Steering witnesses for unknown Gaussian quantum states

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.


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 deviceindependent 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 ≥ 0 such that Tr[Zγ s ] ≥ 1 holds for all separable covariance matrices γ s , while Tr[Zγ] < 1 for some entangled covariance matrix γ [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.

Gaussian states
A continuous variable (CV) system of N bosonic modes is described by the canonical operators of position and momentum xk and pk in the Hilbert space H = N k=1 H k , where H k is the infinite-dimentional Hilbert space of mode k [10,12,13].Defining the vector of canonical operators RT ≡ ( R1 , ..., R2N ) = (x 1 , p1 , ..., xN , pN ) one obtains the commutation relation written as (we assume = 1): where Î is the identity matrix, and Ω ij are the elements of the symplectic matrix Gaussian states are fully described by the first and second order statistical moments, namely the displacement vector d = Tr[ Rρ], and the covariance matrix (CM) γ with its elements defined as [14,21]: 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): In the following we will mostly consider a bipartite Gaussian quantum state ρ AB of N modes with the CM of the following block structure: where γ A and γ B are the CMs of the subsystems of Alice and Bob with N A and N B modes, respectively, γ 12 is the correlation matrix between the two parties, and It can be readily shown that the uncertainty relation in Eq. ( 4) directly implies γ AB > 0, γ A > 0 and γ 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: where r 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 ρ in Hilbert space [10]: from where the characteristic function is readily defined as χ ρ (r) = Tr[ D † (r)ρ].If ρ represents a Gaussian state then its characteristic function takes a particular form [10]: where the first moments are considered to be zero and γ AB is the CM associated with the Gaussian density operator ρ.A Gaussian measurement [22] applied on Alice's subsystem can be described by a positive Gaussian operator Â with a CM given by T A satisfying the CMC: where ÎB 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 ρA B after measurement form a Gaussian state with CM given by γ A B = γ B − γ T 12 (γ A + T A ) −1 γ 12 , which represents the Schur complement of the matrix γ AB + T A ⊕ 0 B with respect to the submatrix γ 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 Then H > 0 if and only if A > 0 and H/A = B − X † A −1 X > 0, where H/A is the Schur complement of block A of the matrix H.
Given that γ AB is a CM satisfying the uncertainty relation in Eq. ( 4) we have γ AB > 0 and γ A > 0, and hence the Schur complement of γ AB with respect to γ A is also a positive matrix γ AB /γ A := γ B − γ T 12 γ −1 A γ 12 > 0. Let us consider the matrix γ AB − 0 A ⊕ σ B , with σ B ≥ 0, and its Schur complement with respect to γ A .Applying the positivity conditions from Lemma 1 leads to the Schur complement [10,24] 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 ≥ 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.

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: γ = SγS T .The singular value decomposition of a real symplectic matrix gives [25]: where S(r i ) is a one-mode squeezing matrix (symplectic and nonorthogonal) with r i the squeezing parameter: 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]: where the corresponding symplectic matrices are given by: According to the Williamson theorem [27] every real symmetric matrix M ≥ 0 can be brought to a diagonal form through symplectic transformations as follows: where s 1 , . . ., s N ≥ 0 are called symplectic eigenvalues of M .By we denote the symplectic trace of the matrix M .

Gaussian quantum steering
Consider the situation where Alice and Bob are two distant parties sharing a common state ρ AB , and Alice performs local measurements Â 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 η ρ η } of preexisting local hidden states ρ η with probabilities p η , and a stochastic map P (a| Â, η) ‡ from the hidden variable η to a, such that Bob's conditioned state after Alice's measurement is given by [7]: This represents the local hidden state (LHS) model, where Bob checks if Alice can simulate the state ρA B based on her knowledge of the parameter η, by drawing the states ρ η according to the distribution p η .Conversely, if Bob cannot find any ensemble F and map P (a| Â, η) satisfying Eq. ( 17), then Bob must admit that Alice can steer his system.
In the Gaussian realm where the joint state ρ AB is a Gaussian state with CM γ 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 Â with a Gaussian characteristic function as in Eq. ( 8) with CM T A , satisfying T A + iΩ N A ≥ 0. When Alice performs a measurement Â and gets an outcome a, Bob's conditioned state ρA B is a Gaussian state with CM given by γ [10].In Ref. [7] a criterion for Gaussian steerability of CM γ AB was derived: A Gaussian state with covariance matrix γ AB with the block form defined in Eq. ( 5) is Alice → Bob non-steerable by Gaussian measurements if and only if holds, where γ AB /γ A := γ B − γ T 12 γ −1 A γ 12 is the Schur complement of γ AB with respect to submatrix γ A and Ω 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 γ 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 ρ AB with covariance matrix γ AB with blocks defined in Eq. ( 5) is Alice → Bob non-steerable by means of Gaussian measurements if and only if there exists a covariance matrix corresponding to Bob's system σ B satisfying σ B + iΩ N B ≥ 0, such that: (20) ‡ P (a| Â, η) denotes the probability distribution of Alice to obtain outcome a when measuring Â and given a local hidden variable η.
§ This follows from applying the positivity conditions from Lemma 1 to the total matrix in Eq. ( 18).
Proof.⇒ Theorem 1 states that the Schur complement γ AB /γ A of a non-steerable CM γ AB is also a CM, i.e. γ AB /γ A + iΩ N B ≥ 0. Based also on the definition of the Schur complement in Eq. ( 10) we state that if a CM γ AB is non-steerable then there exists a CM σ B fulfilling Eq. ( 20), and it is the Schur complement γ AB /γ A .⇐ Conversely, if the relation in Eq. ( 20) is fulfilled for some CM σ B , satisfying CMC σ B + iΩ N B ≥ 0, then by the definition of the Schur complement in Eq. ( 10) it follows that there exists a positive semi-definite matrix P ≥ 0 such that: σ B + P = γ AB /γ A .Therefore, the Schur complement γ AB /γ A also has to fulfill the CMC, since σ B + P + iΩ N B ≥ P ≥ 0, which based on Theorem 1, means that ρ 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 γ is separable with respect to parties A, B if and only if there exist local CMs γ A and γ B for each partition, such that γ ≥ γ A ⊕γ B [26].Hence, it is obvious that any separable Gaussian state is also non-steerable by Gaussian measurements since γ A ⊕ γ B ≥ 0 A ⊕ γ B holds.Thus, Gaussian non-steerability represents a stronger condition than separability of Gaussian states.

