Network mechanism for generating genuinely correlative Gaussian states

Generating a long-distance quantum state with genuine quantum correlation (GQC) is one of the most essential functions of quantum networks to support quantum communication. Here, we provide a deterministic scheme for generating multimode Gaussian states with certain GQC (including genuine entanglement). Efficient algorithms of generating multimode states are also proposed. Our scheme is useful for resolving the bottleneck in generating some multimode Gaussian states and may pave the way towards real world applications of preparing multipartite quantum states in current quantum technologies.


I. INTRODUCTION
The existence of multipartite quantum states that cannot be prepared locally is at the heart of many communication protocols in quantum information science, including quantum teleportation [1], dense coding [2], entanglement-based quantum key distribution [3], and the violation of Bell inequalities [4,5].Therefore, preparing a desired multipartite quantum state from some available resource states under certain quantum operations is of great foundational and practical interest.Among the quantum correlation, the entanglement is used firstly as a physical resource, so preparing bipartite entangled states under the class of local operations and classical communication have been studied extensively [6][7][8][9][10].However, recent study has undergone a major development to multipartite scenarios featuring several independent sources that each distributes a resource state [11].The independence of sources reflects a network structure over which parties are connected.This is not only due to researcher's interests in understanding quantum theory and its relationship in more sophisticated and qualitative scenarios [12][13][14][15][16] but also technological developments towards scalable quantum Quantum networks are of high interest nowadays, which are the way how quantum sources distribute particles to different parties in the network.Quantum networks play a fundamental role in the long-distance secure communication [20,21], exponential gains in communication complexity [22], clock synchronization [23] and distributed quantum computing [24].Most importantly, for the last two decades, generating a multipartite state via appropriate quantum operations from states having lesser number of parties with the assurance of multipartite correlation has been regarded as a benchmark in the development of quantum networking test beds [25][26][27][28].The network mechanism has been used to generate special multipartite states which play an important role for quantum computation and quantum communication tasks [29][30][31][32][33][34].
In this research direction, the infinite dimensional counterpart of the above-mentioned state preparation method should be explored.In particular, Gaussian states constitute a wide and important class of quantum states, which serve as the basis for various types of continuous-variable quantum information processing [35].The goal of this paper is to find the Gaussian networks [36] mechanism for generating multimode Gaussian states.
We provide a protocol for generating multimode Gaussian states with certain amount of genuine Gaussian quantum correlation (GGQC) over a large quantum Gaussian network.This provides a generic method to deterministically generate multimode Gaussian states arXiv:2402.07489v1[quant-ph] 12 Feb 2024 with GQC.
Precisely, we consider quantum Gaussian networks in continuous-variable (CV) systems consisting of spatially separated nodes (parties) P 1 , P 2 , . . ., P N , s (s ≤ N ) independent sources, each generating an n i -mode Gaussian state |ϕ l ⟩ (l = 1, 2, . . ., s).And each node P i consists of m i modes [11].If the nodes share more than one source with other nodes, we call them intermediate nodes.Other nodes are called extremal nodes.Our protocol is to apply 2−mode Gaussian unitary operations U i at intermediate parties and the 2 modes are from different sources.Define Gaussian operation and Φ = Π i Φ i , where I i denotes the identity operator acting on the rest of the modes except modes acted by U i (see Figure 1).We examine the relation between GGQC of resultant state Φ( l |ϕ l ⟩) and GGQC of the source states {|ϕ l ⟩, l = 1, 2, • • • , s}.And show that to make a quantum network having certain amount of GGQC, one needs to create source states containing at least the same amount of GGQC, since the minimum GGQC among the source states coincides with the GGQC of the resultant state Φ( l |ϕ l ⟩), obtained after applying optimal Gaussian unitary operations on the initial state l |ϕ l ⟩.
We note that all Gaussian unitary operations that maximize the GGQC of source states in our scheme are called optimal Gaussian unitary operations.The paper is organized as follows.After reviewing detailed definitions and notations of continuousvariable systems in Sec.II.We provide a GGQC measure in Sec.III.We then give our protocol for generating multimode Gaussian states with certain amount of GGQC in Sec.IV.The last section is a summary of our findings.The Appendix gives the proof of our results.

