# Steering witnesses for unknown Gaussian quantum states

Tatiana Mihaescu (1,2), Hermann Kampermann (3), Aurelian Isar (1,2), Dagmar
Bruß (3)  (1) Department of Theoretical Physics, National Institute of Physics
and Nuclear Engineering, RO-077125 Bucharest-Magurele, Romania (2) Faculty of
Physics, University of Bucharest, RO-077125 Bucharest-Magurele, Romania (3)
Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik III,
D-40225 Düsseldorf, Germany<EMAIL_ADDRESS>
###### Abstract

We define and fully characterize the witnesses based on second moments
detecting steering in Gaussian states by means of Gaussian measurements. All
such tests, which arise from linear combination of variances or second moments
of canonical operators, are easily implemented in experiments. We propose also
a set of linear constraints fully characterizing steering witnesses when the
steered party has one bosonic mode, while in the general case the constraints
restrict the set of tests detecting steering. Given an unknown quantum state
we implement a semidefinite program providing the appropriate steering test
with respect to the number of random measurements performed. Thus, it is a
”repeat-until-success” method allowing for steering detection with less
measurements than in full tomography. We study the efficiency of steering
detection for two-mode squeezed vacuum states, for two-mode general unknown
states, and for three-mode continuous variable GHZ states. In addition, we
discuss the robustness of this method to statistical errors.

##  1 Introduction

The study of quantum correlations is a cornerstone of quantum information
theory enriching the foundational understanding of quantum theory and allowing
applications which outperform any classical approach in certain tasks such as
computation [1], secure communication [2] and metrology [3]. Schrödinger [4,
5] discussed the enchanting phenomenon where one party, Alice, is able to
”steer” the state of a distant party, Bob, by means of entanglement they
share. The implied ”action at a distance” was the core argument in the
Einstein, Podolsky and Rosen (EPR) paper [6] against the completeness of
quantum theory, but only lately quantum steering was conceptualized as a
particular type of nonlocality [7].
In a quantum steering scenario two distant parties (Alice and Bob) share a
common quantum state, where one of the parties, say Alice, is able to convince
Bob that the state they share is entangled. She does so by performing local
measurements and using classical communication, whereas Bob verifies whether
the joint probability distribution can be explained by a local hidden state
(LHS) model [7], in which case it rules out the possibility that Alice is
steering Bob’s state by her choice of local measurement settings. This
definition presents quantum steering as an intermediate form of correlation
between entanglement and Bell nonlocality. It has been intensively studied
leading to some experimental applications such as subchannel discrimination
[8] and one-sided device-independent cryptography [9].
In the context of continuous variable states [10] quantum steering is
extensively investigated starting with Gaussian states and using Gaussian
measurements [7, 11], that have a distinct role in the infinite-dimensional
Hilbert space, being also readily available in experiments [12, 13, 14]. In
this particular case, the necessary and sufficient criterion for steerability
is given in terms of the covariance matrix of the state [11, 15, 16], which
comprises the variances of the canonical operators. Therefore, the detection
of steering in a general unknown Gaussian state requires the full knowledge of
the covariance matrix, which might be excessive and resource-consuming.
In this paper first we prove a criterion for Gaussian steerability that is
equivalent to known criteria in the literature, but it unfolds very nice
properties of the covariance matrix resembling the known covariance matrix
criterion of entanglement (see Section 3, Theorem 2). A necessary condition
proof of this theorem is presented in Ref. [15], using local uncertainty
relations (LUR). Based on this result we define the set of linear tests
detecting Gaussian steering, or witnesses, which arise as linear combinations
of the second moments of the canonical observables. Such tests are commonly
used for the detection of entanglement, known as entanglement witnesses [17,
18, 19]. An entanglement witness based on second moments is a real symmetric
matrix $Z\geq 0$ such that ${\rm Tr}[Z\gamma_{s}]\geq 1$ holds for all
separable covariance matrices $\gamma_{s}$, while ${\rm Tr}[Z\gamma]<1$ for
some entangled covariance matrix $\gamma$ [17]. An analogous definition holds
also for steering witnesses [20] due to the fact that the set of non-steerable
covariance matrices is convex and closed. We introduce constraints fully
characterizing the set of Gaussian steering witnesses, which are shown to be
stronger than analogous constraints on entanglement witnesses.
In addition, we propose a set of linear constraints that are stronger than the
constraints fully characterising the steering witnesses, however in the
particular case, when the steered party has just one bosonic mode, these new
constraints fully characterize the set of steering witnesses. This allows us
to write a semidefinite program finding the optimal steering test for a given
state. We analyze the efficiency of steering detection in unknown covariance
matrices with respect to the number of random measurements required for this
task. For entanglement detection based on covariance matrices an analogous
method was developed in Ref. [19], and therefore, the results in this article
provide us with a framework of comparison between the detection of Gaussian
steering and entanglement.
The paper is organized as follows: Section 2 gives an introduction to the
notions of Gaussian states, Gaussian measurements and symplectic
transformations. In Section 3 quantum Gaussian steering is defined and the
covariance matrix criterion for Gaussian steering is proven. In Section 4 the
steering witnesses based on second moments are introduced and fully
characterized. In Section 5 we construct steering witnesses from random
measurements acquired by homodyne detection. Section 6 presents the results of
steering detection in two-mode squeezed vacuum states and in two-mode general
unknown states. Also, we illustrate the example of steering detection in
three-mode continuous variable GHZ states, where the steered party has two
modes. In Section 7 the statistical analysis of our method is provided. The
summary and conclusions are expanded in Section 8.

