Necessary and sufficient fully separable criterion and entanglement of three-qubit Greenberger–Horne–Zeilinger diagonal states

Abstract

We analytically prove that the necessary criterion coincides with the sufficient criterion for the full separability of three-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. Our result closes the discussion on the separability of three-qubit GHZ diagonal states. Based on the criterion, the corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others in terms of the relative entropy of entanglement.

Appendices

Appendix 1: Proof of Eqs. (17) and (18)

The definition of $C(\overrightarrow{X})$ in Eq. (12) leads to the solutions of the angles $a,b,c$ as functions of the parameters $\delta ,\alpha ,\beta ,\gamma ,$ namely [19],

$$\begin{aligned} \sin a= & {} \frac{1}{2\alpha }\frac{\sqrt{Q}}{\sqrt{R}},\sin b=\frac{1}{2\beta } \frac{\sqrt{Q}}{\sqrt{R}}, \\ \sin c= & {} \frac{1}{2\gamma }\frac{\sqrt{Q}}{\sqrt{R}}, \end{aligned}$$

with $Q=-(\alpha \beta \delta +\alpha \beta \gamma -\alpha \gamma \delta -\beta \gamma \delta ) (\alpha \beta \delta -\alpha \beta \gamma +\alpha \gamma \delta -\beta \gamma \delta ) (\alpha \beta \delta -\alpha \beta \gamma -\alpha \gamma \delta +\beta \gamma \delta ) (\alpha \beta \delta +\alpha \beta \gamma +\alpha \gamma \delta +\beta \gamma \delta ), R=\alpha \beta \gamma \delta (\alpha \beta -\gamma \delta ) (\alpha \gamma -\beta \delta ) (\alpha \delta -\beta \gamma )$. Let us consider the derivative $\frac{\partial C(\overrightarrow{X})}{\partial \alpha },$ we have

$$\begin{aligned} \frac{\partial C\left( \overrightarrow{X}\right) }{\partial \alpha }= & {} -\delta \sin d \frac{\partial d}{\partial \alpha }-\alpha \sin a\frac{\partial a}{\partial \alpha }+\cos a \\&-\beta \sin b\frac{\partial b}{\partial \alpha }-\gamma \sin c\frac{ \partial c}{\partial \alpha } \end{aligned}$$

Using (13), we have $\frac{\partial C(\overrightarrow{X})}{\partial \alpha }=\alpha \sin a(\frac{\partial d}{\partial \alpha }-\frac{\partial a}{ \partial \alpha }-\frac{\partial b}{\partial \alpha }-\frac{\partial c}{ \partial \alpha })+\cos a$. Since $d\equiv a+b+c,$ we have

$$\begin{aligned} \frac{\partial C\left( \overrightarrow{X}\right) }{\partial \alpha }=\cos a, \end{aligned}$$

similarly, $\frac{\partial C(\overrightarrow{X})}{\partial \beta }=\cos b, \frac{\partial C(\overrightarrow{X})}{\partial \gamma }=\cos c$ and $\frac{ \partial C(\overrightarrow{X})}{\partial \delta }=\cos d$.

Appendix 2: Closest separable state of GHZ diagonal state

Up to local Hadamard transformations, the $\left| \hbox {GHZ}_k\right\rangle$ basis is equivalent to the graph state basis $\left| G_\mu \right\rangle$ ($\mu =\mu _1\mu _2\mu _3$ with $\mu _i=0,1$). For any state $\sigma =\sum _{\mu ,\nu }q_{\mu \nu }\left| G_\mu \right\rangle \left\langle G_\nu \right|$, which may not be diagonal in the graph state basis, it is always possible to convert $\sigma$, via local probabilistic operations, into a graph diagonal state $\sigma _d=\sum _\mu q_{\mu \mu }\left| G_\mu \right\rangle \left\langle G_\mu \right|$with the same diagonal elements [21]. So, if $\sigma$ is fully separable, then $\sigma _d$ is also fully separable since the local conversion to a diagonal state cannot introduce entanglement.

On the other hand, we use the fact that the function $-\log x$ is convex which results in

$$\begin{aligned} -\log (\left\langle \phi \right| A\left| \phi \right\rangle )\le -\left\langle \phi \right| \log (A)\left| \phi \right\rangle \end{aligned}$$

for any operator $A$ and any normalized state $\left| \phi \right\rangle$ (see Ref. [20]). In the derivation, the spectrum decomposition of operator $A$ is utilized. The relative entropy of a graph diagonal state $\rho =\sum _\mu p_\mu \left| G_\mu \right\rangle \left\langle G_\mu \right|$ with respect to a fully separable state $\sigma$ is

$$\begin{aligned} S(\rho ||\sigma )= & {} Tr\rho \log _2\rho -Tr\rho \log _2\sigma \\= & {} Tr\rho \log _2\rho -\sum _\mu p_\mu \left\langle G_\mu \right| \log _2\sigma \left| G_\mu \right\rangle \\\ge & {} Tr\rho \log _2\rho -\sum _\mu p_\mu \log _2\left( \left\langle G_\mu \right| \sigma \left| G_\mu \right\rangle \right) \\= & {} S(\rho ||\sigma _d) \end{aligned}$$

Thus, for any fully separable state $\sigma$, we can always find a graph diagonal fully separable state $\sigma _d$ such that the relative entropy $S(\rho ||\sigma )$ is lower bounded by the relative entropy $S(\rho ||\sigma _d)$. The closest fully separable state of an entangled graph diagonal state is a graph diagonal state.