Comparison of quantum discord and fully entangled fraction of two classes of $d\otimes d^2$ states

Abstract

The quantumness of a generic state is the resource of many applications in quantum information theory, and it is interesting to survey the measures which are able to detect its trace in the properties of the state. In this work, we study the quantum discord and fully entangled fraction of two classes of bipartite states and compare their behaviors. These classes are complements to the $d\otimes d$ Werner and isotropic states, in the sense that each class possesses the same purification as the corresponding complemental class of states. Our results show that maximally entangled mixed states are also maximally discordant states, leading to a generalization of the well-known fact that all maximally entangled pure states have also maximum quantum discord. Moreover, it is shown that the separability-entanglement boundary of a Werner or isotropic state is manifested as an inflection point in the diagram of quantum discord of the corresponding complemental state.

Appendix: Proof for Lemma 1

Proof

We provide a proof for $2\otimes 4$ complemental state of a $2\otimes 2$ Werner state. The generalization to higher dimensions is straightforward. The states (46) and (47) can be rewritten as follows

$$\begin{aligned} |\Psi _1\rangle= & {} \cos \alpha |11\rangle +\sin \alpha |2\rangle \left( \cos \theta |3\rangle -\sin \theta |4\rangle \right) , \end{aligned}$$

(57)

$$\begin{aligned} |\Psi _2\rangle= & {} \cos \alpha |22\rangle +\sin \alpha |1\rangle \left( \cos \theta |3\rangle +\sin \theta |4\rangle \right) . \end{aligned}$$

(58)

Having a glance of the above states, one can easily infer that the two maximally entangles states which maximize the fully entangled fraction of the state (45) should be in the following form

$$\begin{aligned} |\psi _1^{\max }\rangle= & {} \frac{1}{\sqrt{2}}|1\rangle \left( \cos \gamma |1\rangle -\sin \gamma |2\rangle \right) +\frac{1}{\sqrt{2}}|2\rangle \left( \cos \gamma ^\prime |3\rangle -\sin \gamma ^\prime |4\rangle \right) ,\end{aligned}$$

(59)

$$\begin{aligned} |\psi _2^{\max }\rangle= & {} \frac{1}{\sqrt{2}}|2\rangle \left( \cos \gamma |1\rangle +\sin \gamma |2\rangle \right) +\frac{1}{\sqrt{2}}|1\rangle \left( \cos \gamma ^\prime |3\rangle +\sin \gamma ^\prime |4\rangle \right) . \end{aligned}$$

(60)

Now using the Eq. (2) we have

$$\begin{aligned} \mathcal {F}=\max _{\{\gamma ,\gamma ^\prime \}}\left\{ \langle \psi _1^{\max }|\rho |\psi _1^{\max }\rangle +\langle \psi _2^{\max }|\rho |\psi _2^{\max }\rangle \right\} , \end{aligned}$$

(61)

which results to

$$\begin{aligned} \mathcal {F}=\frac{1}{4}\max _{\{\gamma ,\gamma ^\prime \}}\left( f_1+f_2\right) , \end{aligned}$$

(62)

where

$$\begin{aligned} f_1= & {} \left( \cos \gamma \cos \alpha +\sin \alpha \cos \theta \cos \gamma ^\prime +\sin \alpha \sin \theta \sin \gamma ^\prime \right) ^2, \end{aligned}$$

(63)

$$\begin{aligned} f_2= & {} \left( \cos \gamma \cos \alpha +\sin \alpha \cos \theta \sin \gamma ^\prime +\sin \alpha \sin \theta \cos \gamma ^\prime \right) ^2. \end{aligned}$$

(64)

It is easy to show that the maximum will be reached at $\gamma =0$ and $\gamma ^\prime =\frac{\pi }{4}$ for an arbitrary $\alpha$ and $\theta$. The proof is completed and could be generalized to the higher dimensions in a straightforward procedure.