# Minimal criteria for continuous-variable genuine multipartite entanglement

Olga Leskovjanová,1,∗ and Ladislav Mišta, Jr.1 1Department of Optics, Palacký
University, 17. listopadu 12, 779 00 Olomouc, Czech Republic  ∗Corresponding
author; E-mail<EMAIL_ADDRESS>
###### Abstract

We derive a set of genuine multi-mode entanglement criteria for second moments
of the quadrature operators. The criteria have a common form of the
uncertainty relation between sums of variances of position and momentum
quadrature combinations. A unique feature of the criteria is that the sums
contain the least possible number of variances of at most two-mode
combinations. The number of second moments we need to know to apply the
criteria thus scales only linearly with the number of modes, as opposed to the
quadratic scaling of the already existing criteria. Each criterion is
associated with a tree graph, which allowed us to develop a direct method of
construction of the criteria based solely on the structure of the underlying
tree. The practicality of the proposed criteria is demonstrated by finding a
number of examples of Gaussian states of up to six modes, whose genuine multi-
mode entanglement is detected by them. The designed criteria are particularly
suitable for verification of genuine multipartite entanglement in large multi-
mode states or when only a set of two-mode nearest-neighbour marginal
covariance matrices of the investigated state is available.

###### pacs:

03.67.-a

##  I Introduction

At the very foundations of the quantum information theory lies the effort to
explain possible relationships between a pair of quantum systems. These
relationships manifest itself through various forms of correlations which we
try to capture by intuitive models complying better with our idea of how the
nature ought to work. Then we can classify states of a composite system into a
nested family of sets according to whether the state admits a separable model
[1], local hidden state model [2] or local hidden variable model [3]. The
absence of the model then points at a closer relationship between the
subsystems, which is referred to as entanglement, steering and non-locality,
and which can be used as a resource for quantum communication and computing
[4].

The complexity of the relationships is significantly increased in systems
consisting of three or more subsystems, called commonly as the multipartite
systems. Here, a particularly important role is played by the genuine
multipartite entanglement (GME) [5], which is a synonym for truly multipartite
inseparability of the subsystems. This means, in other words, that a state
carrying GME cannot be modelled as a statistical mixture of product states.
Physically, preparation of a GME state requires interaction of all subsystems
involved [6].

The significance of GME states follows from the fact that they stay behind
violation of the statistical [7] and non-statistical [8] constraints imposed
by local hidden variable models, complex behaviour of strongly correlated
systems [9] and tests of multiparticle decoherence models [10]. Besides being
of fundamental interest, the GME states also find applications in measurement-
based quantum computing [11], quantum simulations [12], and quantum metrology
[13].

Motivated by both the fundamental questions of many-body physics and the
desire to demonstrate the advantage of quantum computers compared to classic
ones, experimenters have been trying to create ever-larger GME states. An
important part of characterizing these states is obviously the ability to
detect the GME. There are at least three reasons for this. First,
certification of GME confirms the presence of a key resource for measurement-
based model of quantum computing [11]. Second, detection of the GME is crucial
for understanding of the entanglement dynamics in a simulated system [14].
Finally, on a general level successful generation of GME evidences a high
degree of control which was gained over the used experimental platform [15].

Attempts to prepare a large GME state are proceeding in two directions. One
uses systems with a two-dimensional Hilbert state space (qubits), while the
other relies on systems with an infinite-dimensional Hilbert state space, the
so-called continuous-variable (CV) systems. Nowadays, preparation of the qubit
GME states is limited to the medium-scale systems. This includes preparation
of the 14 ion qubits in the Greenberger-Horne-Zeilinger (GHZ) state [10],
18-qubit photonic GHZ state [16] or 51-qubit cluster state on a
superconducting quantum processor [17] to name at least few of them.

A more auspicious framework for preparation of a large GME state is provided
by Gaussian states of CV systems [18]. These systems are realized typically by
optical modes and they are characterized by canonically conjugate position $x$
and momentum $p$ quadrature operators. Gaussian states possess a Gaussian-
shaped Wigner function and they are easily prepared, manipulated and measured
in the laboratory. Additionally, entanglement properties of an $N$-mode
Gaussian state are completely characterized by $N(2N+1)$ independent elements
of its covariance matrix and thus the number of parameters determining the
state grows quadratically with the number of modes. Although so far only
three-mode [19, 20] and four-mode [21, 22] Gaussian GME states have been
demonstrated in the laboratory, medium-scale to large-scale Gaussian states
possessing a weaker form of global entanglement, known as the full
inseparability, have been already observed experimentally. Specifically, a
fully inseparable 10-mode [21] and 60-mode [23] state has been prepared using
optical frequency comb, and 10 000-mode [24] or even one-million-mode [25]
fully inseparable state has been generated by means of the time-domain
multiplexing.

There are two basic ways in which the CV GME can be detected. The first one
utilizes the GME witness in the space of covariance matrices [26]. It requires
knowledge of the entire covariance matrix, or its entire position and momentum
part, and so far it has been used exclusively for verification of the GME in
the theoretical states [27, 28]. The second approach is based on simpler CV
GME criteria [19, 29, 22, 30], which are more transparent and more economical
in terms of the number of needed measurements. All the criteria are built on
the van Loock-Furusawa criterion of full inseparability [31], which
generalizes Duan’s et. al. criterion of two-mode entanglement [32] to the
multi-mode case. The considered criteria are usually designed to detect in a
simple way some paradigmatic GME Gaussian states and therefore they are of the
form of a single inequality for the second moments of the quadrature
operators. As an example can serve us the sum version of the criterion [19],
according to which a three-mode state is GME if the inequality

$\displaystyle\langle\left[\Delta\left(x_{1}-x_{2}\right)\right]^{2}\rangle+\langle\left[\Delta\left(x_{2}-x_{3}\right)\right]^{2}\rangle$
$\displaystyle+\langle\left[\Delta\left(p_{1}+p_{2}+p_{3}\right)\right]^{2}\rangle\geq
2$ (1)

is violated. The criterion exhibits a common feature of the overwhelming
majority of the practically used GME criteria. Namely, some (more often all
[29, 20]) occurring variances contain a global combination of all position or
momentum quadratures. For the three-mode criterion (I) the combination in
question is the total momentum quadrature $p_{1}+p_{2}+p_{3}$, whereas more
general $N$-mode criteria typically contain the weighted combination
$\sum_{i=1}^{N}g_{i}p_{i}$. Consequently, utilization of such the criterion
requires knowledge of all independent elements of the momentum part of the
covariance matrix, the number of which scales quadratically with $N$. Guided
by the effort to detect GME of the largest possible state, we can ask
ourselves whether this quadratic growth can be slowed down, e.g., just to the
linear growth. Finding a positive answer to this question would undoubtedly be
an important result because then the GME certification of some CV states would
require a significantly smaller number of measurements. However, such a result
would be rather counterintuitive because the respective criterion then could
not contain any global combination mentioned above, which might seem to be
indispensable for the detection of the global property of the GME.

Interestingly, this intuition is false. Namely, there are two similar criteria
of GME, which nevertheless contain variances of combinations of only pairs of
position and momentum quadratures. More precisely, one criterion is $N$-mode
and it is given by [22, Eq. (25)],

$\displaystyle\sum_{1\leq i<j\leq
N}\langle\left(x_{i}+x_{j}\right)^{2}+\left(p_{i}-p_{j}\right)^{2}\rangle\geq
f(N),$ (2)

where $f(N)$ is a function of $N$, whereas the other is four-mode and employs
variances of the same combinations (up to the sign) of quadrature operators
[30, Eqs. (44) and (47)]. Importantly, both the criteria contain combinations
for all possible pairs of the modes. Thus as before, the number of the
elements of the covariance matrix one needs for application of both the latter
criteria grows quadratically with the number of modes $N$.

Continuing the same line of thoughts, one can then ask whether the latter
criteria are already simplest criteria capable of detecting the GME or their
structure can be simplified even further. It is obvious that one cannot
simplify the form of the combinations, because they are just two-mode. But
since the criteria use the entire covariance matrix of the investigated state,
what could be reduced in principle is the number of appearing combinations.
Recent results on witnessing qubit GME from separable nearest-neighbour
marginals [33, 34] and their extension to the CV systems [35] indicate that
this simplification should be indeed possible.

In this paper we derive such the simplified CV GME criteria for $N$ modes.
Similar to the known criteria also the proposed criteria have the standard
form akin to the product and sum uncertainty relation for elements of the
covariance matrix and like the criteria [22, Eq. (25)] and [30, Eqs. (44) and
(47)] they include at most two-mode quadrature combinations. However, unlike
any of the existing CV GME criteria they contain the minimum number of the
combinations. The number is minimal because it is equal to the number of
elements in the minimal set of two-mode marginal covariance matrices [33, 35],
which is needed for detection of GME. Since the number of independent
covariance matrix elements contained in the minimal set is equal to $7N-4$
[35] and thus grows linearly with $N$, the number of elements of the
covariance matrix one needs to know for application of the proposed criteria
also scales only linearly with $N$. As every minimal set can be represented by
a tree graph, each of the presented criterion corresponds to a tree. This
allowed us to developed a direct method of construction of the criteria, which
is based solely on the structure of the underlying tree. We also demonstrate
practicality of our criteria by finding analytical as well as numerical
examples of Gaussian states whose GME is detected by the criteria for almost
all configurations of up to six modes.

The proposed criteria are most economic as far as the number of required
measurements is concerned and thus they can serve for verification of GME in
large multi-mode states. They are suitable for detection of the CV GME in
instances when only elements of two-mode marginal covariance matrices
belonging to some minimal set are know. Viewed from a different perspective,
the derived criteria also represent a sufficient condition for a set of two-
mode marginals to be compatible with a GME state.

The paper is structured as follows. In Sec. II we give a brief introduction
into the concept of GME, whereas Sec. III is dedicated to the entanglement in
CV systems. Section IV contains derivation of the minimal criteria for CV GME
and the Gaussian GME states which are detected by the criteria are constructed
in Sec. V. Conclusions are in Sec. VI.

##  II Genuine multipartite entanglement

We deal with $N$ quantum systems $1,2,\ldots,N$, which we collect into the set
$\mathcal{M}=\\{1,2,\ldots,N\\}$. Consider further the split of the set
$\mathcal{M}$ into two disjoint nonempty subsets
$I_{k}=\\{i_{1},i_{2},\ldots,i_{l}\\}$, $i_{n}\in\mathcal{M}$,
$n=1,2,\ldots,l$, $0<l<N$, and $J_{k}=\mathcal{M}\setminus I_{k}$. Here, the
index $k=1,2,\ldots,K_{N}$ labels all different inequivalent splits of the set
$\mathcal{M}$ and $K_{N}=2^{N-1}-1$ is the number of the splits. The splits
shall be henceforth called as bipartite splits and denoted as $I_{k}|J_{k}$.
We then say that a state $\rho$ of the considered system is separable with
respect to the split $I_{k}|J_{k}$ if it can be expressed as

$\displaystyle\rho=\rho_{I_{k}|J_{k}}^{\rm
sep}=\sum_{i}p_{i}\rho_{I_{k}}^{(i)}\otimes\rho_{J_{k}}^{(i)},$ (3)

where $\rho_{I_{k}}^{(i)}$ and $\rho_{J_{k}}^{(i)}$ are local states of
subsystems belonging to the set $I_{k}$ and $J_{k}$, respectively, and $p_{i}$
are probabilities. The states which cannot be expressed in the form (3) are
then called as entangled with respect to the split and this sort of
entanglement is referred to as the bipartite entanglement.

States of systems made up of only two subsystems ($N=2$) are either entangled
or separable, but in systems consisting of more subsystems ($N>2$) different
types of entanglement exist [36, 37]. Various forms of the multipartite
entanglement can be illustrated on the system of just three subsystems $1,2$
and $3$. In this case, there are three bipartite splits $1|23,2|31$ and
$3|12$, as well as a tripartite split $1|2|3$, and one can classify all states
into five separability classes [36, 37] depending on the number of the splits
with respect to which a given state is separable. Clearly, from the point of
view of this classification the strongest form of multipartite entanglement is
carried by the so-called fully inseparable states, which are entangled across
all three bipartite splits.

A specific feature of most important fully inseparable states is that their
preparation requires interaction of all three subsystems. However, this is not
a common property of all fully inseparable states. Namely, consider a class of
states which can be prepared by letting to interact randomly only pairs of
subsystems prepared in a triple product state. In this way, one can prepare
only a convex mixture of states which are separable across some bipartite
split, which is commonly denoted as a biseparable state. Interestingly, there
are qubit biseparable states which are fully inseparable [38]. To distinguish
truly multipartite entangled states requiring a global operation on the whole
system from other multipartite states, the concept of GME has been introduced
as follows. Generalizing the notion of a biseparable state to $N$-partite
system,

$\displaystyle\rho^{\rm bisep}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K_{N}}\lambda_{k}\rho_{I_{k}|J_{k}}^{\rm sep},$ (4)

where $\lambda_{j}$ are probabilities, we say that a state $\rho$ is GME, if
it cannot be expressed in the form (4).

Genuine multipartite entanglement finds applications in many branches of
quantum information science. For this reason, it is desirable to develop
efficient GME criteria. Because the set of biseparable states is convex, we
can use the machinery of entanglement witnesses [40, 39] for this purpose. In
the next section we give a brief introduction into the GME of states of
systems with infinite-dimensional Hilbert state spaces and its detection via
entanglement witnesses based on second moments.

