CV entanglement criteria from squeezing coefficients and Fisher information

We consider an N-mode CV system with a vector of phase-space operators \(\widehat {\mathbf{r}} = (\hat r_1, \ldots ,\hat r_{2N}) = (\hat x_1,\hat p_1, \ldots ,\hat x_N,\hat p_N)\). Any real vector \({\mathbf{g}} = (g_1, \ldots ,g_{2N})\) defines a multi-mode quadrature \(\hat q({\mathbf{g}}) = {\mathbf{g}} \cdot \widehat {\mathbf{r}}\), which generates displacements of the form \(\hat D(\theta ) = {\mathrm{exp}}( - i\hat q({\mathbf{g}})\theta )\). The sensitivity of a Gaussian quantum state \(\hat \rho\) under such displacements is determined by the quantum Fisher information27,28,29

$$F_Q[\hat \rho ,\hat q({\mathbf{g}})] = {\mathbf{g}}^T{\mathbf{\Omega }}^T{\mathbf{\Gamma }}_{\hat \rho }^{ - 1}{\mathrm{\Omega }}{\mathbf{g}},$$ (1)

where \({\mathbf{\Omega }} = \oplus _{i = 1}^N\left( {\begin{array}{*{20}{c}} 0 & 1 \\ { - 1} & 0 \end{array}} \right)\) is the symplectic form and \({\mathbf{\Gamma }}_{\hat \rho }^{ - 1}\) is the inverse of the covariance matrix with elements \(({\mathbf{\Gamma }}_{\hat \rho })_{ij} = \frac{1}{2}\langle \hat r_i\hat r_j + \hat r_j\hat r_i\rangle _{\hat \rho } - \langle \hat r_i\rangle _{\hat \rho }\langle \hat r_j\rangle _{\hat \rho }\). By means of the quantum Cramér–Rao inequality, the quantum Fisher information directly determines the precision bound for a quantum parameter estimation of θ. It was shown in ref. 18 that an upper limit for the sensitivity of mode-separable states is given in terms of the single-mode variances of the same state:

$$F_Q[\hat \rho _{{\mathrm{sep}}},\hat q({\mathbf{g}})] \le 4{\mathbf{g}}^T{\mathbf{\Gamma }}_{{\Pi}(\hat \rho _{{\mathrm{sep}}})}{\mathbf{g}},$$ (2)

where \({\mathbf{\Gamma }}_{{\Pi}(\hat \rho _{{\mathrm{sep}}})}\) is the covariance matrix after all elements except the central 2 × 2 blocks have been set to zero, effectively removing all mode correlations. This corresponds to the covariance matrix of the product state of the reduced density matrices \({\Pi}(\hat \rho ) = \otimes _{i = 1}^N\hat \rho _i\). Any violation of inequality (2) indicates the presence of entanglement between the modes. To identify the contribution of specific subsystems in a multipartite system, this criterion can be generalized for a microscopic analysis of the entanglement structure.25 A witness for entanglement in a specific partition of the full system into subsystems \(\Lambda = {\cal A}_1| \ldots |{\cal A}_M\), where \({\cal A}_l\) describes an ensemble of modes, is obtained from Eq. (2) by replacing the fully separable product state \({\Pi}(\hat \rho )\) on the right-hand side by a product state on the partition \({\cal A}_1| \ldots |{\cal A}_M\). More precisely, any \({\cal A}_1| \ldots |{\cal A}_M\)-separable quantum state, i.e., any state that can be written as \(\hat \rho _\Lambda = \mathop {\sum}

olimits_\gamma p_\gamma \hat \rho _{{\cal A}_1}^{(\gamma )} \otimes \ldots \otimes \hat \rho _{{\cal A}_M}^{(\gamma )}\), where p γ is a probability distribution, must satisfy25

$$F_Q[\hat \rho _\Lambda ,\hat q({\mathbf{g}})] \le 4{\mathbf{g}}^T{\mathbf{\Gamma }}_{{\Pi}_\Lambda (\hat \rho _\Lambda )}{\mathbf{g}},$$ (3)