II. BACKGROUND ON GAUSSIAN SYSTEMS
We now review some definitions and notations concerning Gaussian quantum information theory ( [35,37,38]).Recall that an n-mode Gaussian system is deter- Denote by S(H) the set of all quantum states in a system described by H (the positive operators on H with trace 1).The characteristic function χ ρ for any state ρ ∈ S(H) is defined as , ) < ∞ for all r = 1, 2, . . ., 2n.For ρ ∈ FS(H), its first moment vector = (tr(ρ R1 ), tr(ρ R2 ), . . ., tr(ρ R2n )) T ∈ R 2n and its second moment matrix defined by γ kl = tr[ρ(∆ Rk ∆ Rl +∆ Rl ∆ Rk )] with ∆ Rk = Rk − ⟨ Rk ⟩ ( [39]) are called the mean and the covariance matrix (CM) of ρ respectively.Here M k (R) stands for the algebra of all k × k matrices over the real field R.
Note that a CM Γ must be real symmetric and satisfy the uncertainty condition Γ + iΩ ≥ 0. A Gaussian state ρ ∈ FS(H) is such a state of which the characteristic function χ ρ (z) is of the form For an n-mode CV system determined by

III. A GGQC MEASURE
An amazing feature of quantum mechanics is the existence of quantum correlations.Various methods for quantifying quantum correlations are one of the most actively researched subjects in the past few decades [9,35,40].Measurements of quantum correlations have played an important role in understanding the properties of quantum many-body systems and their nonclassical behaviors.
In the following, we will propose a definition of GGQC measure.To the best of our knowledge, this is the first thought to define multimode genuine Gaussian quantum correlation measure beyond entanglement.In addition, a pure Gaussian state with genuine Gaussian quantum correlation under our GGQC measure is also genuine entanglement [26,41].
For any n−mode Gaussian state ρ A1,A2,...,An on (H A1 ⊗ H A2 ⊗ • • • ⊗ H An ), its CM can be represented as where encode the intermodal correlations between subsystems A i and A j .For any the quantity is discussed in [38,42,43].It is evident that, for any 2partition P of n-mode system A 1 A 2 . . .A n , there exists a permutation π of (1, 2, . . .n) and positive integers n 1 , n 2 with n 1 + n 2 = n such that One can compute the M(ρ) with respect to P denoted by M ρ (P).Now we provide the definition of our GGQC measure.
Definition 1 For any n-mode Gausian state ρ, define the quantity GM(ρ) = min P M ρ (P), here P runs over all 2partitions.
Note that any 2-partition P corresponds a subset α of {1, . . ., n}.Let D ρ (α) be the principle minor that lies in the rows and columns of Γ ρ indexed by α and α denotes its complement set.Then M ρ (P) is also written as M ρ (α) and In fact, GM has the following properties which satisfy the basics of Gaussian quantum correlation measure [38,40,[42][43][44][45].
(2) GM(ρ) = 0 if and only if ρ is a product state with respect to at least one modal bipartition.
It is evident that if GM(|ϕ⟩) ̸ = 0, then |ϕ⟩ is not a product state with respect to any 2-partition of {1, 2, • • • , n}, so we say |ϕ⟩ is genuinely correlative.The property is harmonic with the key generalized geometric measure of genuine entanglement which is defined as the shortest distance of a given multimode state from a nongenuinely multimode entangled state [41].This implies |ϕ⟩ is genuinely correlative if and only if |ϕ⟩ is genuinely entangled.Genuine correlation and genuine entanglement [26,41] are not coincident for mixed states since GM(ρ) ̸ = 0 if and only if ρ is not a product state with respect to any 2-partition of {1, 2, • • • , n}.Compared with some known entanglement measures, such as the distillable entanglement, the entanglement of formation, the entropy of entanglement and the generalized geometric measure [35,41], GM is more easy to calculate since all 2-partitions of {1, 2, • • • , n} are finite and no optimization process is involved.In the next paragraph, we will compute the value of GM for some important Gaussian states.To the best of our knowledge, GM is the only known multimode genuine Gaussian quantum correlation measure beyond entanglement.Since genuine multipartite entanglement has become a standard for quantum many-body experiments [46][47][48][49], GM may become one of the best prospects for unveiling essential Gaussian quantum correlation of multimode systems.
For any 2-mode Gaussian pure state |ϕ⟩, under some suitable local Gaussian unitary operation, its CM can be reduced to the standard form [50] γ ≥ 1 is the single-mode mixedness factor and I 2 is the 2 × 2 unit matrix.A direct computation shows In fact, using the standard form of CM for any 2-mode Gaussian state ρ, 51,52], one can obtain For the case of 3-mode, we analyse a pure state |ϕ γ ⟩ prepared by combining three single-mode squeezed states in a tritter (a three-mode generalization of a beam splitter), which possesses the CM, given by [53], where R ± = cosh(2γ) ± 1 3 sinh(2γ) and S = − 2 3 sinh(2γ).By a direct computation, Therefore we provide a formula of GM as a function of the squeezing strength γ.It is evident that the GM approaches its maximum value 1 as γ → ∞.Combining this and computing formula of the generalized geometric entanglement measure G(.) on |ϕ γ ⟩ [41], we can find an interesting fact for pure states |ϕ γ1 ⟩, |ϕ γ2 ⟩.This tells that the measures GM and G have the same order on three single-mode squeezed states in a tritter.