##  III Continuous-variable entanglement

We assume that the considered systems $1,2,\ldots,N$ are the CV systems. Each
of the systems is then fully characterized by the position and momentum
operators $x_{j}$ and $p_{j}$ ($[x_{j},p_{k}]=i\delta_{j,k}$),
$j=1,2,\ldots,N$, respectively. From now on, we work with the optical
platform, where the systems are realized by modes and the position and
momentum operators by the respective quadrature operators. It is further
convenient to introduce the column vector
$\xi=\left(\xi_{x}^{T},\xi_{p}^{T}\right)^{T}$, where
$\xi_{x}=\left(x_{1},x_{2},\dots,x_{N}\right)^{T}$ and
$\xi_{p}=\left(p_{1},p_{2},\dots,p_{N}\right)^{T}$, which allows us to express
the quadrature commutation rules in the compact form
$[\xi_{j},\xi_{k}]=i(\Omega_{N})_{jk}$ with

$\displaystyle\Omega_{N}=\begin{pmatrix}\mathbb{O}_{N}&\openone_{N}\\\
-\openone_{N}&\mathbb{O}_{N}\end{pmatrix},$ (5)

where $\openone_{N}$ and $\mathbb{O}_{N}$ is the $N\times N$ identity matrix
and zero matrix, respectively. A particularly important class of states of CV
systems is given by the so-called Gaussian states, which are defined as states
with a Gaussian-shaped Wigner phase-space distribution. For $N$ modes the
states are therefore fully described by the $2N\times 1$ vector
$\langle\xi\rangle=\mathrm{Tr}(\xi\rho)$ of the first moments and by the
$2N\times 2N$ real symmetric covariance matrix (CM) $\gamma$ with entries
$(\gamma)_{jk}=\langle\\{\Delta\xi_{j},\Delta\xi_{k}\\}\rangle$, where
$\\{A,B\\}\equiv AB+BA$ is the anticommutator and $\Delta A\equiv A-\langle
A\rangle$. From the definition of the CM it further follows that it obeys the
uncertainty principle [41]

$\displaystyle\gamma+i\Omega_{N}\geq 0.$ (6)

The CM contains information on correlations of the corresponding state and can
be used to detect GME by means of the biseparability criterion [26]. It says
that if an $N$-mode state with CM $\gamma_{\rm BS}$ is biseparable, then there
exist bipartitions $I_{k}|J_{k}$ and CMs $\gamma_{I_{k}}\oplus\gamma_{J_{k}}$
which are block diagonal with respect to the bipartition $I_{k}|J_{k}$, and
probabilities $\lambda_{k}$ such that

$\gamma_{\rm
BS}-\sum_{k=1}^{K_{N}}\lambda_{k}\left(\gamma_{I_{k}}\oplus\gamma_{J_{k}}\right)\geq
0.$ (7)

If the criterion (7) is violated, the corresponding state is GME. An example
of the Gaussian GME state is provided by the CV GHZ-like state [42] with CM

$\gamma^{\rm GHZ}=\begin{pmatrix}a_{+}&c_{+}&c_{+}\\\ c_{+}&a_{+}&c_{+}\\\
c_{+}&c_{+}&a_{+}\end{pmatrix}\oplus\begin{pmatrix}a_{-}&c_{-}&c_{-}\\\
c_{-}&a_{-}&c_{-}\\\ c_{-}&c_{-}&a_{-}\end{pmatrix},$ (8)

where

$a_{\pm}=\frac{e^{\pm 2r}+2e^{\mp 2r}}{3},\quad c_{\pm}=\frac{e^{\pm
2r}-e^{\mp 2r}}{3}$ (9)

and $r\geq 0$ is the squeezing parameter. The GHZ state with CM (8) is also
fully inseparable, because the state is pure and for pure states the concepts
of GME and full inseparability coincide.

The situation changes for mixed states, where similar to qubits [38] one can
find also CV fully inseparable states which are biseparable. One example is
given by the manifestly biseparable state $\left(\rho_{12}^{\rm
TMSV}\otimes\rho_{3}^{\rm sq}+\rho_{1}^{\rm sq}\otimes\rho_{23}^{\rm
TMSV}\right)/2$ [19, 29]. Here, $\rho_{i}^{\rm sq}$ is the single-mode
squeezed state of mode $i$ with the squeezing parameter $r$ and
$\rho_{jk}^{\rm TMSV}$ is the two-mode squeezed vacuum (TMSV) state of modes
$j$ and $k$ with CM

$\gamma_{jk}^{\rm TMSV}=\begin{pmatrix}a&c\\\
c&a\end{pmatrix}\oplus\begin{pmatrix}a&-c\\\ -c&a\end{pmatrix},$ (10)

where $a=\cosh(2r)$ and $c=\sinh(2r)$. An even simpler fully symmetrical
example is provided by the state

$\displaystyle\rho^{\rm test}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\rho^{\rm TMSV}_{12}\otimes|0\rangle_{3}\langle
0|+\rho_{13}^{\rm TMSV}\otimes|0\rangle_{2}\langle 0|\right.$ (11)
$\displaystyle\left.+|0\rangle_{1}\langle 0|\otimes\rho_{23}^{\rm
TMSV}\right),$

where all TMSV states have the same squeezing parameter $r$. Again, the state
is obviously biseparable. Moreover, by applying the positive partial
transposition criterion [41] on CM of the state,

$\gamma^{\rm test}=\frac{1}{3}(\gamma^{\rm
TMSV}_{12}\oplus\openone_{3}+\gamma_{13}^{\rm
TMSV}\oplus\openone_{2}+\openone_{1}\oplus\gamma_{23}^{\rm TMSV}),$ (12)

one can show (see Appendix A for details of the calculation) that for
$r\in(0,1.24)$ the state (11) is at the same time entangled across all three
bipartite splits $1|23,2|31$ and $3|12$, and thus it is fully inseparable as
required.

Both the GHZ-like state with CM (8) as well as the state with CM (11) are
simple states suitable for testing of the CV GME criteria. In the following
sections we develop such criteria based only on variances of two-mode
quadrature combinations and test them, among other things, on these states.

##  IV Minimal criteria for genuine multipartite entanglement

We start by recalling that we look for the GME criteria based on second
moments of quadrature operators. Additionally, we restrict ourselves for a
while to the sum criteria, for which the left-hand side (LHS) is linear in the
second moments. Introducing $M$ quadrature linear combinations

$w_{i}=\sum_{j=1}^{2N}R_{ji}\xi_{j},$ (13)

$i=1,2,\ldots,M$, where $R_{ji}$ are elements of a real $2N\times M$ matrix,
the LHS of the criterion then can be expressed for the state with CM $\gamma$
in the form

$\sum_{i=1}^{M}\langle\left(\Delta
w_{i}\right)^{2}\rangle=\frac{1}{2}\mbox{Tr}\left[\gamma RR^{T}\right],$ (14)

where the symbol Tr stands for the matrix trace. Further, making use of the
structure of biseparable states (4) together with the Cauchy-Schwarz
inequality, we can arrive at the following generic form of the necessary
condition for biseparability:

$\mbox{Tr}[\gamma_{BS}Z]\geq c,$ (15)

for all CMs $\gamma_{BS}$ of biseparable states. Here, $Z=RR^{T}$ is a real,
symmetric $2N\times 2N$ matrix satisfying the condition $Z\geq 0$ and $c$ is a
non-negative constant which depends on matrix $R$. Provided that the matrix
$Z$ has suitable properties and the inequality (15) is violated by some
physical CM, the matrix $Z/c$ can be interpreted as a GME witness in the space
of second moments [43, 26]. Expression of the inseparability criterion in
terms of the witness matrix is advantageous as it allows us to impose
consistently further constraints on its structure, e.g., that the LHS of the
criterion contains only the minimal number of two-mode quadrature
combinations. In the next subsection we derive such the criteria. The
derivation consists of two parts. In the first part, we find the structure of
the LHS, whereas in the second part we derive the right-hand side (RHS) of the
criterion, which is given by the lower bound on the LHS for all biseparable
states.

###  IV.1 Left-hand side of the criterion

At the outset we derive the LHS of the sought GME criterion. Similar to other
entanglement criteria we rule out combinations mixing position and momentum
quadratures by assuming $Z=Z^{x}\oplus Z^{p}$, where $Z^{x}$ and $Z^{p}$ are
real, symmetric, and positive-semidefinite $N\times N$ matrices. Inspired by
the position part of the criterion (I), which involves combinations
$x_{1}-x_{2}$ and $x_{2}-x_{3}$, we further seek the criteria containing the
least number of only two-mode quadrature combinations. We therefore require
the criteria to utilize only the minimal number of two-mode reduced CMs
(marginal CMs), which are needed for detection of GME. The marginal CMs
comprise the so-called minimal set of marginals and its structure is already
known [33, 35]. The set has to contain all modes and one cannot divide it into
a subset and its complement without having a common mode.

A more instructive graphical representation of the minimal set is obtained
using the tools of graph theory [44]. Recall, that an undirected graph of
order $N$ is a pair $G=(V,E)$ of a set $V=\\{1,2,\ldots,N\\}$ of $N$ vertices,
and a set $E\subseteq K\equiv\\{\\{u,v\\}|(u,v)\in V^{2}\wedge u\neq v\\}$ of
edges. In our case a vertex $j$ of the graph represents mode $j$, whereas the
edge connecting adjacent vertices $j$ and $k$ represents marginal CM
$\gamma_{jk}$ of modes $j$ and $k$. The minimal set of marginals is then
represented by an undirected connected graph containing no cycles, also know
as an unlabeled tree [45], which shall be henceforth denoted as $T$. Here, we
are interested in the GME which exists in systems of at least three modes and
we therefore assume $N\geq 3$ from now. A closed formula for the number of
non-isomorphic trees of order $N$ is not known, yet it is known to grow
exponentially [46]. For small $N$ the number can be found in Ref. [47], and in
particular, for $N=3,4,5,6,\ldots,$ there is $1,2,3,6,\ldots,$ non-isomorphic
unlabeled trees.

In the next step we imprint the structure of the respective minimal set of
marginals on the LHS of the required criterion. This can be done elegantly
with the help of the adjacency matrix of a graph $G$, denoted as $A$, which is
an $N\times N$ symmetric matrix in which entry $A_{jk}$ is the number of edges
in $G$ connecting vertices $j$ and $k$. Since a tree is a simple graph with no
loops or multiple edges, its adjacency matrix is a zero-one matrix with zeros
on the diagonal [44]. In order the criterion to use only the minimal set of
marginals corresponding to a tree with adjacency matrix $A$, its matrices
denoted as $Z_{A}^{x}$ and $Z_{A}^{p}$ have to have generally non-zero
diagonal elements and zero off-diagonal elements on the same places as the
matrix $A$. This is simply achieved by the matrices

$Z_{A}^{x,p}=(\openone_{N}+A)\circ Z^{x,p},$ (16)

where $X\circ Y$ is the Hadamard (elementwise) matrix product with entries
$(X\circ Y)_{jk}=X_{jk}Y_{jk}$ [48] and $Z^{x,p}$ are some real symmetric
matrices with no zero entries. The LHS of the required criterion corresponding
to a minimal set of marginal CMs with adjacency matrix $A$ thus possesses the
following structure:

$\mbox{Tr}[\gamma
Z_{A}]=\mbox{Tr}[\gamma^{x}Z_{A}^{x}]+\mbox{Tr}[\gamma^{p}Z_{A}^{p}].$ (17)

Here $Z_{A}^{x,p}$ are some real, symmetric, and positive-semidefinite
matrices of the form (16), and $\gamma^{x}$ and $\gamma^{p}$ is the first and
second $N\times N$ diagonal block of the CM $\gamma$ with entries
$(\gamma^{x})_{jk}=\langle\\{\Delta x_{j},\Delta x_{k}\\}\rangle$ and
$(\gamma^{p})_{jk}=\langle\\{\Delta p_{j},\Delta p_{k}\\}\rangle$,
respectively.

It remains to rewrite the RHS of Eq. (17) in terms of variances of two-mode
quadrature combinations. Since the witness matrices $Z_{A}^{x,p}$ are always
real, symmetric and positive-semidefinite, one can use the Cholesky
decomposition and express them as
$Z_{A}^{\alpha}=L_{A}^{\alpha}(L_{A}^{\alpha})^{T}$, where $L_{A}^{\alpha}$,
$\alpha=x,p$, are real lower triangular $N\times N$ matrices with non-negative
diagonal elements [49, 50, 51]. Like in the case of the matrix
$Z_{A}^{\alpha}$ the structure of the Cholesky matrix $L_{A}^{\alpha}$ also
depends on the respective adjacency matrix $A$, which is expressed by the
lower index $A$. In what follows, the lower index is dropped for brevity,
i.e., from now by the symbol $L^{\alpha}$ we understand a Cholesky matrix
corresponding some adjacency matrix $A$.

Making further use of the cyclic property of the matrix trace and the
definition of a CM we then get

$\displaystyle\mathrm{Tr}[\gamma^{\alpha}Z_{A}^{\alpha}]$ $\displaystyle=$
$\displaystyle\mathrm{Tr}[\gamma^{\alpha}L^{\alpha}(L^{\alpha})^{T}]=2\sum_{i=1}^{N}\langle\\{\Delta[(L^{\alpha})^{T}\xi_{\alpha}]_{i}\\}^{2}\rangle$
(18) $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{N}\langle\left(\Delta
u_{i}^{\alpha}\right)^{2}\rangle\equiv 2U^{\alpha},$

$\alpha=x,p$. Here we introduced the multi-mode position and momentum
variables,

$u_{i}^{x}\equiv\sum_{j=i}^{N}L^{x}_{ji}x_{j},\quad
u_{i}^{p}\equiv\sum_{j=i}^{N}L^{p}_{ji}p_{j},$ (19)

$i=1,\ldots,N$, and sums of their variances,

$U^{x}\equiv\sum_{i=1}^{N}\langle\left(\Delta
u_{i}^{x}\right)^{2}\rangle,\quad U^{p}\equiv\sum_{i=1}^{N}\langle\left(\Delta
u_{i}^{p}\right)^{2}\rangle.$ (20)

Similar to other entanglement criteria we then define the LHS of our criterion
as one half of the expression (17), i.e., $U^{x}+U^{p}$. However, in the
present case each of the quantities $U^{\alpha}$, $\alpha=x,p$, contains $N$
variances of the quadrature combinations of the form (19).

In general, the LHS will not still possess the desired form containing
variances of at most two-mode quadrature combinations. Namely, for
$i=1,2,\ldots,N-2$ the operators (19) contain generally $N-i+1\geq 2$ terms
and not at most two, as required. Interestingly, also all these combinations
can attain the required structure, at least for some trees, if we label
suitably vertices of the tree representing the considered minimal set of
marginal CMs. The labeling can be found in several steps.

At the outset we show how the requirement that quadrature combinations (19)
contain at most two terms manifests itself on the level of the respective
adjacency matrix $A$. Recall first, that we seek a necessary condition for
biseparability of the form $f_{\gamma}(L^{x},L^{p})\geq g(L^{x},L^{p})$. On
the LHS there is a function $f_{\gamma}(L^{x},L^{p})$ which depends on CM
$\gamma$ of the tested state and Cholesky matrices $L^{x,p}$ of some witness
matrices $Z_{A}^{x,p}$. The RHS of the condition, which we derive in the next
section, imposes a lower bound on the LHS for all biseparable states, and it
is formed by a function $g(L^{x},L^{p})$ depending only on the matrices
$L^{x,p}$. The biseparability condition then can be used for testing of the
presence of GME in a given CM $\gamma$ by numerical minimization of the
difference $f_{\gamma}(L^{x},L^{p})-g(L^{x},L^{p})$ over the matrices
$L^{x,p}$. If the minimum is negative, i.e., the biseparability condition is
violated, then the state with CM $\gamma$ is inevitably GME. Here, we are
interested only in a generic form of the biseparability condition capable of
detecting the largest number of GME states from a minimal set of marginal CMs.
Clearly, such a condition is obtained if the function
$f_{\gamma}(L^{x},L^{p})$ depends on the maximum number of independent
variables, i.e., if the Cholesky matrices $L^{x,p}$ corresponding to the
minimal set have the maximum number of non-zero elements. Working with the
positive-semidefinite matrices $Z_{A}^{x,p}$, we would have to take into
account that some diagonal elements of the Cholesky matrices $L^{x,p}$ can be
equal to zero [51, p. 8.3] thereby reducing the number of the variables over
which we will optimize. For this reason we assume from now that the matrices
$Z_{A}^{x,p}$ are positive-definite which implies that the corresponding
Cholesky matrices $L^{x,p}$ possess strictly positive diagonal elements which
can be expressed as [50, Theorem 4.2.5]

$L^{\alpha}_{11}=\sqrt{(Z_{A}^{\alpha})_{11}},\quad
L^{\alpha}_{ii}=\sqrt{(Z_{A}^{\alpha})_{ii}-\sum_{j=1}^{i-1}(L_{ij}^{\alpha})^{2}},$
(21)

$i=2,3,\ldots,N$.

