Characterizing Multipartite Non-Gaussian Entanglement for Three-Mode Spontaneous Parametric Down-Conversion Process

Continuous-variable (CV) systems, where multimode entangled states can be deterministically prepared Weedbrook et al. (2012); Yokoyama et al. (2013); Roslund et al. (2014); Chen et al. (2014); Armstrong et al. (2015); Cavalcanti et al. (2015); Cai et al. (2017); Deng et al. (2017); Larsen et al. (2019); Asavanant et al. (2019); Takeda et al. (2019); Cai et al. (2020); Wang et al. (2020), constitute an important platform for quantum technologies, including quantum teleportation networks Braunstein and Kimble (1998); Yonezawa et al. (2004), quantum key distribution Grosshans and Grangier (2002), quantum secrete sharing Armstrong et al. (2015), boson sampling Hamilton et al. (2017); Paesani et al. (2019), and multi-parameter quantum metrology Guo et al. (2020); Gessner et al. (2018). Non-Gaussian states in CV systems have attracted increasing attentions in recent years Andersen et al. (2015); Walschaers (2021), as they have been proven to be indispensable resources for universal quantum computation Niset et al. (2009); Mari and Eisert (2012), entanglement distillation Eisert et al. (2002); Fiurášek (2002); Takahashi et al. (2010), quantum-enhanced sensing Strobel et al. (2014); Joo et al. (2011), and quantum imaging Liu et al. (2021). Such perspectives have led to a growing interest in the experimental preparation of multimode non-Gaussian quantum states Ra et al. (2020); Douady and Boulanger (2004); Chang et al. (2020).

Despite the fact that substantial progress has been made in the generation of multimode non-Gaussian states, the characterization of their entanglement structure still poses a number of conceptual and practical challenges. The main reason being that the nontrivial correlations appear in higher-order moments of the observables that cannot be sufficiently uncovered by the widely used entanglement criteria based on second-order correlations Simon (2000); van Loock and Furusawa (2003); Adesso and Illuminati (2005); Hillery and Zubairy (2006); He and Reid (2013); Teh and Reid (2014), partially hindering their application for quantum information tasks. To tackle this problem, several approaches have been developed to take higher-order moments into account, such as the entropic entanglement criteria Walborn et al. (2009); Nishioka (2018); Zhang et al. (2021a), the generalized Hillery-Zubairy criteria based on the multimode moments Hillery et al. (2010); Cavalcanti et al. (2011); Agustí et al. (2020), or nonlinear entanglement criteria based on the amplitude-squared squeezing Shen et al. (2015); Zhang et al. (2021b). The quantum Fisher information (QFI) also provides a powerful method for capturing strongly non-Gaussian features of quantum states. It is widely applied to detect multi-particle entanglement in nonclassical spin states Pezze et al. (2018); Strobel et al. (2014); Hauke et al. (2016); Fadel and Gessner (2020); Ren et al. (2021); Yadin et al. (2021); Fadel et al. (2022) and has been recently developed to characterize continuous variables Qin et al. (2019); Gessner et al. (2016, 2017). Moreover, QFI provides a powerful tool to establish a quantitative link between entanglement and quantum metrology Pezzé and Smerzi (2009); Hyllus et al. (2012). However, most of the above mentioned methods are experimentally challenging, as they require full knowledge of the quantum state, which is extremely difficult in the multipartite scenario. Recently, a nonlinear squeezing parameter was proposed for the metrological characterization of non-Gaussian features Gessner et al. (2019); Guo et al. (2021). This can be optimized over a set of observables that are experimentally accessible, and it can ultimately coincide with the state QFI with increasing the order of the measured moments.

Experimentally, the recent advance in the generation of three-photon correlated non-Gaussian states via a direct three-mode spontaneous parametric down-conversion (SPDC) process Chang et al. (2020), opens up novel possibilities for various quantum information processing applications. Thus, it would be interesting to develop experimentally feasible methods to witness different classes of multipartite entanglement in such systems.

