Quantifying entanglement of formation for two-mode Gaussian states: Analytical expressions for upper and lower bounds and numerical estimation of its exact value

Entanglement of formation quantifies the entanglement of a state in terms of the entropy of entanglement of the least entangled pure state needed to prepare it. An analytical expression for this measure exists only for special cases, and finding a closed formula for an arbitrary state remains an open problem. In this work we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for the measure that get tight for several special cases. Further, we show that the problem of calculating the actual value of the entanglement of formation for arbitrary two-mode Gaussian states reduces to a trivial single parameter optimization process, and we provide an efficient algorithm for the numerical calculation of the measure.

Figure 1

(a), (b) Two symplectic transformations ${\mathit{\Sigma }}_{\to }$ and ${\mathit{\Sigma }}_{\leftarrow }$, given in Eqs. (13) and (14), respectively. Both of them are decomposed into a sequence (direct and reverse) of a two-mode squeezing transformation ${\mathit{S}}_2$ and two single-mode squeezing transformations $\mathit{S}$. Every state in the standard form can prepared by applying ${\mathit{\Sigma }}_{\to }$ or ${\mathit{\Sigma }}_{\leftarrow }$ onto a classical state.

Figure 2

The percentile relative difference is depicted between both the upper ${\delta }^{+}$ (blue dots) and lower ${\delta }^{-}$ (red crosses) bound and the actual value of the entanglement of formation, given in Eq. (37), vs the purity of randomly created entangled states. It is apparent that the less the purity the larger the difference between ${\mathcal{E}}_F\left({\mathit{\sigma }}^{\text{sf}}\right)$ and ${\mathcal{E}}_F^{\pm }\left({\mathit{\sigma }}^{\text{sf}}\right)$. We also observe that on average the upper bound is closer to the real value than the lower bound.