Now, let us rewrite the first $N-1$ combinations (19) in the compact form

$\displaystyle u_{i}^{x}$ $\displaystyle=$ $\displaystyle
x_{i}L^{x}_{ii}+(x_{i+1},x_{i+2},\ldots,x_{N})\cdot L^{x}_{i+1:N,i},$
$\displaystyle u_{i}^{p}$ $\displaystyle=$ $\displaystyle
p_{i}L^{p}_{ii}+(p_{i+1},p_{i+2},\ldots,p_{N})\cdot L^{p}_{i+1:N,i},$ (22)

$i=1,2,\ldots,N-1$, where the symbol “$\,\cdot\,$” stands for the scalar
product and

$L^{\alpha}_{i+1:N,i}\equiv(L^{\alpha}_{i+1i},L^{\alpha}_{i+2i},\ldots,L^{\alpha}_{Ni})^{T}$
(23)

are $(N-i)\times 1$ column vectors comprising lower triangular part without
the main diagonal of the matrix $L^{\alpha}$. Hence, we see that the operators
(IV.1) will contain at most two non-zero terms if each of the vectors
$L^{\alpha}_{i+1:N,i}$ will have at most one non-zero component. For a
positive-definite matrix $Z_{A}^{\alpha}$ the vectors can be further expressed
as [50, Theorem 4.2.5]

$\displaystyle
L^{\alpha}_{2:N,1}=\frac{(Z_{A}^{\alpha})_{2:N,1}}{L_{11}^{\alpha}}$ (24)

and

$\displaystyle
L^{\alpha}_{i+1:N,i}=\frac{1}{L_{ii}^{\alpha}}\left[(Z_{A}^{\alpha})_{i+1:N,i}-\sum_{j=1}^{i-1}L_{ij}^{\alpha}L^{\alpha}_{i+1:N,j}\right]$

for $i=2,3,\ldots,N-1$. From Eq. (24) it then follows that the vector
$L^{\alpha}_{2:N,1}$ will contain at most one non-zero component if the vector
$(Z_{A}^{\alpha})_{2:N,1}$ has at most one non-zero component. Moving to the
vector $L^{\alpha}_{3:N,2}$, Eq. (IV.1) with $i=2$, we see that if the vector
$(Z_{A}^{\alpha})_{2:N,1}$ has at most one non-zero component, then the second
term in the square brackets vanishes because it contains products of different
components of the vector $(Z_{A}^{\alpha})_{2:N,1}$, and
$L^{\alpha}_{3:N,2}=(Z_{A}^{\alpha})_{3:N,2}/L^{\alpha}_{22}$ according to Eq.
(IV.1). Again, in order the vector $L^{\alpha}_{3:N,2}$ to have at most one
non-zero component, the vector $(Z_{A}^{\alpha})_{3:N,2}$ has to possess at
most one non-zero component. Repeating the same procedure $(i-1)$-times we see
that the vectors $L^{\alpha}_{j+1:N,j}$, $j=1,2,\ldots,i-1$ will have at most
one non-zero component if the same holds for vectors
$(Z_{A}^{\alpha})_{j+1:N,j}$, $j=1,2,\ldots,i-1$. If this is the case, then
the sum $\sum_{j=1}^{i-1}L_{ij}^{\alpha}L^{\alpha}_{i+1:N,j}$ in Eq. (IV.1)
vanishes, because each term $L_{ij}^{\alpha}L^{\alpha}_{i+1:N,j}$ is a vector
with components given by a product of different components of the vector
$(Z_{A}^{\alpha})_{j+1:N,j}$. The $i$-th vector (IV.1) of the Cholesky matrix
$L^{\alpha}$ then reduces to

$\displaystyle
L^{\alpha}_{i+1:N,i}=\frac{(Z_{A}^{\alpha})_{i+1:N,i}}{L_{ii}^{\alpha}}$ (26)

and again it has at most one non-zero element if the vector
$(Z_{A}^{\alpha})_{i+1:N,i}$ contains at most one non-zero element.

Summarizing the obtained results we see that if we require all the quadrature
combinations (19) to contain at most two non-zero terms, then all vectors
$(Z_{A}^{\alpha})_{i+1:N,i}$, $i=1,2,\ldots,N-1$ have to possess at most one
non-zero element. Making further use of the formula (16) we can express the
latter vectors in the form

$\displaystyle(Z_{A}^{\alpha})_{i+1:N,i}=A_{i+1:N,i}\circ(Z^{\alpha})_{i+1:N,i},$
(27)

which leads, when combined with Eq. (26), to the final formula for the
Cholesky matrices of the sought criterion,

$\displaystyle L^{\alpha}=(\openone_{N}+A)\circ\ell^{\alpha},$ (28)

where $\ell^{\alpha}$ is a lower triangular matrix with non-zero entries and
strictly positive diagonal elements. Ascertaining of the structure of the
adjacency matrix $A$, which leads to the quadrature combinations (19) with at
most two non-zero terms, is now simple. The quadrature combinations (19) then
contain only $4N-2$ elements. From the formula (27) it follows immediately
that this happens if the vectors $A_{i+1:N,i}$, $i=1,2,\ldots,N-1$, contain
only one non-zero element. As the structure of the adjacency matrix is for a
given tree dictated by the labeling of its vertices, the question of the
possibility to have a GME criterion based only on at most two-mode quadrature
combinations boils down to the question of the existence of the tree labeling
which would possess in each vector $A_{i+1:N,i}$ only one non-zero component.
In what follows we show that such a labeling really exists, at least for some
trees.

####  IV.1.1 Linear tree

The first example of a tree for which the labeling is easy to find is an
$N$-vertex linear tree. Consider the standard labeling of the vertices by
natural numbers in an increasing order from the leftmost vertex to the
rightmost vertex as exemplified in Fig. 1 a) (see also labeled trees in the
second, fourth and seventh row of Tab. 3 for examples corresponding to $N=4,5$
and $6$). The corresponding adjacency matrix has elements
$A_{ij}=\delta_{|i-j|,1}$, $i,j=1,2,\ldots,N$, for which the quantities (20)
are given by

$\displaystyle U^{\alpha}_{\rm
lin}=\sum_{i=1}^{N-1}\langle[\Delta(\ell^{\alpha}_{ii}\alpha_{i}+\ell^{\alpha}_{i+1i}\alpha_{i+1})]^{2}\rangle+(\ell^{\alpha}_{NN})^{2}\langle(\Delta\alpha_{N})^{2}\rangle,$

$\alpha=x,p$, where the relations (19), (20) and (28) have been used. Clearly,
the RHS of the latter formula consists of variances of at most two-mode
quadrature combinations as required. Note further, that the presented labeling
is not the only one for which the LHS of the criterion attains the desired
form. For instance, it is easy to show that for the so-called reverse level
order labeling introduced below in this paper and depicted for a linear tree
in Fig. 1 b), the sums (20) also contain variances of at most two-mode
quadrature combinations.

$a)$1234$b)$3421 Figure 1: Four-mode example of the standard labeling $a)$ and
the alternative reverse level order labeling $b)$ of a linear tree, which
yields the quantities (20) with variances of quadrature combinations (19)
containing at most two terms.

####  IV.1.2 Star tree

Let us now move to the $N$-vertex star tree. We see that the labeling we are
looking for is established if we label the central vertex with $N$ and the
other vertices are labeled by the natural numbers $N-1,N-2,\ldots,1$ in a
decreasing order in the anticlockwise direction (see labeled trees in the
third, fifth and eight row of Tab. 3 for examples corresponding to $N=4,5$ and
$6$). For this labeling the only non-zero elements of the adjacency matrix are
$A_{Ni}=A_{iN}=1$, and they yield sums (20) of the following form:

$\displaystyle U^{\alpha}_{\rm
star}=\sum_{i=1}^{N-1}\langle[\Delta(\ell^{\alpha}_{ii}\alpha_{i}+\ell^{\alpha}_{Ni}\alpha_{N})]^{2}\rangle+(\ell^{\alpha}_{NN})^{2}\langle(\Delta\alpha_{N})^{2}\rangle,$

$\alpha=x,p$. Again, the LHS of the corresponding criterion comprise variances
of at most two-mode quadrature combinations.

####  IV.1.3 Generic tree

Interestingly, even generic trees can be labeled such, at least for low
orders, i.e., low $N$, that the corresponding quadrature combinations (19)
contain at most two non-zero terms. This can be done as follows.

Let us start by recalling that the degree of a vertex in a graph is the number
of edges incident to it and that a vertex of degree one is called as a leaf.
Consider now a tree $T$ and imagine an iterative procedure [52, 44] in which
in each step we delete all leaves of the tree obtained in the previous step.
At the end, we are left either with a single vertex or a single edge, which is
the so-called center of the tree $T$ (see Tab. 1 for examples of both the
instances).

Table 1: Examples of the reverse level order labeling for generic trees. The
trees are rooted by selecting a central vertex or one endpoint of the central
edge as a root (red vertices). The vertices are then labeled by natural
numbers in a decreasing order staring by labeling the root with $N$ and then
going from level to level, where in each level the vertices are labeled
consecutively from left to right.

$N$ | Unlabeled tree |  Type of center | Rooted tree
---|---|---|---
| $7$
---
|
---
| vertex
---
| 7643521
---
| $7$
---
|
---
| edge
---
| 7632154
---
| $8$
---
|
---
| edge
---
| 87521643
---

Alternatively, we get the center by taking a vertex or an edge which lies in
the middle of any longest path in the tree [52]. Next, if the center of the
tree is a vertex, we choose it as the so-called root of the tree, whereas if
it is an edge, we choose one of its endpoints as a root (see red vertices in
Tab. 1). Having established a root, we now associate with each vertex a level
which is the number of edges from the vertex to the root and arrange the
vertices into horizontal layers, where in the topmost layer there is only the
root, the second layer is comprised of level-one vertices etc. (see the last
column of Tab. 1). This allows us to introduce a labeling of the tree $T$,
which can be called fittingly as a reverse level order labeling. Here, we
label the vertices by natural numbers in a decreasing order starting by
assigning the label $N$ to the root and continuing level by level going from
left to right at each level (see the last column of Tab. 1, as well as the
third column of Tab. 3 for explicit examples).

We checked by a direct calculation that for all configurations of up to $N=10$
the established reverse level order labeling leads to an adjacency matrix,
where each vector $A_{i+1:N,i}$ contains only one non-zero component and
therefore the respective quadrature combinations (19) contain at most two non-
zero terms. This leads us to a conjecture that this statement holds generally
for an arbitrary $N$-vertex tree.

Note finally that the reverse level order labeling is not the only labeling
yielding vectors $A_{i+1:N,i}$, $i=1,2,\ldots,N-1$, each with only one non-
zero component. Other labeling equipped with the same property is the
“natural” labeling of a linear tree exemplified in the preceding subsubsection
or the labeling of a tree depicted in Fig. 2.

7654321 Figure 2: Example of a tree labeling which is different from the
reverse level order labeling displayed in the second row of Tab. 1, yet it
leads to the GME criterion with at most two-mode quadrature combinations (19).
See text for details.

###  IV.2 Right-hand side of the criterion

In the next step, we find a lower bound for the LHS of the sought criterion,
which we obtained in the preceding subsection. Following the standard approach
[29], we find a lower bound both on the sum $U^{x}+U^{p}$ and on the product
$U^{x}U^{p}$ for all biseparable states. Thus we get two necessary conditions
for biseparability, which we call the sum condition and the product condition,
respectively. Negation of the conditions then yields two sufficient conditions
for GME. Below we give examples of GME states detected by both the criteria,
as well as by only one of them, which demonstrates that they certify GME of
intersecting but generally different sets of states.

In what follows we first derive a product necessary condition for
biseparability and then we continue with derivation of the sum condition.

####  IV.2.1 Product criterion

Our goal is to find a lower bound for the product

$\displaystyle U^{x}U^{p}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{N}\langle\left(\Delta
u_{i}^{x}\right)^{2}\rangle\langle\left(\Delta u_{j}^{p}\right)^{2}\rangle,$
(31)

where

$u_{i}^{\alpha}=\sum_{j=1}^{N}L^{\alpha}_{ji}\alpha_{j},$ (32)

$\alpha=x,p$, for all biseparable states (4). Contrary to Eq. (19), in Eq.
(32) we for simplicity do not take into account explicitly the fact that
$L_{ji}^{\alpha}=0$ for $j<i$.

Note first, that the product (31) is lower bounded as

$\displaystyle U^{x}U^{p}$ $\displaystyle\stackrel{{\scriptstyle 1}}{{\geq}}$
$\displaystyle\left(\sum_{k=1}^{K_{N}}\lambda_{k}U_{k}^{x}\right)\left(\sum_{l=1}^{K_{N}}\lambda_{l}U_{l}^{p}\right)$
(33) $\displaystyle\stackrel{{\scriptstyle 2}}{{\geq}}$
$\displaystyle\left(\sum_{k=1}^{K_{N}}\lambda_{k}\sqrt{U_{k}^{x}U_{k}^{p}}\right)^{2},$

with

$\displaystyle U_{k}^{\alpha}\equiv\sum_{i=1}^{N}\langle\left(\Delta
u_{i}^{\alpha}\right)^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}},$ (34)

$\alpha=x,p$, where the symbol $\langle\,\,\rangle_{\rho}$ stands for the mean
in the state $\rho$. Here, to get inequality $1$ we used the concavity of the
variance with respect to convex mixtures [53], which we applied to the mixture
$\rho^{\rm bisep}$, Eq. (4), whereas inequality $2$ follows from the Cauchy-
Schwarz inequality.

