Total versus quantum correlations in a two-mode Gaussian state
本站小编 Free考研考试/2022-01-02
Jamal El Qars,1,2,31Department of Physics, Faculty of Applied Sciences, Ait-Melloul, Ibn Zohr University, Agadir, Morocco 2EPTHE, Department of Physics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco 3LPHE-MS, Department of Physics, Faculty of Sciences, Mohammed V University, Rabat, Morocco
Abstract 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 ${{ \mathcal E }}_{F,R}^{G}$. Keywords：entanglement;von Neumann and Rényi-2 entropies;quantum mutual information;Gaussian states;spontaneous emission laser
In a quantum bipartite state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$, total correlations ${ \mathcal T }$ can be divided into two parts: classical ${ \mathcal C }$ and quantum ${ \mathcal Q }$, then ${ \mathcal T }={ \mathcal Q }+{ \mathcal C }$ [1]. Besides, adopting the Henderson–Vedral conjecture [1], that is, classical correlations should not be less than quantum ones, i.e. ${ \mathcal C }\geqslant { \mathcal Q }$, it follows that $(1/2){ \mathcal T }\geqslant { \mathcal Q }$, which was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations in bipartite states for arbitrary measures of quantum Q and total ${ \mathcal T }$ correlations [2].
On the one hand, it is generally accepted that the quantum mutual information—denoted ${ \mathcal I }$—is the information-theoretic measure of total correlations ${ \mathcal T }$ in quantum bipartite states [3–5]. Before being introduced in [6], the concept of quantum mutual information has been implicitly used some years ago to study information transfer in quantum measurements [7] and subsequently rediscovered in [8]. From an operational point of view, the quantum mutual information was first interpreted as the amount of randomness noise needed to erase completely the correlations in a quantum state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ by turning it into a product state, i.e. ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}={\hat{\varrho }}_{{ \mathcal X }}\otimes {\hat{\varrho }}_{{ \mathcal Y }}$ [3]. Later, it was rigorously associated with the maximum amount of information that can be securely transmitted between a sender ${ \mathcal X }$ and a receiver ${ \mathcal Y }$, who share a correlated state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ [4].
On the other hand, quantum correlations of mixed states, can arise in diverse incarnations, i.e. Bell nonlocality [9], steering [10], entanglement [11], and discord [5, 12], where all can be exploited for enhancing information processing tasks over any classical approach [13]. In particular, entanglement—usually used as synonymous of quantum correlations—was systematically defined as a kind of nonseparable quantum correlations that cannot be prepared only by means of local operations and classical communication [14]. Nowadays, it becomes evident that entanglement, as a fundamental physical resource for quantum protocols, can play a crucial role in quantum information science [15].
Combining the inequality $(1/2){ \mathcal T }\geqslant { \mathcal Q }$ with the fact that entanglement ${ \mathcal E }$ is one of different manifestations of quantum correlations, i.e. ${ \mathcal E }\subset { \mathcal Q }$, one, consequently, gets $(1/2){ \mathcal I }\geqslant { \mathcal E }$, which must be a necessary requirement for accepting ${ \mathcal E }$ as a genuine measure of entanglement as long as the quantum mutual information ${ \mathcal I }$ is generally used as a useful measure of total correlations ${ \mathcal T }$ [2].
One of the central tasks of quantum information theory is to quantify the amount of entanglement that a quantum state possesses [16]. In pure bipartite states, there is a universal bona fide quantifier of entanglement, i.e. the entropy of entanglement [17]. However, in mixed states—where a more complex scenario emerges—the entropy of entanglement no longer deserves to be a measure of entanglement. Therefore, miscellaneous entanglement measures, which can be distinguished due to their operational meaning or mathematical structures, have been introduced [16].
From a conceptual point of view, Gaussian entanglement measures can exhaustively be classified into two categories. The first one encompasses the negativities [18–20], while, the second is provided by the so-called convex-roof extended measures [21]. Except for the entanglement of formation (EoF) [22]—which physically interpreted as the minimal entanglement needed for preparing an entangled state by mixing pure entangled states, and mathematically computed for two-qubit states [23] as well as for arbitrary two-mode Gaussian states (TMGSs) [24, 25]—no such measure is currently known [26].
Defining the quantum mutual information ${ \mathcal I }$ and the entanglement ${ \mathcal E }$ by means of the von Neumann entropy, it has been shown in [2], within discrete-variable systems, that the inequality $(1/2){ \mathcal I }\geqslant { \mathcal E }$ is well satisfied when substituting ${ \mathcal E }$ by either the distillable entanglement ${{ \mathcal E }}_{D}$ [22] or the squashed entanglement ${{ \mathcal E }}_{\mathrm{sq}}$ [27]. While, it may be violated with respect to the EoF ${{ \mathcal E }}_{F}$ or the entanglement cost ${{ \mathcal E }}_{C}$ [22], where the former may exceed not only half of total correlations, but total correlations itself [2]. In a similar context, it has been shown for a two-qubit system, that quantum correlations quantified by means of the von Neumann EoF ${{ \mathcal E }}_{F}$ may exceed classical ones [28], which violates the Henderson–Vedral conjecture [1]. Consequently—as main conclusion pointed out in [2]—the EoF ${{ \mathcal E }}_{F}$ is too big to be regarded as a genuine measure of entanglement, in the sense that it may exceed total correlations. Attempting to explain the origin of the peculiar behaviors exhibited by the EoF ${{ \mathcal E }}_{F}$ in [2, 28], some doubts around the validity of the Henderson–Vedral conjecture as well as the validity of the pure-state decompositions in the general definition of the EoF ${{ \mathcal E }}_{F}$ have seriously been raised in [2, 28].
In the past decades, a correlated spontaneous emission laser—in which nondegenerate three-level atoms in a cascade configuration are injected into a cavity in a coherent superposition—has attracted a special attention in connection with its potential as a source of a highly correlated two-mode Gaussian light [29, 30]. The quantum correlations between the emitted photons can be induced by preparing the atoms, initially, in a coherent superposition of the upper and lower levels [31–33] or by coupling these two levels by a strong coherent external driving [34] or also by using the two processes simultaneously [35]. Essentially, a correlated spontaneous emission laser is believed to be a source of strong quantum correlations [29, 30], which therefore can be employed for testing quantum nonlocality [36–39]. Here, this system is chosen as a viable and reliable scheme for investigating the Hayden conjecture [40] and the constraint $(1/2){ \mathcal I }\geqslant { \mathcal E }$ in a TMGS.
