Quantum correlations in a family of bipartite separable qubit states

Abstract

Quantum correlations (QCs) in some separable states have been proposed as a key resource for certain quantum communication tasks and quantum computational models without entanglement. In this paper, a family of nine-parameter separable states, obtained from arbitrary mixture of two sets of bi-qubit product pure states, is considered. QCs in these separable states are studied analytically or numerically using four QC quantifiers, i.e., measurement-induced disturbance (Luo in Phys Rev A77:022301, 2008), ameliorated MID (Girolami et al. in J Phys A Math Theor 44:352002, 2011),quantum dissonance (DN) (Modi et al. in Phys Rev Lett 104:080501, 2010), and new quantum dissonance (Rulli in Phys Rev A 84:042109, 2011), respectively. First, an inherent symmetry in the concerned separable states is revealed, that is, any nine-parameter separable states concerned in this paper can be transformed to a three-parameter kernel state via some certain local unitary operation. Then, four different QC expressions are concretely derived with the four QC quantifiers. Furthermore, some comparative studies of the QCs are presented, discussed and analyzed, and some distinct features about them are exposed. We find that, in the framework of all the four QC quantifiers, the more mixed the original two pure product states, the bigger QCs the separable states own. Our results reveal some intrinsic features of QCs in separable systems in quantum information.

Appendix 1 Analyses: \(\rho _{{ab}}\) are associated with \(\varrho _{{ab}}\) through local unitary transformations

State in Eq. (1) can be rewritten as

$$\begin{aligned} \rho _{{ab}}=q |\varphi _1\rangle _{{a}} \langle \varphi _1| |\varphi _2\rangle _{{b}}\langle \varphi _2| +(1-q)|\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|. \end{aligned}$$

(46)

Firstly, by using the following transformations $U_{1a}$ and $U_{2b}$, which are local unitary operations performed on qubits a and b respectively,

$$\begin{aligned} U_{1a} = \left( \begin{matrix} \cos \theta _1 \quad {\text{ e }}^{i\delta _1} \sin \theta _1 \\ {\text{ e }}^{-i\delta _1 } \sin \theta _1 \quad -\cos \theta _1 \\ \end{matrix} \right) ,\ \ \ U_{2b} = \left( \begin{matrix} \cos \theta _2 \quad {\text{ e }}^{i\delta _2} \sin \theta _2 \\ {\text{ e }}^{-i\delta _2 } \sin \theta _2 \quad -\cos \theta _2 \\ \end{matrix}\right) . \end{aligned}$$

(47)

one can get

$$\begin{aligned} \rho _{{ab}}= & {} U_{1a} U_{2b}[q\ |0\rangle _{a} \langle 0| \ |0\rangle _{{b}} \langle 0|\nonumber \\&+ (1-q)U_{1{a}}^\dag U_{2b}^\dag \ |\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|U_{1a} U_{2b}]U_{1a}^\dag U_{2b}^\dag . \end{aligned}$$

(48)

Secondly, by performing the following local unitary transformations,

$$\begin{aligned} M_{{a}}= \left( \begin{matrix} 1 &{} 0 \\ 0 &{} \,\,\,\, {\text{ e }}^{i(\lambda _2-\lambda _1)} \\ \end{matrix}\right) , \ \ N_{{b}}=\left( \begin{matrix} 1 &{} 0 \\ 0 &{} \,\,\,\, {\text{ e }}^{i(\lambda _4-\lambda _3)} \\ \end{matrix}\right) , \end{aligned}$$

(49)

one can further get

$$\begin{aligned} \rho _{{ab}}= & {} U_{1a}M_{ {a}} U_{2 b}N_{ {b}} [ q\ |0\rangle _{{a}} \langle 0| \ |0\rangle _{{b}} \langle 0|\nonumber \\&+ (1-q)M_{{a}}^\dag U_{1a}^\dag N_{{b}}^\dag U_{2b}^\dag \ |\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|U_{1a}M_{ {a}} U_{2b}N_{{b}} ] M_{{a}}^\dag U_{1a}^\dag N_{{b}}^\dag U_{2b}^\dag .\nonumber \\ \end{aligned}$$

(50)

The parameters $\lambda _i$’s are defined as following

$$\begin{aligned} \lambda _i= & {} \arctan \left\{ \frac{\sin (\delta _{i+2}-\delta _i)\tan \theta _i \tan \theta _{i+2}}{1+ \cos (\delta _{i+2}-\delta _i)\tan \theta _i \tan \theta _{i+2}} \right\} . \end{aligned}$$

(51)

Furthermore, defining

$$\begin{aligned} U_{{a}}(\theta _1,\delta _1,\lambda _1,\omega _1 )=U_{1a} M_{{a}}, \ \ U_{{b}}(\theta _2,\delta _2,\lambda _2,\omega _2)=U_{2b}N_{ {b}}, \end{aligned}$$

(52)

one can get

$$\begin{aligned} \rho _{{ab}} = U_{{a}} U_{{b}} [ q\ |0\rangle _{{a}} \langle 0| \ |0\rangle _{{b}} \langle 0| + (1-q)U_{ {a}}^\dag U_{{b}}^\dag \ |\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|U_{{a}} U_{{b}} ]U_{{a}}^\dag U_{{b}}^\dag .\nonumber \\ \end{aligned}$$

(53)

The parameters $\omega _i$’s are defined as following

$$\begin{aligned} \omega _i= & {} \arctan \left\{ \frac{\sin \delta _i \tan \theta _i \cot \theta _{i+2} -\sin \delta _{i+2} }{\cos \delta _i \tan \theta _i \cot \theta _{i+2} -\cos \delta _{i+2}}\right\} , \end{aligned}$$

(54)

Lastly, in Eq. (53), $U_{{a}}^\dag U_{{b}}^\dag \ |\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|U_{{a}} U_{{b}}$ can be written as

$$\begin{aligned} U_{{a}}^\dag U_{{b}}^\dag \ |\varphi _3\rangle _{{a}} \langle \varphi _3| |\varphi _4\rangle _{{b}}\langle \varphi _4|U_{{a}} U_{{b}}= |\xi \rangle _{{a}} \langle \xi | |\zeta \rangle _{{b}}\langle \zeta |, \end{aligned}$$

(55)

where $|\xi \rangle _{{a}}= \sqrt{\alpha } |0\rangle _{{a}} +\sqrt{1-\alpha } |1\rangle _{{a}}$ and $|\zeta \rangle _{{b}}= \sqrt{\beta } |0\rangle _{{a}} +\sqrt{1-\beta } |1\rangle _{{a}}$ with

$$\begin{aligned} \alpha = \cos ^2(\theta _1-\theta _3) - \sin 2\theta _1 \sin 2\theta _3 \sin ^2 \frac{\delta _1 - \delta _3}{2}. \end{aligned}$$

(56)

$$\begin{aligned} \beta = \cos ^2(\theta _2-\theta _4) - \sin 2\theta _2 \sin 2\theta _4 \sin ^2 \frac{\delta _2 - \delta _4}{2}. \end{aligned}$$

(57)

Finally, by defining

$$\begin{aligned} \varrho _{{ab}}= q\ |0\rangle _{{a}} \langle 0| \ |0\rangle _{{b}} \langle 0| + (1-q) |\xi \rangle _{{a}} \langle \xi | |\zeta \rangle _{{b}}\langle \zeta |, \end{aligned}$$

(58)

one can conclude that the concerned state $\rho _{{ab}}$ in this paper are related with a kernel state $\varrho _{{ab}}$ through local unitary transformations.