Steering witnesses based on second moments
Based on Theorem 2 we can define the set of non-steerable CMs as follows: (21) where A → 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 × 2N -matrices where γ AB is the CM of a bipartite system with N = N A + N B modes, A → B denotes Alice to Bob non-steerability, where Γ A →B (R 2N ) is defined in Eq.( 21).All matrices Z ∈ Z A →B (R 2N ) for which there exists γ with Tr[Zγ] < 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 ≥ 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 × 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 ∈ Z A →B (R 2N ), if and only if 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 ≥ 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 ≥ 0 and satisfying also str[Z] ≥ 1 2 is equivalent to obtaining Tr[Zγ] ≥ 1 for all CMs γ.In the following we will prove that condition str[Z B ] ≥ 1 2 guarantees that Tr[Zγ] ≥ 1 for all A → B non-steerable CMs.Therefore, condition str[Z] < 1  2 together with conditions (23,24) assure that there exists a CM γ such that Tr[Zγ] < 1, i.e. it is A → B steerable.Consider the SWs based on second moments of the following block form: In the following we will use an important result discussed in Ref. [18] : where the minimization is performed over the set of CMs σ B .⇒ We have to prove that condition in Eq.( 24) holds for any SW described in Definition 1 (22).Based on Theorem 2 for any γ AB ∈ Γ A →B (R 2N ) there exists a CM corresponding to the quantum state of Bob σ B such that: holds.Moreover, the matrix 0 A ⊕σ B is a non-steerable CM itself, and from the Definition 1 of SWs in Eq. ( 22) we have the following condition: The expression 2 str[Z] represents the value of the ground state energy of the Hamiltonian defined as Ĥ = Z kl Rk Rl , where R = (x 1 , p1 , ..., xN , pN ) T is the vector of canonical operators.
Using Eq. ( 27) we conclude that SWs have to fulfill the condition ⇐ Conversely, we prove that a matrix Z satisfying the conditions in Eq. ( 24) represents a SW with Tr[Zγ AB ] ≥ 1 for any non-steerable CM γ AB ∈ Γ A →B (R 2N ).Starting from Eq. ( 24) and using Eq.(27)  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 γ 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: where Z is a real symmetric 2N × 2N matrix satisfying 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 str , 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.

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: 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 ¶: with , where z j , j = 1, • • • 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: Now, the sum of symplectic eigenvalues gives: 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 1 2N B > 0. This assures that the sum of symplectic eigenvalues exceeds 1  2 , however, it rules out the possibility that some symplectic eigenvalues may satisfy z j ∈ (0, 1 2N B ).An exception represents the case when the steered party has one mode (N B =1) and Z B in Eq. ( 34) is a 2 × 2 matrix with only one symplectic eigenvalue satisfying z ≥ 1 2 , which is equivalent to relation (33).However, the interval of forbidden symplectic eigenvalues z j / ∈ (0, 1 2N B ) is the largest for N B = 2, while in the limit of large N B → ∞ we have 1 2N B → 0, and the constraints in Proposition 1 become equivalent to the constraints fully characterizing SWs in Theorem 4. ¶ Given that Tr[M ] ≥ 2 str[M ] holds for any positive matrix M [28], the symplectic transformations do not only preserve the symplectic eigenvalues, but also positivity.