In this paper, we construct a class of CV nonlinear squeezing parameters for testing fully separable, inseparable, and fully inseparable tripartite non-Gaussian states with practical measurements. Firstly, we analyze the QFI for arbitrary sets of accessible local observables (involving higher-order moments) in three subsystems and find out the optimal combination that maximizes the violation of different entanglement bounds. These bounds are divided into three classes according to the separability properties with respect to the three splittings (fully separable) and particular bipartite splittings (biseparable). To avoid the requirement of full quantum state tomography, we then construct a nonlinear squeezing parameter by analytically determining the optimal measurements within arbitrary accessible third-order observables, that results in an excellent approximation to the QFI. The level of the nonlinear squeezing parameter quantifies the metrological advantage provided by different classes of entangled states. Moreover, by analyzing the accessible conditions in the three-mode SPDC experiment of Ref. Chang et al. (2020), we show that our method is capable of detecting fully inseparable tripartite entangled states by performing a limited number of measurements. Our results lead to an experimentally feasible way to systematically investigate CV multipartite non-Gaussian entanglement, which paves a way for exploiting their potential quantum advantages beyond Gaussian states.

Before considering a specific system, we firstly introduce a general method to detect non-Gaussian entanglement with a two-step optimization.

Step 1.$-$ We characterize the entanglement with choosing optimal operators in QFI. For an arbitrary $N$-partite separable quantum state $\hat{\rho}_{\mathrm{sep}}$, it was shown that the QFI must be less than a bound $B_{n}$ given by the variance Gessner et al. (2016)

Here, $\hat{A}_{j}$ is a local observable in the reduced state $\hat{\rho}_{j}$, and $\mathrm{Var}(\hat{A}_{j})_{\hat{\rho}}=\langle\hat{A}_{j}^{2}\rangle_{\hat{\rho}}-\langle\hat{A}_{j}\rangle^{2}_{\hat{\rho}}$ indicates the variance. $F_{Q}$ denotes the QFI, which describes the sensitivity of the parameter $\theta$ when the state $\hat{\rho}$ is transformed with unitary evolution $\hat{\rho}_{\theta}=e^{-i\sum_{j}\hat{A}_{j}\theta}\hat{\rho}e^{i\sum_{j}\hat{A}_{j}\theta}$ and provides a bound on the accuracy to determine $\theta$ as $(\Delta\theta_{est})^{2}\geq 1/F_{Q}[\hat{\rho},\sum_{j}\hat{A}_{j}]$.

Since Eq. (1) represents a necessary criterion for separability, its violation is sufficient criterion for entanglement. In order to witness the entanglement in the largest possible parameter range, we need to choose an optimal local operator $\hat{A_{j}}$, which can be constructed by analytical optimizing over arbitrary linear combination of accessible operators (namely, $\hat{A}_{j}=\sum_{m=1}c_{j}^{(m)}\hat{A}_{j}^{(m)}=\mathbf{c}_{j}\cdot\mathbf{\hat{A}}_{j}$). In this case, the full operator $\hat{A}(\mathbf{c})=\sum_{j=1}^{N}\mathbf{c}_{j}\cdot\hat{\mathbf{A}}_{j}$ is characterized by the combined vector $\mathbf{c}=(\mathbf{c}_{1},\cdots,\mathbf{c}_{N})^{T}$. According to Eq. (1), the quantity $W[\hat{\rho},\hat{A}(\mathbf{c})]=F_{Q}[\hat{\rho},\hat{A}(\mathbf{c})]-4\sum_{j=1}^{N}\mathrm{Var}(\mathbf{c}_{j}\cdot\mathbf{\hat{A}}_{j})_{\hat{\rho}}$ must be nonpositive for arbitrary choices of $\mathbf{c}$ whenever the state is separable. We can now maximize $W[\hat{\rho},\hat{A}(\mathbf{c})]$ by variation of $\mathbf{c}$ to obtain an optimized entanglement operator $\hat{A}$. According to Ref. Gessner et al. (2016), the quantity can be expressed as $W[\hat{\rho},\hat{A}(\mathbf{c})]=\mathbf{c}^{T}\left(Q_{\hat{\rho}}^{\mathcal{A}}-4\Gamma_{\Pi(\hat{\rho})}^{\mathcal{A}}\right)\mathbf{c}$ and the optimal $\mathbf{c}$ can be obtained by calculating the eigenvector of the maximum eigenvalue of the matrix $Q_{\hat{\rho}}^{\mathcal{A}}-4\Gamma_{\Pi(\hat{\rho})}^{\mathcal{A}}$ (see Appendix A for details).