where \({\Pi}_\Lambda (\hat \rho _\Lambda ) = \otimes _{l = 1}^M\hat \rho _{{\cal A}_l}\) and \(\hat \rho _{{\cal A}_l}\) is the reduced density matrix of \(\hat \rho _\Lambda\) on \({\cal A}_l\). The covariance matrix \({\mathbf{\Gamma }}_{{\Pi}_\Lambda (\hat \rho _\Lambda )}\) can be easily obtained from \({\mathbf{\Gamma }}_{\hat \rho _\Lambda }\) by setting only those off-diagonal blocks to zero which describe correlations between different subsystems \({\cal A}_l\). The fully separable case, Eq. (2), is recovered if each \({\cal A}_l\) contains exactly one mode.

By combining the separability criterion (3) with the expression for the quantum Fisher information of Gaussian states (1), we find the following condition for the covariance matrix of \({\cal A}_1| \ldots |{\cal A}_M\)-separable states:

$${\mathbf{\Gamma }}_{\hat \rho }^{ - 1} - 4{\mathbf{\Omega }}^T{\mathbf{\Gamma }}_{{\Pi}_\Lambda (\hat \rho )}{\mathbf{\Omega }} \le 0,$$ (4)

where we have used that both expressions (1) and (3) are valid for arbitrary g and then multiplied the equation with Ω from both sides using \({\mathbf{\Omega }}^T{\mathbf{\Omega }} = {\Bbb I}_{2N}\) and ΩT = −Ω. Inequality (4) expresses that the matrix on the left-hand side must be negative semidefinite. Hence, if we find a single positive eigenvalue, entanglement in the considered partition is revealed. Thus, it suffices to check whether the maximal eigenvalue λ max is positive. The corresponding eigenvector e max further identifies a 2N-dimensional “direction” in phase space such that the sensitivity under displacements generated by \(\hat q({\mathbf{e}}_{{\mathrm{max}}})\) maximally violates Eq. (3).

A lower bound on the quantum Fisher information of arbitrary states can be found from elements of the covariance matrix using25

$$F_Q[\hat \rho ,\hat q({\mathbf{g}})] \ge \frac{{({\mathbf{h}}^T{\mathbf{\Omega g}})^2}}{{{\mathbf{h}}^T{\mathbf{\Gamma }}_{\hat \rho }{\mathbf{h}}}},$$ (5)

which holds for arbitrary g, h. Choosing h = Ωg with |g|2 = 1 leads with (3) to the separability condition25

$$\xi _\Lambda ^{ - 2}(\hat \rho _{{\mathrm{sep}}}) \le 1,$$ (6)

where

$$\xi _\Lambda ^2(\hat \rho )\,:=\, \begin{array}{*{20}{c}} {} \\ \mathop{min}\limits_{\mathbf {g}} \end{array}\,4\left( {{\mathbf{g}}^T{\mathbf{\Omega }}^T{\mathbf{\Gamma }}_{{\Pi}_\Lambda (\hat \rho )}{\mathbf{\Omega g}}} \right)({\mathbf{g}}^T{\mathbf{\Gamma }}_{\hat \rho }{\mathbf{g}}),$$ (7)

is the bosonic multi-mode squeezing coefficient for the partition Λ. Here, the minimizing g can be interpreted as a direction in phase space that identifies a multi-mode quadrature \(\hat q({\mathbf{g}})\) with a squeezed variance which can be traced back to mode entanglement.25