Next, we lower bound the product

$\displaystyle U_{k}^{x}U_{k}^{p}=\sum_{i,j=1}^{N}\langle\left(\Delta
u_{i}^{x}\right)^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}}\langle\left(\Delta
u_{j}^{p}\right)^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}}.$ (35)

For this purpose, consider a bipartite split $k\equiv I_{k}|J_{k}$, where
$I_{k}=\\{i_{1},i_{2},\ldots,i_{l}\\}$ and
$J_{k}=\\{i_{l+1},i_{l+2},\ldots,i_{N}\\}$, and assume that the system is
prepared in the separable state $\rho_{I_{k}|J_{k}}^{\rm sep}$, Eq. (3). Let
as further decompose the operators (32) into two parts, one belonging to the
subsystem $I_{k}$ and the other to the subsystem $J_{k}$,

$u_{i}^{\alpha}=u_{i,I_{k}}^{\alpha}+u_{i,J_{k}}^{\alpha},$ (36)

where

$u_{i,I_{k}}^{\alpha}\equiv\sum_{n=1}^{l}L^{\alpha}_{i_{n}i}\alpha_{i_{n}},\quad
u_{i,J_{k}}^{\alpha}\equiv\sum_{n=l+1}^{N}L^{\alpha}_{i_{n}i}\alpha_{i_{n}}.$
(37)

Inspired by the approach of Ref. [20] we can then lower bound the variance
product appearing on the RHS of Eq. (35) as

$\displaystyle\langle(\Delta u_{i}^{x})^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm
sep}}\langle(\Delta u_{j}^{p})^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}}$
$\displaystyle\stackrel{{\scriptstyle
1}}{{\geq}}\left[\sum_{l}p_{l}^{(k)}\langle\left(\Delta
u_{i}^{x}\right)^{2}\rangle_{\rho_{I_{k}\otimes
J_{k}}^{(l)}}\right]\left[\sum_{m}p_{m}^{(k)}\langle\left(\Delta
u_{j}^{p}\right)^{2}\rangle_{\rho_{I_{k}\otimes J_{k}}^{(m)}}\right]$
$\displaystyle\stackrel{{\scriptstyle
2}}{{\geq}}\left[\sum_{l}p_{l}^{(k)}\sqrt{\langle\left(\Delta
u_{i}^{x}\right)^{2}\rangle_{\rho_{I_{k}\otimes
J_{k}}^{(l)}}\langle\left(\Delta
u_{j}^{p}\right)^{2}\rangle_{\rho_{I_{k}\otimes J_{k}}^{(l)}}}\right]^{2}$
$\displaystyle\stackrel{{\scriptstyle
3}}{{\geq}}\left\\{\sum_{l}p_{l}^{(k)}\left[\sqrt{\langle(\Delta
u_{i,I_{k}}^{x})^{2}\rangle_{\rho_{I_{k}}^{(l)}}\langle(\Delta
u_{j,I_{k}}^{p})^{2}\rangle_{\rho_{I_{k}}^{(l)}}}\right.\right.$
$\displaystyle\hskip 14.22636pt+\left.\left.\sqrt{\langle(\Delta
u_{i,J_{k}}^{x})^{2}\rangle_{\rho_{J_{k}}^{(l)}}\langle(\Delta
u_{j,J_{k}}^{p})^{2}\rangle_{\rho_{J_{k}}^{(l)}}}\right]\right\\}^{2}$
$\displaystyle\stackrel{{\scriptstyle
4}}{{\geq}}\frac{1}{4}\left\\{\left|\left[(L^{x})^{T}L^{p}\right]_{ij}^{(I_{k})}\right|+\left|\left[(L^{x})^{T}L^{p}\right]_{ij}^{(J_{k})}\right|\right\\}^{2}$
$\displaystyle\stackrel{{\scriptstyle
5}}{{=}}\frac{1}{4}\left\\{\left[(L^{x})^{T}L^{p}\right]_{ij}^{2}+2(|X_{ij}^{(k)}|-X_{ij}^{(k)})\right\\},$
(38)

where we introduced the denotation $\rho_{I_{k}\otimes
J_{k}}^{(l)}\equiv\rho_{I_{k}}^{(l)}\otimes\rho_{J_{k}}^{(l)}$. Here,
inequality $1$ follows from concavity of the variance, inequality $2$ is a
consequence of the Cauchy-Schwarz inequality and inequality $3$ results from
the averaging in the product state $\rho_{I_{k}\otimes J_{k}}^{(l)}$ and again
the Cauchy-Schwarz inequality. Finally, in inequality $4$ we used the
uncertainty relations and the quantities appearing on the RHS of the
inequality and equality $5$ are defined as

$\displaystyle\left[(L^{x})^{T}L^{p}\right]_{ij}^{(I_{k})}$
$\displaystyle\equiv$
$\displaystyle\sum_{n=1}^{l}\left[(L^{x})^{T}\right]_{ii_{n}}L^{p}_{i_{n}j},$
(39) $\displaystyle\left[(L^{x})^{T}L^{p}\right]_{ij}^{(J_{k})}$
$\displaystyle\equiv$
$\displaystyle\sum_{n=l+1}^{N}\left[(L^{x})^{T}\right]_{ii_{n}}L^{p}_{i_{n}j},$
(40) $\displaystyle X_{ij}^{(k)}$ $\displaystyle\equiv$
$\displaystyle\left[(L^{x})^{T}L^{p}\right]_{ij}^{(I_{k})}\left[(L^{x})^{T}L^{p}\right]_{ij}^{(J_{k})}.$

Combining now Eq. (35) with inequality (IV.2.1) we get immediately the
following inequality

$\displaystyle
U_{k}^{x}U_{k}^{p}\geq\frac{1}{4}\left(\mathcal{L}+\mathcal{L}^{(k)}\right),$
(42)

where

$\displaystyle\mathcal{L}$ $\displaystyle\equiv$
$\displaystyle\sum_{i,j=1}^{N}\left[(L^{x})^{T}L^{p}\right]_{ij}^{2},$ (43)
$\displaystyle\mathcal{L}^{(k)}$ $\displaystyle\equiv$ $\displaystyle
2\sum_{i,j=1}^{N}(|X_{ij}^{(k)}|-X_{ij}^{(k)}).$ (44)

Finally, putting inequalities (33) and (42) together, we arrive at the sought
necessary condition for biseparability of the form:

$\displaystyle
U^{x}U^{p}\geq\frac{1}{4}\left\\{\mathcal{L}+\mbox{min}\left[\mathcal{L}^{(1)},\ldots,\mathcal{L}^{(2^{N-1}-1)}\right]\right\\},$
(45)

where we used the fact that $K_{N}=2^{N-1}-1$.

####  IV.2.2 Sum criterion

Derivation of the sum condition is simpler. First, we use concavity of the
variance to get

$\displaystyle U^{x}+U^{p}$ $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{N}\sum_{k=1}^{K_{N}}\lambda_{k}\sum_{\alpha=x,p}\langle\left(\Delta
u_{i}^{\alpha}\right)^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}}.$ (46)

Next, we lower bound the sum of the variances on the RHS as

$\displaystyle\sum_{\alpha=x,p}\langle\left(\Delta
u_{i}^{\alpha}\right)^{2}\rangle_{\rho_{I_{k}|J_{k}}^{\rm sep}}$
$\displaystyle\stackrel{{\scriptstyle
1}}{{\geq}}\sum_{\alpha=x,p}\sum_{l}p_{l}^{(k)}\langle\left(\Delta
u_{i}^{\alpha}\right)^{2}\rangle_{\rho_{I_{k}\otimes J_{k}}^{(l)}}$
$\displaystyle\stackrel{{\scriptstyle
2}}{{=}}\sum_{\alpha=x,p}\sum_{l}p_{l}^{(k)}\left[\langle(\Delta
u_{i,I_{k}}^{\alpha})^{2}\rangle_{\rho_{I_{k}}^{(l)}}+\langle(\Delta
u_{i,J_{k}}^{\alpha})^{2}\rangle_{\rho_{J_{k}}^{(l)}}\right]$
$\displaystyle\stackrel{{\scriptstyle
3}}{{\geq}}\left|\left[(L^{x})^{T}L^{p}\right]_{ii}^{(I_{k})}\right|+\left|\left[(L^{x})^{T}L^{p}\right]_{ii}^{(J_{k})}\right|,$
(47)

where inequality $1$ again follows from concavity of the variance, whereas in
equality $2$ we used the decomposition (36) and the fact that we average over
the product state $\rho_{I_{k}\otimes J_{k}}^{(l)}$. To get inequality $3$ we
used the inequality between the arithmetic and geometric mean, the uncertainty
relations, and the definitions (39) and (40). Finally, if we combine
inequalities (46) and (IV.2.2) we obtain the desired sum criterion in the
following form:

$\displaystyle
U^{x}+U^{p}\geq\mathrm{min}\left[\mathcal{K}^{(1)},\ldots,\mathcal{K}^{(2^{N-1}-1)}\right],$
(48)

where

$\displaystyle\mathcal{K}^{(k)}\equiv\sum_{i=1}^{N}\left\\{\left|\left[(L^{x})^{T}L^{p}\right]_{ii}^{(I_{k})}\right|+\left|\left[(L^{x})^{T}L^{p}\right]_{ii}^{(J_{k})}\right|\right\\}.$
(49)

Testing of the presence of GME in a given quantum state by means of the
criteria (45) and (48) requires to find a set of parameters
$\ell^{\alpha}_{ij}$, for which the respective inequalities (45) and (48) are
violated. This can be done by minimization of the difference

$D_{P}\equiv
U^{x}U^{p}-\frac{1}{4}\left\\{\mathcal{L}+\mbox{min}\left[\mathcal{L}^{(1)},\ldots,\mathcal{L}^{(2^{N-1}-1)}\right]\right\\}$
(50)

or

$D_{S}\equiv
U^{x}+U^{p}-\mathrm{min}\left[\mathcal{K}^{(1)},\ldots,\mathcal{K}^{(2^{N-1}-1)}\right]$
(51)

over the parameters $\ell^{\alpha}_{ij}$. Clearly, if the difference is
negative, the investigated state is GME.

In the next Section we use this approach to verify GME in several multi-mode
quantum states. This requires to select a certain tree and express the
respective differences (50) and (51) in terms of the parameters
$\ell^{\alpha}_{ij}$ by means of the formula (28). This can be rather
cumbersome, in particular, for trees with many vertices. For this reason, it
is desirable to develop a simpler construction of the criteria, which would
ideally utilize only the structure of the considered tree. In the following
subsection we present such a construction for the sum criterion.

###  IV.3 Direct construction of the sum criterion

Practical utility of the proposed criteria would be significantly increased if
we would be able to construct them directly from the underlying tree. While
construction of the LHS for both the criteria is the same, construction of the
RHS is transparent only for the sum criterion. Below, we therefore construct
the RHS for the sum criterion, whereas the construction for the product
criterion is deferred for further research.

####  IV.3.1 Construction of the left-hand side

The quantities $U^{x,p}$, Eq. (20), on the LHS of both criteria are the same
and their construction is easy. Namely, each $N$-vertex tree consists of $N-1$
edges [44]. Each of the edges connects a pair of vertices with labels $i$ and
$j$, where by $i$ we denote the label which is less than the label $j$, i.e.,
$i<j$. With each of the edges we associate a two-mode quadrature linear
combination $u_{i}^{\alpha}$ (see Fig. 3). Sum of the variances
$\langle(\Delta u_{i}^{\alpha})^{2}\rangle$ over all edges then comprises (up
to the term $(\ell^{\alpha}_{NN})^{2}\langle(\Delta\alpha_{N})^{2}\rangle$)
the quantity $U^{\alpha}$.

$i<j$
---
$\ell_{ii}^{\alpha}$ | $\ell_{jj}^{\alpha}$
$\ell_{ji}^{\alpha}$
$i$$j$
$\alpha_{i}$ | $\alpha_{j}$
$u_{i}^{\alpha}=\ell_{ii}^{\alpha}\alpha_{i}+\ell_{ji}^{\alpha}\alpha_{j}$
Figure 3: Construction of the two-mode quadrature combination $u_{i}^{\alpha}$
corresponding to the edge $e_{ij}=ij$ connecting vertices $i$ and $j$. See
text for details.

More precisely, to each vertex $i$ we ascribe the quadrature $\alpha_{i}$, as
well as the diagonal element $\ell_{ii}^{\alpha}$ of the matrix
$\ell^{\alpha}$. Next, to each edge connecting the vertices $i$ and $j$ we
assign the off-diagonal entry $\ell_{ji}^{\alpha}$. The edge is then
represented by the quadrature combination

$u_{i}^{\alpha}=\ell^{\alpha}_{ii}\alpha_{i}+\ell^{\alpha}_{ji}\alpha_{j}.$
(52)

The quantity $U^{\alpha}$ in Eq. (20) is then given by the sum of variances of
all $N-1$ quantities (52), plus the term
$(\ell^{\alpha}_{NN})^{2}\langle(\Delta\alpha_{N})^{2}\rangle$. Making use of
this rule we can easily rederive the quantities $U^{\alpha}$ for the linear
tree and the star tree given in Eqs. (IV.1.1) and (IV.1.2), respectively. In
the next section we also use the rule to construct the criteria corresponding
to the more complex trees in the third column of Tab. 3.

####  IV.3.2 Construction of the right-hand side

Construction of the RHS of the sum criterion is also easy. Recall first, that
in a tree with root $N$ there is for each vertex $i$ a unique $i,N$-path. The
so-called parent of vertex $i$ is an adjacent vertex on the $i,N$-path [44]
and each non-root vertex has exactly one parent. Let $j$ be the label of the
parent of vertex $i$, where inequality $i<j$ always holds in our labeling, and
$e_{ij}\equiv ij$ be the edge joining the vertex $i$ with its parent $j$. Now,
for each non-root vertex $i$ we create a pair $(i,e_{ij})$ and represent it by
the pair $(\ell^{x}_{ii}\ell^{p}_{ii},\ell^{x}_{ji}\ell^{p}_{ji})$ of products
of elements of matrices $\ell^{x,p}$, which are associated with the vertex $i$
and the edge $e_{ij}$ according to Fig. 3. The root $N$ has no parent and thus
there is no edge associated to it. Therefore, it is represented just by the
term $\ell^{x}_{NN}\ell^{p}_{NN}$.

Moving to the construction of the RHS it is obvious that to every bipartite
split $k\equiv I_{k}|J_{k}$ of $N$ modes into two disjoint subsets
$I_{k}=\\{i_{1},i_{2},\ldots,i_{l}\\}$ and
$J_{k}=\\{i_{l+1},i_{l+2},\ldots,i_{N}\\}$ there corresponds exactly the same
bipartite split of the set of all $N$ vertices of the considered graph. The
split divides the set of all $N-1$ pairs $(i,e_{ij})$ into two disjoint
subsets, denoted as $\mathcal{U}$ and $\mathcal{C}$, depending on whether the
edge $e_{ij}$ is uncut by the split or it is cut off by the split. Formally,
$(i,e_{ij})\in\mathcal{U}$ if $i,j\in I_{k}$ or $i,j\in J_{k}$, whereas
$(i,e_{ij})\in\mathcal{C}$ if $i\in I_{k}$ and $j\in J_{k}$ or vice versa.
Now, to each pair $(i,e_{ij})\in\mathcal{U}$ we ascribe the term
$|\ell^{x}_{ii}\ell^{p}_{ii}+\ell^{x}_{ji}\ell^{p}_{ji}|$, whereas with each
pair $(i,e_{ij})\in\mathcal{C}$ we associate a different term
$|\ell^{x}_{ii}\ell^{p}_{ii}|+|\ell^{x}_{ji}\ell^{p}_{ji}|$. The sum of all
the latter terms for all $N-1$ pairs $(i,e_{ij})$ plus the term
$|\ell^{x}_{NN}\ell^{p}_{NN}|$, which corresponds to the root, gives finally
the quantity $\mathcal{K}^{(k)}$, Eq. (49), appearing on the RHS of the sum
criterion (48).