##  2 Gaussian states

A continuous variable (CV) system of $N$ bosonic modes is described by the
canonical operators of position and momentum $\hat{x}_{k}$ and $\hat{p}_{k}$
in the Hilbert space $\mathcal{H}=\bigotimes_{k=1}^{N}\mathcal{H}_{k}$, where
$H_{k}$ is the infinite-dimentional Hilbert space of mode $k$ [10, 12, 13].
Defining the vector of canonical operators $\mathbf{\hat{R}}^{\rm
T}\equiv(\hat{R}_{1},...,\hat{R}_{2N})=(\hat{x}_{1},\hat{p}_{1},...,\hat{x}_{N},\hat{p}_{N})$
one obtains the commutation relation written as (we assume $\hbar=1$):

$[\hat{R}_{i},\hat{R}_{j}]={\rm i}\Omega_{ij}{\rm\hat{I}},\quad i,j=1,...,2N,$
(1)

where ${\rm\hat{I}}$ is the identity matrix, and $\Omega_{ij}$ are the
elements of the symplectic matrix

$\Omega_{N}=\bigoplus_{1}^{N}\left({\begin{array}[]{*{50}c}0&1\\\ -1&0\\\
\end{array}}\right).$ (2)

Gaussian states are fully described by the first and second order statistical
moments, namely the displacement vector $\mathbf{d}={\rm
Tr}[\mathbf{\hat{R}}\rho]$, and the covariance matrix (CM) $\gamma$ with its
elements defined as [14, 21]:

$\gamma_{ij}=\langle\\{\hat{R}_{i}-\langle\hat{R}_{i}\rangle,\hat{R}_{j}-\langle\hat{R}_{j}\rangle\\}_{+}\rangle_{\rho},$
(3)

where $\\{,\\}_{+}$ represents the anticommutator. The CM of a physical
quantum state has to fulfill the Robertson-Schrödinger uncertainty relation,
which we will often refer to as covariance matrix criterion (CMC):

$\gamma+{\rm i}\Omega_{N}\geq 0.$ (4)

In the following we will mostly consider a bipartite Gaussian quantum state
$\rho_{AB}$ of $N$ modes with the CM of the following block structure:

$\gamma_{AB}=\begin{pmatrix}\gamma_{A}&\gamma_{12}\\\ \gamma_{12}^{\rm
T}&\gamma_{B}\end{pmatrix},$ (5)

where $\gamma_{A}$ and $\gamma_{B}$ are the CMs of the subsystems of Alice and
Bob with $N_{A}$ and $N_{B}$ modes, respectively, $\gamma_{12}$ is the
correlation matrix between the two parties, and $N=N_{A}+N_{B}$. It can be
readily shown that the uncertainty relation in Eq. (4) directly implies
$\gamma_{AB}>0$, $\gamma_{A}>0$ and $\gamma_{B}>0$.

A much more general approach in describing any $N-$mode CV state is based on
the completeness of the set of displacement operators in phase space defined
as:

$\hat{D}(\mathbf{r})={\rm e}^{{\rm i}~{}\mathbf{r}^{\rm
T}\Omega_{N}\mathbf{\hat{R}}},$ (6)

where $\mathbf{r}^{\rm T}=(x_{1},p_{1},...,x_{N},p_{N})$ is a real vector of
phase space variables. The connection between Hilbert space and phase space
descriptions is given by the Fourier-Weyl relation for a given density
operator $\hat{\rho}$ in Hilbert space [10]:

$\hat{\rho}=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{2N}}d^{2N}\mathbf{r}~{}{\rm
Tr}[\hat{D}^{\dagger}(\mathbf{r})\hat{\rho}]\hat{D}(\mathbf{r}),$ (7)

from where the characteristic function is readily defined as
$\chi_{\rho}(\mathbf{r})={\rm Tr}[\hat{D}^{\dagger}(\mathbf{r})\hat{\rho}]$.
If $\hat{\rho}$ represents a Gaussian state then its characteristic function
takes a particular form [10]:

$\chi_{\rho}(\mathbf{r})={\rm e}^{-\frac{1}{4}\mathbf{r}^{\rm
T}\Omega_{N}^{\rm T}\gamma_{AB}\Omega_{N}\mathbf{r}},$ (8)

where the first moments are considered to be zero and $\gamma_{AB}$ is the CM
associated with the Gaussian density operator $\hat{\rho}$. A Gaussian
measurement [22] applied on Alice’s subsystem can be described by a positive
Gaussian operator $\hat{A}$ with a CM given by $T^{A}$ satisfying the CMC:
$T^{A}+{\rm i}\Omega_{N_{A}}\geq 0$. The calculation of ${\rm
Tr}_{A}[(\hat{A}\otimes\mathcal{\hat{I}}_{B})\hat{\rho}]$, where
$\mathcal{\hat{I}}_{B}$ is the identity operator defined on Bob’s space,
transforms into a multivariate Gaussian integral on characteristic function
level. The remaining modes of Bob’s conditioned state $\tilde{\rho}_{B}^{A}$
after measurement form a Gaussian state with CM given by
$\gamma_{B}^{A}=\gamma_{B}-\gamma_{12}^{\rm
T}(\gamma_{A}+T^{A})^{-1}\gamma_{12}$, which represents the Schur complement
of the matrix $\gamma_{AB}+T^{A}\oplus 0_{B}$ with respect to the submatrix
$\gamma_{A}+T^{A}$ [10]. The following Lemma of the Schur complement will be
useful in further discussions on Gaussian steering.

Lemma 1 [23]. Consider a Hermitian matrix

$H=\begin{pmatrix}A&X\\\ X^{\dagger}&B\end{pmatrix}.$ (9)

Then $H>0$ if and only if $A>0$ and $H/A=B-X^{\dagger}A^{-1}X>0$, where $H/A$
is the Schur complement of block $A$ of the matrix $H$.

Given that $\gamma_{AB}$ is a CM satisfying the uncertainty relation in Eq.
(4) we have $\gamma_{AB}>0$ and $\gamma_{A}>0$, and hence the Schur complement
of $\gamma_{AB}$ with respect to $\gamma_{A}$ is also a positive matrix
$\gamma_{AB}/\gamma_{A}:=\gamma_{B}-\gamma_{12}^{\rm
T}\gamma_{A}^{-1}\gamma_{12}>0$. Let us consider the matrix
$\gamma_{AB}-0_{A}\oplus\sigma_{B}$, with $\sigma_{B}\geq 0$, and its Schur
complement with respect to $\gamma_{A}$. Applying the positivity conditions
from Lemma 1 leads to the Schur complement [10, 24]

$\gamma_{AB}/\gamma_{A}=\sup\\{\sigma_{B}:\gamma_{AB}\geq
0_{A}\oplus\sigma_{B}\\},$ (10)

which means that the matrix set on the right-hand side has a supremum (i.e. a
minimum upper bound) with respect to the Löwner partial order ($X\geq Y$ if
and only if $X-Y$ is positive semidefinite), and that this supremum is given
by the Schur complement on the left-hand side.