Hence, in a mixed Gaussian state ${\hat{\varrho }}_{12}$ involving two cavity modes of a nondegenerate three-level laser [29, 41–43], and motivated by the fact that the inequality $(1/2){{ \mathcal I }}_{R}^{G}\geqslant {{ \mathcal E }}_{F,R}^{G}$ is proven to hold for general mixed TMGSs [44, 45], where ${ \mathcal I }$ and ${{ \mathcal E }}_{F}$ are defined via the Rényi-2 entropy [46], we give strong evidence hinting that the origin of the counterintuitive behaviors exhibited by the EoF ${{ \mathcal E }}_{F}$ in [2, 28], should intrinsically be related to the von Neumann entropy by which the EoF ${{ \mathcal E }}_{F,V}$ is defined, rather than related to the conceptual definition of the EoF ${{ \mathcal E }}_{F}$ or the Henderson–Vedral conjecture. For this, we first verify the holding of the inequality $(1/2){{ \mathcal I }}_{R}^{G}\geqslant {{ \mathcal E }}_{F,R}^{G}$, next, via the von Neumann entropy, we show that even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which largely violates $(1/2){ \mathcal I }\geqslant {{ \mathcal E }}_{F}$, and further supports the Hayden conjecture [40].
The remainder of this paper is organized as follows. In section 2, we introduce the system at hand, and we derive the master equation for the state ${\hat{\varrho }}_{12}$. Next, we obtain the dynamics of the first and second moments of the two cavity modes variables, and further we evaluate the stationary covariance matrix of the state ${\hat{\varrho }}_{12}$. In section 3, we quantify the EoF as well as the quantum mutual information in the bipartite state ${\hat{\varrho }}_{12}$ using two different entropic measures, i.e. the von Neumann and the Rényi-2 entropies. Also, we present our results. Finally, in section 4, we draw our conclusions.
2. Model and master equation
In a cavity coupled to a vacuum reservoir, we consider an ensemble of nondegenerate three-level atoms resonantly interacting with two cavity modes of the quantized cavity field. The jth cavity mode is specified by its annihilation operator ${\hat{a}}_{j}$, frequency ωj and a decay rate κj (we take for simplicity κ1,2 = κ). The atoms are assumed to be injected into the cavity at a rate ra and removed within a time τ [31]. As shown in figure 1, the upper, intermediate and lower energy levels of a single atom are denoted respectively by ∣a⟩, ∣b⟩ and ∣c⟩.
Figure 1.
New window|Download| PPT slide Figure 1.A nondegenerate three-level laser in a cascade configuration [32]. The transition ∣a⟩ → ∣b⟩(∣b⟩ → ∣c⟩) of frequency ω1(ω2) and spontaneous emission decay rate γab(γbc) is assumed to be dipole-allowed, while, ∣a⟩ → ∣c⟩ is dipole forbidden [29].
In the rotating wave approximation, the interaction between the two cavity modes and a single three-level atom, can be described in the interaction picture by the Hamiltonian [30]$\begin{eqnarray}\begin{array}{l}{\hat{{ \mathcal H }}}_{\mathrm{int}}={\rm{i}}{\hslash }\left({\chi }_{{ab}}{\hat{a}}_{1}| a\rangle \langle b| +{\chi }_{{bc}}{\hat{a}}_{2}| b\rangle \langle c| \right.\\ \ \ \left.-\,{\chi }_{{ab}}| b\rangle \langle a| {\hat{a}}_{1}^{\dagger }-{\chi }_{{bc}}| c\rangle \langle b| {\hat{a}}_{2}^{\dagger }\right),\end{array}\end{eqnarray}$with χab(χbc) being the coupling constant for the transition∣a⟩ → ∣b⟩(∣b⟩ → ∣c⟩) [31]. Also, we assume that the atoms are initially prepared in an arbitrary coherent superposition of the upper ∣a⟩ and the lower ∣c⟩ energy levels with the probabilities ∣α∣2 and ∣β∣2 [33]. The initial state of a single atom writes ∣ψatom(0)⟩ = α∣a⟩ + β∣c⟩, thus, its associated density operator is given by$\begin{eqnarray}{\hat{\varrho }}_{\mathrm{atom}}(0)={\varrho }_{{aa}}^{\left(0\right)}| a\rangle \langle a| +{\varrho }_{{ac}}^{\left(0\right)}| a\rangle \langle c| +{\varrho }_{{ca}}^{\left(0\right)}| c\rangle \langle a| +{\varrho }_{{cc}}^{\left(0\right)}| c\rangle \langle c| ,\end{eqnarray}$where ${\varrho }_{{aa}}^{\left(0\right)}=| \alpha {| }^{2},$${\varrho }_{{cc}}^{\left(0\right)}=| \beta {| }^{2}$ and ${\varrho }_{{ac}}^{\left(0\right)}={\varrho }_{{ca}}^{\left(0\right)* }=\alpha {\beta }^{* }$. In what follows, we take, for convenient, identical spontaneous decay rates, i.e. γab,bc = γ, and identical coupling transitions, i.e. χab,bc = χ.
