Quantum Correlations in Three-Beam Symmetric Gaussian States Accessed via Photon-Number-Resolving Detection and Quantum Universal Invariants

PHYSICAL REVIEW RESEARCH 7, 043292 (2025) Quantum correlations in three-beam symmetric Gaussian states accessed via photon-number-resolving detection and quantum universal invariants Jan Peˇ rina, Jr. ,1,*Nazarii Sudak ,2Artur Barasi´ nski ,2,†and Antonín ˇ Cernoch 3 1Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic 2Institute of Theoretical Physics, University of Wroclaw, Plac Maxa Borna 9, 50-204 Wrocław, Poland 3Joint Laboratory of Optics, Institute of Physics of CAS, 17. listopadu 50a, 779 00 Olomouc, Czech Republic (Received 21 January 2025; accepted 25 November 2025; published 15 December 2025) Quantum correlations of symmetric three-beam Gaussian states are analyzed using their quantum universal invariants. These invariants, one-, two-, and three-beam purities, are expressed in terms of the beams’ intensity moments up to sixth order. The three-beam symmetric Gaussian states with varying amounts of the noise are experimentally generated using entangled photon pairs from down-conversion, their invariants are determined, and their quantum correlations are quantified. The coexistence of bipartite and tripartite entanglement and genuine tripartite entanglement is observed in these states that resemble the noisy GHZ/W states. DOI: 10.1103/vd3h-4pts I. INTRODUCTION The discovery of phase-squeezed light [1], whose observation [2] belongs to the first experiments certifying the existence of nonclassical states of light, triggered the fast development of the wide area of quantum optics devoted to the properties of bosonic modes with their infinitely large Hilbert spaces [3]. Today, this area, known as continuous-variable (CV) quantum optics, represents a particularly promising avenue to develop different types of quantum technologies applicable, e.g., in quantum information processing and quantum key distribution [4]. Compared to other developing quantum technologies (e.g., based on discrete variables), CV states exhibit resilience against decoherence and provide large Hilbert spaces for information encoding. They also allow for purification [5] and entanglement distillation [6]. Moreover, deterministic generation of large entangled states represents the most remarkable attribute of CV fields. CV quantum optics has also become a viable route for quantum computation, where realistic squeezing levels above 10 dB [7] are sufficient to achieve fault tolerance [8]. In addition, squeezed states have been found extraordinarily useful in the detection of the gravitational waves [9]. Although homodyne tomography [10] of Gaussian states can be reduced to just two orthogonal cuts, the use of a local oscillator requires advanced experimental setups. Nevertheless, at present, there exists an alternative promising *Contact author: [email protected] †Contact author: [email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. for practical applications: photon-number-resolving detectors [11]. The fact that they cannot capture the detected-state phase properties does not have to be a drawback. This especially occurs when complete characterization/identification of a measured state does not require phase information due to its properties. Such states can be recognized when we determine their quantum universal invariants (QUIs) [12] and these QUIs are uniquely determined using only photocount measurements. Such states then immediately become extraordinarily promising for practical applications in various CV-based metrology and quantum information processing protocols. Here, we show that symmetric three-beam Gaussian states (STBGSs) belong to this kind of states. We note that these states are also known as three-mode squeezed thermal states and include, among others, the noisy GHZ/W states [13,14]. They have already demonstrated their effectiveness in the implementation of several CV protocols [15–17]. According to theory, their full identification is reached when we know their one-, two-, and three-beam purities. The main result of our investigations—the formula that gives the three-beam purity in terms of intensity moments up to the sixth order, together with the formula for the two-beam purity discovered recently [18]—then allows for this identification. Subsequently, different forms of quantum correlations can be characterized and classified, including the genuine three-beam entanglement quantified by the Gaussian cotangle [19]. Finally, using experimental three-beam Gaussian fields generated effectively with the help of parametric down-conversion [18,20], the practicality of the developed approach is demonstrated. The success in qualitative simplification of the experimental identification of STBGSs poses a natural question: What are the groups of states whose identification does not require the full quantum state reconstruction