Extremal entanglement and mixedness in continuous variable systems

We investigate the relationship between mixedness and entanglement for Gaussian states of continuous variable systems. We introduce generalized entropies based on Schatten $p$ norms to quantify the mixedness of a state and derive their explicit expressions in terms of symplectic spectra. We compare the hierarchies of mixedness provided by such measures with the one provided by the purity (defined as $\mathrm{tr}{\varrho }^2$ for the state $\varrho$) for generic $n$-mode states. We then review the analysis proving the existence of both maximally and minimally entangled states at given global and marginal purities, with the entanglement quantified by the logarithmic negativity. Based on these results, we extend such an analysis to generalized entropies, introducing and fully characterizing maximally and minimally entangled states for given global and local generalized entropies. We compare the different roles played by the purity and by the generalized $p$ entropies in quantifying the entanglement and the mixedness of continuous variable systems. We introduce the concept of average logarithmic negativity, showing that it allows a reliable quantitative estimate of continuous variable entanglement by direct measurements of global and marginal generalized $p$ entropies.

Figure 1

Plot of the curves (solid lines) of maximal and minimal von Neumann entropy at given linear entropy for two-mode Gaussian states. The density of states in the plane of entropies is represented as well, by plotting the distribution of 20 000 randomly generated states (dots).

Figure 2

Plot of the absolute difference between the maximal and the minimal values of the generalized entropies $S_p$ at fixed linear entropy $S_L$ for two-mode Gaussian states, as a function of $p$. Different curves correspond to different values of the linear entropy: $S_L=0.8$ (dashed line), $S_L=0.5$ (solid line), and $S_L=0.1$ (dash-dotted line).

Figure 3

Plot of the maximal logarithmic negativity of Gaussian states for fixed marginal purities. The surface represents GMEMMS, states that saturate inequality (70).

Figure 4

Summary of entanglement properties of two-mode (nonsymmetric) Gaussian states in the space of marginal purities ${\mu }_{1,2}$ ($x$ and $y$ axes) and global purity $\mu$. In fact, on the $z$ axis we plot the ratio $\mu ∕{\mu }_1{\mu }_2$ to gain a better graphical distinction between the various regions. In this space, all physical states lie between the horizontal plane $z=1$ representing product states and the upper limiting surface representing GMEMMS. Separable and entangled states are well separated except for a narrow region of coexistence (depicted in black). Separable states fill the region depicted in dark gray, while in the region containing only entangled states we have depicted the average logarithmic negativity, Eq. (88), growing from white to medium gray. The mathematical relations defining the boundaries between all these regions are collected in Table . The three-dimensional envelope is cut at $z=3.5$.

Figure 5

Possible scheme for the generation of Gaussian least-entangled mixed states (GLEMS). A single-mode squeezed state (obtained, for example, by an optical parametric oscillator or amplifier) interferes with a thermal state through a 50:50 beam splitter. The resulting two-mode state is a minimally entangled mixed Gaussian state at given global and marginal purities.

Figure 6

Plot of the nodal surface which solves the equation ${\kappa }_p=0$ with ${\kappa }_p$ defined by Eq. (80), for (a) $p=3$ and (b) $p=4$. The entanglement of Gaussian states that lie on the leaf-shaped surfaces is fully quantified in terms of the marginal purities and the global generalized entropy (a) $S_3$ or (b) $S_4$. The equations of the surfaces in the space ${\epsilon }_p\equiv \{{\mu }_1,{\mu }_2,S_p\}$ are given by Eqs. (83, 84, 85).

Figure 7

Summary of the entanglement properties for symmetric Gaussian states at fixed global and marginal generalized $p$ entropies, for (a) $p=1$ (von Neumann entropies), (b) $p=2$ (linear entropies), (c) $p=3$, and (d) $p=4$. In the entangled region, the average logarithmic negativity ${\overline{E}}_{\mathcal{N}}(S_{p_i},S_p)$, Eq. (88), is depicted, growing from gray to black. For $p>2$ an additional dashed curve is plotted; it represents the nodal line of inversion. Along it the entanglement is fully determined by the knowledge of the global and marginal generalized entropies $S_{p_i},S_p$, and GMEMS and GLEMS are equally entangled. On the left side of the nodal line GMEMS (GLEMS) are maximally (minimally) entangled Gaussian states at fixed $S_{p_i},S_p$. On the right side of the nodal line they are inverted: GMEMS (GLEMS) are minimally (maximally) entangled states. Also notice how the dashed region of coexistence becomes narrower with increasing $p$. The equations of all boundary curves can be found in Eq. (86).

Figure 8

Upper and lower bounds on the logarithmic negativity of symmetric Gaussian states as functions of the global and marginal generalized $p$ entropies, for (a) $p=1$ (von Neumann entropies), (b) $p=2$ (linear entropies), (c) $p=3$, and (d) $p=4$. The black (gray) surface represents GMEMS (GLEMS). Notice that for $p>2$ GMEMS and GLEMS surfaces intersect along the inversion line (meaning they are equally entangled along that line), and beyond it they interchange their role. The equations of the inversion lines are obtained from Eqs. (83, 84, 85), with the position $S_{p_1}=S_{p_2}\equiv S_{p_i}$.

Figure 9

The relative error $\delta {\overline{E}}_{\mathcal{N}}$, Eq. (89), on the average logarithmic negativity as a function of the ratio $S_{p_i}∕S_p$, for (a) $p=1$, (b) $p=2$, (c) $p=3$, and (d) $p=4$, plotted at (a) $S_V=1$, (b) $S_L=1∕2$, (c) $S_3=1∕4$, and (d) $S_4=1∕6$. Notice how, in general, the error decays exponentially and, in particular, faster with increasing $p$. For $p>2$, notice how the error reaches zero on the inversion node (see the insets), then grows and reaches a local maximum before going back to zero asymptotically.