The master equation for the reduced density operator ${\hat{\varrho }}_{12}\equiv \hat{\varrho }$ of the two cavity modes, writes [47, 48]$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\hat{\varrho } & = & \displaystyle \frac{-{\rm{i}}}{{\hslash }}{\mathrm{Tr}}_{\mathrm{atom}}\left[{\hat{{ \mathcal H }}}_{\mathrm{int}},{\hat{\varrho }}_{\{\mathrm{atom}+\mathrm{field}\}}\right]\\ & & +\,\displaystyle \sum _{j=1,2}\displaystyle \frac{{\kappa }_{j}}{2}{ \mathcal L }\left[{\hat{a}}_{j}\right]\hat{\varrho },\end{array}\end{eqnarray}$where the Lindblad operator ${ \mathcal L }\left[{\hat{a}}_{j}\right]\hat{\varrho }=2{\hat{a}}_{j}\hat{\varrho }{\hat{a}}_{j}^{\dagger }-{\left[{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j},\hat{\varrho }\right]}_{+}$ describes the damping of the jth cavity mode through the vacuum reservoir. Within the linear-adiabatic approximation [30], equation (3) would be$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}\hat{\varrho } & = & \displaystyle \frac{A{\varrho }_{{aa}}^{\left(0\right)}}{2}[2{\hat{a}}_{1}^{\dagger }\hat{\varrho }{\hat{a}}_{1}-{\hat{a}}_{1}{\hat{a}}_{1}^{\dagger }\hat{\varrho }-\hat{\varrho }{\hat{a}}_{1}{\hat{a}}_{1}^{\dagger }]+\displaystyle \frac{A{\varrho }_{{cc}}^{\left(0\right)}}{2}\\ & & \times \,[2{\hat{a}}_{2}\hat{\varrho }{\hat{a}}_{2}^{\dagger }-{\hat{a}}_{2}^{\dagger }{\hat{a}}_{2}\hat{\varrho }-\hat{\varrho }{\hat{a}}_{2}^{\dagger }{\hat{a}}_{2}]\\ & & +\,\displaystyle \frac{A{\varrho }_{{ac}}^{\left(0\right)}}{2}[\hat{\varrho }{\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}^{\dagger }+{\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}^{\dagger }\hat{\varrho }+{\hat{a}}_{1}{\hat{a}}_{2}\hat{\varrho }+\hat{\varrho }{\hat{a}}_{1}{\hat{a}}_{2}\\ & & -\,2{\hat{a}}_{1}^{\dagger }\hat{\varrho }{\hat{a}}_{2}^{\dagger }-2{\hat{a}}_{2}\hat{\varrho }{\hat{a}}_{1}]+\displaystyle \sum _{j=1,2}\displaystyle \frac{{\kappa }_{j}}{2}{ \mathcal L }\left[{\hat{a}}_{j}\right]\hat{\varrho },\end{array}\end{eqnarray}$where ${\varrho }_{{ac}}^{\left(0\right)}$ is assumed to be real, A = 2raχ2/γ2 is the linear gain coefficient [49]. In equation (4), the first(second) term proportional to ${\varrho }_{{aa}}^{\left(0\right)}$(${\varrho }_{{cc}}^{\left(0\right)}$), describes the gain(loss) of the mode ${\hat{a}}_{1}$(${\hat{a}}_{2}$), while, the third term proportional to ${\varrho }_{{ac}}^{\left(0\right)}$ traduces the coupling between the modes ${\hat{a}}_{1}$ and ${\hat{a}}_{2}$ via the atomic coherence induced by the initial superposition of the upper-∣a⟩ and the lower-∣c⟩ levels [30]. Finally, the last term proportional to κ describes the damping of the two cavity modes through the vacuum reservoir. On the basis of equation (4), one can move to the Schrödinger picture using ${\partial }_{t}\langle \hat{{ \mathcal O }}(t)\rangle \,=\mathrm{Tr}({\partial }_{t}\hat{\varrho }\hat{{ \mathcal O }}(t))$. Thus, one gets the dynamics of the first and second moments of the two cavity modes variables [30]$\begin{eqnarray}\begin{array}{l}{\partial }_{t}\langle {\hat{a}}_{j}(t)\rangle =\displaystyle \frac{-{\mu }_{j}}{2}\langle {\hat{a}}_{j}(t)\rangle +\displaystyle \frac{{\left(-1\right)}^{j}A{\varrho }_{{ac}}^{\left(0\right)}}{2}\\ \quad \times \,\langle {\hat{a}}_{3-j}^{\dagger }(t)\rangle ,\ \mathrm{for}\ j=1,2,\end{array}\end{eqnarray}$$\begin{eqnarray}{\partial }_{t}\langle {\hat{a}}_{j}^{2}(t)\rangle =-{\mu }_{j}\langle {\hat{a}}_{j}^{2}(t)\rangle +{\left(-1\right)}^{j}A{\varrho }_{{ac}}^{\left(0\right)}\langle {\hat{a}}_{1}(t){\hat{a}}_{2}^{\dagger }(t)\rangle ,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\partial }_{t}\langle {\hat{a}}_{j}^{\dagger }(t){\hat{a}}_{j}(t)\rangle =-{\mu }_{j}\langle {\hat{a}}_{j}^{\dagger }(t){\hat{a}}_{j}(t)\rangle +\displaystyle \frac{{\left(-1\right)}^{j}A{\varrho }_{{ac}}^{\left(0\right)}}{2}\\ \ \ \times \,\left[\langle {\hat{a}}_{1}^{\dagger }(t){\hat{a}}_{2}^{\dagger }(t)\rangle +\langle {\hat{a}}_{1}(t){\hat{a}}_{2}(t)\rangle \right]+(2-j)A{\varrho }_{{aa}}^{\left(0\right)},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\partial }_{t}\langle {\hat{a}}_{1}(t){\hat{a}}_{2}(t)\rangle =-\displaystyle \frac{{\mu }_{1}+{\mu }_{2}}{2}\langle {\hat{a}}_{1}(t){\hat{a}}_{2}(t)\rangle +\displaystyle \frac{A{\varrho }_{{ac}}^{\left(0\right)}}{2}\\ \ \ \times \,\left[\langle {\hat{a}}_{1}^{\dagger }(t){\hat{a}}_{1}(t)\rangle -\langle {\hat{a}}_{2}^{\dagger }(t){\hat{a}}_{2}(t)\rangle +1\right],\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\partial }_{t}\langle {\hat{a}}_{1}(t){\hat{a}}_{2}^{\dagger }(t)\rangle =-\displaystyle \frac{{\mu }_{1}+{\mu }_{2}}{2}\langle {\hat{a}}_{1}(t){\hat{a}}_{2}^{\dagger }(t)\rangle \\ \ \ +\,\displaystyle \frac{A{\varrho }_{{ac}}^{\left(0\right)}}{2}\left[\langle {\hat{a}}_{1}^{2}(t)\rangle -\langle {\hat{a}}_{2}^{\dagger 2}(t)\rangle \right],\end{array}\end{eqnarray}$where ${\mu }_{1}=\kappa -A{\varrho }_{{aa}}^{\left(0\right)}$ and ${\mu }_{2}=\kappa +A{\varrho }_{{cc}}^{\left(0\right)}.$ Next, we introduce the population inversion η defined as ${\varrho }_{{aa}}^{\left(0\right)}=(1-\eta )/2$ with ≤ η ≤ 1 [31]. Moreover, using the fact that ${\varrho }_{{aa}}^{\left(0\right)}\,+{\varrho }_{{cc}}^{\left(0\right)}=1$ and $| {\varrho }_{{ac}}^{\left(0\right)}| =\sqrt{{\varrho }_{{aa}}^{\left(0\right)}{\varrho }_{{cc}}^{\left(0\right)}}$, we obtain ${\varrho }_{{cc}}^{\left(0\right)}=(1+\eta )/2$ and ${\varrho }_{{ac}}^{\left(0\right)}={\varrho }_{{ca}}^{\left(0\right)}=\sqrt{1-{\eta }^{2}}/2$.