and what parameters (QUIs) are necessary for the state identification? This question gains particular significance in multipartite systems, where 2643-1564/2025/7(4)/043292(12) 043292-1 Published by the American Physical Society JAN PE ˇ RINA, JR. et al. PHYSICAL REVIEW RESEARCH 7, 043292 (2025) the structure of quantum correlations is more intricate and complex [21] and thus potentially more attractive for various applications. Moreover, our experimental results for threebeam fields already provide certain insight into understanding whether and how various types of quantum correlations are distributed over many different systems. The paper is organized as follows. The structure and parametrization of three-beam symmetric Gaussian states are discussed in Sec. II. Their entanglement (steering) is theoretically analyzed in Sec. III (IV). Experimental characterization of STBGSs using intensity moments of different orders is discussed in Sec. V. Also, the experimental one-, two-, and three-beam purities and seralian are given in Sec. V,together with other experimental characteristics including mean photon number, the Fano factor, noise-reduction parameter, and the Kullback-Leibler divergence. The properties of the observed genuine tripartite entanglement are analyzed in Sec. VI, together with those belonging to the observed steering. The properties of the corresponding GHZ/W states are discussed in Sec. VII. Section VIII brings conclusions. Estimates and bounds for the quantities characterizing STBGSs are discussed in Appendix Aassuming knowledge of the intensity moments up to fourth order. The role of the number of measurement repetitions in the determination of experimental errors is addressed in Appendix B. II. STRUCTURE OF THREE-BEAM SYMMETRIC GAUSSIAN STATES We consider a CV three-beam single-mode symmetric system [22] described in the Hilbert space H=3 j=1Hj, where Hjstands for the infinite-dimensional Fock space of beam j. Let ˆxj=(ˆaj+ˆa† j) and ˆpj=−i(ˆaj−ˆa† j) be the quadrature operators with aj(a† j) denoting the photon annihilation (creation) operator acting on Hj. The quadrature operators are then collected in the vector ˆ X=(ˆx1,ˆp1,ˆx2,ˆp2,ˆx3,ˆp3),with the canonical commutation relations expressed as [ ˆ Xj,ˆ Xk]= 2ijk, where =3 j=1(01 −10 ) is the symplectic form. Then, any Gaussian state with its statistical operator ˆρ3 is fully specified, up to local displacement, by its covariance matrix (CM) σ3with the matrix elements given by [σ3]jk =1 2(ˆ Xjˆ Xk+ˆ Xkˆ Xj)−ˆ Xjˆ Xk. The CM must fulfill the Robertson-Schrödinger uncertainty relation σ3+i⩾0 [23–25]. For future convenience, let us write the CM σ3of an STBGS in terms of two-by-two submatrices as σ3=⎛ ⎝ αγγ γαγ γγα ⎞ ⎠,(1) where α=diag(a,a) is the local CM corresponding to the reduced state of one beam and γ=diag(c+,c−). We note that Eq. (1) refers to the so-called standard form [26,27]. In our approach, we rely on an alternative parametrization of σ3that uses three out of four state invariants [12]. They admit direct interpretation for generic Gaussian states and include global purity μ3=Det(σ3)−1/2, marginal k-beam purities μk=Det(σk)−1/2for k=1,2, and two-beam Seralian 2=2Det(α)+2Det(γ). By selecting μ1,μ2, and 2as independent parameters (motivated by Ref. [28]), the following relationship can be introduced: a=1 μ1 ,c±=1 4μ2 12 2−4 μ2 2±, ≡1 4   4μ2 1−μ4 122 μ6 1−4μ2 1 μ2 2 .(2) From Eq. (2), one immediately finds that the necessary conditions for a physical state, c±∈R, and the RobertsonSchrödinger uncertainty relation, are simultaneously expressed as μ2 1⩽μ2⩽μ1and (3) F(μ1,μ 2)⩽2⩽min 4 μ2 1−2 μ2 ,1+1 μ2 2, F(μ1,μ 2)=⎧ ⎨ ⎩ 2 μ2,for μ2 1⩽μ2⩽4μ2 1 3+μ2 1 , 1 32+6 μ2 1−ω,for 4μ2 1 3+μ2 1 <μ 2⩽μ1, (4) where ω=(μ2 1+3)2 μ4 1−12 μ2 2 . Importantly, both purities μ1and μ2are experimentally available [18] using the measured intensity moments that are the normally ordered photon-number moments. They are obtained by the Stirling numbers from the measured photonnumber moments [29]. For the Seralian 2, however, only its upper and lower bounds are derived [12], independently of Eq. (4). Nevertheless, in the analyzed symmetric case, 2can readily be determined from purity μ3, as the above analyzed parameters must satisfy the relation μ−2 3=Det(σ3)=1 2μ2 1μ3 26μ2−3μ3 22 2+μ2 1μ3 23 2 +μ2 22 2−43μ2 22−2μ2 1−μ2 1μ2 22 2.