STATES WITH GGQC
We now introduce a procedure for preparing a Gaussian network to be in a large multimode state with certain amount of GGQC.Let us consider a Gaussian network with N parties (nodes) P 1 , P 2 , . . ., P N , s (s ≤ N ) independent sources, each generating an n i -mode Gaussian state |ϕ i ⟩ (i = 1, 2 . . ., s).Then the quantum Gaussian network is a system involving n = s i=1 n i modes, the initial state is given by ρ = ⊗|ϕ i ⟩.Our main result reads as follows.
Theorem 4.1.For initial state ρ = ⊗|ϕ i ⟩, there exist optimal Gaussian unitary operations such that the resultant states give maximal GGQC by Let us now stress some key points about Theorem 4.1.(i) Theorem 4.1 provides an explicit formula for the maximum GGQC that can be generated by our protocol.We need to prepare a number of low mode source states containing at least the same amount of GGQC in order to create a multimode Gaussian state with certain amount of GGQC.Note that the property of genuine correlation and genuine entanglement [41] is harmonic for any pure Gaussian state, our protocol also supports generation of multipartite genuinely entangled states in continuous-variable systems.This provides an important supply on generation of entangled states in discretevariable systems [6][7][8][9][10].
(ii) Theorem 4.1 tells us that resultant state remains genuine correlation as long as all source states are genuinely correlative.This implies that multiple choices of the set of source states {|ϕ i ⟩} are realistic for creating a multimode Gaussian state with certain genuine correlation.This information is valuable in the situation when one is forced to prepare Gaussian states with lower mode in laboratory in order to generate multimode Gaussian states by our protocol.It is due to the fact that preparing source states like photos in some physical substrates is difficult.Multiple choices also means there are multiple plans information distribution of quantum Gaussian networks.It is wellknown that design of information distribution between multiple nodes is a challenging problem in quantum domains yet [17].In fact, one can compute the mean value and the standard deviation of GM corresponding to different source states.The design of lower mean and lower standard deviation mean lower cost on average and stronger stability of quantum networks.Thus the nonuniqueness of the set of source states is also a crucial point of our protocol.
(iii) For any 0 < c < 1, we can create an n-mode pure Gaussian state ρ with GM(ρ) = c from 2-mode pure Gaussian states and 3-mode pure three single-mode squeezed states in a tritter (see Section III).For example, one can create a 7-mode pure Gaussian state ρ with GM(ρ) = c by applying two 2-mode Gaussian unitary operations over two 2-mode pure Gaussian states and one 3-mode pure three single-mode squeezed state in a tritter.The suitable parameter selection of such source states can guarantee that the resultant state ρ satisfies the condition GM(ρ) = c.By Theorem 4.1, one can see that another critical point in implementing our protocol is to find out the optimal Gaussian unitary operations {U i }.Note that every 2mode Gaussian unitary operation U i is determined by a 4 × 4 symplectic matrix S i (see Section II), we will provide a one-parameter classification of S i in order to identify the optimal Gaussian unitary operations.For fluency of paper, such one-parameter classification is placed in appendix.Based on such one-parameter classification, the optimal Gaussian unitary operations {U i } can be given as follows.