We need to carry out our study in the stationary regime, then, the non-zero steady-state solutions of equations ((5)–(9)) can be expressed as$\begin{eqnarray}\langle {\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}{\rangle }_{{ss}}=\displaystyle \frac{-A{\left(1-\eta \right)}^{2}}{4\eta \left(\kappa +A\eta \right)}+\displaystyle \frac{A\left(1-{\eta }^{2}\right)}{2\eta \left(2\kappa +A\eta \right)},\end{eqnarray}$$\begin{eqnarray}\langle {\hat{a}}_{2}^{\dagger }{\hat{a}}_{2}{\rangle }_{{ss}}=\displaystyle \frac{-A\left(1-{\eta }^{2}\right)}{4\eta \left(\kappa +A\eta \right)}+\displaystyle \frac{A\left(1-{\eta }^{2}\right)}{2\eta \left(2\kappa +A\eta \right)},\end{eqnarray}$$\begin{eqnarray}\langle {\hat{a}}_{1}{\hat{a}}_{2}{\rangle }_{{ss}}=\displaystyle \frac{-A\left(1-\eta \right)\sqrt{1-{\eta }^{2}}}{4\eta \left(\kappa +A\eta \right)}+\displaystyle \frac{A\sqrt{1-{\eta }^{2}}}{2\eta \left(2\kappa +A\eta \right)}.\end{eqnarray}$Notice that the equations ((10)–(12)) are physically meaningful only if η ≥ 0, so that 0 ≤ η ≤ 1, which corresponds to the regime of lasing without population inversion [30].
Since the two cavity modes ${\hat{a}}_{1}$ and ${\hat{a}}_{2}$ involve in a TMGS [29, 36], therefore, they can be fully described by their covariance matrix σ12 of elements ${\vartheta }_{{ij}}=\left(\langle [{\hat{u}}_{i},{\hat{u}}_{j}{]}_{+}\rangle \right)/2\,-\langle {\hat{u}}_{i}\rangle \langle {\hat{u}}_{j}\rangle $, where ${\hat{u}}^{{\rm{T}}}=({\hat{x}}_{1},{\hat{y}}_{1},{\hat{x}}_{2},{\hat{y}}_{2})$ is the vector of the quadrature operators ${\hat{x}}_{j}=({\hat{a}}_{j}^{\dagger }+{\hat{a}}_{j})/\sqrt{2}$ and ${\hat{y}}_{j}={\rm{i}}({\hat{a}}_{j}^{\dagger }\,-{\hat{a}}_{j})/\sqrt{2}$ [20]. With the aid of equations ((10)–(12)), one gets$\begin{eqnarray}{\sigma }_{12}=\left(\begin{array}{cc}{\sigma }_{1} & {\sigma }_{3}\\ {\sigma }_{3}^{{\rm{T}}} & {\sigma }_{2}\end{array}\right),\end{eqnarray}$where the block-matrix σj = ϑjj 1l2 represents the jth cavity mode for j = 1, 2, while σ3 = ϑ12diag(1,-1) describes the correlations between them, with ${\vartheta }_{{jj}}=\langle {\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}{\rangle }_{s}+1/2$ and ${\vartheta }_{12}=(\langle {\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}^{\dagger }{\rangle }_{s}+\langle {\hat{a}}_{1}{\hat{a}}_{2}{\rangle }_{s})/2$.
3. EoF versus total correlations in a two-mode Gaussian state
3.1. Gaussian EoF
Before defining the Gaussian EoF, we briefly recall what is intended by entanglement. A bipartite pure state $| {\psi }_{{ \mathcal X }{ \mathcal Y }}\rangle $ is said to be entangled, if cannot be factorised as $| {\psi }_{{ \mathcal X }{ \mathcal Y }}\rangle \,=| {\phi }_{{ \mathcal X }}\rangle \otimes | \varphi { \mathcal Y }\rangle $. While, a mixed state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ is entangled if cannot be factorised as a convex combinations of product states, i.e. ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}={\sum }_{i}{p}_{i}{\hat{\varrho }}_{{{ \mathcal X }}_{i}}\otimes {\hat{\varrho }}_{{{ \mathcal Y }}_{i}}$ where pi are probabilities with ∑ipi = 1 [11].
In pure bipartite states, the entropy of entanglement is a simple and unique measure of entanglement [17]. However, for mixed states, such a measure no longer deserves for quantifying entanglement [11], and therefore various measures, including the Rényi-α Gaussian EoF ${{ \mathcal E }}_{F,\alpha }^{G}$, were introduced [24, 25, 44].
For a bipartite quantum state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$, the Ré nyi-α EoF ${{ \mathcal E }}_{F,\alpha }$ is defined as the convex-roof of the Rényi-α entropy ${{ \mathcal S }}_{\alpha }{({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})=(1-\alpha )}^{-1}\mathrm{ln}\{\mathrm{Tr}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}^{\alpha })\}$ on pure states [44, 50], i.e.$\begin{eqnarray}\begin{array}{l}{{ \mathcal E }}_{F,\alpha }({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})\\ := \ \mathop{\inf }\limits_{\{{p}_{i}{},| {\psi }_{i}\rangle \}}\displaystyle \sum _{i}{p}_{i}{{ \mathcal S }}_{\alpha }(| {\psi }_{i}\rangle )\ \mathrm{with}\ \alpha \in (0,1)\cup (1,+\infty ),\end{array}\end{eqnarray}$where the minimisation is over all the decompositions of ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}={\sum }_{i}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| $ into set of pure states {∣ψi⟩} with pi ≥ 0 and ∑ipi = 1.
Here, let us pause to briefly recall that—in quantum information theory—the degree of information possessed by a bipartite quantum state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ is conventionally quantified by the von Neumann entropy defined as ${{ \mathcal S }}_{1}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})\,\equiv -\mathrm{Tr}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}\mathrm{ln}{\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})$, that is the direct counterpart to the Shannon entropy in classical information theory [51]. More precisely, such entropy—used under the name of entanglement entropy—quantifies the degree of quantum information contained in an ensemble of a large number of independent and identically distributed copies of the state [52]. In addition, it was widely employed to study entanglement in miscellaneous fields of physics, e.g. ground states of quantum many body systems and lattice systems [53, 54], relativistic quantum field theory [55], and the holographic theory of black holes [56].