(5) This implies that the purities of twoand three-beam systems, along with the single-beam purity, are sufficient to fully identify and thus characterize an STBGS. However, the experimental determination of the three-beam purity using intensity moments is prone to significant errors. This is so since the determination of μ3requires intensity moments up to sixth order [12]. Therefore, following the approach in Ref. [18], it is advisable to parametrize the STBGSs by Eq. (2) and replace the Seralian 2by its minimum and maximum allowed values. In Ref. [18], this method proved useful for obtaining the negativity of two-beam fields with higher degrees of entanglement. Moreover, combining Eq. (5) with Eq. (4), we reveal the upper and lower bounds for μ3in terms of the marginal purities μ1and μ2. Despite their simple form, the three-beam symmetric states σ3exhibit a remarkably rich structure of nonclassical correlations. Whereas their entanglement is addressed in detail 043292-2 QUANTUM CORRELATIONS IN THREE-BEAM SYMMETRIC … PHYSICAL REVIEW RESEARCH 7, 043292 (2025) TABLE I. Conditions for observation of different types of correlations characterizing entanglement and Gaussian steering A→BC and BC →Ain the system with three beams A,B,andCexpressed in terms of purities μ1and μ2. Degrees of purity Correlations Entanglement A−BC μ2 1⩽μ2⩽3μ2 1+√9+62μ2 1−7μ4 1−3 10−2μ2 1 Coexistence Region I 3μ2 1+√9+62μ2 1−7μ4 1−3 10−2μ2 1 <μ 2⩽μ1 √2−μ2 1 Coexistence Region II μ1 √2−μ2 1 <μ 2⩽μ1Entangled states Gaussian steering A→BC μ2 1⩽μ2⩽√16μ2 1+9−3 2Unsteerable states √16μ2 1+9−3 2<μ 2⩽√3μ1 √4−μ2 1 Coexistence region √3μ1 √4−μ2 1 <μ 2⩽μ1Steerable states Gaussian steering BC →A μ2 1⩽μ2⩽4μ2 1 3+μ2 1 Unsteerable states 4μ2 1 3+μ2 1 <μ 2⩽√2μ1 √3−μ2 1 Coexistence region √2μ1 √3−μ2 1 <μ 2⩽μ1Steerable states in Sec. III, their steering properties are deeply analyzed in Sec. IV. III. ENTANGLEMENT IN SYMMETRIC THREE-BEAM GAUSSIAN STATES Positivity of a partially transposed CM, ˜σ, has been established as the necessary and sufficient condition for determining the separability of (1 +N)-beam and bisymmetric (M+N)-beam Gaussian states [26–28]. This so-called positive partial transpose (PPT) criterion provides a clear qualitative characterization of the entanglement present in such states. Specifically, a Gaussian state described by the CM σis separable if and only if all symplectic eigenvalues ˜viof its partially transposed CM ˜σsatisfy ˜vi⩾1. Though the converse does not necessarily hold in a generic case, as bound entangled states may also meet the PPT criterion. For tripartite Gaussian states, a comprehensive qualitative classification of entanglement was presented in Ref. [21], which distinguishes five classes according to their separability under different bipartitions. Among these, only three are relevant for STBGSs: fully inseparable (class 1), bound entangled (class 4), and fully separable (class 5) states. Based on the single-mode and two-mode purities, μ1and μ2,the corresponding classification is summarized in Table I.Notably, only class 1 states (though not all) can be uniquely identified using marginal purities. For the remaining cases, two coexistence regions emerge: Region I, containing states from classes 4 and 5 (depending on the value of 2), and Region II, comprising states from classes 1, 4, and 5. The domain of tripartite entanglement shown in Table Icoincides with that of pairwise entanglement [30], reflecting the promiscuous sharing of CV entanglement [19]. However, a notable discrepancy arises in coexistence Region II, where states exhibiting tripartite entanglement without any bipartite entanglement are found (see below). These results emerge in the following calculations. Without loss of generality, we consider the bipartition of the CM σ3in the form A−BC, where A,B, and Cdenote the field beams. In this case, the symplectic eigenvalues of the partially transposed CM ˜σ=σ, where =diag(1,−1,1,1,1,1), are given by ˜v2 1=(a−c−)(a−c+), ˜v2 ±=1 2(2a2−3c−c++a(c−+c+)±θ),(6) where θ=9a2c2 −+2ac−c+(c−−7a)+(9a−7c−)(a+c−)c2 +. Utilizing Eq. (2), it becomes apparent that ˜v1and ˜v+remain greater than or equal to 1 throughout the entire domain delineated by Eqs. (3) and (4). Consequently, the entanglement characterization of σ3is solely based on ˜v−. Through this analysis, we have determined that the state σ3satisfies the PPT criterion (˜v−⩾1) when μ2 1⩽μ2⩽ 3μ2 1+9+62μ2 1−7μ4 1−3 10 −2μ2 1 ,(7) for 0 <μ 1⩽1 and all 2values specified in Eq. (4). This parameter range is hereafter referred to as Region I. Our analysis indicates, however, that satisfying the PPT criterion within Region I does not necessarily guarantee the separability of the states σ3. To illustrate this, let us consider a specific example of σ3, namely, the noisy GHZ/W state. These states are fully characterized by their oneand two-beam purities μ1and μ2or equivalently by the parameters a= μ−1 1and n=μ1/μ2(cf. Ref. [31]). We examine two noisy GHZ/W states within Region I: {μ1,μ 2}={0.5,0.25}and {μ1,μ 2}={0.8,0.652352}. According to the classification of Refs. [31,32], the former corresponds to a separable state (class 5), while the latter represents a bound entangled state (class 4). Thus, Region I contains both classes 4 and 5, with the specific class determined by 2. On the other hand, one can find that ˜v−<1; i.e., the CMs σ3describe fully inseparable states (class 1), if μ1 2−μ2 1 ⩽μ2⩽μ1,(8) where 0 <μ 1⩽1 and all 2values specified in Eq. (4). Between these two regions (see Fig. 1), we