Step 2.$-$ Computing the QFI is challenging for arbitrary multipartite states, as it requires the full density matrix of the system Braunstein and Caves (1994). For this reason, it is often more practical to find its lower bound, which, regarded as squeezing parameters $\chi^{2}$, involves simple measurements that are usually experimentally feasible. The relation between the QFI and $\chi^{2}$ fulfills Pezzé and Smerzi (2009),

which is saturable for certain measurement operator $\hat{M}$. Often $\hat{M}$ is a complicated operator, of difficult implementation. Therefore, similarly to the analytical optimization of $\hat{A}$, $\hat{M}$ will be optimized over a linear combination of experimentally practical observables (see the details in Appendix B).

It is noticed that the general method introduced here is able to be applied to arbitrary quantum states for detecting the multipartite entanglement, where the optimal operator $\hat{A}$ and $\hat{M}$ will differ for different systems. In the following, we will take a particular non-Gaussian system as an example and obtain the optimal operator $\hat{A}$ and $\hat{M}$ for detecting the entanglement with the above two-step optimization.

Here, we focus on the three-mode SPDC process realized in Ref. Chang et al. (2020), where a pump photon at frequency $\omega_{p}$ is down converted to three nondegenerate photons at frequencies $\omega_{1},~{}\omega_{2},~{}\omega_{3}$, respectively. This process is described by the interaction Hamiltonian

where $\kappa$ is the third-order coupling constant, $\hat{b}$ and $\hat{a}_{i}$ ($i=1,2,3$) are the annihilation operators for the pump and $i$th mode, respectively. With this Hamiltonian, the three-mode non-Gaussian state is obtained by $\partial\hat{\rho}/\partial t=-i[\hat{H},\hat{\rho}]/\hbar$ with considering the initial state as vacuum for the generated triplets and coherent mode $\alpha_{p}$ for the pump.

A three-mode fully separable state allows for a description in three splittings $1|2|3$ ($\hat{\rho}_{\mathrm{sep}}=\sum_{k}p_{k}\hat{\rho}_{1}^{k}\otimes\hat{\rho}_{2}^{k}\otimes\hat{\rho}_{3}^{k}$), which provides the bound $B_{1}$. Thus, observing $F_{Q}>B_{1}$ indicates inseparability among the three modes. Furthermore, we can denote the maximum among the sum of variances for different bipartitions $1|23,~{}2|13,~{}3|12$, as bound $B_{2}$. The state is then confirmed to be fully inseparable if $F_{Q}>B_{2}$ (see Appendix A.2 for details).

A key step to witness tripartite non-Gaussian entanglement is the choice of the local observables $\hat{A}_{j}$, corresponding to the generators of parameter imprinting. To find the optimal operator $\hat{A}_{j}$, we express $\mathcal{A}=\{\hat{\mathbf{A}}_{1},\hat{\mathbf{A}}_{2},\hat{\mathbf{A}}_{3}\}$ through a set of accessible local operators for each mode $\hat{\mathbf{A}}_{j}=\{\hat{A}_{j}^{(1)},\hat{A}_{j}^{(2)},\dots\}$. The optimization of local observable $\hat{A}_{opt}$ to witness entanglement is analyzed over arbitrary linear combinations of the accessible operators. As the generated states are non-Gaussian, their characteristics cannot be sufficiently captured by linear quadratures $\hat{x}_{j},\hat{p}_{j}$, where $\hat{x}_{j}=\hat{a}_{j}+\hat{a}_{j}^{\dagger}$, $\hat{p}_{j}=-i(\hat{a}_{j}-\hat{a}_{j}^{\dagger})$. Therefore, we extend the family of accessible operators to the second-order, i.e. $\mathbf{\hat{A}}_{j}=[\hat{x}_{j},\hat{p}_{j},\hat{x}_{j}^{2},\hat{p}_{j}^{2},(\hat{x}_{j}\hat{p}_{j}+\hat{p}_{j}\hat{x}_{j})/2]$.