Besides, the α-Rényi entropies defined as ${{ \mathcal S }}_{\alpha }({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})\,={(1-\alpha )}^{-1}\mathrm{ln}\{\mathrm{Tr}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}^{\alpha })\}$ with α ∈ (0, 1) ∪ (1, + ∞) [46], were introduced as a generalization of the von Neumann entropy [56, 57]. Their interpretation is essentially related to derivatives of the free energy with respect to temperature [58], and which have found applications, particularly, in the study of channel capacities [59], work value of information [60], and entanglement spectra in many-body systems [61].
Notice that the class of α-Rényi entropies are continuous, non-negatives, invariants under the action of the unitary operations, additive on tensor-product states, and converge to the von Neumann entropy in the limit α → 1 [62].
Given an arbitrary TMGS with covariance matrix ${\sigma }_{{ \mathcal X }{ \mathcal Y }}$, an upper bound of the Rényi-α EoF ${{ \mathcal E }}_{F,\alpha }$ can be obtained by limiting the decomposition in equation (14) only over pure Gaussian states, therefore, we obtain the Rényi-α Gaussian EoF [44, 50]$\begin{eqnarray}{{ \mathcal E }}_{F,\alpha }^{G}({\sigma }_{{ \mathcal X }{ \mathcal Y }}):= \mathop{\inf }\limits_{\{{{\rm{\Xi }}}_{{ \mathcal X }{ \mathcal Y }}| 0\lt {{\rm{\Xi }}}_{{ \mathcal X }{ \mathcal Y }}\leqslant {\sigma }_{{ \mathcal X }{ \mathcal Y }},\det {{\rm{\Xi }}}_{{ \mathcal X }{ \mathcal Y }}=1\}}{{ \mathcal S }}_{\alpha }({{\rm{\Xi }}}_{{ \mathcal X }}),\ \end{eqnarray}$where the minimisation is taken over a pure TMGS with covariance matrix ${{\rm{\Xi }}}_{{ \mathcal X }{ \mathcal Y }}$ smaller than ${\sigma }_{{ \mathcal X }{ \mathcal Y }}$, and the sub-matrix ${{\rm{\Xi }}}_{{ \mathcal X }}$ is the marginal covariance matrix of the first mode obtained from ${{\rm{\Xi }}}_{{ \mathcal X }{ \mathcal Y }}$ by partial tracing over the second mode.
Choosing ${{ \mathcal S }}_{\alpha }$ to be the conventional von Neumann entropy defined, in the limit α → 1, as ${{ \mathcal S }}_{1}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})\,=-\mathrm{Tr}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}\mathrm{ln}{\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})$ [50], then equation (15) defines the usual Gaussian EoF ${{ \mathcal E }}_{F,1}^{G}\equiv {{ \mathcal E }}_{F,V}^{G}$ [23–25]. Finding analytical solution of the optimization problem in equation (15) for generic states is—in general—a nontrivial task, nevertheless, for α → 1, closed formulas were determined for two-qubit states [23], isotropic and Werner states [63, 64], as well as for arbitrary TMGSs [24, 25].
Within the Gaussian framework, it has been demonstrated that the Rényi-2 entropy ${{ \mathcal S }}_{2}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})=-\mathrm{ln}\{\mathrm{Tr}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}^{2})\}$, is operationally linked to the Shannon entropy of Gaussian Wigner distributions sampling by homodyne detections in phase space [50], in addition, it fulfills the strong subadditivity inequality, i.e. ${{ \mathcal S }}_{2}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})+{{ \mathcal S }}_{2}({\hat{\varrho }}_{{ \mathcal Y }{ \mathcal Z }})\geqslant {{ \mathcal S }}_{2}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }{ \mathcal Z }})+{{ \mathcal S }}_{2}({\hat{\varrho }}_{{ \mathcal Y }})$ [50], a key requirement for quantum information theory, therefore, it can be used for defining bona fide measures of correlations, such as the Rényi-2 Gaussian EoF [44, 50] and the Rényi-2 Gaussian quantum mutual information [45, 50].
In equation (15), replacing ${{ \mathcal S }}_{\alpha }$ with the Rényi-2 entropy ${{ \mathcal S }}_{2}$, we thus obtain the Rényi-2 Gaussian entanglement ${{ \mathcal E }}_{2}^{G}$ [50], which recently dubbed more fittingly as the Rényi-2 Gaussian EoF ${{ \mathcal E }}_{2}^{G}\equiv {{ \mathcal E }}_{F,R}^{G}$ [44].
For two-mode squeezed thermal state with the covariance matrix σ12 (13), the Gaussian EoF ${{ \mathcal E }}_{F,V}^{G}$ reads [24, 25]$\begin{eqnarray}{{ \mathcal E }}_{F,V}^{G}=\left\{\begin{array}{c}f\left(\upsilon \right)\ \mathrm{if}\ {\tilde{\nu }}_{-}\lt 1/2,\\ 0\,\mathrm{otherwise},\end{array}\right.\end{eqnarray}$where $f(x)=(x+\tfrac{1}{2})\mathrm{ln}(x+\tfrac{1}{2})-(x-\tfrac{1}{2})\mathrm{ln}(x-\tfrac{1}{2})$ and $\upsilon \,=\tfrac{\left({\vartheta }_{11}+{\vartheta }_{22}\right)({\vartheta }_{11}{\vartheta }_{22}-{\vartheta }_{12}^{2}+\tfrac{1}{4})-{\vartheta }_{12}\sqrt{\det ({\sigma }_{12}+\tfrac{{\rm{i}}}{2}{\rm{\Omega }})}}{{\left({\vartheta }_{11}+{\vartheta }_{22}\right)}^{2}-4{\vartheta }_{12}^{2}}$, with ${\rm{\Omega}}={\mathop{\oplus}\limits^{2}}_{1}\left(\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right)$ being the symplectic form, and ${\tilde{\nu }}_{-}\,=\sqrt{(\tilde{{\rm{\Delta }}}-\sqrt{{\tilde{{\rm{\Delta }}}}^{2}-4\det {\sigma }_{12}})/2}$ is the minimum symplectic eigenvalue of the partially transposed covariance matrix (13 ) with $\tilde{{\rm{\Delta }}}=\det {\sigma }_{1}+\det {\sigma }_{2}-2\det {\sigma }_{3}$ [65]. While, the Rényi-2 Gaussian EoF writes [44, 50, 65]$\begin{eqnarray}{{ \mathcal E }}_{F,R}^{G}=\left\{\begin{array}{cc}\tfrac{1}{2}\mathrm{ln}\left[\tfrac{{\left[(4g+1){s}_{+}-\sqrt{\left[{\left(4g-1\right)}^{2}-16{s}_{-}^{2}\right]\left[{s}_{+}^{2}-{s}_{-}^{2}-g\right]}\right]}^{2}}{16{\left({s}_{-}^{2}+g\right)}^{2}}\right] & \ \mathrm{if}\ 4| {s}_{-}| +1\leqslant 4g\lt 4{s}_{+}-1,\\ 0 & 4\,g\geqslant 4{s}_{+}-1,\end{array}\right.\end{eqnarray}$where s± = (ϑ11 ± ϑ22)/2 and $g=\left({\vartheta }_{11}{\vartheta }_{22}-{\vartheta }_{12}^{2}\right)$.