find the states that belong to one of three classes: class 1, class 4, or class 5. For example, a separable state in this region is represented by the noisy GHZ/W state with parameters {μ1,μ 2}={0.5,0.28}, while a bound entangled state in this region is exemplified by the noisy GHZ/W state with parameters {μ1,μ 2}= {0.5,1/(2√3)}. In addition to the state classification outlined above, it is also feasible to determine the lower and upper bounds of the two-beam and three-beam logarithmic negativities EN,2 and EN,3[33–35]. They are defined in general as EN≡ ln ||˜ρ||1, where ||·||1denotes the trace norm. For Gaussian states, this measure can be determined in terms of the symplectic spectrum ˜vof the partially transposed CM ˜σ, 043292-3 JAN PE ˇ RINA, JR. et al. PHYSICAL REVIEW RESEARCH 7, 043292 (2025) FIG. 1. Twoand three-beam entanglement analyzed in the plane spanned by oneand two-beam purities μ1and μ2. Panel (a) highlights areas differing in three-beam entanglement properties: Region I (coexistence of separable and bound entangled states) is shown in blue, Region II (coexistence of separable, bound entangled, and fully entangled states) is depicted in orange, and the fully entangled region is drawn in green. Panel (b) focuses on two-beam entanglement: Area with two-beam separable (entangled) states is shown in light blue (green). In the light-orange area, separable and entangled states coexist. These regions are classified using the criteria of Ref. [30]. Isolated black boxes (black triangles) refer to the experimental data discussed for M=6.7(M=40) independent modes. Upper and lower bounds on three-beam (two-beam) logarithmic negativity EN,3[EN,2] are shown in panels (c) [(d)] and (e) [(f)], respectively, as they depend on “rotated” marginal purities (μ1+μ2)/2and(μ1−μ2)/2. In panels (c) and (e), the black solid curves indicate contour levels of EN,3equal to 0.5, 1, and 2. In panels (d) and (f), the black solid curves correspond to EN,2=0.5. EN=max{0,−i:˜vi<1ln ˜vi}. In this case, only one eigenvalue can take values smaller than 1, and the logarithmic negativity is defined as EN,3=max{0,−ln ˜v−}. Moreover, since ˜v−in Eq. (6) is monotonic with respect to 2,thelower and upper bounds of logarithmic negativity are determined by the extremal values of 2for the fixed oneand two-beam purities μ1and μ2. Specifically, the bounds are given by Emin N,3=EN(max{2}) and Emax N,3=EN(min{2}). Given the complexity of the resulting expressions, we do not present them explicitly. Instead, we draw Emin N,3and Emax N,3for the symmetric three-beam Gaussian states as they depend on the “rotated” marginal purities (μ1+μ2)/2 and (μ1−μ2)/2in Figs. 1(c)–1(f). Finally, the bounds on the logarithmic negativity can be employed to characterize genuine tripartite entanglement through the residual cotangle [19], defined as Tπ=E2 N,3−2E2 N,2,(9) where EN,2=−1 2ln(a2−a|cm−cp|−cmcp). Using Eq. (2) and following a series of lengthy but straightforward calculations, one can demonstrate that the residual cotangle Tπis a monotonic function of 2. Consequently, it follows that Tπ attains its maximum and minimum values at the respective boundary values of 2. IV. STEERING IN SYMMETRIC THREE-BEAM GAUSSIAN STATES Gaussian steering is a stronger form of quantum correlation than entanglement [36], playing a key role in asymmetric quantum information tasks such as one-sided device-independent protocols. Due to its directional nature, it exhibits a rich and structured hierarchy of correlations that depend on state purity and symmetry. A detailed classification of the Gaussian-steering properties of STBGSs, based on their marginal purities, is presented in Table I. Interestingly, STBGSs show no Gaussian steering between individual beams. To demonstrate this, recall that the measure of Gaussian steerability from Mto one-beam subsystems simplifies to [36] GM→1(σ)=max 0,ln μM+1 μ1,(10) where μ1denotes the purity of a single beam and μM+1the purity of the joint (M+1)-beam subsystem. From Eq. (10), it follows that fully symmetric states σ3cannot exhibit Gaussian steering of type 1 →1, as the condition μ2>μ 1violates the constraint given in Eq. (3). However, Gaussian steering emerges between the twobeam and single-beam subsystems. In the 2 →1 scenario, the ratio of purities is given by μ3 μ1=μ2 1μ2(μ22+γ) 3μ2(−μ22+γ)+μ2 1(4 +μ22(μ22−γ)), (11) where γ=μ2 22 2−4. Equation (11) is monotonic in 2,so its extrema occur at the boundaries of the allowed 2domain. 