Experimental setup

In the following, we analyze experimentally generated N-mode Gaussian states with N = 2, 3, 4, subject to asymmetric loss using the two entanglement criteria defined by the quantum Fisher information, Eq. (4), and the multi-mode squeezing coefficient, Eq. (6). The graph representations of the three classes of Gaussian multi-mode entangled states considered here are shown in Fig. 1. They are often referred to as CV two-mode Gaussian entangled state (N = 2, Fig. 1a), three-mode CV Greenberger–Horne–Zeilinger (GHZ) state (N = 3, Fig. 1b), and four-mode square Gaussian cluster state (N = 4, Fig. 1c). The experimental generation of the states is described in detail in the Methods section and refs. 30,31 In all cases, the CV entangled states are generated by nondegenerate optical parametric amplifiers (NOPAs) with −3 dB squeezing at the sideband frequency of 3 MHz. The two-mode Gaussian entangled state is prepared directly by a NOPA. The three-mode GHZ state is obtained by combining a phase-squeezed and two amplitude-squeezed states using two beam splitters with transmissivities of T 1 = 1/3 and T 2 = 1/2, respectively, as shown in Fig. 1d.30 Similarly, the four-mode square Gaussian cluster state is prepared by coupling two phase-squeezed and two amplitude-squeezed states on a beam-splitter network consisting of three beam splitters with T 3 = 1/5 and T 4 = T 5 = 1/2, respectively, as shown in Fig. 1e.31

Fig. 1 Graph representation of multipartite CV entangled states and their preparation. a CV two-mode Gaussian entangled state. b Three-mode GHZ state. c Four-mode square Gaussian cluster state, respectively. d, e show the beam-splitter network used to generate the three-mode GHZ state and four-mode square Gaussian cluster state, respectively. The phase shift (PS) is realized by locking the relative phase of two light beams at the corresponding beam splitter. f–h show the schematics of preparation and measuring the two-mode Gaussian entangled state, three-mode GHZ state, and four-mode square Gaussian cluster state, respectively. PS phase shift, NOPA nondegenerate optical parametric amplifier, HWP half-wave plate, PBS polarizing beam splitter, LO local oscillator, HD 1–4 homodyne detectors, DM dichroic mirror Full size image

To study the robustness of multipartite entanglement under transmission losses, a lossy quantum channel for mode A is simulated using a half-wave plate (HWP) and a polarizing beam splitter (PBS). The output mode is given by \(\hat A{^\prime } = \sqrt \eta \hat A + \sqrt {1 - \eta } \hat \upsilon\), where η and \(\hat \upsilon\) represent the transmission efficiency of the quantum channel and the vacuum mode induced by loss into the quantum channel, respectively, as shown in Fig. 1f–h. Let us now turn to the characterization of CV entanglement based on the experimentally generated data.

Experimental results

Figure 2a shows the inverse squeezing coefficient (7) \(\xi _{A|B}^{ - 2}\) for a CV two-mode Gaussian entangled state in a lossy channel (LC) for the only possible partition A|B of the bipartite system. The coefficient \(\xi _{A|B}^{ - 2}\) decreases as the transmission efficiency η decreases but it always violates the separability condition (6) unless η = 0, i.e., when mode A is completely lost. This confirms that CV two-mode Gaussian entanglement only decreases but never fully disappears due to particle losses, i.e., CV two-mode Gaussian entanglement is robust to loss.32 We observe the same behavior for the criterion Eq. (4), which makes use of the Gaussian quantum Fisher information. Figure 2b shows the maximum eigenvalue λ max of the matrix \({\mathbf{\Gamma }}_{\hat \rho }^{ - 1} - 4{\mathbf{\Omega }}^T{\mathbf{\Gamma }}_{\hat \rho _A \otimes \hat \rho _B}{\mathbf{\Omega }}\). According to Eq. (4), a positive value indicates entanglement. Both coefficients attain their two-fold degenerate maximal value for the phase space directions g = (sinϕ, 0, cosϕ,0) and g = (0, −sinϕ, 0, cosϕ), where ϕ is a function of η (for η = 1 we have ϕ = π/4 25). These directions indicate strong correlations in the momentum quadratures and anti-correlations in the position quadratures, allowing us to relate the entanglement to the squeezing of the collective variances \({\mathrm{\Delta }}(\hat x_A{\mathrm{sin}}\phi + \hat x_B{\mathrm{cos}}\phi )^2\) and \({\mathrm{\Delta }}(\hat p_A{\mathrm{sin}}\phi - \hat p_B{\mathrm{cos}}\phi )^2\). It should be noted that \(\xi _{A|B}^{ - 2}\) and \(\xi _{B|A}^{ - 2}\) (\(\lambda _{A|B}\) and \(\lambda _{B|A}\)) are identical because the entanglement coefficients only depend on the partition and not on the order in which the subsystems are labeled.