Explicit construction of all the quantities $\mathcal{K}^{(k)}$ for the
simplest case of $N=3$ modes is presented in Fig. 4.

full | 123$l^{x}_{11}l^{p}_{11}$$l^{x}_{21}l^{p}_{21}$$l^{x}_{22}l^{p}_{22}$$l^{x}_{32}l^{p}_{32}$$l^{x}_{33}l^{p}_{33}$
---|---
$1|23$ |
| 123
$\mathcal{K}^{(1)}=|\ell^{x}_{11}\ell^{p}_{11}|+|\ell^{x}_{21}\ell^{p}_{21}|+|\ell^{x}_{22}\ell^{p}_{22}+\ell^{x}_{32}\ell^{p}_{32}|+|\ell^{x}_{33}\ell^{p}_{33}|$
$2|31$ |
| 123
$\mathcal{K}^{(2)}=|\ell^{x}_{11}\ell^{p}_{11}|+|\ell^{x}_{21}\ell^{p}_{21}|+|\ell^{x}_{22}\ell^{p}_{22}|+|\ell^{x}_{32}\ell^{p}_{32}|+|\ell^{x}_{33}\ell^{p}_{33}|$
$3|12$ |
| 123
$\mathcal{K}^{(3)}=|l^{x}_{11}l^{p}_{11}+l^{x}_{21}l^{p}_{21}|+|l^{x}_{22}l^{p}_{22}|+|l^{x}_{32}l^{p}_{32}|+|l^{x}_{33}l^{p}_{33}|$
Figure 4: Graphical construction of the quantities $\mathcal{K}^{(k)}$,
$k=1,2,3$, appearing on the RHS of the sum criterion (48) for the $3$-vertex
tree with the standard labeling (uppermost graph). The lower graphs correspond
consecutively to the full tree after the cuts with respect to bipartite splits
$1\equiv 1|23$, $2\equiv 2|31$ and $3\equiv 3|12$. The uncut edges are
represented by solid lines, whereas the cut off edges by the dashed lines. See
text for details.

##  V Applications

###  V.1 Three-mode states

First, we test functionality of the product criterion (45) and sum criterion
(48) for the simplest case of $N=3$ modes. Then there is only one linear tree
which for the standard labeling yields the adjacency matrix given above Eq.
(IV.1.1). Hence, one gets using the formula (43)

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle(\ell^{x}_{11}\ell^{p}_{11}+\ell^{x}_{21}\ell^{p}_{21})^{2}+(\ell^{x}_{21}\ell^{p}_{22})^{2}+(\ell^{x}_{22}\ell^{p}_{21})^{2}$
(53) $\displaystyle+$
$\displaystyle(\ell^{x}_{22}\ell^{p}_{22}+\ell^{x}_{32}\ell^{p}_{32})^{2}+(\ell^{x}_{32}\ell^{p}_{33})^{2}+(\ell^{x}_{33}\ell^{p}_{32})^{2}$
$\displaystyle+$ $\displaystyle(\ell^{x}_{33}\ell^{p}_{33})^{2}.$

The three modes can be divided into three bipartite splits which we denote as
$1\equiv 1|23$, $2\equiv 2|31$ and $3\equiv 3|12$, and for the splits we
further obtain

$\displaystyle\mathcal{L}^{(1)}$ $\displaystyle=$ $\displaystyle
2\left(|\ell^{x}_{11}\ell^{p}_{11}\ell^{x}_{21}\ell^{p}_{21}|-\ell^{x}_{11}\ell^{p}_{11}\ell^{x}_{21}\ell^{p}_{21}\right),$
$\displaystyle\mathcal{L}^{(3)}$ $\displaystyle=$ $\displaystyle
2\left(|\ell^{x}_{22}\ell^{p}_{22}\ell^{x}_{32}\ell^{p}_{32}|-\ell^{x}_{22}\ell^{p}_{22}\ell^{x}_{32}\ell^{p}_{32}\right),$
$\displaystyle\mathcal{L}^{(2)}$ $\displaystyle=$
$\displaystyle\mathcal{L}^{(1)}+\mathcal{L}^{(3)},$ (54)

by Eq. (44). Making use of the expressions for the quantities $U_{\rm
lin}^{x,p}$, Eq. (IV.1.1), we get finally the minimal three-mode product
criterion for GME,

$\displaystyle\prod_{\alpha=x,p}\left\\{\sum_{i=1}^{2}\langle[\Delta(\ell^{\alpha}_{ii}\alpha_{i}+\ell^{\alpha}_{i+1i}\alpha_{i+1})]^{2}\rangle+\left(\ell^{\alpha}_{33}\right)^{2}\langle(\Delta\alpha_{3})^{2}\rangle\right\\}$
$\displaystyle\geq\frac{1}{4}\left\\{\mathcal{L}+\mbox{min}\left[\mathcal{L}^{(1)},\mathcal{L}^{(2)},\mathcal{L}^{(3)}\right]\right\\}.$
(55)

Moving to the sum criterion (48) for three modes, the product on the LHS of
the latter inequality is replaced with the sum, whereas the quantities
$\mathcal{K}^{(k)}$ appearing on the RHS were derived in Fig. 4. Additionally,
since $\mathcal{K}^{(1)},\mathcal{K}^{(3)}\leq\mathcal{K}^{(2)}$, it is
sufficient to take only $\mbox{min}\\{\mathcal{K}^{(1)},\mathcal{K}^{(3)}\\}$
on the RHS of the considered criterion. Putting it all together the minimal
three-mode sum criterion for GME reads as

$\displaystyle\sum_{\alpha=x,p}\left\\{\sum_{i=1}^{2}\langle[\Delta(\ell^{\alpha}_{ii}\alpha_{i}+\ell^{\alpha}_{i+1i}\alpha_{i+1})]^{2}\rangle+\left(\ell^{\alpha}_{33}\right)^{2}\langle(\Delta\alpha_{3})^{2}\rangle\right\\}$
$\displaystyle\geq\mathrm{min}\left(|\ell^{x}_{11}\ell^{p}_{11}|+|\ell^{x}_{21}\ell^{p}_{21}|+|\ell^{x}_{22}\ell^{p}_{22}+\ell^{x}_{32}\ell^{p}_{32}|+|\ell^{x}_{33}\ell^{p}_{33}|,\right.$
$\displaystyle\left.|\ell^{x}_{11}\ell^{p}_{11}+\ell^{x}_{21}\ell^{p}_{21}|+|\ell^{x}_{22}\ell^{p}_{22}|+|\ell^{x}_{32}\ell^{p}_{32}|+|\ell^{x}_{33}\ell^{p}_{33}|\right).$

At the outset we confirm on a particular example that a fully inseparable but
not GME state really does not violate the inequalities (V.1) and (V.1). As an
example we use the family of fully inseparable biseparable states (11) with CM
$\gamma^{\rm test}$, Eq. (12), where $r\in(0,1.24)$. For all states with the
squeezing parameter $r_{i}=0.05\times i$, $i=0,1,\ldots,24$, we have minimized
numerically the differences $D_{P}$ and $D_{S}$, Eqs. (50) and (51) with
$N=3$, i.e., the difference of the LHS and the RHS of inequalities (V.1) and
(V.1). Our analysis reveals that for all considered values of the squeezing
the differences are very small but always strictly positive numbers, which
confirms correctness of the proposed criteria.

Next, we demonstrate the ability of the proposed criteria to detect GME.
First, we show this on two sets of basic GME Gaussian states, including the CV
GHZ-like state. Then, we also present two methods allowing us to find
numerical examples of Gaussian states whose GME can be verified by our
criteria.

####  V.1.1 Analytical examples

Initially, we apply our criteria to the basic Gaussian states, which are
widely used for testing of GME criteria. This includes the family of three-
mode CV GHZ-like states with CM (8) and the split squeezed states [54] created
by splitting of a single squeezed state on two balanced beam splitters, which
possesses the following CM:

$\gamma^{\rm
sss}=\frac{1}{4}\left[\left(\begin{array}[]{ccc}a_{+}&b_{+}&-b_{+}\\\
b_{+}&c_{+}&d_{+}\\\
-b_{+}&d_{+}&c_{+}\end{array}\right)\oplus\left(\begin{array}[]{ccc}a_{-}&b_{-}&-b_{-}\\\
b_{-}&c_{-}&d_{-}\\\ -b_{-}&d_{-}&c_{-}\end{array}\right)\right],$ (57)

where

$\displaystyle a_{\pm}$ $\displaystyle=$ $\displaystyle 2(1+e^{\pm 2r}),\quad
b_{\pm}=\sqrt{2}(1-e^{\pm 2r}),$ $\displaystyle c_{\pm}$ $\displaystyle=$
$\displaystyle 3+e^{\pm 2r},\quad d_{\pm}=1-e^{\pm 2r}.$ (58)

Again, we minimized the difference $D_{P}$, Eq. (50), over the parameters
$\ell^{\alpha}_{ij}$. To speed up the numerical search we further imposed on
the parameters the following constraint $-1\leq\ell^{\alpha}_{ij}\leq 1$. The
corresponding minimal differences for the CMs (8) and (57) with squeezing
parameters $r_{i}=0.05\times i$, $i=0,1,\ldots,40$ are depicted in Fig. 5. The
figure as well as the numerics reveal that for the GHZ-like state except for
the case $r=0$ for all other considered values of the squeezing the minimized
difference is negative. Similarly, in the case of the split squeezed state the
minimized difference becomes negative as soon as the considered value of
squeezing exceeds the value of $r=0.6$. These two examples clearly demonstrate
the ability of the proposed product criterion (V.1) to detect GME of basic
three-mode Gaussian states.

Figure 5: Difference $D_{P}$, Eq. (50), minimized over the elements
$\ell^{\alpha}_{ij}\in[-1,1]$ for the CV GHZ-like state with CM (8) (red
circles) and the split squeezed state with CM (57) (blue diamonds) versus the
squeezing parameter $r$. See text for details.

Interestingly, the sum criterion (V.1) fails to detect GME of any of the
previous three-mode states detected by the product criterion. Nevertheless,
one can find examples of three-mode GME Gaussian states, which are detected by
both criteria or only by the sum criterion itself.

####  V.1.2 Numerical examples

We find examples of Gaussian states whose GME can be detected by the sum
criterion (V.1) using the Gaussian analog [35] of the iterative algorithm
[33]. Initially, a pure-state CM $\gamma^{(0)}$ is randomly generated. In the
first step of the algorithm an optimal witness $Z^{(1)}$ detecting GME from
minimal set of two-mode marginal CMs of CM $\gamma^{(0)}$ is found. The second
step consists of finding an optimal CM $\gamma^{(1)}$ minimizing
$\mbox{Tr}(\gamma Z^{(1)})$. In the next iteration the output CM
$\gamma^{(1)}$ is taken as an input to the first step which produces an
optimal witness $Z^{(2)}$. The new witness is then used in the second step to
get optimal CM $\gamma^{(2)}$ for the witness, etc. By performing few
iterations of the algorithm on a randomly generated seed CMs we produced one
hundred CMs of GME states, which were then tested by our product and sum GME
criteria (V.1) and (V.1). In total, our criteria were capable of detecting GME
for nearly 80% of the CMs. One of the obtained CMs is given by

$\gamma_{1}=\left(\begin{array}[]{ccc}6.3&-6.6&2.8\\\ -6.6&10&-5.6\\\
2.8&-5.6&5.4\\\ \end{array}\right)\oplus\left(\begin{array}[]{ccc}5.4&5.3&4\\\
5.3&5.5&4.2\\\ 4&4.2&3.5\\\ \end{array}\right),$ (59)

where the elements were rounded for brevity to one decimal place in which case
physicality of the rounded CM remained preserved. Minimization of the
differences $D_{P}$ and $D_{S}$ over the parameters
$\ell^{\alpha}_{ij}\in[-1,1]$ yields $D_{P}=-0.245$ and $D_{S}=-0.165$, where
the respective rounded optimal values of the parameters $\ell^{\alpha}_{ij}$
can be found in Appendix B. As both the differences are negative the GME of
the state with CM (59) is detected both by the product criterion (V.1), as
well as sum criterion (V.1).

Analogously as in the case of CM (59) we can find a CM of a Gaussian state
whose GME is detected only by the sum criterion. The CM in question has
integer entries and it is of the following form:

$\gamma_{2}=\left(\begin{array}[]{ccc}5&-5&-2\\\ -5&7&3\\\ -2&3&2\\\
\end{array}\right)\oplus\left(\begin{array}[]{ccc}3&4&-3\\\ 4&6&-5\\\
-3&-5&6\\\ \end{array}\right).$ (60)

Indeed, while the product criterion (V.1) yields a non-negative minimal
difference $D_{P}$ and thus it does not detect GME, for the sum criterion
(V.1) one gets $D_{S}=-0.070$, which clearly confirms the presence of GME in
the considered state. The rounded optimal values of the parameters
$\ell^{\alpha}_{ij}$ attaining the latter difference are given explicitly in
Appendix B.

####  V.1.3 Example derived from a guessed witness

There is yet another method of how one can find examples of three-mode
Gaussian states violating inequality (V.1). What is more, below we use the
method to find examples of GME states detected by the sum criteria for more
than three modes.

Typically, the inseparability criteria based on uncertainty relations are
tailored to the state whose entanglement is to be detect, which is know. Here,
on the other hand, we know the criteria and seek states, which are detected by
them. This task can be solved relatively easily if we set the parameters
$\ell^{\alpha}_{ij}$ to suitable values. Namely, by using Eq. (18) and
dividing the LHS of the criterion with the RHS, we can express the inequality
(48) in the form $\mbox{Tr}[\gamma Z]\geq 1$, where $Z=Z^{x}\oplus Z^{p}$ and
$\gamma=\gamma^{x}\oplus\gamma^{p}$. Here, $Z$ is a real $2N\times 2N$
symmetric positive-semidefinite matrix and provided that the parameters
$\ell^{\alpha}_{ij}$ are chosen properly, it can detect GME. In fact, the
matrix $Z$ is nothing but the already mentioned GME witness in the space of
covariance matrices [26].

In order to learn whether the guessed matrix $Z$ will detect GME, we have to
perform its symplectic diagonalization. Since typically $Z>0$, there is a
symplectic matrix $S$, i.e., a real $2N\times 2N$ matrix satisfying the
symplectic condition $S\Omega_{N}S^{T}=\Omega_{N}$, which transforms the
matrix as [55]

$\displaystyle
SZS^{T}=\mbox{diag}(\nu_{1},\nu_{2},\ldots,\nu_{N},\nu_{1},\nu_{2},\ldots\nu_{N})\equiv
Z_{W}.$

Here, $Z_{W}$ is a diagonal matrix, where the entries $\nu_{i}>0$,
$i=1,2,\ldots,N$, are the so-called symplectic eigenvalues. Now, if
$\mbox{Tr}[Z_{W}]<1$ there will be always a CM $\gamma$, for which
$\mbox{Tr}(\gamma Z)<1$, which evidences the presence of GME. To see this,
note that $\mbox{Tr}[Z_{W}]=\mbox{Tr}[SZS^{T}]=\mbox{Tr}[S^{T}SZ]$, so if we
set $\gamma=S^{T}S$, we have $\mbox{Tr}[\gamma Z]<1$ and thus the witness
matrix $Z$ detects GME carried by the CM $\gamma$.

The GME witness $Z$ can be obtained, e.g., by setting some parameters
$\ell^{\alpha}_{ij}$ equal to $+1$ or $-1$, often small parameters, like
$\ell^{\alpha}_{33}$, equal to $10^{-2}$, and by guessing the remaining free
parameters. For example, consider the values of the parameters
$\ell^{\alpha}_{ij}$ in Tab. 2.