043292-4 QUANTUM CORRELATIONS IN THREE-BEAM SYMMETRIC … PHYSICAL REVIEW RESEARCH 7, 043292 (2025) (c) (e) (f) (d) FIG. 2. Three-mode Gaussian steering: Steerable region, coexistence region, and unsteerable region for steering parameters (a) G2→1and (b) G1→2are drawn in turn in green, orange, and blue colors in the plane spanned by oneand two-beam purities μ1and μ2. Isolated black boxes (black triangles) refer to the experimental data for M=6.7(M=40). Upper and lower bounds G2→1 max [G1→2 max ]andG2→1 min [G1→2 min ]ofthe steering parameter as they depend on “rotated” marginal purities (μ1+μ2)/2and(μ1−μ2)/2 are drawn in panels (c) [(d)] and (e) [(f)], respectively. In panels (c)–(f), black solid curves correspond to G2→1 max ,G1→2 max ,G2→1 min ,andG1→2 min equal to 0.5, 1, and 2. Consequently, G2→1(σ3)>0for √2μ1 3−μ2 1 ⩽μ2⩽μ1,(12) where 0 <μ 1<1 and for all attainable 2values. The coexistence region, where steering occurs only for certain 2, is 4μ2 1 3+μ2 1 ⩽μ2<√2μ1 3−μ2 1 ,(13) as illustrated in Fig. 2(a). In the opposite configuration, 1 →2, the measure of Gaussian steerability is defined as [36] G1→2(σ3)=max ⎧ ⎨ ⎩ 0,− j:¯νj<1 ln ¯νj⎫ ⎬ ⎭ ,(14) where ¯νjare the symplectic eigenvalues of the Schur complement Mσto single-beam CM [37]. For σ3, the relevant symplectic eigenvalue is ¯ν2=3μ2(−μ22+γ)+μ2 1(4 +μ22(μ22−γ)) 2μ2 2 , (15) with γdefined below Eq. (11). Since ¯νis also monotonic in 2, one finds G1→2(σ3)>0for √3μ1 4−μ2 1 ⩽μ2⩽μ1,(16) and all allowed 2. The coexistence region, where steering occurs only for specific 2,is 16μ2 1+9−3 2⩽μ2<√3μ1 4−μ2 1 ,(17) as shown in Fig. 2(b). Both measures, G2→1(σ3) and G1→2(σ3), are monotonic in 2and thus reach their bounds for max{2}and min{2}, respectively. The corresponding lower and upper bounds of G2→1and G1→2are plotted in Figs. 2(c)–2(f) as functions of the “rotated” marginal purities (μ1+μ2)/2 and (μ1−μ2)/2. V. EXPERIMENTAL INVESTIGATION OF QUANTUM CORRELATIONS IN THREE-BEAM SYMMETRIC GAUSSIAN STATES The above theoretical characterization enables experimentally reliable estimation of CV nonclassical correlations using intensity moments up to fourth order through determination of marginal purities, offering robust sufficient conditions and analytical bounds. Furthermore, since CV logarithmic negativity and Gaussian steering measures are monotonic functions of 2, the experimental marginal purities also facilitate a reliable estimation of entanglement. This result further enables the estimation of genuine tripartite entanglement via the residual cotangle defined in Eq. (9). Accurate classification of entanglement in the coexistence regions requires the knowledge of 2, which can be obtained either directly from μ3using intensity moments up to sixth order or estimated from the experimental data (see Ref. [12]). Using the experimental photocount technique, a group of STBGSs including those with genuine three-beam 043292-5 JAN PE ˇ RINA, JR. et al. PHYSICAL REVIEW RESEARCH 7, 043292 (2025) FIG. 3. Structure of three-beam Gaussian states containing photon-pair (red ∞) and single-photon (green →) fields. Both fields can be symmetrized by alternating the signal (s) and the idler (i) beams (double ∞and →). entanglement was prepared and investigated. The state preparation is based on the fact that the Gaussian states are fully characterized just by their firstand second-order amplitude moments and so they can in principle be constructed from pairwise correlations embedded in twin beams. The underlying principal scheme drawn in Fig. 3uses three twin beams whose constituting beams contribute differently to beams 1, 2, and 3. Additional beams with increasing intensities then form the noisy parts of STBGSs. Using the technique of compound beams suggested in Ref. [20], all needed beams are derived from a sequence of measurements of photocount distributions of identical weak twin beams by two singlephoton avalanche photodiodes (APDs). The experimental data characterizing the analyzed STBGSs are then obtained by suitably concatenating the data representing the signal and the idler photocounts of these weak twin beams (for details, see Sec. I and Fig. 1 in the Supplemental Material [38]). We note that, when constructing the states, we alternate the signal and the idler beams to arrive at the Gaussian states symmetric in both intensities and correlations. Whereas the correlated parts of all analyzed states are composed of 12 weak twin beams, up to 180 weak twin beams are used to form the noise parts of the analyzed states. The experimental data were transformed into photocount histograms f(c1,c2,c3), giving the probability of measuring simultaneously (c1,c2,c3) photocounts in beams (1,2,3). Knowing the parameters of the detectors used, the corresponding photon-number distribution p(n1,n2,n3) was reconstructed using the maximum-likelihood approach [39,40]. For the model accompanying the experimental results (see Sec. II in the Supplemental Material [38]), parameters of weak twin beams were determined (see Ref. [20]). The applied technique provided us with 30 compound beams each having around 0.8 mean correlated photons per beam and mean noise photon numbers per beam increasing from 0 to 6. Their main characteristics are quantified using one-beam mean photon number nand Fano factor F, two-beam noise-reduction parameter R, and two-beam Kullback-Leibler divergence per mode H2/M. The