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 = 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: where c is the vector of coefficients c j and m = Tr[Pγ], with 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 ϕ between the vertical and horizontal polarization components, a polarization rotator on angle φ and a polarizing beam splitter, for mixing the initial modes denoted by â and b [29].The mode k arriving at the detector has the following expression: The generalized quadrature operator is given by: which covers the whole continuum of quadratures in terms of initial modes â and b for θ ∈ [0, π], φ ∈ [0, π] and ϕ ∈ [0, 2π).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: where P is the measurement matrix of the variances we use in Eq. ( 38): The N −mode CM is a symmetric, real 2N × 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 θ, φ and ϕ.This scheme can be extended to N modes with a single homodyne detector by using the same twomode combination scheme N − 1 times.Starting with three initial modes â, b and ĉ, the generalized mode arriving at the detector would be: k = exp(iϕ 1 ) cos φ â + exp(iϕ 2 ) sin φ cos ψ b + sin φ sin ψ ĉ. (43)

Detection of Gaussian steering
Our method of detection relies on the SDP described in Eq. ( 38), 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 θ, φ and ϕ.
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 μj ≥ 1 for all j = 1, . . ., N B , where μj represent the symplectic eigenvalues of the Schur complement γ AB /γ A .The Gaussian A → B steerability can then be quantified via [11]: 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]: Steering can also be quantified by a minimal SW, where Z min corresponds to the smallest possible value Tr[Z min γ AB ] = w min for a CM γ AB .The program proposed in Eq. (38) 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 γ AB our method shows that the minimal SW Z min is related to the steerability measure from Ref. [11] as follows:

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: where r is the squeezing parameter.For this particular case, using formula (45), the expression quantifying steering is a monotonic function in r: 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).

Random two-mode covariance matrices
For the case of general unknown CMs we test our method by generating random twomode CMs and using random measurements.We start with a CM of a thermal state, which is a diagonal matrix with the symplectic eigenvalues ν i ≥ 1 for every mode i = 1, ..., N , related to the thermal photon number n i as ν i = 2n i + 1 [14]: where ν 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: 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].In Figure 2 we illustrate the data from running the algorithm for 5 × 10 5 different randomly generated CMs of thermal states with symplectic eigenvalues randomly generated from a finite interval ν i ∈ [0, 5] and symplectic eigenvalues with squeezing parameters r i ∈ [0, 2].It shows the efficiency of steering detection by calculating the SW (see Eq. ( 38)) 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).

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 momentumsqueezed 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]: where a is related to the squeezing parameters in momentum r m and position r p as [33]: In fact, a proper (unnormalized) CV GHZ state is a simultaneous eigenstate with zero eigenvalues of total momentum p1 + p2 + p3 and relative positions xi − xj (i, j = 1, 2, 3), whereas the CV state described in Eq. ( 51) approaches the CV GHZ state in the limit of infinite squeezing (a → ∞) [34].Fraction of steering detection for three-mode CV GHZ states with a = 2, 3, ..., 26, see Eq. ( 51).The data are obtained from 4.5 × 10 5 runs of the algorithm, and is normalized such that they sum up to 1. Steering is quantified by (G A→B (γ GHZ )) defined in Eq. (44).
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. ( 51)) 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. (44).
In Figure 3 we present the efficiency of steering detection in GHZ states described by the CM given in Eq. (51) using the method of SWs as a function of the number of random measurements.The algorithm was applied to 4.5 × 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.

Statistical analysis
In real experiments the homodyne data is obtained by n repetitions of a measurement with direction θ i , giving rise to a collection of outcomes X ij = xθ i j , (j = 1, . . ., n).In the case of Gaussian states these outcomes are governed by the normal probability distribution N i (µ i , m i ) with the mean µ i = xθ i , and variance m i = Tr[P i γ] = (∆x θ i ) 2 (see Sec. 5).The variances m i are estimated by the sample variances, which follow the χ 2 n i −1 distribution [35].From the statistical error carried out by a χ 2 n i −1 distribution and using standard error propagation the resulting error for Z = Tr[Zγ] becomes [19]: 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. (55) 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 Z and ∆ Z [36].
In Figure 4 we show the estimated value of the SW Z with the 3σ−confidence for a steerable squeezed vacuum state CM γ with Tr[Z min γ] = 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 (≈ 3 × 10 3 ) than in the case of 7 different measurement settings (≈ 4.6 × 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.

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 → ∞ 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.

Figure 1 .
Figure 1.Fraction of steering detection for two-mode squeezed vacuum states with r ∈ [0, 2], see Eq. (47): 5 × 10 5 samples.The data are normalized such that they sum up to 1 for each column.The measure of Gaussian Steering (G A→B (γ AB )) is defined in Eqs.(45) and (48).

Figure 2 .
Figure 2. Fraction of steering detection for two-mode randomly generated CMs of thermal states with symplectic eigenvalues ν i ∈ [0, 5] and symplectic transformations with squeezing parameter r i ∈ [0, 2].It represents 5 × 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 (G A→B (γ AB )) such that they sum up to 1.
Figure 3.Fraction of steering detection for three-mode CV GHZ states with a = 2, 3, ..., 26, see Eq. (51).The data are obtained from 4.5 × 10 5 runs of the algorithm, and is normalized such that they sum up to 1. Steering is quantified by (G A→B (γ GHZ )) defined in Eq. (44).

Figure 4 .
Figure 4.The statistical estimate of Z for a squeezed vacuum state covariance matrix γ with the maximum of 3σ confidence interval according to Eq. (55).The horizontal black dashed line indicates the minimal value of the witness for the considered CM Tr[Z min γ] = 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.