Fig. 2 Experimental results for the CV two-mode Gaussian entangled state in a lossy channel with transmission efficiency η. a Inverse multi-mode squeezing coefficients (7). The plot shows the squeezing coefficient \(\xi _{A|B}^{ - 2}\) obtained by numerically minimizing in Eq. (7), using the experimentally measured covariance matrices (blue dots) and the theoretical prediction based on the state preparation schemes described in Fig. 1 (blue line). Values above one violate (6) and indicate entanglement. b Gaussian quantum Fisher information entanglement criterion, expressed by the maximum eigenvalue of the matrix on the left-hand side (l.h.s.) of Eq. (4). Positive values violate the separability condition (4). The error bars represent one standard deviation and are obtained from the statistics of the measured data Full size image

The entanglement structure becomes more interesting for the three-mode GHZ state, exhibiting four non-trivial partitions of the system, as well as three reduced two-mode states. The squeezing coefficient (7), as well as the Gaussian Fisher information entanglement criterion (4), are plotted in Fig. 3 for all four partitions. Both show that at η = 1, the three bi-separable partitions A|BC, B|AC, and C|AB are equivalent due to the symmetry of the state, but as η is decreased, the entanglement in the partition A|BC is more strongly affected by the losses than that of the other two partitions. In the extreme case where mode A is fully lost (η = 0) there is still some residual entanglement between B and C.33 In this case, all partitions are equivalent to the bi-partition B|C. The data shown in Fig. 3 confirms this: In both cases, the entanglement witness for all partitions coincide at η = 0, except A|BC which, as expected, yields zero.

Fig. 3 Experimental results for the three-mode GHZ state in a lossy channel with transmission efficiency η. a Inverse multi-mode squeezing coefficients. b Gaussian Fisher information entanglement criterion. Shown are numerically optimized coefficients for the partitions A|B|C (blue circles), A|BC (red squares), B|AC (black diamonds), and C|AB (purple triangles) from experimentally obtained covariance matrices and the curves represent the theoretical prediction Full size image

We further notice a discontinuity for the theoretical predictions of both witnesses regarding the fully separable partition A|B|C as a function of η (blue lines in Fig. 3). This can be explained by analyzing the corresponding optimal phase space direction g. In the presence of only moderate losses, the maximal correlations and squeezing are identified along the direction g = (0, c 1 , 0, c 2 , 0, c 2 ) with \(c_1^2 + 2c_2^2 = 1\), i.e., the multi-mode quadrature \(\hat q({\mathbf{g}}) = c_1\hat p_A + c_2\hat p_B + c_2\hat p_C\) which involves all three modes. The squeezing along this phase-space direction diminishes with increasing losses. When the losses of mode A become dominant, the squeezing along the phase space direction \({\mathbf{g}} = (0,0,1,0, - 1,0)/\sqrt 2\), i.e., of the quadrature \(\hat q({\mathbf{g}}) = (\hat x_B - \hat x_C)/\sqrt 2\) is more pronounced as it does not decay with η, being independent of mode A. The discontinuity is therefore explained by a sudden change of the optimal squeezing direction due to depletion of mode A. We remark that the experimentally prepared states are the same, except for the variable η. The change of the squeezing direction simply implies that when the local noise exceeds a critical value, the entanglement is more easily revealed by analyzing the quantum state from a different “perspective” in phase space. Notice that having access to the full covariance matrix, we can analyze both entanglement witnesses for arbitrary directions.