one-beam mean photon number nis plotted in Fig. 4(a) for the onebeam mean noise photon number nnincreasing from 0 to 3. Increasing nn, the Fano factor Fof a beam, F1= 1+(W1)2/W1, is kept roughly constant [see Fig. 4(b)]. The increase of noise also relatively weakens the pairwise beams correlations contained in the correlated units, which are the source of entanglement of the whole state. Twobeam quantum correlations are quantified in Fig. 4(c) by the FIG. 4. (a) One-beam mean photon number n,(b)itsFano factor F, (c) two-beam noise-reduction parameter R, and two-beam Kullback-Leibler divergence per one typical spatiotemporal mode H2/Mas they depend on one-beam mean noise photon number nn. Used symbols and curves are described in the caption to Fig. 5.In panel (b) [(c)], the horizontal dashed line F=1[R=1] defines the classical-quantum border. noise-reduction parameter Rdefined as R12 =1+[(W1− W2)]2/(W1+W2). For the analyzed states, the noisereduction parameter Rincreases from 0.5 to 0.9 because of the noise. The increase of mean noise photon number nnalso results in lowering inseparability of the two-beam reduced states [see Fig. 4(d)]. Inseparability of the state is quantified by the two-beam Kullback-Leibler divergence H2≡SR,1+SR,2− SR,12 defined as the difference between the one-beam and two-beam Renyi entropies [SR=−ln(μ)]. We note that, according to Fig. 4(d), inseparability of the states as determined by the intensity moments up to sixth order is considerably larger than the idealized one derived only from second-order intensity moments (see below). To discuss quantum correlations, we determine up to sixthorder intensity moments reduced per mode in each beam (for the reduction, see Sec. III in the Supplemental Material [38]), derive the fields’ one-, two-, and three-beam purities μ1,μ2, and μ3, and implement the above theoretical formulas. However, before analyzing the experimental data, we have to address the problem of experimental precision required to reliably reveal the needed purities. This leads us to three approaches to the data analysis that differ by the number of required measurement realizations. The usual number of measurement realizations gives the intensity moments up to fourth order with sufficient precision. This gives the purities μ1and μ2and constraints on possible values of the seralian 2. The purities μ1and μ2identify in the single-mode theoretical model a group of compatible 043292-6 QUANTUM CORRELATIONS IN THREE-BEAM SYMMETRIC … PHYSICAL REVIEW RESEARCH 7, 043292 (2025) STBGSs whose analysis sets the upper (green curves and symbols) and lower (red curves and symbols) bounds for the seralian 2. Alternatively, we can estimate the value of seralian 2directly from the experimental intensity moments [see Eq. (5) and Appendix A]. Nevertheless, the obtained high-quality data (6.95 ×108measurements on weak twin beams provided ≈6.95/6×108realizations of the analyzed states) allow us to reliably determine the intensity moments up to sixth order. They uniquely identify the state and so give all fields’ characteristics (black curves and symbols). On the other hand, having technical applications in mind and assuming the specific form of STBGSs imposed by the used weak twin beams, the second-order intensity moments suffice in the state identification (blue curves and symbols; see Ref. [41]). Our experimental data suggest that about 108measurements are needed to completely identify the state (for more details, see Appendix B). About 106measurements allow us to safely characterize the state using the lower and upper bounds through fourth-order intensity moments. Finally, only about 104state realizations suffice when we rely only on second-order intensity moments. We note that the analyzed states can directly be emitted in the experiment once the APDs are replaced by photon-number-resolving detectors (see, e.g., in Ref. [42]). We note here that the structure of the experimental fields is richer than that of the single-mode theoretical model and so the above three approaches to the experimental data analysis do not necessarily give compatible predictions for the fields’ properties. The more elaborated approach is used; i.e., the higher the intensity moments are involved, the more reliable and more detailed the predictions are. However, the experimental errors must be kept reasonably low. Considering the intensity moments wk1 1wk2 2wk3 3reduced per one typical spatiotemporal mode in each beam, the formula for one-beam purity μ1is simple, μ1=1 1+4w1+12w12−4w2 1 .