Table 2: Guessed values of parameters $\ell^{\alpha}_{ij}$, $\alpha=x,p$, used to get a GME witness $Z_{3}$. $\ell^{x}_{11}$ | 8 |  | $\ell^{p}_{11}$ | 10
---|---|---|---|---
$\ell^{x}_{21}$ | -1 |  | $\ell^{p}_{21}$ | 2
$\ell^{x}_{22}$ | 1 |  | $\ell^{p}_{22}$ | 5
$\ell^{x}_{32}$ | -1 |  | $\ell^{p}_{32}$ | 1
$\ell^{x}_{33}$ | 0.01 |  | $\ell^{p}_{33}$ | 0.01

Then using Fig. 4 we get $\mathcal{K}^{(1)}\doteq 86$ and
$\mathcal{K}^{(3)}\doteq 84$. Hence, the witness matrix
$Z_{3}=[L^{x}(L^{x})^{\rm T}\oplus L^{p}(L^{p})^{\rm T}]/(2\mathcal{K}^{(3)})$
is

$\displaystyle Z_{3}=\frac{1}{168}\left[\left(\begin{array}[]{ccc}64&-8&0\\\
-8&2&-1\\\
0&-1&1.0001\end{array}\right)\oplus\left(\begin{array}[]{ccc}100&20&0\\\
20&29&5\\\ 0&5&1.0001\end{array}\right)\right].$ (68)

Next, we find numerically the matrix $S_{3}$ symplectically diagonalizing
$Z_{3}$, calculate the CM $\gamma_{3}=S_{3}^{T}S_{3}$, round it for brevity to
two decimal places and add to the rounded matrix $10^{-2}\times\openone$ to
get a physical CM (see Appendix C.1 for details of the derivation). Thus we
arrive at the CM

$\displaystyle\gamma_{3}=\left(\begin{array}[]{ccc}1.40&1.15&0.30\\\
1.15&7.64&2.48\\\ 0.30&2.48&1.49\\\
\end{array}\right)\oplus\left(\begin{array}[]{ccc}0.84&-0.15&0.09\\\
-0.15&0.33&-0.50\\\ 0.09&-0.50&1.49\\\ \end{array}\right).$ (76)

To be able to compare the strength of violation of the criterion (48) by CM
$\gamma_{3}$ with previous examples, we again optimize the difference (51)
over $\ell^{\alpha}_{ij}\in[-1,1]$, which yields $D_{S}=-0.15$. This once
again confirms that the proposed sum criterion can really detect GME from
minimal set of two-mode marginals.

In the next section we use the latter method to derive Gaussian GME states of
$N=4,5$ and $6$ modes, which are detected by the sum criteria (48)
corresponding to different trees.

###  V.2 Multi-mode states

It remains to clarify whether the proposed criteria are also capable to
certify GME in states with more than three modes. Below we use the method of
previous section to construct examples of Gaussian states of up to six modes,
whose GME is detected by our criteria. As the method is based on the sum
criterion (48), the obtained examples violate the sum criterion pronouncedly
stronger than the product criterion which often fails to detect the GME of the
constructed state. For this reason, the following discussion is restricted to
the sum criterion (48).

Initially, we apply the criterion to the four-mode CV GHZ-like state [31].
However, for $r\in(0,2]$, and both configurations $4a$ and $4b$ (second and
third row of Tab. 3), we did not find any set of parameters
$\ell^{\alpha}_{ij}$, for which the sum criterion (48) (as well as the product
criterion (45)) would be violated. This indicates that the proposed criteria
are not suitable for detection of the GME carried by multi-mode GHZ-like
states. This can be attributed to the specific form of GME carried by the GHZ-
like states, which manifests itself, as far as the momentum quadratures is
concerned, by squeezing in the global quadrature combination
$\sum_{i=1}^{4}p_{i}$, whereas the proposed criteria contain only selected
two-mode momentum quadrature combinations which are for the studied GHZ-like
state contaminated by noise.

Nevertheless, using the method of previous section we found examples of GME
states of $N=4,5$ and $6$ modes detectable by the sum criterion for almost all
configurations. The exception being criteria corresponding to the five-mode
and six-mode linear tree and the six-mode star tree, for which we so far did
not succeed in guessing the GME witness $Z$. The best results obtained by us
are summarized in Tab. 3. The table contains all non-isomorphic trees with up
to six vertices, which are rooted and labelled such that the respective sum
criterion contains variances of at most two-mode quadrature combinations. The
values of the corresponding difference $D_{S}$, Eq. (51), are for given $N$
ordered decreasingly.

Table 3: All non-isomorphic trees with $N=3,4,5$ and $6$ vertices. The root of
the tree is depicted by a red circle. The linear trees are labeled according
to the standard labeling, whereas the other trees are covered by the reverse
level order labeling, which lead to the GME criteria involving variances of at
most two-mode quadrature combinations. The last column contains the difference
$D_{S}$, Eq. (51), for the displayed trees and CMs constructed by means of the
method of Subsec. V.1.3. The values of $D_{S}$ are calculated for the values
of parameters $\ell^{\alpha}_{ij}$ found by a numerical minimization over
$\ell^{\alpha}_{ij}\in[-10,10]$. The cases for which we did not found an
example are denoted by a cross. Explicit form of all the CMs as well as other
details of derivation of $D_{S}$ can be found in Appendix C. See text for
details.

$N$ | Index | Labelled tree | Difference $D_{S}$
---|---|---|---
3 | $3$ |  123 | $-17.87$
4 | $4a$ |  1 2 3 4 | $-20.31$
| $4b$ |  3412 | $-5.36$
5 | $5a$ |  12345 | $\bigtimes$
| $5b$ |  52431 | $-3.64$
| $5c$ |  12543 | $-2.30$
6 | $6a$ |  1 2 3 4 5 6 | $\bigtimes$
| $6b$ |  632154 | $\bigtimes$
| $6c$ |  2 5 6 3 1 4 | $-1.00$
| $6d$ |  3 5 6 4 1 2 | $-0.91$
| $6e$ |  5 6 1 2 4 3 | $-0.65$
| $6f$ |  5 6 4 2 3 1 | $-0.13$

Since optimization over $\ell^{\alpha}_{ij}\in[-1,1]$ yields typically for
$N>3$ small values of $D_{S}$, the values of the difference in Tab. 3 were
obtained for optimization over an extended interval
$\ell^{\alpha}_{ij}\in[-10,10]$. This is why the value of $D_{S}$ for the
three-mode case is roughly hundred-times larger compared to the examples
presented in previous section. The Tab. 3 indicates that up to the
configurations $3$ and $4a$ the value of $D_{S}$ decreases with increasing
$N$, which is a typical behaviour of GME states (cf. Tab. I of Ref. [33]). The
exception is likely to be attributed to the fact that in the case $3$ we
guessed a little worse witness than for configuration $4a$. All the guessed
witnesses, the CMs of the states, and other details of derivation of the
results in Tab. 3 are given explicitly in Appendix C.

We accomplished the present analysis by an independent check of the presence
of GME verifiable only from the minimal set of two-mode marginal CMs in the
Gaussian states used in Tab. 3. For this purpose, we used the already
mentioned method of entanglement witnesses in the space of CMs [26], which
detect the GME from the minimal set of two-mode marginal CMs [35]. More
precisely, for each CM and all considered configurations we found numerically
such a witness, which independently confirms the presence of the GME
detectable only from the respective minimal set of marginals in the
constructed examples.

##  VI Conclusions

In this paper we derived a set of product and sum criteria for GME of
continuous-variable systems. The criteria are based on the second moments of
quadrature operators and unlike the overwhelming majority of the known GME
criteria contain only the variances of at most two-mode quadrature
combinations. Moreover, the presented criteria are also minimal in the sense
that they contain only the minimum number of the latter variances. In other
words, the criteria require only the knowledge of those elements of the CM of
the investigated state, which belong to the minimal set of two-mode marginal
CMs. Since our criteria do not take into account the $x-p$ correlations, the
number of the elements is $4N-2$ and thus it scales only linearly with the
number of modes. This distinguishes our criteria from other GME criteria which
always utilize a strictly larger number of elements of the CM, which scales
quadratically with the number of modes. Since any minimal set of marginal CMs
can be represented by a special kind of a graph known as a tree, each
criterion is intimately associated to a particular tree. This allowed us to
develop a method of construction of all the sum criteria directly from the
underlying tree.

Further, we have shown that the product criterion detects GME of the three-
mode CV GHZ-like state and the split squeezed state, but neither of the states
is detected by the sum criterion. We have also constructed numerical examples
of three-mode Gaussian GME states which are simultaneously detected by both
the criteria as well as only by the sum criterion. To get examples of GME
states detected by our criteria for more than three modes, we developed a
method based on symplectic diagonalization of a guessed witness in the space
of CMs. Thus we found states detected by the sum criterion for nearly all
configurations of up to six modes.

Interestingly, our criteria do not detect the GME in three- and four-mode GME
states with all two-mode marginals separable, which were recently found in
Ref. [35]. Therefore, we tested the criteria for another more than hundred
numerically generated GME states with separable marginals and we always got a
negative result. This rises a question of how to modify the present criteria
so as to make them capable of detecting the GME from separable marginals,
which is deferred for further research. Likewise, inspired by the existence of
the CV GME criteria for higher-order moments [56, 57] we could try to
generalize the proposed criteria to the higher-order moments, which is thus
another open problem. Finally, one may ask whether the present algorithm can
be used to derive analogous minimal criteria for genuine multipartite quantum
steering [58, 54]. Since the steering criteria are typically obtained only by
small modifications of derivation of the standard GME criteria [29, 20], we
expect that the same approach will be viable also in the case of our criteria,
which will be addressed elsewhere.

The present paper provides a rigorous method of construction of minimal
criteria of GME. The method relies on the concept of witness matrix in the
space of CMs and graph theory, and gives a recipe of how to systematically
construct similar minimal criteria for other types of multipartite quantum
correlations. The GME criteria presented here are also experimentally friendly
as they are simple and immediately reveal how to measure them. What is more,
the criteria do not require to measure entire CM but only its strictly smaller
part which is needed for detection of GME. This feature provides an advantage
compared to other GME criteria by saving the number of measurements needed for
testing of GME of the investigated state. We believe that our results will
stimulate theoretical research of minimal criteria of multipartite quantum
properties and find applications in characterization of the properties in
experimentally prepared quantum states.

###### Acknowledgements.

O.L. acknowledges support from IGA-PrF-2022-005 and the European Union’s 2020
research and innovation program (CSA - Coordination and support action,
H2020-WIDESPREAD-2020-5) under grant agreement No. 951737 (NONGAUSS). L.M.
acknowledges the project 8C22002 (CVStar) of MEYS of the Czech Republic, which
has received funding from the European Union’s Horizon 2020 Research and
Innovation Programme under Grant Agreement no. 731473 and 101017733.

## Author contribution

Both authors contributed equally to the results and the manuscript writing.

##  Appendix A Fully inseparable biseparable Gaussian state

This section is dedicated to the proof that the three-mode biseparable state
(11) with CM (12) is in a certain region of squeezing parameters fully
inseparable, i.e., it is entangled across all three bipartite splits
$1|23,2|31$ and $3|12$. The state possesses a fully symmetric CM of the form

$\gamma^{\rm test}=\begin{pmatrix}b&e&e\\\ e&b&e\\\
e&e&b\end{pmatrix}\oplus\begin{pmatrix}b&-e&-e\\\ -e&b&-e\\\
-e&-e&b\end{pmatrix},$ (78)

where

$b=\frac{2\cosh(2r)+1}{3},\quad e=\frac{\sinh(2r)}{3}.$ (79)

Due to the symmetry it is sufficient to verify entanglement of the state with
respect to just one split, e.g., $3|12$, which we will do using the results of
Ref. [59, 60]. Interference of modes $1$ and $2$ on a balanced beam splitter
does not change entanglement properties across the split and decouples mode
$2$ from modes $1$ and $3$. Hence, we are left with the CM

$\gamma^{\rm test}_{13}=\begin{pmatrix}b+e&\sqrt{2}e\\\
\sqrt{2}e&b\end{pmatrix}\oplus\begin{pmatrix}b-e&-\sqrt{2}e\\\
-\sqrt{2}e&b\end{pmatrix}$ (80)

of modes $1$ and $3$. Entanglement of the state with the latter CM then can be
proved easily by using positive partial transposition criterion [41] expressed
in terms of the symplectic eigenvalues [60]. The partial transposition
operation with respect to mode $3$ transforms the CM (80) as
$\gamma_{13}^{{\rm test}\,(T_{3})}=(\openone\oplus\sigma_{z})\gamma_{13}^{\rm
test}(\openone\oplus\sigma_{z})$ and its symplectic eigenvalues $\mu_{\pm}$
are obtained from the spectrum $\\{\pm i\mu_{+},\pm i\mu_{-}\\}$ of the matrix
$\Omega_{2}\gamma_{13}^{{\rm test}\,(T_{3})}$ [60] in the form

$\mu_{\pm}=\sqrt{\frac{\alpha\pm\sqrt{\alpha^{2}-4\beta}}{2}},$ (81)

where $\alpha=2b^{2}+3e^{2}$ and $\beta=(b^{2}-2e^{2})^{2}-(be)^{2}$.
According to the partial transposition criterion the state with CM (80) is
entangled if the minimal symplectic eigenvalue $\mu_{-}$ satisfies
$\mu_{-}<1$. Thus we are looking for an interval of squeezing parameters
$r\geq 0$, for which $\mu_{-}<1$. In Fig. 6 we plot the dependence of the
symplectic eigenvalue $\mu_{-}$ on the squeezing parameter $r$.

Figure 6: Lower symplectic eigenvalue $\mu_{-}$, Eq. (81), versus the
squeezing parameter $r$ for the state with CM $\gamma^{\rm test}_{13}$, Eq.
(80).

The figure shows that as $r$ increases from $r=0$ when $\mu_{-}=1$ the
symplectic eigenvalue satisfies $\mu_{-}<1$ for up to $r=r_{\rm max}$ when
$\mu_{-}=1$ and then $\mu_{-}>1$ with further increase of $r$. The threshold
value of $r_{\rm max}$ can be calculated from the condition $\mu_{-}=1$, which
simplifies to $\alpha-\beta=1$ and leads further to the following equation:

$3e^{12r}-34e^{10r}-35e^{8r}+132e^{6r}-35e^{4r}-34e^{2r}+3=0.$ (82)

Upon solving the latter equation numerically one finally finds $r_{\rm
max}\doteq 1.24$ and thus the state with CM (80) is entangled for
$r\in(0,1.24)$. As a consequence, the state with CM (78) is fully inseparable
for $r\in(0,1.24)$.

##  Appendix B Optimal parameters

This section contains optimal values of the parameters $\ell^{\alpha}_{ij}$
minimizing the differences (50) and (51) between the LHS and the RHS of the
proposed minimal GME criteria.

