Computable genuine multimode entanglement measure: Gaussian versus non-Gaussian

Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state space, namely, via the generalized geometric measure (GGM). It is defined by the shortest distance of a given multimode state from a nongenuinely multimode entangled state. For the multimode Gaussian states, we derive a closed-form expression of GGM in terms of the symplectic eigenvalues of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three and four modes and find that both addition and subtraction of photons enhance the genuine multimode entanglement of the state compared to its Gaussian counterpart. Our analysis reveals that, when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction, while the scenario reverses when one considers the four mode non-Gaussian states. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition to both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe that GGM freezes under subtraction of photons involving multiple modes, in some specific cases. This feature does not appear while adding photons. Finally, we relate the enhancements of GGM with the distance-based non-Gaussianity measure.

Figure 1

Schematic representation of symplectic diagonalization. Any arbitrary Gaussian state can be written as a product of thermal states obtained by applying the appropriate unitary operation.

Figure 2

GGM of $|{\psi }_3^b\left(t\right)\rangle$ vs time. The state is generated according to the Hamiltonian $H_I$, given in Eq. (27). Solid, dotted, and dashed lines are for different choices of ${\gamma }_1$ and ${\gamma }_2$ values.

Figure 3

GGM of the Gaussian as well as non-Gaussian states against time. In particular, we consider a three-mode Gaussian state $|{\psi }_3^b\left(t\right)\rangle$ (green dot-dashed) and non-Gaussian states, $|{\psi }_3^b{\left(t\right)}^{\text{add}\left\{m_i\right\}}\rangle$ (red solid) and $|{\psi }_3^b{\left(t\right)}^{\text{sub}\left\{m_i\right\}}\rangle$ (blue dashed), given in Eqs. (35) and (36). Here ${\gamma }_1=0.8$ and ${\gamma }_2=0.5$. In the construction of non-Gaussian states $|{\psi }_3^b{\left(t\right)}^{\text{add}\left\{m_i\right\}}\rangle$ and $|{\psi }_3^b{\left(t\right)}^{\text{sub}\left\{m_i\right\}}\rangle$, we choose $m_1=5,m_2=0$, and $m_3=0$.

Figure 4

GGM against added (solid circles) and subtracted (hollow circles) photons from the first mode, $m_1$. In (a), we set the squeezing strength, $r=0.4$, while we choose $r=0.8$ in (b). In both the cases, subtraction yields a higher value of GGM than that of the addition.

Figure 5

Difference between GGM of the photon-subtracted and -added states against the $(m_1,m_2)$ plane in (a) and $(m_1,m_3)$ plane in (b). In particular, we study the behavior of $\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{sub}\{m_1,m_2\}}\rangle )-\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{add}\{m_1,m_2\}}\rangle )$ and $\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{sub}\{m_1,m_3\}}\rangle )-\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{add}\{m_1,m_3\}}\rangle )$. Positive values guarantee that subtraction is always better than that of the addition. We set $r=0.4$.

Figure 6

Map of GGM in the $(m_1,n)$ plane. (a) $m_1$ and $m_{2\left(3\right)}=n$ photons are added in modes. To understand whether adding photons in adjacent modes can generate more multimode entanglement than in the alternate mode, we plot $\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{add}\{m_1,m_3=n\}}\rangle \left)\right)-\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{add}\{m_1,m_2=n\}}\rangle )$. We find that adding photons in the alternate modes is more beneficial than in the adjacent modes. In (b), subtraction of photons is considered. A negative value of the quantity, $\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{sub}\{m_1,m_3=n\}}\rangle \left)\right)-\mathcal{G}\left(\right|{\psi }_{\text{FMSV}}^{\text{sub}\{m_1,m_2=n\}}\rangle )$, indicates that the picture is opposite to that of the addition, thereby specifying that subtracting photons in the alternate mode is an advantageous enterprise. Here $r$ is fixed to 0.4.

Figure 7

GGM vs $m_1$ with $m_1+m_2=20$. (a) GGM is measured after addition of 20 photons in the adjacent, $m_1$ and $m_2$ (solid circles), and in the alternate, $m_1$ and $m_3$ (solid squares), modes. (b) GGM is plotted after subtracting photons from the adjacent (hollow circles) and the alternate (hollow squares) modes. Here $r=0.4$. When photons are subtracted from the alternate modes, GGM remains constant with $m_1$; we call such a phenomenon freezing, as proven in Sec. 4 C.

Figure 8

GGM with respect to $m_1$ and $m_2$, when total number of photons added (subtracted) in three modes is fixed to 20, i.e., $m_1+m_2+m_3=20$. Panel (a) corresponds to addition of photons. GGM is generated when an equal number of photons are added in all the three modes. In (b), subtraction of photons is plotted with $m_1$ and $m_2$. For a fixed $m_2$, we find that the GGM remains constant with $m_1+m_3$, the freezing phenomenon, similar to the one obtained in Fig. 7 (see also Theorem 2 in Sec. IV C). Here $r=0.4$.

Figure 9

Difference between non-Gaussianity, ${\delta }_{\text{NG}}\left(\right|{\psi }_{\text{FMSV}}^{\text{add}\{m_1,m_2\}}\rangle )-{\delta }_{\text{NG}}\left(\right|{\psi }_{\text{FMSV}}^{\text{sub}\{m_1,m_2\}}\rangle )$, of the photon-added and -subtracted states against the $(m_1,m_2)$ plane. We observe that for single mode operations addition and subtraction of photons lead to the same amount of non-Gaussianity, while for two mode operations photon addition leads to greater non-Gaussianity. The squeezing parameter is fixed to 0.4.