher than that of benzonitrile. This value
is on the order of that found in TMC-1, where the abundance ratio C9H8/C6H5CN
is 13 (Cernicharo et al., 2021a, c). It is also interesting to note that the
column density of indene in L1495B is of the same order as that of the single
five-membered cycle cyclopentadiene (the C9H8/C5H6 ratio is 1.4; see Table 1),
while the same is also found in TMC-1, where the C9H8/C5H6 abundance ratio is
1.3 (Cernicharo et al., 2021a). Finally, it is also interesting to examine the
abundance ratio between a pure hydrocarbon cycle such as cyclopentadiene and
its CN derivative. In L1495B, the abundance of the isomer 1-C5H5CN is 0.8 % of
that of C5H6 (see Table 1), while in TMC-1 this isomer is present with an
abundance of 2.6 % of that of cyclopentadiene. In summary, the abundance
ratios between different cycles in L1495B are consistent with those found in
TMC-1 within a factor of three.

The positive correlation between the abundance of C6H5CN and HC7N found here,
together with the coincident spatial distribution of benzonitrile and carbon
chains found in TMC-1 by Cernicharo et al. (2023), suggest that there is a
chemical link between aromatic cycles and carbon chains; that is, the same
kind of chemistry that is responsible for the formation of linear chains is
also behind the synthesis of aromatic rings. However, while the synthesis of
carbon chains can be explained by current gas-phase chemical models of cold
dense clouds (e.g., Agúndez & Wakelam 2013), it is not yet well understood how
large hydrocarbon cycles are formed in these environments.

While it is nowadays generally accepted that aromatic rings are formed in situ
in the cold clouds where they are detected, there is still debate over which
processes drive the cyclization from small hydrocarbon species to single
cycles such as cyclopentadiene and benzene. It has been proposed that benzene
can be formed through ion-neutral reactions that lead to the C6H${}_{7}^{+}$
ion (McEwan et al., 1999; Woods, 2011; Agúndez et al., 2021), although there
are important uncertainties; to name one, the branching ratios of the
dissociative recombination of the ion C6H${}_{7}^{+}$ that is thought to lead
to benzene. Some neutral–neutral reactions have been studied and are promising
routes to cyclopentadiene and benzene in cold clouds. For example, the
reaction between 1,3-butadiene and the radical CH can yield cyclopentadiene
(He et al., 2020), which upon further reaction with CH can produce benzene
(Caster et al., 2021). Benzene can also be formed in the reaction between C2H
and 1,3-butadiene (Jones et al., 2011). However, when this chemical scheme
(see Appendix D in Cernicharo et al. 2021c) is implemented in chemical models
of cold dense clouds, the calculated abundances of benzonitrile and
cyclopentadiene fall below the observed values (Burkhardt et al., 2021b;
Cernicharo et al., 2022). There are still many reactions that need to be
studied before understanding how cyclization occurs in cold clouds. A recent
example is the study of the reaction between the propargyl and vinyl radicals
by García de la Concepción et al. (2023). Moreover, we also need to understand
how single cycles such as cyclopentadiene and benzene convert into fused rings
such as indene and naphthalene. In this context, Doddipatla et al. (2021)
recently found that indene can be formed in the reaction of vinyl benzene with
CH.

In addition to the challenge of explaining how aromatic cycles are formed in
cold clouds, constraining how abundant and widespread they are is of use in
evaluating their role in the physical and chemical evolution of the clouds.
For example, cycles such as indene and naphthalene are little polar and have
large sizes, which makes them strong candidates to be deposited as ices on
dust grains. In a recent study, Maté et al. (2023) find that the abundance of
indene observed in TMC-1 implies that it should be present on ices with an
exceedingly large abundance, accounting for up to 2 % of the carbon, which
seems unrealistically high. It is clear that we still need to understand the
interplay between gas and dust regarding these very big cycles. Astronomical
observations able to constrain the type of region where aromatic rings are
present and their abundances, combined with laboratory and theoretical
studies, will eventually allow us to obtain a good understanding of the
chemistry and physics of large aromatic rings in cold clouds.

##  5 Conclusions

We searched for large hydrocarbon cycles in four cold dense clouds. Using the
line stack technique, we detected benzonitrile in three sources (L1495B,
Lupus-1A, and L483), while it was not detected toward the fourth source
(L1527). In addition, we detected three other cycles toward L1495B, namely
indene, cyclopentadiene, and 1-cyano cyclopentadiene. By analyzing the
abundances of benzonitrile derived here, together with values derived in the
literature toward other sources, we find a positive correlation between the
column density of C6H5CN and that of HC7N. This result indicates that aromatic
rings are favored in those regions where long carbon chains are abundant. The
various cycles are likely to have similar relative abundances in cold dense
clouds, as suggested by the comparison between L1495B and TMC-1.

###### Acknowledgements.

We acknowledge funding support from Spanish Ministerio de Ciencia e Innovación
through grants PID2019-106110GB-I00, PID2019-107115GB-C21, and
PID2019-106235GB-I00. We thank the anonymous referee for a report that helped
to improve this paper.

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