(18) Contrary to this, the formulas for twoand three-beam purities μ2and μ3are rather complex and they can be found in Refs. [12,18], respectively. In the analyzed experiment described in detail in Ref. [20], the detectors were characterized by detection efficiencies ηs= 0.274, ηi=0.324 and dark-count rates ds=2.8×10−3,di= 3.8×10−3. The application of the above-mentioned formulas gave us the purities μ1,μ2, and μ3as shown in Fig. 5. They naturally decrease with the increasing mean noise photon number nn. According to the graphs in Figs. 5(a)– 5(c),wehaveμ1>μ 2>μ 3for all experimental STBGSs. Moreover, as the seralian 2is an important parameter for characterizing the state, we apply Eqs. (4) and (2) to determine its values from those of the purities [see Fig. 5(d)]. Formula (2) also allows us to derive the coefficients a,c+, and c−from the above purities, as shown in Fig. 6. To arrive at the values of coefficients c+and c−, we need to know the three-beam purity μ3, which is experimentally demanding. Alternatively, we may rely only on the experimental values of purities μ1and μ2and set the lower and upper bounds for the coefficients c+and c−. FIG. 5. (a) One-, (b) two-, and (c) three-beam purities μ1,μ2, and μ3and (d) seralian 2as they depend on one-beam mean noise photon number nn. Isolated symbols with error bars give experimental data, and curves originate from the general Gaussian model. Black curves and symbols ◦denote exact values obtained from the intensity moments up to sixth order. The use of moments only up to the second order gives blue curves and symbols ∗. Fourthorder moments provide the lower (green curves and symbols +)and upper (red curves and symbols ) estimates with respect to quantum correlations for the determined quantities. VI. OBSERVED QUANTUM CORRELATIONS: GENUINE TRIPARTITE ENTANGLEMENT The three-beam correlations in the analyzed states are established by two-beam correlations (using photon pairs) covering all three beams. This results in the fields’ three-beam entanglement (tripartite entanglement A−BC), quantified by the logarithmic negativity EN,3, that naturally decreases with the increasing mean noise photon number nn[see Fig. 7(a)]. FIG. 6. State coefficients (a) aand (b) c−and c+,c−⩽c+,as they depend on one-beam mean noise photon number nn. Symbols and curves are described in the caption to Fig. 5. 043292-7 JAN PE ˇ RINA, JR. et al. PHYSICAL REVIEW RESEARCH 7, 043292 (2025) FIG. 7. (a) Tripartite negativity per mode EN,3/M, (b) bipartite negativity per mode EN,2/M, (c) cotangle per mode Tπ/M,and (d) three-beam Kullback-Leibler divergence per mode H3/Mas they depend on one-beam mean noise photon number nn. Symbols and curves are described in the caption to Fig. 5. In panels (a) and (b), the point where the green (red) curve nullifies gives the border between entangled (coexistence II) and coexistence II (coexistence I) regions defined in Table I. In panel (c), the navy blue curve and symbols  refer to noisy GHZ/W states identified by μ1and μ2, as discussed in Sec. VII. The simplest approach for data processing as well as the most elaborated one suggest that the tripartite entanglement disappears for nn≈2, i.e., when the noise is approximately 2.5 times larger than the correlated signal. Tripartite entanglement includes contributions from both bipartite entanglement (between beams A−Band A−C) and genuine tripartite entanglement. By quantifying bipartite entanglement using the logarithmic negativity EN,2, the genuine tripartite entanglement can be inferred by calculating the residual cotangle [19] introduced in Eq. (9). Following the curves in Fig. 7(b), the bipartite negativity EN,2is lost already for nnaround 1. As shown in Figs. 7(a)–7(c), the fields with nnbelow 2 exhibit the genuine tripartite entanglement. In particular, for nnbetween 1 and 2, this entanglement persists even in the absence of the bipartite entanglement. Interestingly, inside this region, the values of cotangle Tπare significantly lower compared to those observed for nn<1, which highlights the promiscuous nature of CV entanglement sharing [19]. We note that the properties of the analyzed states are close to those of the noisy GHZ/W states, as evidenced by the corresponding values of cotangle Tπin Fig. 7(c) (for details, see Sec. VII). The approach based on up to fourth-order intensity moments confirms this by determining the upper and lower bounds of these quantities, though it guarantees the genuine tripartite entanglement only for the fields with FIG. 8. Three-beam steering parameters per mode (a) G1→2/M and (b) G2→1/Mas they depend on one-beam mean noise photon number nn. Used symbols and curves are described in the caption to Fig. 5. The point where the green (red) curve nullifies identifies the border between steerable (coexistence) and coexistence (unsteerable) regions defined in Table I. nnsmaller than 0.4. We note that, according to Figs. 7(a) and 7(b), tripartite entanglement is more resistant against the noise than the bipartite entanglement. Inseparability of three-beam fields to their oneand two-beam counterparts, quantified by three-beam Kullback-Leibler divergence H3≡SR,A+SR,BC −SR,ABC defined in terms of the Rényi-2 entropies SRand plotted in Fig. 7(d), naturally decreases with the increasing nn. Steering represents a