Analytical expression of genuine tripartite quantum discord for symmetrical X-states

Abstract

The study of classical and quantum correlations in bipartite and multipartite systems is crucial for the development of quantum information theory. Among the quantifiers adopted in tripartite systems, the genuine tripartite quantum discord (GTQD), estimating the amount of quantum correlations shared among all the subsystems, plays a key role since it represents the natural extension of quantum discord used in bipartite systems. In this paper, we derive an analytical expression of GTQD for three-qubit systems characterized by a subclass of symmetrical X-states. Our approach has been tested on both GHZ and maximally mixed states reproducing the expected results. Furthermore, we believe that the procedure here developed constitutes a valid guideline to investigate quantum correlations in form of discord in more general multipartite systems.

Appendices

Appendix 1: Relative entropies definition

Following Zhao et al. [42], we can define the relative entropy $S(\rho _{A|BC})$ for tripartite systems as:

$$\begin{aligned} S(\rho _{A|BC})=\underset{\{E_{ij}^{BC}\}}{\min }\sum _{ij}p_{ij} S(\rho _{A|E_{ij}^{BC}}), \end{aligned}$$

(33)

where:

$$\begin{aligned} \rho _{A|E_{ij}^{BC}}&= \frac{\tilde{\rho }_{A|E_{ij}^{BC}}}{p_{ij}} =\frac{1}{p_{ij}}\mathrm {Tr}_{B,C}\left[ \left( I^{A}\otimes E_{ij}^{BC}\right) \rho \right] ,\end{aligned}$$

(34)

$$\begin{aligned} p_{i,j}&= \mathrm {Tr}_{A,B,C}\left[ \left( I^{A}\otimes E_{ij}^{BC}\right) \rho \right] . \end{aligned}$$

(35)

In the previous expressions, the operators $E_{ij}^{BC}$ are positive-operator-valued measures (POVMs) that act on parties $B$ and $C$ (i.e., in the Hilbert space $\mathcal {H}_{BC}=\mathcal {H}_{B}\otimes \mathcal {H}_{C}$), and whose outcomes are labeled with two indices ($i,j$). For sake of simplicity, we will replace the global POVM $E_{ij}^{BC}$ with the external product of two local POVMs, acting separately on parties $B$ and $C$, using the same procedure given in [14]. Moreover, following the convention in literature [12, 16, 17, 30, 42]), we use orthogonal projection-valued measures (PVMs) to optimize entropy in Eq. (33), since they are easier to implement in the numerical minimization process.⁴ Then, the measurement operators are:

$$\begin{aligned} E_{ij}^{BC}&\rightarrow \Pi _{i}^{B}\otimes \Pi _{j}^{C}=\left| \beta _{i}\right\rangle \left\langle \beta _{i}\right| \otimes \left| \gamma _{j}\right\rangle \left\langle \gamma _{j}\right| , \end{aligned}$$

(36)

where $\left| \beta _{i}\right\rangle$ and $\left| \gamma _{j}\right\rangle$ are orthogonal normalized basis states of the Hilbert spaces $\mathcal {H}_{B}$ and $\mathcal {H}_{C}$, respectively.

A possible parametrization of the basis vectors $\left| \beta _{i}\right\rangle$ and $\left| \gamma _{j}\right\rangle$ with respect to the standard basis $\{\left| 0\right\rangle ,\left| 1\right\rangle \}$ can be found in literature (see [3]; for a full derivation of the basis vectors see [40]):

$$\begin{aligned} \left| \beta _{1}\right\rangle&=\cos \theta _{1}\left| 0_{B}\right\rangle +e^{+i\phi _{1}}\sin \theta _{1}\left| 1_{B}\right\rangle ,\end{aligned}$$

(37)

$$\begin{aligned} \left| \beta _{2}\right\rangle&=\sin \theta _{1}\left| 0_{B}\right\rangle -e^{+i\phi _{1}}\cos \theta _{1}\left| 1_{B}\right\rangle , \end{aligned}$$

(38)

$$\begin{aligned} \left| \gamma _{1}\right\rangle&=\cos \theta _{2}\left| 0_{C}\right\rangle +e^{+i\phi _{2}}\sin \theta _{2}\left| 1_{C}\right\rangle , \end{aligned}$$

(39)

$$\begin{aligned} \left| \gamma _{2}\right\rangle&=\sin \theta _{2}\left| 0_{C}\right\rangle -e^{+i\phi _{2}}\cos \theta _{2}\left| 1_{C}\right\rangle , \end{aligned}$$

(40)

where the angles $\theta _{i}$ and $\phi _{i}$ belong to the interval $[0;2\pi )$.

