Quantifying entanglement in two-mode Gaussian states

Entangled two-mode Gaussian states are a key resource for quantum information technologies such as teleportation, quantum cryptography, and quantum computation, so quantification of Gaussian entanglement is an important problem. Entanglement of formation is unanimously considered a proper measure of quantum correlations, but for arbitrary two-mode Gaussian states no analytical form is currently known. In contrast, logarithmic negativity is a measure that is straightforward to calculate and so has been adopted by most researchers, even though it is a less faithful quantifier. In this work, we derive an analytical lower bound for entanglement of formation of generic two-mode Gaussian states, which becomes tight for symmetric states and for states with balanced correlations. We define simple expressions for entanglement of formation in physically relevant situations and use these to illustrate the problematic behavior of logarithmic negativity, which can lead to spurious conclusions.

Figure 1

State decompositions. Any state, ${\mathbf{\sigma }}^{\text{sf}}$, can be constructed by applying a sequence of (a) two-mode squeezing $S_2\left(r\right)$ followed by local squeezing $S\left(r_i\right)$ to a classical state ${\mathbf{\sigma }}_{\text{c}}$ or, reversely, (b) local squeezing $S\left({\tilde{r}}_i\right)$ followed by two-mode squeezing $S_2\left(\tilde{r}\right)$ to a classical state ${\tilde{\mathbf{\sigma }}}_{\text{c}}$.

Figure 2

Lower bound for entanglement of formation. We plot with black dots the optimum symplectic eigenvalue ${\tilde{\nu }}_{o_{-}}=e^{-2r_o}$ (calculated numerically [24]) versus the corresponding value based on ${\tilde{r}}_{-}$, i.e., $e^{-2{\tilde{r}}_{-}}$, for randomly generated states. The symplectic eigenvalue is a bounded value $\in (0,1]$, which shows (a) that ${\tilde{\mathcal{E}}}_F\left(\mathbf{\sigma }\right)\le {\mathcal{E}}_F\left(\mathbf{\sigma }\right)$ and (b) that the bound is also tight for separable and infinite entangled states. We also depict, with blue squares [27] and red triangles [28], the corresponding values we get from the previously known lower bounds. The closer the dots are to the diagonal, the smaller the deviation from the real value of entanglement. It is clear that our bound is, on average, tighter than previous bounds. All quantities plotted are dimensionless.

Figure 3

Comparison between entanglement of formation (solid blue line) and logarithmic negativity (dashed red line). Assuming that a two-mode squeezed vacuum state with $r=1$ is sent through a lossy channel of transmissivity $0\le \tau \le 1$, we compare the two measures. The deterministic upper bounds (upper lines), i.e., the amount of entanglement assuming that an infinitely squeezed state is sent through the same channel, are also depicted, since they provide further insight regarding the qualitative differences between entanglement of formation and logarithmic negativity. The deterministic bound for logarithmic negativity can be found in Ref. [29]. Specifically, for logarithmic negativity, the deterministic bound of a state with transmissivity value ${\tau }_a$ can also be reached by sending the squeezed state $(r=1)$ through a channel of transmissivity ${\tau }_b$, with ${\tau }_b>{\tau }_a$. However, in contrast, entanglement of formation predicts that we cannot reach the deterministic bound with a squeezed state $(r=1)$ regardless of how much we raise the transmissivity. This is a critical difference, since the two quantifiers disagree on whether or not a physical upper bound has been reached. All quantities plotted are dimensionless.