Total versus quantum correlations in a two-mode Gaussian state

In Li and Luo (2007 Phys. Rev. A 76 032327), the inequality $(1/2){ \mathcal T }\geqslant { \mathcal Q }$ was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of total ${ \mathcal T }$ and quantum ${ \mathcal Q }$ correlations in bipartite quantum states. Besides, Hayden et al (2006 Commun. Math. Phys. 265 95) have conjectured that, in some conditions within systems endowed with infinite-dimensional Hilbert spaces, quantum correlations may dominate not only half of total correlations but total correlations itself. Here, in a two-mode Gaussian state, quantifying ${ \mathcal T }$ and ${ \mathcal Q }$ respectively by the quantum mutual information ${{ \mathcal I }}^{G}$ and the entanglement of formation (EoF) ${{ \mathcal E }}_{F}^{G}$, we verify that ${{ \mathcal E }}_{F,R}^{G}$ is always less than $(1/2){{ \mathcal I }}_{R}^{G}$ when ${{ \mathcal I }}^{G}$ and ${{ \mathcal E }}_{F}^{G}$ are defined via the Rényi-2 entropy. While via the von Neumann entropy, ${{ \mathcal E }}_{F,V}^{G}$ may even dominate ${{ \mathcal I }}_{V}^{G}$ itself, which partly consolidates the Hayden conjecture, and partly, provides strong evidence hinting that the origin of this counterintuitive behavior should intrinsically be related to the von Neumann entropy by which the EoF ${{ \mathcal E }}_{F,V}^{G}$ is defined, rather than related to the conceptual definition of the EoF ${{ \mathcal E }}_{F}$. The obtained results show that-in the special case of mixed two-mode Gaussian states-quantum entanglement can be faithfully quantified by the Gaussian Rényi-2 EoF ${{ \mathcal E }}_{F,R}^{G}$.