To find the optimal linear combination, we make use of the matrix form $W[\hat{\rho},\hat{A}(\mathbf{c})]=\mathbf{c}^{T}\left(Q_{\hat{\rho}}^{\mathcal{A}}-4\Gamma_{\Pi(\hat{\rho})}^{\mathcal{A}}\right)\mathbf{c}$ with the non-Gaussian state $\hat{\rho}$ generated in the three-mode SPDC process. Then by calculating the maximum eigenvalue and eigenvectors of the matrix $Q_{\hat{\rho}}^{\mathcal{A}}-4\Gamma_{\Pi(\hat{\rho})}^{\mathcal{A}}$, the optimal local operator $\hat{A}_{opt}=\sum_{j=1}^{3}(x_{j}^{2}+p_{j}^{2})$ is obtained. It can also be rewritten as $\hat{A}_{opt}=\sum_{j=1}^{3}\hat{a}_{j}^{\dagger}\hat{a}_{j}=\hat{N}$, because the constant term in $\hat{A}_{opt}$ has no effects on the results (see the detailed analytical optimization in Appendix A.2).

By comparing the QFI $F_{Q}[\hat{\rho},\hat{A}_{opt}]$ to different bounds, such as the fully separable bound $B_{1}$ and the biseparable bound $B_{2}$, the three-mode non-Gaussian entanglement can be classified. In Fig. 1, we plot the QFI and the different bounds (solid curves) as functions of the effective coupling strength $\alpha_{p}\kappa t$, where $t$ denotes the effective interaction time. We find that the QFI $F_{Q}[\hat{\rho},\hat{A}_{opt}]$ is higher than the biseparable bound $B_{2}$, which means that the non-Gaussian states generated in the three-mode SPDC dynamics Eq. (3) are fully inseparable.

It has been clarified that only a specific class of entangled states enable quantum-enhanced parameter estimation Pezze et al. (2018), where a sufficient condition for quantum-enhanced metrology is Rivas and Luis (2010),

where $\hat{N}=\sum_{j}^{3}\hat{a}_{j}^{\dagger}a_{j}$ is the total number operator for the three modes. The right hand side of the inequality give a bound $B_{0}$ for QFI (the derivation is given in Appendix C). It is obviously seen from Fig. 1 that both of the entanglement bounds ($B_{1}$ and $B_{2}$) are above the metrological bound $B_{0}$ (the lowest solid curve), which indicates that the entanglement witnessed in the three-mode SPDC dynamics is useful for quantum metrology.

In this section, we will take the Step 2 optimization to obtain the squeezing parameter $\chi^{2}$ with optimal measurement $\hat{M}$. By extending the family of accessible observables to third order, the optimal $\hat{M}$ we obtained is $\hat{M}_{opt}=\hat{x}_{1}\hat{x}_{2}\hat{x}_{3}-\hat{x}_{1}\hat{p}_{2}\hat{p}_{3}-\hat{p}_{1}\hat{x}_{2}\hat{p}_{3}-\hat{p}_{1}\hat{p}_{2}\hat{x}_{3}$ (see the details in Appendix B). As shown in Fig. 1 by the red dashed curve, the nonlinear squeezing parameter with optimal choices of measurements $\chi^{-2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{opt})$ approximates extremely well the exact QFI $F_{\mathrm{Q}}[\hat{\rho},\hat{A}_{opt}]$. Hence, the squeezing parameter $\chi^{-2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{opt})$ can act as a faithful substitute for the QFI in the left-hand side of inequalities (1) and (4), avoiding to perform quantum state tomography.