For the three-mode CM (59) and the product criterion (V.1) the optimal
parameters $\ell^{\alpha}_{ij}$ are summarized in Tab. 4

Table 4: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-1,1]$, $\alpha=x,p$, used to calculate the difference $D_{P}=-0.245$ for the product criterion (V.1) and the CM $\gamma_{1}$, Eq. (59). $\ell^{x}_{11}$ | 0.947 |  | $\ell^{p}_{11}$ | 0.989
---|---|---|---|---
$\ell^{x}_{21}$ | 0.760 |  | $\ell^{p}_{21}$ | -0.987
$\ell^{x}_{22}$ | 0.860 |  | $\ell^{p}_{22}$ | 0.817
$\ell^{x}_{32}$ | 1 |  | $\ell^{p}_{32}$ | -1
$\ell^{x}_{33}$ | $1\times 10^{-5}$ |  | $\ell^{p}_{33}$ | $1.8\times 10^{-6}$

In the case of CM (59) and the sum criterion (V.1) the optimal parameters
$\ell^{\alpha}_{ij}$ are summarized in Tab. 5

Table 5: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-1,1]$, $\alpha=x,p$, used to calculate the difference $D_{S}=-0.165$ for the sum criterion (V.1) and the CM $\gamma_{1}$, Eq. (59). $\ell^{x}_{11}$ | 0.425 |  | $\ell^{p}_{11}$ | 1
---|---|---|---|---
$\ell^{x}_{21}$ | 0.253 |  | $\ell^{p}_{21}$ | -0.954
$\ell^{x}_{22}$ | 0.305 |  | $\ell^{p}_{22}$ | 0.792
$\ell^{x}_{32}$ | 0.505 |  | $\ell^{p}_{32}$ | -1
$\ell^{x}_{33}$ | $2\times 10^{-4}$ |  | $\ell^{p}_{33}$ | $4\times 10^{-4}$

Optimal parameters $\ell^{\alpha}_{ij}$ for the CM (60) and the sum criterion
(V.1) are summarized in Tab. 6

Table 6: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-1,1]$, $\alpha=x,p$, used to calculate the difference $D_{S}=-0.070$ for the sum criterion (V.1) and the CM $\gamma_{2}$, Eq. (60). $\ell^{x}_{11}$ | 0.514 |  | $\ell^{p}_{11}$ | 1
---|---|---|---|---
$\ell^{x}_{21}$ | 0.314 |  | $\ell^{p}_{21}$ | -0.652
$\ell^{x}_{22}$ | 0.419 |  | $\ell^{p}_{22}$ | 0.489
$\ell^{x}_{32}$ | -0.908 |  | $\ell^{p}_{32}$ | 0.559
$\ell^{x}_{33}$ | $3\times 10^{-5}$ |  | $\ell^{p}_{33}$ | $8.7\times 10^{-7}$

##  Appendix C Derivation of examples from a guessed witness

In this section we present a complete derivation of Gaussian states of $N\geq
3$ modes violating minimal GME criteria (45) and (48), based on a guessed
witness in the space of CMs.

###  C.1 N=3

We start with the guessed three-mode witness matrix $Z_{3}$, Eq. (68). Making
use of the method [61], we can find numerically the matrix $S_{3}$
symplectically diagonalizing $Z_{3}$ in the form

$S_{3}=\begin{pmatrix}\mathbb{O}_{3}&X_{3}\\\
Y_{3}&\mathbb{O}_{3}\end{pmatrix},$ (83)

where

$\displaystyle X_{3}=\left(\begin{array}[]{ccc}-0.0445813&0.222904&-1.11455\\\
0.0879785&-0.503613&0.492602\\\
0.904559&-0.110546&-0.00252406\end{array}\right)$ (87)

and

$\displaystyle Y_{3}=\left(\begin{array}[]{ccc}-0.139315&-1.11452&-1.11455\\\
-0.309427&-2.52072&-0.491755\\\
1.12874&0.190238&-0.00710211\end{array}\right).$ (92)

From the symplectic matrix (83) we then get the following CM:

$\displaystyle\gamma_{3}^{\prime}=S_{3}^{T}S_{3}=\left(\begin{array}[]{ccc}1.38922&1.14998&0.299419\\\
1.14998&7.63236&2.48042\\\
0.299419&2.48042&1.4841\end{array}\right)\oplus\left(\begin{array}[]{ccc}0.827955&-0.15424&0.0907433\\\
-0.15424&0.315532&-0.49624\\\ 0.0907433&-0.49624&1.48488\end{array}\right).$
(100)

Next, for simplicity we round all elements of the latter CM to two decimal
places thereby getting the matrix $\bar{\gamma}_{3}$, which slightly violates
the Heisenberg uncertainty principle $\gamma+i\Omega_{3}\geq 0$ [41], Eq. (6)
with $N=3$ of the main text. Therefore, we have added to the CM a small amount
of thermal noise thereby creating a physical CM
$\gamma_{3}=\bar{\gamma}_{3}+0.01\times\openone$ given explicitly in Eq. (76).
Now, if we optimize numerically the difference (51) for the sum criterion
(V.1) over the parameters $\ell^{\alpha}_{ij}\in[-1,1]$, we arrive at
$D_{S}=-0.15$ and the optimal parameters are listed in Tab. 7.

Table 7: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-1,1]$, $\alpha=x,p$, used to calculate the difference (51) for the sum criterion (V.1) and the CM $\gamma_{3}$, Eq. (76). $\ell^{x}_{11}$ | 0.90 |  | $\ell^{p}_{11}$ | 1
---|---|---|---|---
$\ell^{x}_{21}$ | -0.23 |  | $\ell^{p}_{21}$ | 0.98
$\ell^{x}_{22}$ | 0.23 |  | $\ell^{p}_{22}$ | 1
$\ell^{x}_{32}$ | -1 |  | $\ell^{p}_{32}$ | 1
$\ell^{x}_{33}$ | 0.01 |  | $\ell^{p}_{33}$ | 0.01

The difference (51) in the first row of Tab. 3 of the main text has been
calculated for the unrounded CM $\gamma_{3}^{\prime}$, Eq. (100). If we
optimize numerically the difference (51) for the sum criterion (V.1) over the
parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at $D_{S}=-17.87$ and
the optimal parameters are listed in Tab. 8.

Table 8: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, $\alpha=x,p$, used to calculate the difference $D_{S}$, Eq. (51), given in the first row of Tab. 3 for the CM $\gamma_{3}^{\prime}$, Eq. (100). $\ell^{x}_{11}$ | 9.11 |  | $\ell^{p}_{11}$ | 10
---|---|---|---|---
$\ell^{x}_{21}$ | -2.31 |  | $\ell^{p}_{21}$ | 10
$\ell^{x}_{22}$ | 2.31 |  | $\ell^{p}_{22}$ | 10
$\ell^{x}_{32}$ | -10 |  | $\ell^{p}_{32}$ | 10
$\ell^{x}_{33}$ | 0.01 |  | $\ell^{p}_{33}$ | 0.01

###  C.2 N=4

####  C.2.1 Linear tree 4a

We start with the guessed four-mode witness matrix $Z_{4a}$ of the linear tree
$4a$,

$Z_{4a}=\frac{1}{5.5002}\left(\begin{array}[]{cc}Z_{4a}^{x}&\mathbb{0}_{4}\\\
\mathbb{0}_{4}&Z_{4a}^{p}\end{array}\right),$ (102)

where

$Z_{4a}^{x}=\left(\begin{array}[]{cccc}1&-0.5&0&0\\\ -0.5&0.5&-0.5&0\\\
0&-0.5&1.25&-0.5\\\ 0&0&-0.5&1.0001\\\ \end{array}\right)\\\ $ (103)

and

$Z_{4a}^{p}=\left(\begin{array}[]{cccc}1&0.5&0&0\\\ 0.5&0.5&0.5&0\\\
0&0.5&1.25&0.5\\\ 0&0&0.5&1.0001\\\ \end{array}\right).\\\ $ (104)

Next, we find numerically the matrix $S_{4a}$ symplectically diagonalizing
$Z_{4a}$ in the form:

$S_{4a}=\begin{pmatrix}\mathbb{O}_{4}&X_{4a}\\\
Y_{4a}&\mathbb{O}_{4}\end{pmatrix},$ (105)

where

$\displaystyle
X_{4a}=\left(\begin{array}[]{cccc}-0.666646&1.33331&-0.666635&0.333288\\\
-1.05853&0.895417&-0.870291&0.275906\\\
0.0468749&-0.178567&-0.191418&1.00347\\\
0.571074&0.0886146&-0.850668&0.204715\\\ \end{array}\right)$ (110)

and

$\displaystyle
Y_{4a}=\left(\begin{array}[]{cccc}-0.666646&-1.33331&-0.666635&-0.333288\\\
1.05853&0.895417&0.870291&0.275906\\\ 0.0468749&0.178567&-0.191418&-1.00347\\\
-0.571074&0.0886146&0.850668&0.204715\\\ \end{array}\right).$ (116)

From the symplectic matrix (105) we then get the following CM:

$\displaystyle\gamma_{4a}=S_{4a}^{T}S_{4a}=\left(\begin{array}[]{cccc}1.89322&1.79444&0.87087&0.350295\\\
1.79444&2.61924&1.70931&0.530383\\\ 0.87087&1.70931&1.96209&0.828527\\\
0.350295&0.530383&0.828527&1.23606\\\
\end{array}\right)\oplus\left(\begin{array}[]{cccc}1.89322&-1.79444&0.87087&-0.350295\\\
-1.79444&2.61924&-1.70931&0.530383\\\ 0.87087&-1.70931&1.96209&-0.828527\\\
-0.350295&0.530383&-0.828527&1.23606\\\ \end{array}\right).$ (126)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-20.31$ and the optimal parameters are listed in Tab. 9.

Table 9: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, $\alpha=x,p$, used to calculate the difference (51) for the sum criterion (48) and the CM $\gamma_{4a}$, Eq. (126). $\ell^{x}_{11}$ | 10 |  | $\ell^{p}_{11}$ | 10
---|---|---|---|---
$\ell^{x}_{21}$ | -5.08 |  | $\ell^{p}_{21}$ | 5.08
$\ell^{x}_{22}$ | 5.09 |  | $\ell^{p}_{22}$ | 5.08
$\ell^{x}_{32}$ | -9.03 |  | $\ell^{p}_{32}$ | -9.03
$\ell^{x}_{33}$ | 5.07 |  | $\ell^{p}_{33}$ | 5.10
$\ell^{x}_{43}$ | -10 |  | $\ell^{p}_{43}$ | 10
$\ell^{x}_{44}$ | 0.01 |  | $\ell^{p}_{44}$ | 0.01

####  C.2.2 Star tree 4b

We guess the four-mode witness matrix $Z_{4b}$ corresponding to the star tree
$4b$ in the following form:

$Z_{4b}=\frac{1}{68.02}\left(\begin{array}[]{cc}Z_{4b}^{x}&\mathbb{0}_{4}\\\
\mathbb{0}_{4}&Z_{4b}^{p}\end{array}\right),$ (128)

where

$Z_{4b}^{x}=\left(\begin{array}[]{cccc}4&0&0&-2\\\ 0&1&0&-1\\\ 0&0&4&-2\\\
-2&-1&-2&3.01\\\ \end{array}\right)\\\ $ (129)

and

$Z_{4b}^{p}=\left(\begin{array}[]{cccc}100&0&0&10\\\ 0&25&0&5\\\ 0&0&25&5\\\
10&5&5&3.01\\\ \end{array}\right).\\\ $ (130)

The matrix $S_{4b}$ symplectically diagonalizing $Z_{4b}$ is of the form:

$S_{4b}=\begin{pmatrix}V_{4b}&X_{4b}\\\ Y_{4b}&W_{4b}\end{pmatrix},$ (131)

where

$\displaystyle V_{4b}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0.258041&-0.339431&1.61423&0.280456\\\ 0&0&0&0\\\ \end{array}\right),$ (136)
$\displaystyle
X_{4b}=\left(\begin{array}[]{cccc}-0.123786&-0.247003&-0.247382&1.23881\\\
-0.0795257&-0.506179&-0.206184&0.647292\\\ 0&0&0&0\\\
0.454241&-0.0135947&-0.0420425&-0.192404\\\ \end{array}\right),$ (142)
$\displaystyle
Y_{4b}=\left(\begin{array}[]{cccc}0.61893&1.23502&0.618456&1.23881\\\
-0.397629&-2.53089&-0.515461&-0.647292\\\ 0&0&0&0\\\
2.27121&-0.0679731&-0.105106&0.192405\\\ \end{array}\right),$ (148)

and

$\displaystyle W_{4b}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0.051608&-0.067886&0.645693&-0.280457\\\ 0&0&0&0\\\ \end{array}\right).$ (154)

From the symplectic matrix (131) we then get the following CM:

$\displaystyle\gamma_{4b}=S_{4b}^{T}S_{4b}=\left(\begin{array}[]{cccc}5.76616&1.52878&0.765563&1.53348\\\
1.52878&8.05052&1.5276&3.0599\\\ 0.765563&1.5276&3.26498&1.5323\\\
1.53348&3.0599&1.5323&2.06932\\\
\end{array}\right)\oplus\left(\begin{array}[]{cccc}0.230645&0.061151&0.0612449&-0.306695\\\
0.061151&0.322021&0.122208&-0.611981\\\
0.0612449&0.122208&0.522397&-0.612921\\\
-0.306695&-0.611981&-0.612921&2.06932\\\ \end{array}\right).$ (164)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-5.36$ and the optimal parameters are listed in Tab. 9.

Table 10: Rounded values of the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, $\alpha=x,p$, used to calculate the difference (51) for the sum criterion (48) and the CM $\gamma_{4b}$, Eq. (164). $\ell^{x}_{11}$ | 2.35 |  | $\ell^{p}_{11}$ | 10
---|---|---|---|---
$\ell^{x}_{41}$ | -2.27 |  | $\ell^{p}_{41}$ | 2.13
$\ell^{x}_{22}$ | 2.17 |  | $\ell^{p}_{22}$ | 10
$\ell^{x}_{42}$ | -2.42 |  | $\ell^{p}_{42}$ | 2.00
$\ell^{x}_{33}$ | 4.09 |  | $\ell^{p}_{33}$ | 9.99
$\ell^{x}_{43}$ | -2.28 |  | $\ell^{p}_{43}$ | 2.13
$\ell^{x}_{44}$ | 0.12 |  | $\ell^{p}_{44}$ | 0.09

###  C.3 N=5

####  C.3.1 Star tree 5b

We guess the five-mode witness matrix $Z_{5b}$ corresponding to the star tree
$5b$,

$Z_{5b}=\frac{1}{2.4801}\left(\begin{array}[]{cc}Z_{5b}^{x}&\mathbb{0}_{5}\\\
\mathbb{0}_{5}&Z_{5b}^{p}\end{array}\right),$ (166)

where

$Z_{5b}^{x}=\left(\begin{array}[]{ccccc}1&0&0&0&-0.1\\\ 0&0.25&0&0&-0.05\\\
0&0&1&0&-0.1\\\ 0&0&0&0.25&-0.05\\\ -0.1&-0.05&-0.1&-0.05&0.0401\\\
\end{array}\right)\\\ $ (167)

and

$\displaystyle Z_{5b}^{p}=\left(\begin{array}[]{ccccc}1&0&0&0&0.1\\\
0&0.25&0&0&0.05\\\ 0&0&1&0&0.1\\\ 0&0&0&0.25&0.05\\\
0.1&0.05&0.1&0.05&0.0401\\\ \end{array}\right).$

By finding numerically the matrix $S_{5b}$ symplectically diagonalizing
$Z_{5b}$, we then get the following CM:

$\gamma_{5b}=S_{5b}^{T}S_{5b}=\left(\begin{array}[]{cc}\gamma_{5b}^{x}&\mathbb{0}_{5}\\\
\mathbb{0}_{5}&\gamma_{5b}^{p}\end{array}\right),$ (169)

where

$\displaystyle\gamma_{5b}^{x}=\left(\begin{array}[]{ccccc}1.02221&0.0444158&0.0222153&0.0444158&0.222178\\\
0.0444158&1.0888&0.0444158&0.0888027&0.444208\\\
0.0222153&0.0444158&1.02221&0.0444158&0.222178\\\
0.0444158&0.0888027&0.0444158&1.0888&0.444208\\\
0.222178&0.444208&0.222178&0.444208&1.22203\\\ \end{array}\right),$ (175)
$\displaystyle\gamma_{5b}^{p}=\left(\begin{array}[]{ccccc}1.02221&0.0444158&0.0222153&0.0444158&-0.222178\\\
0.0444158&1.0888&0.0444158&0.0888027&-0.444208\\\
0.0222153&0.0444158&1.02221&0.0444158&-0.222178\\\
0.0444158&0.0888027&0.0444158&1.0888&-0.444208\\\
-0.222178&-0.444208&-0.222178&-0.444208&1.22203\\\ \end{array}\right).$ (181)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-3.64$.