Notice that the asymptotic regularization of the EoF ${{ \mathcal E }}_{F}$ (i.e. $\mathop{\mathrm{lim}}\limits_{n\to \infty }{{ \mathcal E }}_{F}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}^{\otimes n})/n$) coincides with the entanglement cost ${{ \mathcal E }}_{C}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})$ [22, 66], an entanglement measure which was interpreted as the minimum amount of singlets (i.e. maximally entangled antisymmetric two-qubit states) needed for generating an entangled bipartite state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ by means of local operations and classical communication [65, 66]. In fact, obtaining analytical expression of the entanglement cost ${{ \mathcal E }}_{C}$ for an arbitrary state, is a difficult task [66]. However, since the additivity of the EoF ${{ \mathcal E }}_{F,V}^{G}$ for TMGSs is proven to be true [25], it consequently follows that ${{ \mathcal E }}_{C,V}^{G}\equiv {{ \mathcal E }}_{F,V}^{G}$ [67].
Finally, we recall some of interesting properties of the Rényi-2 Gaussian EoF ${{ \mathcal E }}_{F,R}^{G}$ [45, 50]: (i) it does not increase under all Gaussian local operations and classical communication, thus it is a proper measure of Gaussian entanglement; (ii) it is additive on tensor product states; and (iii) it satisfies both the Coffman–Kundu–Wootters-type monogamy inequality [68] and the Koashi–Winter monogamy relation [69].
3.2. Gaussian quantum mutual information
Based on the Landauer’s erasure principle [70], Groisman et al [3] have defined the quantum mutual information as the amount of noise required to erase completely the correlations contained in a joint density operator ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ by turning it into a product state, i.e. ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}={\hat{\varrho }}_{{ \mathcal X }}\otimes {\hat{\varrho }}_{{ \mathcal Y }}$. Essentially, a strong argument in favor for accepting the quantum mutual information as a measure of total correlations in bipartite states is given in [4], i.e. if two parties ${ \mathcal X }$ and ${ \mathcal Y }$ share a quantum correlated state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$, the maximum amount of information that ${ \mathcal X }$(${ \mathcal Y }$) can securely send to ${ \mathcal Y }$(${ \mathcal X }$) is exactly equal to the quantum mutual information of the shared state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$.
For a bipartite state ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$, the quantum mutual information ${{ \mathcal I }}_{1}\equiv {{ \mathcal I }}_{V}$ defined via the von Neumann entropy ${{ \mathcal S }}_{1}$ reads [5]$\begin{eqnarray}{{ \mathcal I }}_{V}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }})={{ \mathcal S }}_{1}({\hat{\varrho }}_{{ \mathcal X }})+{{ \mathcal S }}_{1}({\hat{\varrho }}_{{ \mathcal Y }})-{{ \mathcal S }}_{1}({\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}),\end{eqnarray}$where ${\hat{\varrho }}_{{ \mathcal X }({ \mathcal Y })}={\mathrm{Tr}}_{{ \mathcal Y }({ \mathcal X })}{\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ is the marginal state of the party ${ \mathcal X }({ \mathcal Y })$ obtained by partial tracing over ${ \mathcal Y }({ \mathcal X })$.
From an operational point of view, the von Neumann quantum mutual information given by equation (18) quantifies the amount of the information extracted on ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}$ by looking at the system in its entirety, minus the information that can be obtained from separate observations of the subsystems with the marginal states ${\hat{\varrho }}_{{ \mathcal X }}$ and ${\hat{\varrho }}_{{ \mathcal Y }}$. It is always positive and vanishes only if ${\hat{\varrho }}_{{ \mathcal X }{ \mathcal Y }}={\hat{\varrho }}_{{ \mathcal X }}\otimes {\hat{\varrho }}_{{ \mathcal Y }}$ [3–5].