Although more accessible in experiments comparing to the QFI, the nonlinear squeezing parameter $\chi^{2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{opt})$ can still be of demanding implementation, as (for the three-mode SPDC process) it requires measuring four collective observables in $\hat{M}_{opt}$. Therefore, we show that this can be further simplified by a measurement $\hat{M}$ which is still capable of detecting fully tripartite inseparable states but contains less number of observables. For this purpose, an experimentally feasible coincidence measurement is evaluated, which can be written as $\hat{Q}=[\hat{x}_{1}\sin(\theta_{1})+\hat{p}_{1}\cos(\theta_{1})]\times[\hat{x}_{2}\sin(\theta_{2})+\hat{p}_{2}\cos(\theta_{2})]\times[\hat{x}_{3}\sin(\theta_{3})+\hat{p}_{3}\cos(\theta_{3})]$, where $\theta_{j}\in[0,2\pi)$. The alternative observable $\hat{M}$ can be obtained by varying $\theta_{j}$. For the simplest case of $\hat{M_{1}}$ that contains only one collective observable, by optimizing $\theta_{1,2,3}$ we find that the nonlinear parameter $\chi^{-2}(\hat{M}_{1})=|\langle[\hat{A}_{opt},\hat{M}_{1}]\rangle_{\hat{\rho}}|^{2}/\mathrm{Var}(\hat{M}_{1})_{\hat{\rho}}$ is maximized by any one of four terms in $\hat{M}_{opt}$, e.g. $\hat{M}_{1}=\hat{x}_{1}\hat{x}_{2}\hat{x}_{3}$. We find that this performs as an excellent proxy for the QFI when $\alpha_{p}\kappa t\leq 0.1$, as indicated by the blue dashed line in Fig. 1.

To test entanglement, we need to compare the simplified nonlinear parameter $\chi^{-2}(\hat{M}_{1})$ with different bound $B_{n}$. The simplified nonlinear parameter has the form of,

In experiments, $\chi^{-2}(\hat{M}_{1})$ can be obtained by detecting $\hat{p}_{1}\hat{x}_{2}\hat{x}_{3}$, $\hat{x}_{1}\hat{p}_{2}\hat{x}_{3}$, $\hat{x}_{1}\hat{x}_{2}\hat{p}_{3}$, and $\hat{x}_{1}\hat{x}_{2}\hat{x}_{3}$ with collective measurements Chang et al. (2020); Bouwmeester et al. (1999); Wu et al. (2019); Yuan et al. (2020). And the different bounds $B_{n}=4\sum_{j}\mathrm{Var}(\hat{N}_{j})$ can be obtained by number-resolving detectors Gansen et al. (2007); Kardynał et al. (2008); Divochiy et al. (2008).

Besides, we also provide the results of simplified $\hat{M}_{2}$ and $\hat{M}_{3}$ that involve two and three collective observables, receptively, as shown by the two middle dashed curves in Fig. 1 (the detailed expressions and analysis can be found in Appendix D). As expected, the more observables are included in $M$, the larger range of coupling strengths over which non-Gaussian multipartite entanglement can be witnessed.

In this section we use the QFI $F_{Q}$ and the nonlinear parameter $\chi^{-2}$ to examine the three-mode non-Gaussian entanglement for a realistic scenario. In experiments, thermal noise in the initial state $\hat{\rho}_{j}(n_{th}^{j})$ is inevitable, with average thermal photon number $\langle n_{th}^{j}\rangle=1/(e^{\beta\omega_{j}}-1)$, and $\beta\omega_{j}=\hbar\omega_{j}/(k_{B}T)$. Here, we refer to the reported tripartite states produced by three-mode SPDC in the superconducting experiment Chang et al. (2020), where the noise temperature is $T=25$ mK and the frequencies of three modes are $\omega_{1}=2\pi\times 4.2$GHz, $\omega_{2}=2\pi\times 6.1$GHz, $\omega_{3}=2\pi\times 7.5$GHz, respectively. In this case, the simplified squeezing parameter $\chi^{2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{1})$ is sufficient to characterize the entanglement, since the achieved effective coupling strength is quite small.