The change of the optimal direction is observed for both entanglement coefficients, whereas the transition occurs at a larger value of η for the Fisher information criterion (4) (see Supplementary Information). There we also show the two-mode entanglement properties after tracing over one of the modes in an analysis of the reduced density matrices, which show that two-mode entanglement persists after tracing over one of the subsystems, in stark contrast to GHZ states of discrete variables.34

Finally, we analyze the four-mode square Gaussian cluster state in Fig. 4. We find that the decoherence of entanglement depends on the cluster state’s geometric structure. As shown in Fig. 4a, the inverse multi-mode squeezing coefficient \(\xi _{A|B|C|D}^{ - 2}\) for the fully separable partition is not sensitive to transmission loss on mode A, while decoherence affects the coefficients for other partitions shown in Fig. 4b–d. For 1⊗1⊗2 partitions, only the results of \(\xi _{C|D|AB}^{ - 2}\), \(\xi _{A|B|CD}^{ - 2}\), \(\xi _{B|D|AC}^{ - 2}\), and \(\xi _{A|D|BC}^{ - 2}\) are shown in Fig. 4b (\(\xi _{B|C|AD}^{ - 2}\) and \(\xi _{A|C|BD}^{ - 2}\) are shown in Fig. S3 in Supplementary Information). The discontinuity for the A|B|CD partition is again explained by a transition of the optimal squeezing direction at a critical value of the transmission η for the isolated mode A (see Supplementary Information). The two coefficients \(\xi _{C|D|AB}^{ - 2}\) and \(\xi _{A|B|CD}^{ - 2}\) (\(\xi _{B|D|AC}^{ - 2}\) and \(\xi _{A|D|BC}^{ - 2}\)) are equal for η = 1 because of the symmetric roles of these modes in these partitions. As shown in Fig. 4b, c, the most sensitive coefficients to transmission losses of mode A are those where mode A is an individual subsystem. The coefficients \(\xi _{C|ABD}^{ - 2}\) and \(\xi _{D|ABC}^{ - 2}\) overlap due to the symmetric roles of modes C and D.

Fig. 4 Experimental results for the four-mode square Gaussian cluster state in a lossy channel with transmission efficiency η. a–d Inverse multi-mode squeezing coefficients ξ−2 for the partitions of classes 1⊗1⊗1⊗1, 1⊗1⊗2, 1⊗3, and 2⊗2, respectively. e–h The corresponding data for Gaussian Fisher information entanglement criterion. The data points are numerically optimized coefficients from experimentally obtained covariance matrices and the curves represent the corresponding numerically optimized predictions from the theoretical model Full size image

Figure 4d shows the inverse multi-mode squeezing coefficients for 2⊗2 partitions. It is interesting that the coefficient \(\xi _{AC|BD}^{ - 2}\) \(\left( {\xi _{AD|BC}^{ - 2}} \right)\) is immune to transmission loss of mode A. This indicates that the collective coefficients for 2⊗2 partitions, where each partition is composed by two neighboring modes (recall the graph representation in Fig. 1c), are not sensitive to the loss of any one mode. In contrast, the coefficient \(\xi _{AB|CD}^{ - 2}\), where each subsystem is composed by two diagonal modes, is still sensitive to transmission loss. As before, we find that the qualitative behavior of the squeezing coefficient ξ−2 coincides with that of λ max of the Gaussian Fisher information criterion (4), see Fig. 4e–h.

A further understanding of the entanglement structure is provided by an analysis of the three-mode and two-mode reduced density matrices of the state as well as of the optimal directions. A detailed analysis reveals that the loss-robustness is drastically reduced for all partitions if either mode C or D is traced out (see Supplementary Information). Moreover, for very small values of η, the entanglement in the partitions A|CD, D|AB, and C|AB in the reduced three-mode states is revealed by the criterion (4) but not by the squeezing approximation (7), where we assumed h = Ωg to simplify the optimization (see Supplementary Information).