Since we are studying a system whose state is symmetrical under any permutation of its subsystems, any subscript or superscript referring to a particular subsystem in the relative entropy expression (33) can be dropped. Now, recalling the sum rule $\sum _{l=1}^{2}\tilde{\lambda }_{l}^{(ij)}=p_{ij}$ for the eigenvalues $\tilde{\lambda }_{l}^{(ij)}$ of $\tilde{\rho }_{ij}=\tilde{\rho }_{A|E_{ij}^{BC}}$ (crf. Eqs. (34, 35)), we can simplify Eq. (33) as follows:

$$\begin{aligned} S(\rho _{A|BC})=\underset{\{E_{ij}^{BC}\}}{\min }\left[ -H(p) +\sum _{i,j}S(\tilde{\rho }_{ij})\right] , \end{aligned}$$

(41)

where $H(p)=-\sum _{i,j}p_{ij}\log _{2}(p_{ij})$ is the Shannon Entropy of the probability ensemble$\{p_{ij}\}$.

Now, using Eqs. (37)–(40) to write the PVMs—together with the change of variables ($\phi _{1}-\phi _{2}\rightarrow \phi _{1}$, $\phi _{1}+\phi _{2}\rightarrow \phi _{2}$), which simplifies our calculations—the relative entropy in (41) can be written as a function of four angular variables:

$$\begin{aligned} S(\rho _{A|BC})&=\underset{\theta _{i},\phi _{i}}{\min }\, S_{rel}(\theta _{1},\theta _{2},\phi _{1},\phi _{2}). \end{aligned}$$

(42)

The final expression for $S(\rho _{A|BC})$, with all terms written explicitly, is given in Sect. 4.2.

Appendix 2: Analytical study of $S_{rel}(\theta _{1},\theta _{2},\phi _{1},\phi _{2})$

The relative entropy $S_{rel}(\theta _{1},\theta _{2},\phi _{1},\phi _{2})$ of Eq. (22)

$$\begin{aligned} S_{rel}(\theta _{1},\theta _{2},\phi _{1},\phi _{2})=1 +\frac{1}{6}\left[ \lambda _{A} \log _{2}\lambda _{A}+\lambda _{B}\log _{2}\lambda _{B}\right] -\frac{1}{12}\sum _{i=1}^{4}\lambda _{i}\log _{2}\lambda _{i} \end{aligned}$$

(43)

can be studied in a simplified form setting $\theta _{1}=\theta _{2}=\theta$. Under this condition, the $\lambda _{j}$ of Eq. (23) become:

$$\begin{aligned} \lambda _{A}&=3+a_{1}\cos ^{2}(2\theta ),\nonumber \\ \lambda _{B}&=3-a_{1}\cos ^{2}(2\theta ),\nonumber \\ \lambda _{C}&=\frac{9}{16}\sin ^{4}(2\theta )\, f(\phi _{1},\phi _{2}),\\ f(\phi _{1},\phi _{2})&=\left[ \left( c_{1}-c_{2}\right) ^{2}+4c_{2} \left( \cos (\phi _{1})+\cos (\phi _{2})\right) \left( c_{2} \cos (\phi _{1})+c_{1}\cos (\phi _{2})\right) \right] \nonumber \\ \lambda _{1,2}&=\lambda _{B}\pm \sqrt{4a_{1}^{2}\cos ^{2}(2\theta ) +\lambda _{C}},\nonumber \\ \lambda _{3,4}&=\lambda _{A}\pm \sqrt{\lambda _{C}}.\nonumber \end{aligned}$$

(44)

The minima of $S_{rel}$ must satisfy the equation:

$$\begin{aligned} \frac{\partial S_{rel}(\theta ,\theta ,\phi _{1},\phi _{2})}{\partial \theta }=0, \end{aligned}$$

(45)

which can be rewritten as follows:

$$\begin{aligned} \frac{\partial S_{rel}}{\partial \lambda _{A}}\frac{\partial \lambda _{A}}{\partial \theta }+\frac{\partial S_{rel}}{\partial \lambda _{B}} \frac{\partial \lambda _{B}}{\partial \theta }+\sum _{i=1}^{4} \frac{\partial S_{rel}}{\partial \lambda _{i}} \frac{\partial \lambda _{i}}{\partial \theta }=0. \end{aligned}$$