Specifically, the experimental result for $\alpha_{p}\kappa t\approx 0.016$ is marked as a red star in Fig. 2, where the parameters are verified by reproducing the reported results in Ref. Chang et al. (2020). As shown in Fig. 2(a), the simplified nonlinear parameter $\chi^{-2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{1})$ (red dashed line) approximates well the QFI $F_{Q}(\hat{\rho},\hat{A}_{opt})$ (solid black line). Thus, $\chi^{-2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{1})$ is adequate to witness entanglement in such range of coupling strengths. Besides this, different separable bounds to characterise entanglement are illustrated in the figure, showing that the metrological bound $B_{0}$ coincides with the separable bound $B_{1}$ in this small coupling regime.

In summary, Fig. 2(b) shows that we can characterize the entanglement of the final state over a wide range of both coupling strength and environmental temperature. Full tripartite inseparability is witnessed with larger coupling and lower temperature (top right area) by violating the bound $B_{2}$ of all three possible $i|jk$ bipartitions. The intermediate area, where $\chi^{2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{1})$ violates the tripartite $a|b|c$ bound $B_{1}$ but doesn’t violate $B_{2}$, indicates there is entanglement among three modes. States in the bottom left area are not detected as useful for quantum metrology, since $\chi^{2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{1})$ is lower than the metrological bound $B_{0}$. In addition, as the red star is located in the top right area, a metrological useful full tripartite inseparability is witnessed in the experimentally reported three-mode states Chang et al. (2020).

In this section we compare our criterion to the widely used Hillery-Zubairy criterion Hillery et al. (2010), where full inseparability is witnessed if $I_{\mathrm{HZ}}=\mathrm{min}\{I_{1},I_{2},I_{3}\}>0$, where $I_{i}=|\langle\hat{a}_{1}\hat{a}_{2}\hat{a}_{3}\rangle|-\sqrt{\langle\hat{N}_{i}\rangle\langle\hat{N}_{j}\hat{N}_{k}\rangle}$ and $\hat{N}_{i}=\hat{a}_{i}^{\dagger}\hat{a}$, $i\neq j\neq k\neq i$. Similarly, we define $I_{\chi}=\chi^{-2}(\hat{\rho},\hat{A}_{opt},\hat{M}_{opt})-B_{2}$ based on the optimal measurements we proposed, such that a state is fully inseparable if $I_{\chi}>0$. The values of $I_{\mathrm{HZ}}$ and $I_{\chi}$ are presented in Fig. 3 as functions of the effective coupling strength and the inverse temperature, where the three-mode states are generated under the interaction Hamiltonian (3). It is shown that $I_{\mathrm{HZ}}$ fails to detect tripartite inseparability when $\alpha_{p}\kappa t>0.66$, while our witness $I_{\chi}$ works well in a larger coupling regime. This can be explained by the fact that detecting entanglement for large coupling strength requires observing correlations of higher order, which are not considered in the witness $I_{\mathrm{HZ}}$.

As it is shown in Fig. 3(a), $I_{\chi}$ fails to witness fully inseparable tripartite entangled states in the area below the dashed black line at the region with very weak coupling and high temperature, where $I_{\mathrm{HZ}}$ has a better performance [Fig. 3(b)]. This result can be explained by the fact that our entanglement conditions are tailored to detect metrologically useful entanglement, and that the region where the HZ criterion outperforms our criterion might correspond to states less useful for parameter-estimation tasks.

Based on the widely used quantifiers for metrological sensitivity, we have obtained an experimentally practical entanglement witness for CV tripartite non-Gaussian states generated by three-mode SPDC. First, the optimal phase-space observable that maximizes the QFI has been analyzed to detect the tripartite non-Gaussian entanglement, where fully inseparable states, biseparable entangled states, and fully separable states can be distinguished. Then, in order to avoid the full quantum state tomography required by the QFI, we constructed a nonlinear squeezing parameter by analytically determining the optimal measurements within arbitrary accessible observables, and demonstrate that it performs as well as the QFI by involving up to third-order moments of phase-space measurements. Moreover, by considering the experimental conditions in a recent three-mode SPDC experiment, we show that our method can detect fully inseparable tripartite non-Gaussian entanglement with only a limited number of measurements. Notably, the level of the nonlinear squeezing parameter quantifies the metrological advantage provided by the examined entangled states. Our results provide an approach to understand and characterize multipartite non-Gaussian entanglement, and paves the way to harness their potential applications in quantum metrology experiments.