###  2.1 Symplectic transformations

The equivalent of unitary operators acting on the quantum state space are the
symplectic transformations $S$ on CMs, which are symmetric real matrices
acting by congruence on CMs: $\gamma^{\prime}=S\gamma S^{\rm T}$. The singular
value decomposition of a real symplectic matrix gives [25]:

$S=K\Big{[}\bigoplus_{i=1}^{N}S(r_{i})\Big{]}L,$ (11)

where $S(r_{i})$ is a one-mode squeezing matrix (symplectic and nonorthogonal)
with $r_{i}$ the squeezing parameter:

$S(r_{i})=\left({\begin{array}[]{*{20}c}{\rm e}^{-r_{i}}&0\\\ 0&{\rm
e}^{r_{i}}\end{array}}\right),$ (12)

and $K$, $L$ are symplectic and orthogonal matrices. Denote by $K(2N)$ the
group of orthogonal symplectic matrices isomorphic to the group of complex
unitary matrices $U(N)$ [25]:

$K(2N)=\\{S(X,Y)|X-{\rm i}Y\in U(N)\\},$ (13)

where the corresponding symplectic matrices are given by:

$S(X,Y)=\left({\begin{array}[]{*{20}c}X&Y\\\ -Y&X\end{array}}\right).$ (14)

According to the Williamson theorem [27] every real symmetric matrix $M\geq 0$
can be brought to a diagonal form through symplectic transformations as
follows:

$SMS^{T}={\rm diag}(s_{1},s_{1},\dotsc,s_{N},s_{N}),$ (15)

where $s_{1},\dotsc,s_{N}\geq 0$ are called symplectic eigenvalues of $M$. By

${\rm str}[M]:=\sum_{i=1}^{N}s_{i}$ (16)

we denote the symplectic trace of the matrix $M$.

##  3 Gaussian quantum steering

Consider the situation where Alice and Bob are two distant parties sharing a
common state $\rho_{AB}$, and Alice performs local measurements $\hat{A}$ on
her state, with eigenvalues $a$. A formal definition of steering says that
Alice is not able to steer Bob’s state if there exists an ensemble
$F=\\{p_{\eta}\rho_{\eta}\\}$ of preexisting local hidden states $\rho_{\eta}$
with probabilities $p_{\eta}$, and a stochastic map
$P(a|\hat{A},\eta)$111$P(a|\hat{A},\eta)$ denotes the probability distribution
of Alice to obtain outcome $a$ when measuring $\hat{A}$ and given a local
hidden variable $\eta$. from the hidden variable $\eta$ to $a$, such that
Bob’s conditioned state after Alice’s measurement is given by [7]:

$\tilde{\rho}_{B}^{A}=\sum_{\eta}P(a|\hat{A},\eta)\rho_{\eta}p_{\eta}.$ (17)

This represents the local hidden state (LHS) model, where Bob checks if Alice
can simulate the state $\tilde{\rho}_{B}^{A}$ based on her knowledge of the
parameter $\eta$, by drawing the states $\rho_{\eta}$ according to the
distribution $p_{\eta}$. Conversely, if Bob cannot find any ensemble $F$ and
map $P(a|\hat{A},\eta)$ satisfying Eq. (17), then Bob must admit that Alice
can steer his system.
In the Gaussian realm where the joint state $\rho_{AB}$ is a Gaussian state
with CM $\gamma_{AB}$, we consider that Alice’s measurement is also Gaussian
(i.e. mapping Gaussian states into Gaussian states). As discussed in the
previous section a Gaussian measurement is represented by a positive operator
$\hat{A}$ with a Gaussian characteristic function as in Eq. (8) with CM
$T^{A}$, satisfying $T^{A}+{\rm i}\Omega_{N_{A}}\geq 0$. When Alice performs a
measurement $\hat{A}$ and gets an outcome $a$, Bob’s conditioned state
$\tilde{\rho}_{B}^{A}$ is a Gaussian state with CM given by
$\gamma_{B}^{A}=\gamma_{B}-\gamma_{12}^{\rm
T}(\gamma_{A}+T^{A})^{-1}\gamma_{12}$ [10]. In Ref. [7] a criterion for
Gaussian steerability of CM $\gamma_{AB}$ was derived:

Theorem 1. [7] A Gaussian state with covariance matrix $\gamma_{AB}$ with the
block form defined in Eq. (5) is Alice $\to$ Bob non-steerable by Gaussian
measurements if and only if

$\gamma_{AB}+0_{A}\oplus{\rm i}\Omega_{N_{B}}\geq 0$ (18)

or, equivalently222This follows from applying the positivity conditions from
Lemma 1 to the total matrix in Eq. (18).

$\gamma_{A}\geq 0\quad\text{and}\quad\gamma_{AB}/\gamma_{A}+{\rm
i}\Omega_{N_{B}}\geq 0$ (19)

holds, where $\gamma_{AB}/\gamma_{A}:=\gamma_{B}-\gamma_{12}^{\rm
T}\gamma_{A}^{-1}\gamma_{12}$ is the Schur complement of $\gamma_{AB}$ with
respect to submatrix $\gamma_{A}$ and $\Omega_{N_{B}}$ is the symplectic
matrix of $N_{B}$ modes as defined in Eq. (2).

From the previous section we know that the Schur complement of a CM is a
positive matrix. However, for a non-steerable Alice to Bob Gaussian state the
Schur complement of its CM with respect to $\gamma_{A}$ has to satisfy a
stronger condition than positivity, namely the CMC from Eq. (4). An important
result of this article is given by the following theorem on Gaussian non-
steerability.

Theorem 2. A bipartite quantum Gaussian state $\rho_{AB}$ with covariance
matrix $\gamma_{AB}$ with blocks defined in Eq. (5) is Alice $\to$ Bob non-
steerable by means of Gaussian measurements if and only if there exists a
covariance matrix corresponding to Bob’s system $\sigma_{B}$ satisfying
$\sigma_{B}+{\rm i}\Omega_{N_{B}}\geq 0$, such that:

$\gamma_{AB}\geq 0_{A}\oplus\sigma_{B}.$ (20)

###### Proof.

$\Rightarrow$ Theorem 1 states that the Schur complement
$\gamma_{AB}/\gamma_{A}$ of a non-steerable CM $\gamma_{AB}$ is also a CM,
i.e. $\gamma_{AB}/\gamma_{A}+{\rm i}\Omega_{N_{B}}\geq 0$. Based also on the
definition of the Schur complement in Eq. (10) we state that if a CM
$\gamma_{AB}$ is non-steerable then there exists a CM $\sigma_{B}$ fulfilling
Eq. (20), and it is the Schur complement $\gamma_{AB}/\gamma_{A}$.
$\Leftarrow$ Conversely, if the relation in Eq. (20) is fulfilled for some CM
$\sigma_{B}$, satisfying CMC $\sigma_{B}+{\rm i}\Omega_{N_{B}}\geq 0$, then by
the definition of the Schur complement in Eq. (10) it follows that there
exists a positive semi-definite matrix $P\geq 0$ such that:
$\sigma_{B}+P=\gamma_{AB}/\gamma_{A}$. Therefore, the Schur complement
$\gamma_{AB}/\gamma_{A}$ also has to fulfill the CMC, since $\sigma_{B}+P+{\rm
i}\Omega_{N_{B}}\geq P\geq 0$, which based on Theorem 1, means that
$\rho_{AB}$ is a non-steerable quantum state. ∎

