Genuine multipartite entanglement of symmetric Gaussian states: Strong monogamy, unitary localization, scaling behavior, and molecular sharing structure

We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of Adesso and Illuminati [Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows us to quantify the genuine $N$-partite entanglement not encoded into $2,\dots ,K,\dots ,(N-1)$-partite quantum correlations. Strong monogamy is numerically verified, and the explicit expression of the measure of residual genuine multipartite entanglement is analytically derived, by a recursive formula, for a subclass of Gaussian states. These are fully symmetric (permutation-invariant) states that are multipartitioned into blocks, each consisting of an arbitrarily assigned number of modes. We compute the genuine multipartite entanglement shared by the blocks of modes and investigate its scaling properties with the number and size of the blocks, the total number of modes, the global mixedness of the state, and the squeezed resources needed for state engineering. To achieve the exact computation of the block entanglement, we introduce and prove a general result of symplectic analysis: Correlations among $K$ blocks in $N$-mode multisymmetric and multipartite Gaussian states, which are locally invariant under permutation of modes within each block, can be transformed by a local (with respect to the partition) unitary operation into correlations shared by $K$ single modes, one per block, in effective nonsymmetric states where $N-K$ modes are completely uncorrelated. Due to this theorem, the above results, such as the derivation of the explicit expression for the residual multipartite entanglement, its nonnegativity, and its scaling properties, extend to the subclass of non-symmetric Gaussian states that are obtained by the unitary localization of the multipartite entanglement of symmetric states. These findings provide strong numerical evidence that the distributed Gaussian entanglement is strongly monogamous under and possibly beyond specific symmetry constraints, and that the residual continuous-variable tangle is a proper measure of genuine multipartite entanglement for permutation-invariant Gaussian states under any multipartition of the modes.

Figure 1

(Color online) Pictorial representation of a quantum system multipartitioned into $K$ subsystems ($K=6$ in the figure), each containing in general one or more elementary units (the total number of elementary units, depicted as spheres, is $N=16$ in the figure). The “weak” monogamy of bipartite quantum correlations imposes that the entanglement between a reference (yellow) subsystem and the rest of the system exceeds the sum of the bipartite entanglements between the reference subsystem and each other (blue) individual subsystem. In Ref. 28, we further decomposed this excess between the global block entanglement and the total bipartite entanglement into a sum of genuine multipartite entanglements involving $3,4,\dots ,K$ parties at a time. The last contribution quantifies the truly $K$-partite quantum correlations shared among all parties. It is natural to expect that this term is non-negative: such an argument yields a stronger monogamy constraint on the simultaneous distribution of both bipartite and multipartite entanglement. In Ref. 28, we proved the validity of such a decomposition and hence the existence of such a strong monogamy, for a system of $N$ harmonic oscillators, one per party, in a globally permutation-invariant Gaussian state. Here we extend the strong monogamy framework to multipartitions comprising an arbitrary number of modes per party, retrieving a measure of genuine multipartite entanglement for larger classes of Gaussian states beyond symmetry constraints.