(46)

The derivatives of the $\lambda _{j}$ appearing in Eq. (46) are given by:

$$\begin{aligned} \frac{\partial \lambda _{A}}{\partial \theta }&=-4a_{1}\cos (2\theta )\sin (2\theta )\nonumber \\ \frac{\partial \lambda _{B}}{\partial \theta }&=+4a_{1}\cos (2\theta )\sin (2\theta )\nonumber \\ \frac{\partial \lambda _{C}}{\partial \theta }&=\frac{9}{2}\sin ^{3}(2\theta )\cos (2\theta )f(\phi _{1},\phi _{2})\\ \frac{\partial \lambda _{1,2}}{\partial \theta }&=\frac{\partial \lambda _{B}}{\partial \theta }\pm \frac{16a_{1}^{2}\cos (2\theta )\sin (2\theta )+\frac{\partial \lambda _{C}}{\partial \theta }}{2\sqrt{4a_{1}^{2} \cos ^{2}(2\theta )+\lambda _{C}},},\nonumber \\ \frac{\partial \lambda _{3,4}}{\partial \theta }&=\frac{\partial \lambda _{A}}{\partial \theta }\pm \frac{\frac{\partial \lambda _{C}}{\partial \theta }}{2\sqrt{\lambda _{C}},}\nonumber \end{aligned}$$

(47)

and furthermore:

$$\begin{aligned} \frac{\partial S_{rel}}{\partial \lambda _{A,B}}&=+\frac{1}{6\ln 2}(\ln \lambda _{A,B}+1)\end{aligned}$$

(48)

$$\begin{aligned} \frac{\partial S_{rel}}{\partial \lambda _{i}}&=-\frac{1}{12\ln 2}(\ln \lambda _{i}+1)\quad (i=1,2,3,4) \end{aligned}$$

(49)

Equation (46) then becomes:

$$\begin{aligned}&\cos (2\theta )\sin (2\theta )\left[ -4a_{1}\frac{\partial S_{rel}}{\partial \lambda _{A}}+4a_{1}\frac{\partial S_{rel}}{\partial \lambda _{B}}\right. \nonumber \\&\quad +\left( 4a_{1}+\frac{16a_{1}^{2}+\frac{9}{2}\sin ^{2}(2\theta ) f(\phi _{1},\phi _{2})}{2\sqrt{4a_{1}^{2}\cos ^{2}(2\theta ) +\lambda _{C}},}\right) \frac{\partial S_{rel}}{\partial \lambda _{1}}\nonumber \\&\quad +\left( 4a_{1}-\frac{16a_{1}^{2} +\frac{9}{2}\sin ^{2}(2\theta )f(\phi _{1},\phi _{2})}{2\sqrt{4a_{1}^{2}\cos ^{2}(2\theta )+\lambda _{C}},}\right) \frac{\partial S_{rel}}{\partial \lambda _{2}}\nonumber \\&\quad +\left. \left( -4a_{1}+\frac{\frac{9}{2}\sin ^{2}(2\theta ) f(\phi _{1},\phi _{2})}{2\sqrt{\lambda _{C}},}\right) \frac{\partial S_{rel}}{\partial \lambda _{3}}\right. \nonumber \\&\quad +\left. \left( -4a_{1} -\frac{\frac{9}{2}\sin ^{2}(2\theta )f(\phi _{1}, \phi _{2})}{2\sqrt{\lambda _{C}},}\right) \frac{\partial S_{rel}}{\partial \lambda _{4}}\right] =0 \end{aligned}$$

(50)

This expression shows that $\frac{\partial S_{rel}}{\partial \theta }=0$ when $\sin (2\theta )=0$ or $\cos (2\theta )=0$, that is, the function can attain its minimum value for a value in the set $\theta =0+k\frac{\pi }{2}$ or $\theta =\frac{\pi }{4}+k\frac{\pi }{2}$, where $k\in \mathbb {Z}$. Indeed, it can be shown that the whole l.h.s. of Eq. (50) goes to zero when $\theta$ approaches in the limit the values listed before.