In particular, for a TMGS ${\hat{\varrho }}_{12}$ with the covariance matrix σ12 (13), equation (18) writes [12]$\begin{eqnarray}{{ \mathcal I }}_{V}^{G}=f(\sqrt{{I}_{1}})+f(\sqrt{{I}_{2}})-f({\nu }_{+})-f({\nu }_{-}),\end{eqnarray}$where f(x) is defined above, ${I}_{k}=\det {\sigma }_{k}$ and ${\nu }_{\pm }=\sqrt{({\rm{\Delta }}\pm \sqrt{{{\rm{\Delta }}}^{2}-4\det {\sigma }_{12}})/2}$ are respectively the symplectic invariants and the symplectic eigenvalues of the covariance matrix σ12 (13) with Δ = I1 + I2 + 2I3. Whereas, by means of the Gaussian Rényi-2 entropy ${{ \mathcal S }}_{2}({\hat{\varrho }}_{12})\,=\tfrac{1}{2}\mathrm{ln}\left(\det {\sigma }_{12}\right)$, the Rényi-2 quantum mutual information ${{ \mathcal I }}_{2}^{G}\equiv {{ \mathcal I }}_{R}^{G}$ reads [50]$\begin{eqnarray}{{ \mathcal I }}_{R}^{G}={{ \mathcal S }}_{2}({\hat{\varrho }}_{1})+{{ \mathcal S }}_{2}({\hat{\varrho }}_{2})-{{ \mathcal S }}_{2}({\hat{\varrho }}_{12})\end{eqnarray}$$\begin{eqnarray}=\displaystyle \frac{1}{2}\mathrm{ln}\left(\displaystyle \frac{\det {\sigma }_{1}\det {\sigma }_{2}}{\det {\sigma }_{12}}\right),\end{eqnarray}$which shown to coincide exactly with the Shannon continuous mutual information of the Wigner function of the TMGS ${\hat{\varrho }}_{12}$ [50]. Operationally, the Rényi-2 Gaussian quantum mutual information ${{ \mathcal I }}_{R}^{G}({\hat{\varrho }}_{12})$ can be interpreted as the amount of extra discrete information that needs to be transmitted through a continuous variable channel for reconstructing the complete joint Wigner function of the TMGS ${\hat{\varrho }}_{12}$ rather than just the two marginal Wigner functions of the subsystems [50, 71]. In other words, ${{ \mathcal I }}_{R}^{G}({\hat{\varrho }}_{12})$ quantifies the total quadrature correlations of the state ${\hat{\varrho }}_{12}$ [50]. Finally, we notice that the Rényi-2 quantum mutual information ${{ \mathcal I }}_{R}({\hat{\varrho }}_{12})$ is always non-negative for Gaussian states, and vanishes only if ${\hat{\varrho }}_{12}={\hat{\varrho }}_{1}\otimes {\hat{\varrho }}_{2}$, however, it can be negative for more general states (e.g. non-Gaussian ones), then, it does not admit an operational interpretation beyond the Gaussian framework.
In what follows, we show that—under various circumstances—the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is well satisfied via the Rényi-2 entropy, while, it can largely be violated via the von Neumann entropy.
In figure 2, we plot $(1/2){{ \mathcal I }}_{R}^{G}-{{ \mathcal E }}_{F,R}^{G}$ (panel (a)) and ${{ \mathcal I }}_{V}^{G}-{{ \mathcal E }}_{F,V}^{G}$ (panel (b)) against the population inversion η for various values of the linear gain coefficient A, with a cavity decay rate κ = 1. Remarkably, figure 2(a) shows that for a density values of η and A, $(1/2){{ \mathcal I }}_{R}^{G}-{{ \mathcal E }}_{F,R}^{G}$ is always positive meaning that the inequality $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is well satisfied when the quantum mutual information ${{ \mathcal I }}_{\alpha }^{G}$ and the EoF ${{ \mathcal E }}_{F,\alpha }^{G}$ are defined via the Rényi-2 entropy. Whereas, in spite of the fact that the entanglement degree is generally accounts as a portion of total correlations, figure 2(b) shows that ${{ \mathcal I }}_{V}^{G}-{{ \mathcal E }}_{F,V}^{G}$ may be negative, then ${{ \mathcal E }}_{F,V}^{G}$ may exceed ${{ \mathcal I }}_{V}^{G}$, which constitutes a large violation of the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$. Such counterintuitive behavior (i.e. ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$) supporting the Hayden conjecture [40], clearly evidences that the von Neumann EoF ${{ \mathcal E }}_{F,V}^{G}$ may not be the best choice for estimating entanglement in mixed TMGSs, which is quite consistent with the conclusion pointed out in [20].
Figure 2.
New window|Download| PPT slide Figure 2.(a): $\tfrac{1}{2}{{ \mathcal I }}_{R}^{G}-{{ \mathcal E }}_{F,R}^{G}$ and (b): ${{ \mathcal I }}_{V}^{G}-{{ \mathcal E }}_{F,V}^{G}$ versus the parameters A and η for κ = 1. (a) shows that the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is satisfied via the Rényi-2 entropy (i.e. α = 2), in contrast, it is largely violated—in (b)—via the von Neumann entropy (i.e. α = 1), where even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which consolidates the Hayden conjecture [40], in addition, undermines the interpretation of the von Neumann EoF ${{ \mathcal E }}_{F,V}$ as just a fraction of the total correlations.
Next, in figure 3, we plot $(1/2){{ \mathcal I }}_{R}^{G}-{{ \mathcal E }}_{F,R}^{G}$ (panel (a)) and ${{ \mathcal I }}_{V}^{G}-{{ \mathcal E }}_{F,V}^{G}$ (panel (b)) against the parameters κ and A for η = 0.2. Similarly to the results illustrated in figure 2, figure 3(a) shows within a density values of κ and A that the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is well satisfied with respect to the Rényi-2 entropy. While, another violation of $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ via the von Neumann entropy is shown in figure 3(b), where even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which undermines the interpretation of the von Neumann EoF ${{ \mathcal E }}_{F,V}$ as just a fraction of the total correlations.
Figure 3.
New window|Download| PPT slide Figure 3.(a): $\tfrac{1}{2}{{ \mathcal I }}_{R}^{G}-{{ \mathcal E }}_{F,R}^{G}$ and (b): ${{ \mathcal I }}_{V}^{G}-{{ \mathcal E }}_{F,V}^{G}$ versus the parameters A and κ for η = 0.2. (a) shows the holding of the inequality $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ via the Rényi-2 entropy (i.e. α = 2 ), while, (b) shows that even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which supports the Hayden conjecture [40], and further evidences that the von Neumann entropy (i.e. α = 1) may not be the best choice for estimating entanglement in mixed TMGSs
Here, it is interesting to emphasize that the holding of the constraint $(1/2){{ \mathcal I }}_{R}^{G}\geqslant {{ \mathcal E }}_{F,R}^{G}$ for a wide range of the parameters κ, A and η in figures 2(a)–3(a), is not a curious coincidence since, the inequality $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is proven to be true as long as α ≥ 2 for general mixed TMGSs [44, 45]. However—as can be seen from figures 2(b)–3(b)—it may be largely violated with respect to the von Neumann entropy (α = 1), where even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which therefore provides strong evidence hinting that the origin of this counterintuitive behavior should be related to the von Neumann entropy by which the EoF ${{ \mathcal E }}_{F,V}$ is defined, rather than related to the conceptual definition of the EoF ${{ \mathcal E }}_{F}$ or the Henderson–Vedral conjecture.
In quantum information theory, the additivity is a very desirable property that can largely reduce the evaluation of entanglement [20]. Indeed, since quantum mechanics is statistical, often physical meaning of entanglement measures is acquired only in the asymptotic regime of many copies of a given state, which can be reduced to a single copy for additive measures [72], such as the conditional entanglement of mutual information [72], the logarithmic negativity [19], the squashed entanglement [27], the EoF ${{ \mathcal E }}_{F,V}$ of arbitrary TMGSs [25], and the entanglement cost [66].