Figure 2

(Color online) Pictorial representation of the process of unitary localization and delocalization of multipartite entanglement in multisymmetric Gaussian states. The left picture represents a (pure or mixed) Gaussian state of $N$ modes ($N=16$ in the figure), all of them in general correlated, described by a covariance matrix $\sigma$. The $N$-mode system is multipartitioned into $K<N$ “molecules” ${\mathcal{M}}_1,\dots ,{\mathcal{M}}_K$ ($K=6$ in the figure). Each molecule ${\mathcal{M}}_j$ consists in a block of $m_j$ modes, each of them having the same reduced covariance matrix (the same color in the figure). By definition, multisymmetric states are invariant under mode permutations within a single molecule. There exists a symplectic transformation $S_{\mathrm{loc}}\equiv W_1\oplus W_2\oplus \cdots \oplus W_K$, acting locally on each molecule, which transforms the original state into a new one, described by a covariance matrix ${\sigma }^{\prime }=S_{\mathrm{loc}}\sigma S_{\mathrm{loc}}^T$, such that all correlations are concentrated among $K$ modes only (“nuclei”), one per molecule, as illustrated in the right picture. Precisely, ${\sigma }^{\prime }$ corresponds to a generally entangled state of the $K$ nuclei, tensor $(N-K)$ single-mode thermal states, which do not bear any correlation with each other and with the nuclei. Any bipartite and multipartite form of block entanglement shared by the molecules in the state $\sigma$ is then converted (localized) into an effective entanglement among single modes (realizing the transition from left to right in the figure). The localization is reversible, as $S_{\mathrm{loc}}$ amounts to a local unitary operation at the level of Hilbert space, hence the entanglement can be redistributed (delocalized) among all modes again by applying the inverse symplectic transformation $S_{\mathrm{loc}}^{-1}$ to ${\sigma }^{\prime }$ (realizing the transition from right to left in the figure).

Figure 3

(Color online) Genuine multipartite entanglement $G_{\tau }^{\mathrm{res}}$ of pure, fully symmetric $N$-mode Gaussian states ${\sigma }_{N:\{1,\dots ,1\}}^{\left(N\right)}$, plotted as a function of $N$ (up to ${10}^3$) and of the average squeezing $r$ needed in the preparation of the state. All the quantities plotted are dimensionless.

Figure 4

(Color online) Genuine multipartite entanglement $G_{\tau }^{\mathrm{res}}$ in the state ${\sigma }_{4:\{1,2,3,4\}}^{\left(N\right)}$ as a function of the squeezing $r$. The plot reports the entanglement shared by four molecules made, respectively, of 1, 2, 3, and modes in 10-mode fully symmetric Gaussian states, obtained, respectively, as reductions of a pure, fully symmetric $N$-mode Gaussian state with $N=10$ (solid red line, no modes are traced out), $N=11$ (dashed magenta line), $N=12$ (dot-dashed green line), $N=13$ (dot-dot-dashed cyan line), and $N=14$ (dotted blue line). The genuine multipartite molecular entanglement decreases with increasing $N$, i.e., with the number of discarded modes. In the pure-state instance (no modes being discarded) it diverges as $r\to \infty$. All the quantities plotted are dimensionless.

Figure 5

(Color online) Molecular entanglement hierarchy in a pure fully symmetric $M$-mode Gaussian state with $M=12$. From left to right: the bipartite entanglement between two 6-mode molecules (yellow line) is strictly stronger than the genuine tripartite entanglement between three 4-mode molecules (magenta line), and so on, while the atomic genuine 12-partite entanglement (red line) is the weakest one. The bipartite entanglement between any two modes in the reduced mixed state obtained by tracing over 10 modes is plotted as well for reference (dashed black line). All the quantities plotted are dimensionless.