In Ref. [15] a one way proof of this theorem based on local uncertainty
relations (LURs) is provided. The non-steerability criterion formulated in
Theorem 2 shows a strong similarity to the separability of Gaussian states. A
continuous variable state with CM $\gamma$ is separable with respect to
parties $A,B$ if and only if there exist local CMs $\gamma_{A}$ and
$\gamma_{B}$ for each partition, such that
$\gamma\geq\gamma_{A}\oplus\gamma_{B}$ [26]. Hence, it is obvious that any
separable Gaussian state is also non-steerable by Gaussian measurements since
$\gamma_{A}\oplus\gamma_{B}\geq 0_{A}\oplus\gamma_{B}$ holds. Thus, Gaussian
non-steerability represents a stronger condition than separability of Gaussian
states.

##  4 Steering witnesses based on second moments

Based on Theorem 2 we can define the set of non-steerable CMs as follows:

$\Gamma_{A\not\to
B}(\mathbb{R}^{2N}):=\\{\gamma_{AB}|~{}\gamma_{AB}=0_{A}\oplus\sigma_{B}+P,~{}\text{with}~{}\sigma_{B}+{\rm
i}\Omega_{N_{B}}\geq 0~{}\text{and}~{}P\geq 0\\},$ (21)

where $A\not\to B$ denotes Alice to Bob non-steerability, and $P$ is a
positive-semidefinite matrix. The set of non-steerable CMs by Gaussian
measurements forms a closed convex subset of the space of all covariance
matrices, similarly to the set of all CMs and the set of separable CMs. This
allows to completely describe the set of non-steerable CMs by a family of
linear inequalities representing the steering witnesses (SWs).

Definition 1. We define the set of real symmetric $2N\times 2N$-matrices

$\mathcal{Z}_{A\not\to B}(\mathbb{R}^{2N})=\\{Z|Z\geq
0,\forall\gamma_{AB}\in\Gamma_{A\not\to B}(\mathbb{R}^{2N}):{\rm
Tr}[Z\gamma_{AB}]\geq 1\\},$ (22)

where $\gamma_{AB}$ is the CM of a bipartite system with $N=N_{A}+N_{B}$
modes, $A\not\to B$ denotes Alice to Bob non-steerability, where
$\Gamma_{A\not\to B}(\mathbb{R}^{2N})$ is defined in Eq.(21). All matrices
$Z\in\mathcal{Z}_{A\not\to B}(\mathbb{R}^{2N})$ for which there exists
$\gamma$ with ${\rm Tr}[Z\gamma]<1$ will be called steering witnesses (SWs).

This Definition 1 is very similar to the Definition 6.4 in Ref. [18] for
entanglement witnesses (EWs) since it relies solely on the fact that the set
of non-steerable CMs and the set of separable CMs are both convex and closed
sets. There it is also shown that for any such hyperplane $Z\geq 0$ holds,
which follows from the positivity of CMs. We proceed, in analogy with the
methods used in Ref. [18], to characterise the set of SWs and prove some of
their properties.

Theorem 3. Let $Z$ be a real symmetric $2N\times 2N$ matrix on a phase space
of $N=N_{A}+N_{B}$ modes, with $Z_{A}$ and $Z_{B}$ denoting the block diagonal
submatrices of $Z$ corresponding to subsystems of Alice and Bob, respectively.
Then $Z$ is a steering witness, namely $Z\in\mathcal{Z}_{A\not\to
B}(\mathbb{R}^{2N})$, if and only if

$\displaystyle Z$ $\displaystyle\geq 0,$ (23) $\displaystyle{\rm str}[Z_{B}]$
$\displaystyle\geq\frac{1}{2},$ (24) $\displaystyle{\rm str}[Z]$
$\displaystyle<\frac{1}{2}.$ (25)

###### Proof.

Below Theorem 6.2 in Ref. [18] it is proven that any hyperplane cutting the
set of CMs is represented by a positive semi-definite matrix, and therefore,
condition $Z\geq 0$ holds for any EW and SW as well. In addition, in Theorem
6.3 in the same reference it is proven that such a test represented by $Z\geq
0$ and satisfying also ${\rm str}[Z]\geq\frac{1}{2}$ is equivalent to
obtaining ${\rm Tr}[Z\gamma]\geq 1$ for all CMs $\gamma$. In the following we
will prove that condition ${\rm str}[Z_{B}]\geq\frac{1}{2}$ guarantees that
${\rm Tr}[Z\gamma]\geq 1$ for all $A\rightarrow B$ non-steerable CMs.
Therefore, condition ${\rm str}[Z]<\frac{1}{2}$ together with conditions (23,
24) assure that there exists a CM $\gamma$ such that ${\rm Tr}[Z\gamma]<1$,
i.e. it is $A\rightarrow B$ steerable.
Consider the SWs based on second moments of the following block form:

$Z=\begin{pmatrix}Z_{A}&Z_{12}\\\ Z_{12}^{{\rm T}}&Z_{B}\end{pmatrix}.$ (26)

In the following we will use an important result discussed in Ref. [18] 333The
expression $2\,{\rm str}[Z]$ represents the value of the ground state energy
of the Hamiltonian defined as $\hat{H}=Z\sum_{kl}\hat{R}_{k}\hat{R}_{l}$,
where
$\mathbf{\hat{R}}=(\hat{x}_{1},\hat{p}_{1},...,\hat{x}_{N},\hat{p}_{N})^{{\rm
T}}$ is the vector of canonical operators.:

$\min_{\sigma_{B}}{\rm Tr}[Z_{B}\sigma_{B}]=2\,{\rm str}[Z_{B}],$ (27)

where the minimization is performed over the set of CMs $\sigma_{B}$.
$\Rightarrow$ We have to prove that condition in Eq.(24) holds for any SW
described in Definition 1 (22). Based on Theorem 2 for any
$\gamma_{AB}\in\Gamma_{A\not\to B}(\mathbb{R}^{2N})$ there exists a CM
corresponding to the quantum state of Bob $\sigma_{B}$ such that:

${\rm Tr}[Z\gamma_{AB}]\geq{\rm Tr}[Z(0_{A}\oplus\sigma_{B})]$ (28)

holds. Moreover, the matrix $0_{A}\oplus\sigma_{B}$ is a non-steerable CM
itself, and from the Definition 1 of SWs in Eq. (22) we have the following
condition:

${\rm Tr}[Z(0_{A}\oplus\sigma_{B})]={\rm Tr}[Z_{B}\sigma_{B}]\geq 1.$ (29)

Using Eq. (27) we conclude that SWs have to fulfill the condition

${\rm str}[Z_{B}]\geq\frac{1}{2}.$ (30)

$\Leftarrow$ Conversely, we prove that a matrix $Z$ satisfying the conditions
in Eq. (24) represents a SW with ${\rm Tr}[Z\gamma_{AB}]\geq 1$ for any non-
steerable CM $\gamma_{AB}\in\Gamma_{A\not\to B}(\mathbb{R}^{2N})$. Starting
from Eq. (24) and using Eq. (27) we have that

${\rm Tr}[Z_{B}\sigma_{B}]\geq 1$ (31)

holds for any valid CM $\sigma_{B}$. Now, any non-steerable CM $\gamma_{AB}$
fulfills $\gamma_{AB}\geq 0_{A}\oplus\sigma_{B}$ (Theorem 2) such that the
condition ${\rm Tr}[Z\gamma_{AB}]\geq{\rm Tr}[Z_{B}\sigma_{B}]$ holds.
Therefore we get for any non-steerable state ${\rm Tr}[Z\gamma_{AB}]\geq 1$. ∎

We are now in the position to summarize the results for the detection of
Gaussian steering with witnesses based on second moments.

Theorem 4. [Steerability] A CM $\gamma_{AB}$ of two parties consisting of
$N=N_{A}+N_{B}$ modes is Alice to Bob steerable by means of Gaussian
measurements if and only if there exists a $Z$ such that:

${\rm Tr}[Z\gamma_{AB}]<1,$ (32)

where $Z$ is a real symmetric $2N\times 2N$ matrix satisfying

$Z\geq 0\quad and\quad{\rm str}[Z_{B}]\geq\frac{1}{2},$ (33)

where $Z_{B}$ denotes the principal submatrix of $Z$ belonging to the
subsystem of Bob. Matrices $Z$ are called steering witnesses based on second
moments.

The EWs defined in Refs. [17, 18] that detect bipartite entanglement fulfill
the condition ${\rm str}[Z_{A}]+{\rm str}[Z_{B}]\geq\frac{1}{2}$, where $A,B$
denote the two parties, and $Z_{A},Z_{B}$ are the block diagonal matrices of
the EW associated with each party. SWs are characterised by stronger
constraints than these and therefore, any SW based on second moments is also
an EW. This is consistent with the fact that any steerable state necessarily
contains also entanglement.

##  5 Linear constraints for steering witnesses

In the following we present linear conditions for the SWs that are stronger
than Eq. (33) in Theorem 4, such that the SWs can be calculated by a
semidefinite program (SDP) to detect steering in a given CM. An analogous idea
for EWs was developed in Ref. [19].

Proposition 1. For the steering witness $Z$ of an $N-$mode covariance matrix
steerable from Alice to Bob, with $N=N_{A}+N_{B}$, the inequalities (33) are
satisfied if (if and only if for $N_{B}=1$) the following conditions are
fulfilled:

$\displaystyle Z\geq 0,$ $\displaystyle Z_{B}+{\rm
i}\frac{1}{2N_{B}}\Omega_{N_{B}}\geq 0.$ (34)

###### Proof.

Since $Z$ is a positive semidefinite matrix, also the principal submatrix
$Z_{B}$ is positive semidefinite. For every such matrix there exists a
symplectic transformation $S$ that brings it to its Williamson normal form as
follows 444Given that ${\rm Tr}[M]\geq 2\ {\rm str}[M]$ holds for any positive
matrix $M$ [28], the symplectic transformations do not only preserve the
symplectic eigenvalues, but also positivity.:

$S(Z_{B}+{\rm i}\frac{1}{2N_{B}}\Omega_{N_{B}})S^{T}=Z_{B}^{w}+{\rm
i}\frac{1}{2N_{B}}\Omega_{N_{B}},$ (35)

with $Z_{B}^{w}={\rm diag}(z_{1},z_{1},\cdots,z_{N_{B}},z_{N_{B}})$, where
$z_{j}$, $j=1,\cdots N_{B}$ are positive symplectic eigenvalues of $Z_{B}$. By
imposing the positivity condition on the eigenvalues of the total matrix in
Eq. (35) we arrive at the following inequality for the symplectic eigenvalues:

$z_{j}\geq\frac{1}{2N_{B}},\quad j=1,\cdots,N_{B}.$ (36)

Now, the sum of symplectic eigenvalues gives:

$\sum_{j=1}^{N_{B}}z_{j}\geq\sum_{j=1}^{N_{B}}\frac{1}{2N_{B}}=\frac{1}{2}.$
(37)

∎

Relation (36) explains why the conditions on SWs given in the Proposition 1
are stronger than Eq. (33) from Theorem 4: it imposes a lower bound on the
symplectic eigenvalues that is strictly greater than zero, as
$\frac{1}{2N_{B}}>0$. This assures that the sum of symplectic eigenvalues
exceeds $\frac{1}{2}$, however, it rules out the possibility that some
symplectic eigenvalues may satisfy $z_{j}\in(0,\frac{1}{2N_{B}})$. An
exception represents the case when the steered party has one mode ($N_{B}$=1)
and $Z_{B}$ in Eq. (5) is a $2\times 2$ matrix with only one symplectic
eigenvalue satisfying $z\geq\frac{1}{2}$, which is equivalent to relation
(33). However, the interval of forbidden symplectic eigenvalues
$z_{j}\notin(0,\frac{1}{2N_{B}})$ is the largest for $N_{B}=2$, while in the
limit of large $N_{B}\rightarrow\infty$ we have $\frac{1}{2N_{B}}\rightarrow
0$, and the constraints in Proposition 1 become equivalent to the constraints
fully characterizing SWs in Theorem 4.