####  C.3.2 Tree 5c

We start with the guessed five-mode witness matrix $Z_{5c}$ corresponding to
the tree $5c$,

$Z_{5c}=\frac{1}{2.4801}\left(\begin{array}[]{cc}Z_{5c}^{x}&\mathbb{0}_{5}\\\
\mathbb{0}_{5}&Z_{5c}^{p}\end{array}\right),$ (182)

where

$Z_{5c}^{x}=\left(\begin{array}[]{ccccc}1&-0.1&0&0&0\\\ -0.1&0.26&0&0&-0.05\\\
0&0&1&0&-0.1\\\ 0&0&0&0.25&-0.05\\\ 0&-0.05&-0.1&-0.05&0.0301\\\
\end{array}\right)\\\ $ (183)

and

$Z_{5c}^{p}=\left(\begin{array}[]{ccccc}1&0.1&0&0&0\\\ 0.1&0.26&0&0&0.05\\\
0&0&1&0&0.1\\\ 0&0&0&0.25&0.05\\\ 0&0.05&0.1&0.05&0.0301\\\
\end{array}\right).\\\ $ (184)

Next, we find numerically the matrix $S_{5c}$ symplectically diagonalizing
$Z_{5c}$ and from that symplectic matrix we then get the following CM:

$\gamma_{5c}=S_{5c}^{T}S_{5c}=\left(\begin{array}[]{cc}\gamma_{5c}^{x}&\mathbb{0}_{5}\\\
\mathbb{0}_{5}&\gamma_{5c}^{p}\end{array}\right),$ (185)

where

$\displaystyle\gamma_{5c}^{x}=\left(\begin{array}[]{ccccc}1.01364&0.16921&0.00397589&0.00844686&0.035641\\\
0.16921&1.10058&0.0438767&0.0877648&0.438485\\\
0.00397589&0.0438767&1.02196&0.0439097&0.219645\\\
0.00844686&0.0877648&0.0439097&1.08779&0.439145\\\
0.035641&0.438485&0.219645&0.439145&1.19671\\\ \end{array}\right),$ (191)
$\displaystyle\gamma_{5c}^{p}=\left(\begin{array}[]{ccccc}1.01364&-0.16921&-0.00397589&-0.00844686&0.035641\\\
-0.16921&1.10058&0.0438767&0.0877648&-0.438485\\\
-0.00397589&0.0438767&1.02196&0.0439097&-0.219645\\\
-0.00844686&0.0877648&0.0439097&1.08779&-0.439145\\\
0.035641&-0.438485&-0.219645&-0.439145&1.19671\\\ \end{array}\right).$ (197)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-2.30$.

###  C.4 N=6

####  C.4.1 Tree 6c

The guessed six-mode witness matrix $Z_{6c}$ for the tree $6c$ is of the form:

$Z_{6c}=\frac{1}{3.2301}\left(\begin{array}[]{cc}Z_{6c}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&Z_{6c}^{p}\end{array}\right),$ (198)

where

$\displaystyle Z_{6c}^{x}=\left(\begin{array}[]{cccccc}1&0&-0.1&0&0&0\\\
0&1&0&0&-0.1&0\\\ -0.1&0&1.01&0&0&-0.1\\\ 0&0&0&0.25&0&-0.05\\\
0&-0.1&0&0&0.02&-0.01\\\ 0&0&-0.1&-0.05&-0.01&0.0301\\\ \end{array}\right)$

and

$\displaystyle Z_{6c}^{p}=\left(\begin{array}[]{cccccc}1&0&0.1&0&0&0\\\
0&1&0&0&0.1&0\\\ 0.1&0&1.01&0&0&0.1\\\ 0&0&0&0.25&0&0.05\\\
0&0.1&0&0&0.02&0.01\\\ 0&0&0.1&0.05&0.01&0.0301\\\ \end{array}\right).$

From the numerically calculated matrix $S_{6c}$ symplectically diagonalizing
$Z_{6c}$ we then get the following CM:

$\gamma_{6c}=S_{6c}^{T}S_{6c}=\left(\begin{array}[]{cc}\gamma_{6c}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&\gamma_{6c}^{p}\end{array}\right),$ (201)

where

$\displaystyle\gamma_{6c}^{x}=\left(\begin{array}[]{cccccc}1.00601&0.00996299&0.110751&0.0220412&0.0996522&0.102581\\\
0.00996299&1.1082&0.0996528&0.198639&1.0828&0.99682\\\
0.110751&0.0996528&1.11751&0.224225&0.996235&1.12667\\\
0.0220412&0.198639&0.224225&1.44671&1.98457&2.2454\\\
0.0996522&1.0828&0.996235&1.98457&9.83666&9.97538\\\
0.102581&0.99682&1.12667&2.2454&9.97538&10.2866\\\ \end{array}\right),$ (208)
$\displaystyle\gamma_{6c}^{p}=\left(\begin{array}[]{cccccc}1.00601&0.00996299&-0.110751&-0.0220412&-0.0996522&0.102581\\\
0.00996299&1.1082&-0.0996528&-0.198639&-1.0828&0.99682\\\
-0.110751&-0.0996528&1.11751&0.224225&0.996235&-1.12667\\\
-0.0220412&-0.198639&0.224225&1.44671&1.98457&-2.2454\\\
-0.0996522&-1.0828&0.996235&1.98457&9.83666&-9.97538\\\
0.102581&0.99682&-1.12667&-2.2454&-9.97538&10.2866\\\ \end{array}\right).$
(215)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-1.00$.

####  C.4.2 Tree 6d

For the tree $6d$ we guessed the witness matrix $Z_{6d}$ as follows:

$Z_{6d}=\frac{1}{2.7301}\left(\begin{array}[]{cc}Z_{6d}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&Z_{6d}^{p}\end{array}\right),$ (216)

where

$Z_{6d}^{x}=\left(\begin{array}[]{cccccc}1&0&0&-0.1&0&0\\\ 0&1&0&-0.1&0&0\\\
0&0&1&0&-0.1&0\\\ -0.1&-0.1&0&0.27&0&-0.05\\\ 0&0&-0.1&0&0.02&-0.01\\\
0&0&0&-0.05&-0.01&0.0201\\\ \end{array}\right)\\\ $ (217)

and

$Z_{6d}^{p}=\left(\begin{array}[]{cccccc}1&0&0&0.1&0&0\\\ 0&0.25&0&0.05&0&0\\\
0&0&1&0&0.1&0\\\ 0.1&0.05&0&0.27&0&0.05\\\ 0&0&0.1&0&0.02&0.01\\\
0&0&0&0.05&0.01&0.0201\\\ \end{array}\right).\\\ $ (218)

In the next step, we find numerically the matrix $S_{6d}$ symplectically
diagonalizing $Z_{6d}$, which yields

$\gamma_{6d}=S_{6d}^{T}S_{6d}=\left(\begin{array}[]{cc}\gamma_{6d}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&\gamma_{6d}^{p}\end{array}\right),$ (219)

where

$\displaystyle\gamma_{6d}^{x}=\left(\begin{array}[]{cccccc}1.01727&0.0150336&0.0200197&0.206136&0.200098&0.21641\\\
0.0150336&0.513163&0.0200209&0.178237&0.200154&0.211026\\\
0.0200197&0.0200209&1.10944&0.200099&1.09521&1.00374\\\
0.206136&0.178237&0.200099&1.47744&1.99935&2.24369\\\
0.200098&0.200154&1.09521&1.99935&9.96135&10.0452\\\
0.21641&0.211026&1.00374&2.24369&10.0452&10.2858\\\ \end{array}\right),$ (226)
$\displaystyle\gamma_{6d}^{p}=\left(\begin{array}[]{cccccc}1.01727&0.0300672&0.0200197&-0.206136&-0.200098&0.21641\\\
0.0300672&2.05265&0.0400418&-0.356474&-0.400308&0.422052\\\
0.0200197&0.0400418&1.10944&-0.200099&-1.09521&1.00374\\\
-0.206136&-0.356474&-0.200099&1.47744&1.99935&-2.24369\\\
-0.200098&-0.400308&-1.09521&1.99935&9.96135&-10.0452\\\
0.21641&0.422052&1.00374&-2.24369&-10.0452&10.2858\\\ \end{array}\right).$
(233)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-0.91$.

####  C.4.3 Tree 6e

In the case of tree $6e$ we guessed the witness matrix $Z_{6e}$ as

$Z_{6e}=\frac{1}{3.1801}\left(\begin{array}[]{cc}Z_{6e}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&Z_{6e}^{p}\end{array}\right),$ (234)

where

$\displaystyle Z_{6e}^{x}=\left(\begin{array}[]{cccccc}1&0&0&0&-0.1&0\\\
0&1&0&0&-0.1&0\\\ 0&0&0.01&0&0&-0.01\\\ 0&0&0&0.01&0&-0.01\\\
-0.1&-0.1&0&0&0.03&-0.01\\\ 0&0&-0.01&-0.01&-0.01&0.0301\\\
\end{array}\right)$

and

$Z_{6e}^{p}=\left(\begin{array}[]{cccccc}4&0&0&0&0.2&0\\\ 0&1&0&0&0.1&0\\\
0&0&1&0&0&0.1\\\ 0&0&0&1&0&0.1\\\ 0.2&0.1&0&0&0.03&0.01\\\
0&0&0.1&0.1&0.01&0.0301\\\ \end{array}\right).\\\ $ (236)

From the numerically found symplectic matrix $S_{6e}$ symplectically
diagonalizing $Z_{6e}$ we then construct the following CM:

$\gamma_{6e}=S_{6e}^{T}S_{6e}=\left(\begin{array}[]{cc}\gamma_{6e}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&\gamma_{6e}^{p}\end{array}\right),$ (237)

where

$\displaystyle\gamma_{6e}^{x}=\left(\begin{array}[]{cccccc}2.07887&0.0788541&0.729928&0.729928&0.788929&0.749595\\\
0.0788541&1.07884&0.730827&0.730827&0.788736&0.749621\\\
0.729928&0.730827&19.0655&9.06554&7.28942&9.32218\\\
0.729928&0.730827&9.06554&19.0655&7.28942&9.32218\\\
0.788929&0.788736&7.28942&7.28942&6.89127&7.49556\\\
0.749595&0.749621&9.32218&9.32218&7.49556&8.5861\\\ \end{array}\right),$ (244)
$\displaystyle\gamma_{6e}^{p}=\left(\begin{array}[]{cccccc}0.519719&0.0394271&-0.0364964&-0.0364964&-0.394465&0.374798\\\
0.0394271&1.07884&-0.0730827&-0.0730827&-0.788736&0.749621\\\
-0.0364964&-0.0730827&0.190655&0.0906554&0.728942&-0.932218\\\
-0.0364964&-0.0730827&0.0906554&0.190655&0.728942&-0.932218\\\
-0.394465&-0.788736&0.728942&0.728942&6.89127&-7.49556\\\
0.374798&0.749621&-0.932218&-0.932218&-7.49556&8.5861\\\ \end{array}\right).$
(251)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-0.65$.

####  C.4.4 Tree 6f

In the case of the tree $6f$ we guess the six-mode witness matrix $Z_{6f}$ to
be

$Z_{6f}=\frac{1}{75.4801}\left(\begin{array}[]{cc}Z_{6f}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&Z_{6f}^{p}\end{array}\right),$ (252)

where

$\displaystyle Z_{6f}^{x}=\left(\begin{array}[]{cccccc}25&0&0&-0.5&0&0\\\
0&25&0&-0.5&0&0\\\ 0&0&25&-0.5&0&0\\\ -0.5&-0.5&-0.5&0.04&0.&-0.01\\\
0&0&0&0&25&-0.5\\\ 0&0&0&-0.01&-0.5&0.0201\\\ \end{array}\right)$

and

$\displaystyle Z_{6f}^{p}=\left(\begin{array}[]{cccccc}25&0&0&5&0&0\\\
0&25&0&0.5&0&0\\\ 0&0&25&0.5&0&0\\\ 5&0.5&0.5&2.02&0&0.1\\\
0&0&0&0&0.01&0.01\\\ 0&0&0&0.1&0.01&0.0201\\\ \end{array}\right).$

From the symplectic matrix $S_{6f}$ symplectically diagonalizing $Z_{6f}$ we
get the CM of the following form:

$\gamma_{6f}=S_{6f}^{T}S_{6f}=\left(\begin{array}[]{cc}\gamma_{6f}^{x}&\mathbb{0}_{6}\\\
\mathbb{0}_{6}&\gamma_{6f}^{p}\end{array}\right),$ (255)

where

$\displaystyle\gamma_{6f}^{x}=\left(\begin{array}[]{cccccc}1.00888&0.0070959&0.0070959&0.44554&0.000907054&0.045205\\\
0.0070959&1.00531&0.00530564&0.265838&0.00090708&0.0452774\\\
0.0070959&0.00530564&1.00531&0.265838&0.00090708&0.0452774\\\
0.44554&0.265838&0.265838&12.3541&0.0453557&2.26827\\\
0.000907054&0.00090708&0.00090708&0.0453557&0.020908&0.0454074\\\
0.045205&0.0452774&0.0452774&2.26827&0.0454074&1.27089\\\ \end{array}\right),$
(262)
$\displaystyle\gamma_{6f}^{p}=\left(\begin{array}[]{cccccc}1.00889&0.00267199&0.00267199&-0.0446333&-0.0453539&0.0452056\\\
0.00267199&1.00044&0.000437932&-0.0224336&-0.00453415&0.00444848\\\
0.00267199&0.000437932&1.00044&-0.0224336&-0.00453415&0.00444848\\\
-0.0446333&-0.0224336&-0.0224336&0.124334&0.226785&-0.226828\\\
-0.0453539&-0.00453415&-0.00453415&0.226785&52.2702&-2.27039\\\
0.0452056&0.00444848&0.00444848&-0.226828&-2.27039&1.27089\\\
\end{array}\right).$ (269)

Now, if we optimize numerically the difference (51) for the sum criterion (48)
over the parameters $\ell^{\alpha}_{ij}\in[-10,10]$, we arrive at
$D_{S}=-0.13$.

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