Coexistence of unlimited bipartite and genuine multipartite entanglement: Promiscuous quantum correlations arising from discrete to continuous-variable systems

Quantum mechanics imposes “monogamy” constraints on the sharing of entanglement. We show that, despite these limitations, entanglement can be fully “promiscuous,” i.e., simultaneously present in unlimited two-body and many-body forms in states living in an infinite-dimensional Hilbert space. Monogamy just bounds the divergence rate of the various entanglement contributions. This is demonstrated in simple families of $N$-mode $(N⩾4)$ Gaussian states of light fields or atomic ensembles, which therefore enable infinitely more freedom in the distribution of information, as opposed to systems of individual qubits. Such a finding is of importance for the quantification, understanding, and potential exploitation of shared quantum correlations in continuous variable systems. We discuss how promiscuity gradually arises when considering simple families of discrete variable states, with increasing Hilbert space dimension towards the continuous variable limit. Such models are somehow analogous to Gaussian states with asymptotically diverging, but finite, squeezing. In this respect, we find that non-Gaussian states (which in general are more entangled than Gaussian states) exhibit also the interesting feature that their entanglement is more shareable: in the non-Gaussian multipartite arena, unlimited promiscuity can be already achieved among three entangled parties, while this is impossible for Gaussian, even infinitely squeezed states.

Figure 1

(Color online) (a) Preparation of the four-mode Gaussian states $\mathbf{\gamma }$ of Eq. (2) via two-mode squeezers (boxes) applied on vacuum modes (yellow beams). (b) Structure of bipartite entanglement.

Figure 2

(Color online) Upper bound ${\tau }^{\mathit{\text{bound}}}\left({\mathbf{\gamma }}_{1\mid 2\mid 3}\right)$, Eq. (5), on the tripartite entanglement between modes 1, 2, and 3 (and equivalently 2, 3, and 4) of the four-mode Gaussian state defined by Eq. (2), plotted as a function of the squeezing parameters $s$ and $a$. Focusing on the tripartition 1∣2∣3, the bipartite contangle $\tau \left({\mathbf{\gamma }}_{i\mid \left(jk\right)}\right)$ (with $i,j,k$ a permutation of 1,2,3) is bounded from above by the corresponding bipartite contangle $\tau \left({\mathbf{\sigma }}_{i\mid \left(jk\right)}^p\right)$ in any pure, three-mode Gaussian state with CM ${\mathbf{\sigma }}_{i\mid \left(jk\right)}^p⩽{\mathbf{\gamma }}_{i\mid \left(jk\right)}$. The state ${\mathbf{\sigma }}^p=S_{1,2}\left(a\right)S_{2,3}\left(t\right)S_{2,3}^T\left(t\right)S_{1,2}^T\left(a\right)$ of the three modes 1, 2 and 3, with $t=\frac{1}{2}\mathrm{arccosh}\left[\frac{1+{\mathrm{sech}}^{2}a{\tanh }^{2}s}{1-{\mathrm{sech}}^{2}a{\tanh }^{2}s}\right]$, satisfies this condition and from Eq. (3) we have $\tau \left({\gamma }_{i\mid \left(jk\right)}\right)⩽g\left[{\left(m_{i\mid \left(jk\right)}^{\mathit{\text{bound}}}\right)}^2\right]$. This leads to Eq. (5), where the quantity $g\left[{\left(m_{2\mid \left(13\right)}^{\mathit{\text{bound}}}\right)}^2\right]-g\left[m_{1\mid 2}^2\right]-g\left[m_{2\mid 3}^2\right]$ with $m_{2\mid \left(13\right)}^{\mathit{\text{bound}}}={\sinh }^{2}a+m_{3\mid \left(12\right)}^{\mathit{\text{bound}}}{\cosh }^{2}a$ is not included in the minimization, being always larger than the other terms.

Figure 3

(Color online) Residual multipartite entanglement ${\tau }^{\mathit{res}}\left(\mathbf{\gamma }\right)$ [Eq. (4)], which in the regime of large squeezing $a$ is completely distributed in the form of genuine four-partite quantum correlations.

Figure 4

(Color online) A tripartite pure state of three qudits ($d=2N$ with $N$ even) $A$, $B$, and $C$, is constructed as the tensor product of $N∕2$ copies of a three-qubit GHZ state, and $N∕2$ copies of a three-qubit $W$ state. In the limit $d\to \infty$, $ABC$ constitute a tripartite CV system whose (non-Gaussian) state exhibits full promiscuity, i.e., unlimited genuine tripartite entanglement and unlimited bipartite entanglement between any two parties.