###  5.1 Constructing witnesses

We are now able to define an SDP by constructing the witnesses from given
(random) measurements using the linear constraints in the Proposition 1. Given
the repeated independent measurements $P_{j}$ on the CM, the witness operator
is represented by $Z=\sum_{j}c_{j}P_{j}$, where the coefficients $c_{j}$ are
the output of the optimization algorithm. According to the Proposition we can
detect the Gaussian steering from Alice to Bob in two-mode CMs with the
following SDP:  c⋅m Z=∑_j c_j P_j  Z=(Z_AZ_CZ_C^TZ_B)≥0 Z_B+i 12 Ω_N_B ≥0 ,
where $\bf{c}$ is the vector of coefficients $c_{j}$ and $\bf{m}={\rm
Tr}[P\gamma]$, with $\bf{P}$ being the vector of measurement matrices $P_{j}$,
which represent the homodyne detection measurements as constructed in Ref.
[19].

Consider an experimental scheme detecting two-mode CMs using a single homodyne
detector, a phase shifter of angle $\varphi$ between the vertical and
horizontal polarization components, a polarization rotator on angle $\phi$ and
a polarizing beam splitter, for mixing the initial modes denoted by $\hat{a}$
and $\hat{b}$ [29]. The mode $\hat{k}$ arriving at the detector has the
following expression:

$\hat{k}=\exp({\rm i}\varphi)\cos\phi\ \hat{a}+\sin\phi\ \hat{b}.$ (38)

The generalized quadrature operator is given by:

$\hat{x}_{\theta}=\frac{\exp{(-{\rm i}\theta)}\hat{k}+\exp{({\rm
i}\theta)}\hat{k}^{{\dagger}}}{\sqrt{2}},$ (39)

which covers the whole continuum of quadratures in terms of initial modes
$\hat{a}$ and $\hat{b}$ for $\theta\in[0,\pi]$, $\phi\in[0,\pi]$ and
$\varphi\in[0,2\pi)$. With homodyne detection one can measure the statistical
moments of the quadrature up to the second order [29], and therefore, one can
readily calculate the variance of the quadrature as follows:

$\langle\hat{x}_{\theta}^{2}\rangle-\langle\hat{x}_{\theta}\rangle^{2}=\Tr[P\gamma],$
(40)

where $P$ is the measurement matrix of the variances we use in Eq. (5.1):

$P=uu^{\rm T},\quad u^{\rm
T}=\begin{pmatrix}\cos\phi\cos(\theta-\varphi)&\cos\phi\sin(\theta-\varphi)&\sin\phi\cos\theta&\sin\phi\sin\theta\end{pmatrix}.$
(41)

The $N-$mode CM is a symmetric, real $2N\times 2N$ matrix with $N(2N+1)$
independent parameters. Therefore, for the CM reconstruction of a two-mode
Gaussian state one needs $10$ distinct measurement directions given by the
angles $\theta$, $\phi$ and $\varphi$. This scheme can be extended to $N$
modes with a single homodyne detector by using the same two-mode combination
scheme $N-1$ times. Starting with three initial modes $\hat{a}$, $\hat{b}$ and
$\hat{c}$, the generalized mode arriving at the detector would be:

$\hat{k}=\exp({\rm i}\varphi_{1})\cos\phi\ \hat{a}+\exp({\rm
i}\varphi_{2})\sin\phi\cos\psi\ \hat{b}+\sin\phi\sin\psi\ \hat{c}.$ (42)

##  6 Detection of Gaussian steering

Our method of detection relies on the SDP described in Eq. (5.1), where we run
the optimisation starting with two measurement settings, and then adding other
measurements, one by one, until steering is detected. In this way we are able
to evaluate the efficiency of steering detection in terms of the number of
required measurement settings. In the case of general quantum states of
unknown origin the best strategy could be to perform random measurements,
which in our case reduces to choosing random values for angles $\theta$,
$\phi$ and $\varphi$.
A measure for quantifying steering of a bipartite Gaussian state with CM
defined in Eq. (5) was proposed in Ref. [11]. In particular, the
nonsteerability condition in Eq. (19) is equivalent to $\tilde{\mu}_{j}\geq 1$
for all $j=1,\dots,N_{B}$, where $\tilde{\mu}_{j}$ represent the symplectic
eigenvalues of the Schur complement $\gamma_{AB}/\gamma_{A}$. The Gaussian
$A\rightarrow B$ steerability can then be quantified via [11]:

$\mathcal{G}^{A\to
B}(\gamma_{AB})=\max\bigg{\\{}0,-\sum_{j:\tilde{\mu}_{j}<1}\ln(\tilde{\mu}_{j})\bigg{\\}}.$
(43)

This expression is invariant under local symplectic transformations at the CM
level, and has a formal similarity to the logarithmic negativity [30], which
for Gaussian states quantifies how much the partially transposed CM fails to
fulfill CMC in Eq. (20). The quantification of Gaussian steerability takes a
simpler form when the steered party, e.g. Bob, has one mode only $(N_{B}=1)$
[11]:

$\mathcal{G}^{A\to
B}(\gamma_{AB})=\max\\{0,\frac{1}{2}\ln\frac{\det\gamma_{A}}{\det\gamma_{AB}}\\}.$
(44)

Steering can also be quantified by a minimal SW, where $Z_{min}$ corresponds
to the smallest possible value ${\rm Tr}[Z_{min}\gamma_{AB}]=w_{min}$ for a CM
$\gamma_{AB}$. The program proposed in Eq. (5.1) may reach the minimal value
$w_{min}$ with more measurements than in full tomography, since the linear
constraints used in the optimization for finding the appropriate SWs are
stronger than required in Theorem 4. For two-mode CMs $\gamma_{AB}$ our method
shows that the minimal SW $Z_{min}$ is related to the steerability measure
from Ref. [11] as follows:

$\mathcal{G}^{A\to B}(\gamma_{AB})=\ln{\frac{1}{{\rm
Tr}[Z_{min}\gamma_{AB}]}}.$ (45)

###  6.1 Two-mode squeezed vacuum states

First we test our method on the class of two-mode squeezed vacuum states
(SVS), which are easily accessible in experiments [14]. The CM is of the
following form:

$\gamma=\left({\begin{array}[]{*{20}c}\cosh 2r&0&\sinh 2r&0\cr 0&\cosh
2r&0&-\sinh 2r\cr\sinh 2r&0&\cosh 2r&0\cr 0&-\sinh 2r&0&\cosh
2r\end{array}}\right),$ (46)

where $r$ is the squeezing parameter. For this particular case, using formula
(44), the expression quantifying steering is a monotonic function in $r$:

$\mathcal{G}^{A\to B}(\gamma_{sv})=\ln{\cosh^{2}{(2r)}}.$ (47)

In Fig. 1 we show the fraction of SVS certified with steering with respect to
the number of measurements performed randomly. First, we notice the similarity
between the detection of SVS with higher steering and the detection of high
entanglement [19], since both require $8-9$ measurements on average. However,
SVS with less entanglement can be certified in many cases with $6-7$
measurements, on the other side the detection of low steering mostly requires
$10$ measurements (full tomography). The demand on more measurements for
steering detection in comparison with entanglement is consistent with the fact
that SWs based on second moments satisfy stronger constraints than EWs (see
discussion below Theorem 4).