Theorem 4.2. If the CM of |ϕ i ⟩ reads as
then the optimal Gaussian unitary operation U i can always be designed as Table I, here λ i is one-parameter classification of symplectic matrix S i determining U i .) 2 + (γ To identify optimal Gaussian unitary operations by Theorem 4.2, we consider a simple scenario of a chain or a star network consisting of three identical three singlemode squeezed states in a tritter |ϕ γ ⟩ (Fig. 2).In Section III, it is shown In the case of a chain network (Fig. 2(a)), we apply 2mode Gaussian unitary operation U 1 , U 2 on party P 2 and P 3 respectively.The resultant state denoted by |ψ U1,U2 ⟩ is a 9−mode state, In the case of a star network (Fig. 2(b)), we apply 2−mode unitary U 1 on P 51 and P 52 , U 2 on P 51 and P 53 respectively.Φ 1 , Φ 2 is defined as following: If the maximum is reached at some U 1 , U 2 , we say U 1 , U 2 are optimal Gaussian unitary operations.We will find optimal Gaussian unitary operations of type I in Theorem 4.2.Note that there is a local Gaussian unitary U such that U |ϕ γ ⟩ has the CM The table I of Theorem 4.2 shows that symplectic matrices of type I determining U i (i = 1, 2) satisfy the condition Hence U 1 , U 2 are 2−mode squeezing operation.Recall that a 2−mode squeezing operation is an active transformation which models the physics of optical parametric amplifiers and is routine to create CV entanglement.It acts on the pair of modes i and j via the unitary are optimal Gaussian unitary operations.Additionally, the table I of Theorem 4.2 also provides some other possible choices of optimal Gaussian unitary operations.

V. CONCLUSION
Gaussian networks are fundamental in network information theory.Here senders and receivers are connected through diverse routes that extend across intermediate sender-receiver pairs that act as nodes.The quantum network is Gaussian if the operations at the nodes and the final state shared by end-users are Gaussian.Although classical Gaussian networks is established rigorously, the quantum analogue is far from mature [36].Therefore, it is interesting to find the Gaussian network mechanism for creating a multimode state having certain amount of genuine correlation.
In this paper, we present a deterministic scheme for generating Gaussian states with certain amount of GGQC and distribute them in the form of Gaussian quantum networks.Given limited amount of sources, our scheme can generate genuinely correlative Gaussian states (including genuinely entangled Gaussian states) with the application of optimal Gaussian unitary operations.An explicit description of optimal Gaussian unitary operations is also provided.
Our choice for generating Gaussian states with certain amount of GGQC is not unique since there are multiple choices of optimal Gaussian unitary operations and source states.This raises naturally one interesting question whether all these choices are equivalent, or a subset of these choices are more beneficial.It is key to comprehend the mechanism of information distribution in quantum Gaussian networks [17].
Proof of Proposition From the singular value decomposition, we have Next, we divide four cases according to the value of det(S 11 ).
From Equations ( 4) and ( 5), one gets S 22 = 0, det(S 12 ) = det(S 21 ) = 1.Applying the singular value decomposition to S 12 and S 21 , we can find Then L 9 SR 7 has the form S V .
To prove Theorem 4.1 and Theorem 4.2, we consider a simple scenario having three parties and two sources [see Fig. 3].Here the first two parties share a m-mode state ρ 1 , the second and third parties share a n-mode state ρ 2 , and the central party is performed 2-mode The general protocol can be reduced to this scenario.Taking the example in Fig. 1 again, here If Theorem 4.1 holds true in the simple scenario, then Thus max {Ui} GM(Φ(ρ)) = min i {GM(|ϕ⟩ i )}.
We can obtain that Eq.( 9) is equivalent to Apply a similar process, we can get other Types of S (Table I).

FIG. 2 :
FIG. 2: Protocols in a chain or a star network with three identical 3−mode squeezed vacuum states as source states.
FIG. 3: Schematic representation of 2-mode Gaussian unitary operations acting on two sources.