Therefore, since the additivity of the von Neumann EoF ${{ \mathcal E }}_{F,V}^{G}$ is proven to be true, i.e. ${{ \mathcal E }}_{F,V}^{G}(\rho \otimes {\rho }^{{\prime} })\,={{ \mathcal E }}_{F,V}^{G}(\rho )\,+{{ \mathcal E }}_{F,V}^{G}({\rho }^{{\prime} })$, where ρ and ${\rho }^{{\prime} }$ are two entangled TMGSs [25], it follows that the entanglement cost ${{ \mathcal E }}_{C,V}^{G}(\rho ):= \mathop{\mathrm{lim}}\limits_{n\to \infty }{{ \mathcal E }}_{F,V}^{G}({\rho }^{\otimes n})/n$ coincides with the EoF ${{ \mathcal E }}_{F,V}^{G}$ [66, 67], and consequently, both of them cannot be regarded as genuine measures of entanglement, in the sense that they may exceed total correlations [40]. By contrast, the Rényi-2 EoF ${{ \mathcal E }}_{F,R}^{G}$ seems more appropriate for estimating entanglement in mixed TMGSs, in the sense that, it satisfies the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$, in addition, it is additive and monogamous, which are two strong points for considering it as an entanglement measure [45, 50].
Finally, we notice that the constraint $(1/2){ \mathcal I }\geqslant {{ \mathcal E }}_{F}$ has also been violated in systems endowed with finite-dimensional Hilbert spaces [2, 28], where the quantum mutual information ${ \mathcal I }$ and the EoF ${{ \mathcal E }}_{F}$ are defined only via the conventional von Neumann entropy. Attempting to explain the origin of such a violation, some doubts around the validity of the pure-state decompositions in the general definition of the EoF ${{ \mathcal E }}_{F}$ and the validity of the Henderson–Vedral conjecture have seriously been raised in [2, 28]. However, the comparative study accomplished here, provides strong evidence suggesting that the origin of the counterintuitive behavior exhibited by the EoF ${{ \mathcal E }}_{F}$ in figures 2(b)–3(b) as well as in [2, 28] should intrinsically be related to the von Neumann entropy by which the EoF ${{ \mathcal E }}_{F,V}$ is defined, rather than related to the conceptual definition of the EoF ${{ \mathcal E }}_{F}$ or the Henderson–Vedral conjecture. More generally, the results obtained here evidence that the Gaussian EoF ${{ \mathcal E }}_{F,V}^{G}$ defined via the von Neumann entropy may not be the best choice for estimating entanglement in mixed TMGSs, which is quite consistent with [20].
4. Conclusion
In a TMGS ${\hat{\varrho }}_{12}$, we studied the inequality $(1/2){ \mathcal T }\geqslant { \mathcal Q }$ that was considered 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 [2]. Using the Gaussian EoF ${{ \mathcal E }}_{F,\alpha }^{G}$ and the Gaussian quantum mutual information ${{ \mathcal I }}_{\alpha }^{G}$ respectively as measures of quantum ${ \mathcal Q }$ and total correlations ${ \mathcal T }$ for α = 1, 2, we verified that $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$ is well satisfied when ${{ \mathcal E }}_{F,\alpha }^{G}$ and ${{ \mathcal I }}_{\alpha }^{G}$ are defined via the Rényi-2 entropy (i.e. α = 2). In contrast, via the von Neumann entropy (i.e. α = 1), even ${{ \mathcal E }}_{F,V}^{G}\geqslant {{ \mathcal I }}_{V}^{G}$ may happen, which consolidates the Hayden conjecture [40], in addition, clearly evidences that the von Neumann EoF ${{ \mathcal E }}_{F,V}$ may not be the best choice for estimating entanglement in mixed TMGSs. Moreover, since the additivity of the Gaussian EoF ${{ \mathcal E }}_{F,V}^{G}$ is proven to be true [25], which implies that the entanglement cost ${{ \mathcal E }}_{C,V}^{G}(\hat{\varrho }):= \mathop{\mathrm{lim}}\limits_{n\to \infty }{{ \mathcal E }}_{F,V}^{G}({\hat{\varrho }}^{\otimes n})/n$ and the EoF ${{ \mathcal E }}_{F,V}^{G}$ are identical [67], it follows that both ${{ \mathcal E }}_{F,V}^{G}$ and ${{ \mathcal E }}_{C,V}^{G}$ can not be regarded as genuine measures of entanglement in mixed TMGSs, in the sense that, they may dominate total correlations.
The comparative study accomplished here, provides strong evidence hinting that the origin of the peculiar behavior exhibited by the EoF ${{ \mathcal E }}_{F,V}$ in figures 2(b)–3(b) as well as in [2, 28] , should intrinsically be related to the von Neumann entropy by which the EoF ${{ \mathcal E }}_{F,V}$ is defined, rather than related to the conceptual definition of the EoF ${{ \mathcal E }}_{F}$ or the Henderson–Vedral conjecture [2, 28]. Moreover, it shows that the Rényi-2 EoF ${{ \mathcal E }}_{F,R}^{G}$ is more faithful—than the von Neumann EoF ${{ \mathcal E }}_{F,V}^{G}$—for quantifying entanglement in mixed TMGSs, in the sense that, it satisfies the constraint $(1/2){{ \mathcal I }}_{\alpha }^{G}\geqslant {{ \mathcal E }}_{F,\alpha }^{G}$, furthermore, it is additive and monogamous, which are two strong points for considering it as an entanglement measure [45, 50].
Finally, it is interesting to mention that apart from total correlation—that can be written as sum of quantum and classical parts in bipartite states—there exist similar equalities in the contexts of coherence and randomness, where the total amount of a property can also be decomposed into classical and quantum parts [73, 74]. In this respect, we believe that the full understanding of the relationships between quantum and classical quantities contained in bipartite mixed states is of critical importance, which represents a relevant step towards efficient characterization and quantification of quantum entanglement. This therefore would open up the application of the theoretical work on quantum information processing and communication.
Acknowledgments
I am particularly indebted to an anonymous referee for constructive critiques and insightful comments.
,1,2,31Department of Physics, Faculty of Applied Sciences, Ait-Melloul, 
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