Figure 1: Fraction of steering detection for two-mode squeezed vacuum states
with $r\in[0,2]$, see Eq. (46): $5\times 10^{5}$ samples. The data are
normalized such that they sum up to $1$ for each column. The measure of
Gaussian Steering $(\mathcal{G}^{A\to B}(\gamma_{AB}))$ is defined in Eqs.
(44) and (47).

###  6.2 Random two-mode covariance matrices

For the case of general unknown CMs we test our method by generating random
two-mode CMs and using random measurements. We start with a CM of a thermal
state, which is a diagonal matrix with the symplectic eigenvalues $\nu_{i}\geq
1$ for every mode $i=1,...,N$, related to the thermal photon number $n_{i}$ as
$\nu_{i}=2n_{i}+1$ [14]:

$\gamma_{th}=\bigoplus_{i=1}^{N}\left({\begin{array}[]{*{20}c}\nu_{i}&0\\\
0&\nu_{i}\\\ \end{array}}\right),$ (48)

where $\nu_{i}$ are randomly generated from a uniform distribution in a finite
interval $[1,t]$, $t>1$. Any general CM can be obtained from a thermal state
CM that undergoes a symplectic transformation:

$\gamma=S\gamma_{th}S^{\rm T}.$ (49)

For random symplectic matrices we use the singular value decomposition (see Eq
(11)). We randomly generate unitary matrices $X$ and $Y$ from the Haar
distribution, while for one-mode squeezers $S(r_{i})$ we create random
parameters $r_{i}$ by a uniform distribution in a finite interval. The MATLAB
code generating symplectic matrices as described above, was developed in Ref.
[31].

Figure 2: Fraction of steering detection for two-mode randomly generated CMs
of thermal states with symplectic eigenvalues $\nu_{i}\in[0,5]$ and symplectic
transformations with squeezing parameter $r_{i}\in[0,2]$. It represents
$5\times 10^{5}$ runs of the algorithm where measurement directions are added
successively and the SW is evaluated at every round until steering is
certified. The data are normalised for every value of steering
$(\mathcal{G}^{A\to B}(\gamma_{AB}))$ such that they sum up to 1.

In Figure 2 we illustrate the data from running the algorithm for $5\times
10^{5}$ different randomly generated CMs of thermal states with symplectic
eigenvalues randomly generated from a finite interval $\nu_{i}\in[0,5]$ and
symplectic eigenvalues with squeezing parameters $r_{i}\in[0,2]$. It shows the
efficiency of steering detection by calculating the SW (see Eq. (5.1)) for
every newly added measurement direction until it finds steering. Comparing to
the particular case of SVS discussed in Figure 1 the detection of steering in
general CMs shows slight improvement for low steering where only for a
fraction of $0.45$ of the CMs steering was certified by the 10th measurement.
For high steering one may need most of the time $8-9$ measurements for the
detection of steering. The detection of two-mode Gaussian steering shows a
similar behaviour as the detection of entanglement in randomly generated CMs
[19]. The stronger the correlation the easier steering or entanglement can be
detected ( i.e. it requires fewer measurements).

###  6.3 Three-mode continuous variable GHZ states

The CV counterpart for the GHZ three qubit states is experimentally created
from three squeezed beams (two position-squeezed beams with squeezing $r_{p}$
and one momentum-squeezed beam with squeezing $r_{m}$) which are mixed in a
double beam splitter [32]. The CM obtained in this manner corresponds to a
pure, symmetric, genuine multipartite entangled three-mode Gaussian state,
also known as CV GHZ state [33]:

$\gamma_{GHZ}=\begin{pmatrix}a&0&c&0&c&0\\\ 0&b&0&-c&0&-c\\\ c&0&a&0&c&0\\\
0&-c&0&b&0&-c\\\ c&0&c&0&a&0\\\ 0&-c&0&-c&0&b\end{pmatrix},$ (50)

where $a$ is related to the squeezing parameters in momentum $r_{m}$ and
position $r_{p}$ as [33]:

$a=\frac{1}{3}\sqrt{4\cosh{2(r_{m}+r_{p})}+5},$ (51)

and

$\displaystyle b$ $\displaystyle=\frac{1}{4}(5a-\sqrt{9a^{2}-8}),$ (52)
$\displaystyle c$ $\displaystyle=\frac{1}{4}(a-\sqrt{9a^{2}-8}).$ (53)

In fact, a proper (unnormalized) CV GHZ state is a simultaneous eigenstate
with zero eigenvalues of total momentum $\hat{p}_{1}+\hat{p}_{2}+\hat{p}_{3}$
and relative positions $\hat{x}_{i}-\hat{x}_{j}$ $(i,j=1,2,3)$, whereas the CV
state described in Eq. (50) approaches the CV GHZ state in the limit of
infinite squeezing $(a\rightarrow\infty)$ [34].

Figure 3:  Fraction of steering detection for three-mode CV GHZ states with
$a=2,3,...,26$, see Eq. (50). The data are obtained from $4.5\times 10^{5}$
runs of the algorithm, and is normalized such that they sum up to 1. Steering
is quantified by $(\mathcal{G}^{A\to B}(\gamma_{GHZ}))$ defined in Eq. (43).

Let us consider the partition where Alice has one mode (it is not important
which mode, due to the special form of the CM in Eq. (50)) and Bob has the
other two modes. By calculating the symplectic eigenvalues of the Schur
complement corresponding to the situation where Alice is trying to steer Bob’s
state by her choice of measurements (see Theorem 1), we obtain that the CV GHZ
state is steerable for $a>1$, and the amount of steering is given by the
measure in Eq. (43).
In Figure 3 we present the efficiency of steering detection in GHZ states
described by the CM given in Eq. (50) using the method of SWs as a function of
the number of random measurements. The algorithm was applied to $4.5\times
10^{5}$ samples, where for each sample the number of measurement settings to
detect steering was recorded. For three-mode CMs there are $21$ independent
measurement settings required for full tomography, whereas with our method we
detect steering mostly with $19$ measurements. This is despite the fact that
in our case $N_{B}=2$, and the linear constraints for the SWs defined in
Proposition 1 are stronger than required in Theorem 3. However, a small
fraction of $3\%$ of steerable GHZ states are detected by more measurements
than needed in full tomography.