Figure 6

(Color online) Plot, as a function of the squeezing $r$, of the genuine $K$-partite entanglement $G_{\tau }^{\mathrm{res}}$ in $N$-mode pure fully symmetric Gaussian states, multipartitioned into $K$ molecules of varying sizes $\{m_1,\dots ,m_K\}$, with ${\sum }_j{m}_j=M$ (see plot legends for a pictorial representation of the molecular groupings). Panel (a) depicts the instance $M=N=6$, $K=3$; the molecules are made of, from bottom to top, {1,1,4} modes (solid red line), {1,2,3} modes (dashed green line), and {2,2,2} modes (dotted blue line), respectively. Panel (b) illustrates the instance $M=N=8$, $K=4$; the molecules are made of, from bottom to top, {1,1,1,5} modes (solid red line), {1,1,2,4} modes (dashed magenta line), {1,1,3,3} modes (dot-dashed green line), {1,2,2,3} modes (dot-dot-dashed cyan line), and {2,2,2,2} modes (dotted blue line), respectively. Panel (c) depicts the instance $M=N=15$, $K=5$; the molecules are made of, from bottom to top, {1,1,1,1,11} modes (solid red line), {1,1,1,3,7} modes (dashed magenta line), {1,2,3,4,5} modes (dot-dashed green line), {2,2,2,3,6} modes (dot-dot-dashed cyan line), and {3,3,3,3,3} modes (dotted blue line), respectively (only 5 out of 30 possible molecular partitions are shown). Panel (d) illustrates a mixed-state instance: the total number of modes in the pure state is $N=28$ but only $M=24$ of them are partitioned into $K=6\text{molecules}$ (i.e., we are tracing over the extra 4 modes to be left with the multipartite molecular entanglement in a 24-mode fully symmetric mixed state); the molecules are made of, from bottom to top, {1,1,1,1,1,19} modes (solid red line), {1,1,2,3,8,9} modes (dashed magenta line), {2,2,2,6,6,6} modes (dot-dashed green line), {3,3,3,3,4,8} modes (dot-dot-dashed cyan line), and {4,4,4,4,4,4} modes (dotted blue line), respectively (only 5 out of 199 possible molecular partitions are shown). Notice how, for a fixed total number of modes and parties (molecules), the distributed multipartite entanglement increases with the ranking of the sorted list of partitions, i.e., it is bigger whenever the individual molecule sizes are more similar to each other. Notice also how the genuine multipartite entanglement in any partition increases unboundedly with the squeezing $r⪢0$ for pure states [panels (a)–(c)], while it eventually saturates for mixed states [panel (d)]. All the quantities plotted are dimensionless.

Figure 7

(Color online) Plot, as a function of the squeezing $r$, of the genuine multipartite entanglement $G_{\tau }^{\mathrm{res}}$ in 10 000 random fully symmetric $N$-mode Gaussian states. Each point corresponds to the $K$-partite entanglement among $K$ molecules ${\mathcal{M}}_1,\dots ,{\mathcal{M}}_K$, each comprising $m_j$ modes, with ${\sum }_{j=1}^K{m}_j=M$. For each instance, the real parameter $r$ is randomized between 0 and 2, the integer $N$ (total number of modes of a pure fully symmetric Gaussian state with squeezing $r$) is randomized between 2 and 100, the integer $M$ (effective number of modes partitioned in molecules) is randomized between 2 and $N$, and the integer $K$ (number of molecules) is randomized between 2 and $M$. For any $M<N$, the multipartite entanglement is being computed in a mixed state [as $(N-M)$ modes are traced out]; furthermore, for any $K<M$, the molecules have in general different sizes. The partition $\{m_1,\dots ,m_K\}$ is chosen at random as well among all the possible partitions. Upon application of the unitary localization, each point corresponds equivalently to the genuine $K$-partite entanglement of generally mixed, generally nonsymmetric random $K$-mode Gaussian states belonging to the class of multi-GLEMS. In up to two million states (not shown here) randomly generated according to this prescription, with $N$ up to ${10}^3$, all values of $G_{\tau }^{\mathrm{res}}$ have been found non-negative. This numerical evidence supports the conjecture that the strong monogamy decomposition holds for all Gaussian states beyond symmetry constraints under permutation of the modes, and that the positive-defined residual contangle is a proper quantifier of genuine multipartite entanglement for general states on harmonic lattices. The solid line in the figure corresponds to the instance $M=N=2$, $K=2$, $m_1=m_2=1$, i.e., to the bipartite entanglement in two-mode squeezed states. As the genuine multipartite entanglement decreases with $N$, such a bipartite entanglement stands as an absolute upper limit for any $(N>2)$-mode instance. This bound can only be attained for $N=M=2s$, $K=2$, $m_1=m_2=s$ (with $s$ an integer scale factor) by virtue of the scale invariance discussed in Sec. 5C. All the quantities plotted are dimensionless.