stronger form of quantum correlations. For our STBGSs, two-beam steering does not occur. On the other hand, both one beam can steer the remaining two beams (steering parameter G1→2) and vice versa (G2→1), as evidenced and quantified in Figs. 8(a) and 8(b). Steering by one beam is stronger than its two-beam counterpart, and they are both observed for the states with low amount of the noise nn<1. VII. COMPARISON WITH THEORETICAL NOISY GHZ/W STATES In the last section, we provide detailed comparison of the properties of the generated states with those of the noisy three-mode GHZ/W states. These states belong to the family of fully symmetric mixed Gaussian states, also known as three-mode squeezed thermal states [32]. The properties of these states were extensively studied in the literature [35]. The noisy GHZ/W states are characterized by two independent parameters, which define a fully symmetric covariance matrix σ3 that was introduced in Eq. (1). Adopting the parametrization from Ref. [35], the following relations are derived: μ3=μ2 μ13 , 2=1 μ2 1+μ2 1 μ2 2 .(19) The values of purity μ3and seralian 2of the GHZ/W states are compared with those of the generated states in Figs. 9(a) and 9(b), respectively. Whereas the purities μ3of GHZ/W states are close to the upper bound of μ3of the generated states from the bottom [compare Fig. 5(c)], the seralians 2 043292-8 QUANTUM CORRELATIONS IN THREE-BEAM SYMMETRIC … PHYSICAL REVIEW RESEARCH 7, 043292 (2025) FIG. 9. (a) Three-beam purity μ3, (b) seralian 2,(c)twobeam negativity per mode EN,2/M, (d) three-beam negativity per mode EN,3/M, (e) cotangle per mode Tπ/M, and (f) three-beam Kullback-Leibler divergence per mode HGHZ 3/Mfrom the corresponding GHZ/W states as they depend on one-beam mean noise photon number nn. For GHZ/W states, navy blue curves originate in the general Gaussian model and isolated navy blue symbols with error bars give experimental data. The remaining symbols and curves are described in the caption to Fig. 5(M=6.7). are close to the lower bound of 2of the generated states from the upper [compare Fig. 5(d)]. This means that the GHZ/W states belong to the states with the greatest entanglement among the states sharing the purities μ1and μ2. The corresponding values of two-beam negativity per mode EN,2/M, three-beam negativity per mode EN,3/M, and cotangle per mode Tπ/Mas they vary with the increasing mean noise photon number nnare plotted by navy blue symbols and curvesinturninFigs.9(c)–9(e). The vicinity of the GHZ/W states to the generated states is judged using the Kullback-Leibler divergence HGHZ 3that is determined for the generated states and the GHZ/W states along the formula: HGHZ 3=−3√h1+h2 8μ4 1μ2+ln √2μ4 1 μ2 2√h3,(20) h1= −4μ4 1+−4+2μ2 12μ2 2! μ8 1−10μ4 1μ2 2+9μ4 2!, h2=3−μ4 1+μ2 28μ2+μ2 1−4+2 2μ2 2, h3=6+2 2−3+2μ2 1μ2 2−2μ2 1−4+2 2μ2 2/μ2 +23−2μ2 1μ2−4+2 2μ2 2.(21) As evidenced by the values of the Kullback-Leibler divergence per mode HGHZ 3/Mplotted in Fig. 9(f), the generated states and the corresponding GHZ/W states are rather close to each other: The greater the mean noise photon number nnis, the closer the states are. Similarity of the two kinds of states suggests to apply the entanglement classification developed for the GHZ/W states in Refs. [32,35] to the generated states. The corresponding classification is written as μ3 1 2−μ1 <μ 2→class 1,     5μ4 1+3μ6 18+μ2 1 18 −4μ2 1 <μ 2⩽μ3 1 2−μ1→class 4, μ2 1<μ 2⩽    5μ4 1+3μ6 18+μ2 1 18 −4μ2 1→class 5. (22) According to this classification, the corresponding noisy GHZ/W states fall into class 1 for nn<1.67 and class 5 for nn>1.88. Class 4 was not observed, probably owing to the lower number of measurements in the intermediate range of nn. Furthermore, the coexistence of bipartite and tripartite CV entanglement occurs when the conditions μ1(1+μ2 1 3−μ2 1 )1/2<μ 2 and μ1<√3μ2are fulfilled, as discussed in Ref. [31]. According to our experimental data, this regime is observed for nn<1.04. VIII. CONCLUSION Three-beam symmetric Gaussian states were uniquely identified through their universal quantum invariants— one-, two-, and three-beam purities. These invariants were experimentally determined using second-, fourth-, and sixthorder intensity moments obtained through photon-numberresolving detection. The states exhibiting genuine tripartite entanglement and the state featuring the coexistence of twoand three-beam entanglement were observed this way. While the generated states closely resemble the noisy GHZ/W states, our method reveals subtle differences in their properties that demonstrate its sensitivity and reliability. Due to its conceptual simplicity, experimental feasibility, and strong diagnostic power, the presented method has significant potential for broad applications. It can be readily extended to the study of more complex families of Gaussian states. Foreseen applications span a variety of areas involving multipartite 043292-9