##  7 Statistical analysis

In real experiments the homodyne data is obtained by $n$ repetitions of a
measurement with direction $\theta_{i}$, giving rise to a collection of
outcomes $X_{ij}=\langle\hat{x}_{\theta_{i}}\rangle_{j}$, ($j=1,\ldots,n$). In
the case of Gaussian states these outcomes are governed by the normal
probability distribution $\mathcal{N}_{i}(\mu_{i},m_{i})$ with the mean
$\mu_{i}=\langle\hat{x}_{\theta_{i}}\rangle$, and variance $m_{i}={\rm
Tr}[P_{i}\gamma]=(\Delta\hat{x}_{\theta_{i}})^{2}$ (see Sec. 5). The variances
$m_{i}$ are estimated by the sample variances, which follow the
$\chi_{n_{i}-1}^{2}$ distribution [35]. From the statistical error carried out
by a $\chi_{n_{i}-1}^{2}$ distribution and using standard error propagation
the resulting error for $\bar{Z}={\rm Tr[Z\gamma]}$ becomes [19]:

$\Delta\bar{Z}=\sqrt{\sum_{i}\Big{(}\frac{d\bar{Z}}{d\bar{P}_{i}}\Big{)}^{2}(\Delta\bar{P}_{i})^{2}}=\sqrt{\frac{2}{n-1}}\sqrt{\sum_{i}c_{i}^{2}m_{i}^{2}}.$
(54)

Figure 4: The statistical estimate of $\bar{Z}$ for a squeezed vacuum state
covariance matrix $\gamma$ with the maximum of $3\sigma$ confidence interval
according to Eq. (54). The horizontal black dashed line indicates the minimal
value of the witness for the considered CM $\Tr[Z_{min}\gamma]=0.7477$. The
vertical dashed lines indicate the number of measurement repetitions required
to detect Gaussian steering with $7$ (blue), $8$ (orange) and $9$ (green)
measurement settings.

Our method of steering detection is based on the SDP presented in Section 5
where the coefficients $c_{i}$ are calculated using $m_{i}$ as input data.
However, the formula in Eq. (54) does not take into account that these two
variables are not independent. We overcome this difficulty by using two sets
of homodyne data, from the first one we derive the coefficients $c_{i}$ and
the second one is used to evaluate $\bar{Z}$ and $\Delta\bar{Z}$ [36].
In Figure 4 we show the estimated value of the SW $\bar{Z}$ with the
$3\sigma-$confidence for a steerable squeezed vacuum state CM $\gamma$ with
${\rm Tr}[Z_{min}\gamma]=0.7477$, as a function of measurement repetitions
$n$. We can see that detecting steering with $7-8$ measurement settings would
require much more repetitions of the measurements compared to the case of $9$
measurement settings. As a result, the total number of measurements is one
order of magnitude less for $9$ measurement settings ($\approx 3\times
10^{3}$) than in the case of $7$ different measurement settings ($\approx
4.6\times 10^{4}$). Therefore, statistical analysis may help the
experimentalist to decide whether it is more advantageous to add new
measurement directions or to increase the number of repetitions of the
measurements in order to detect steering.

##  8 Summary and conclusions

Non-steerable Gaussian states have covariance matrices (CMs) forming a closed
and convex set, which gives rise to hyperplanes or steering witnesses (SWs) ,
separating any other point from this set. This is easily seen from Theorem 2
which provides a criterion for non-steerable CMs. Based on this result we
characterized the SWs for Gaussian states by providing a set of constraints to
be satisfied by any such witness.

The method of detecting quantum correlation by witnesses has proven to be very
useful and accessible in experiments with entangled states [37], while an
extensive study about the efficiency of entanglement detection with random
measurements in continuous variable states was presented in Ref. [19]. In this
article we have developed similar optimization tools for detecting steering in
Gaussian states with similar complexity and scaling behaviour.

We proposed a set of stronger linear constraints on steering witness operators
in comparison to entanglement witnesses (EWs) and studied the efficiency of
steering detection with respect to the number of measurement settings. These
linear constraints are proven to fully characterize the set of steering
witnesses when the steered party has only one bosonic mode. The SDP we
developed in Section 5 uses random measurements of variances from the homodyne
detection, as the building blocks for constructing the optimal steering test
for a given unknown state.

In the case of two-mode squeezed vacuum states we noticed that detection of
steering in Gaussian states requires more measurements on average than for
entanglement detection [19]. This lies in agreement with the fact that SWs
satisfy stronger conditions than EWs (see the discussion above Theorem 4).

We applied our method also for general random two-mode Gaussian states. In
this case, the detection of both types of quantum correlations, i. e.
entanglement and steering, has a behaviour confirming the general idea that
higher quantum correlations are easier to detect. In particular, high steering
and entanglement in two-mode states are typically detected by $8-9$
measurement settings using our method, while full tomography requires $10$
measurement settings.
In addition, we provided an example of steering detection when the steered
party consists of more than one mode, namely the three-mode continuous
variable GHZ states. In this case the linear constraints used in the
optimization are stronger compared to the SW constraints in Theorem 4,
reducing the set of possible SWs. Nevertheless, the result shows that in most
of the cases the GHZ states are detected with steering by $19$ measurements or
less, which is two measurements fewer than in full tomography. For $N$-mode
CMs when the steered party (Bob) has $N_{B}\rightarrow\infty$ number of modes
we have shown that the linear constraints tend to be equivalent to the exact
constraints fully characterizing the set of SWs. Therefore, our method of
steering detection may become better for larger number of modes. We provided
also a statistical analysis of our method showing a good robustness to
statistical errors.

#### Acknowledgements.

D.B. and H.K. acknowledge financial support by the QuantERA project QuICHE via
the German Ministry of Education and Research (BMBF Grant No. 16KIS1119K) and
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for
Quantum Computing (ML4Q) EXC 2004/1 - 390534769. T.M. and A.I. acknowledge
financial support received from the Romanian Ministry of Research, Innovation
and Digitization, through the Project PN 23 21 01 01/2023.

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