When we make the further assumption that $\theta =\frac{\pi }{4}$ and $\phi _{1}=0$ (as suggested by numerical calculations), we get:

$$\begin{aligned} \lambda _{A}&=3,\nonumber \\ \lambda _{B}&=3,\nonumber \\ \lambda _{C}&=\frac{9}{16}\, f(0,\phi _{2}),\\ f(0,\phi _{2})&=\left[ \left( c_{1}-c_{2}\right) ^{2}+4c_{2} \left( 1+\cos (\phi _{2})\right) \left( c_{2}+c_{1} \cos (\phi _{2})\right) \right] ,\nonumber \\ \lambda _{1,2}&=\lambda _{3,4}=3\pm \sqrt{\lambda _{C}},\nonumber \end{aligned}$$

(51)

and

$$\begin{aligned} S_{rel}({\textstyle \frac{\pi }{4}},{\textstyle \frac{\pi }{4}},0, \phi _{2})=1+\log _{2}3-\frac{1}{6}\sum _{i=1}^{2}\lambda _{i}\log _{2}\lambda _{i}. \end{aligned}$$

(52)

Therefore, the minimum is reached when

$$\begin{aligned} \frac{\partial S_{rel}({\textstyle \frac{\pi }{4}},{\textstyle \frac{\pi }{4}},0,\phi _{2})}{\partial \phi _{2}}=0, \end{aligned}$$

(53)

that is,

$$\begin{aligned} \frac{\partial }{\partial \phi _{2}}\left( \lambda _{1}\log _{2}\lambda _{1}+ \lambda _{2}\log _{2}\lambda _{2}\right) =0. \end{aligned}$$

(54)

With further simplifications, we get:

$$\begin{aligned} \left[ \ln \frac{\left( 3+\sqrt{\lambda _{C}}\right) }{\left( 3-\sqrt{\lambda _{C}} \right) }\frac{1}{2\ln 2\sqrt{\lambda _{C}}}\right] \frac{\partial \lambda _{C}}{\partial \phi _{2}}=0. \end{aligned}$$

(55)

The expression in the square brackets is always greater than 0, since the argument of the logarithm is always greater than 1 if $\lambda _{C}>0$, and when $\lambda _{C}\rightarrow 0$ the limit is finite, positive and different from zero. Therefore the extremum can be found only for:

$$\begin{aligned} \frac{\partial \lambda _{C}}{\partial \phi _{2}}=0 \quad \Longleftrightarrow \quad \frac{\partial f(0,\phi _{2})}{\partial \phi _{2}}=0, \end{aligned}$$

(56)

which leads to the final equation:

$$\begin{aligned} \sin (\phi _{2})\left( c_{2}+c_{1}+2c_{1}\cos (\phi _{2})\right) =0. \end{aligned}$$

(57)

The solutions are:

$$\begin{aligned} \sin (\phi _{2})=0\quad \text {or}\quad \cos (\phi _{2}) =\left( -\frac{c_{1}+c_{2}}{2c_{1}}\right) , \end{aligned}$$

(58)

that is,

$$\begin{aligned} \phi _{2}=0+k\pi \quad \text {or}\quad \phi _{2}=\pm \arccos \left( -\frac{c_{1}+c_{2}}{2c_{1}}\right) +2k\pi , \end{aligned}$$

(59)

where $k\in \mathbb {Z}$. Clearly, the second set of extrema exists only if:

$$\begin{aligned} -1\le -\frac{c_{1}+c_{2}}{2c_{1}}\le +1\quad \Rightarrow \quad {\left\{ \begin{array}{ll} c_{1}>0\quad \text {and}\quad &{} -3c_{1}\le c_{2}\le c_{1}\\ c_{1}<0\quad \text {and}\quad &{} c_{1}\le c_{2}\le -3c_{1} \end{array}\right. }. \end{aligned}$$

(60)

Appendix 3: Comparison between $S_{2}$ and $S_{3}$

When $\theta ={\textstyle \frac{\pi }{4}}$ and $\phi _{1}=0$, the expression for $S_{rel}$ can be written as:

$$\begin{aligned} S_{rel}({\textstyle \frac{\pi }{4}},{\textstyle \frac{\pi }{4}},0,\phi _{2})&= 1+\log _{2}3-\frac{1}{6}\left[ \left( 3+\sqrt{\lambda _{C}}\right) \log _{2}\left( 3+\sqrt{\lambda _{C}}\right) \right. \nonumber \\&+\left. \left( 3-\sqrt{\lambda _{C}}\right) \log _{2}\left( 3-\sqrt{\lambda _{C}}\right) \right] \nonumber \\&= 1-\frac{1}{2}\left[ \left( 1+\frac{1}{3}\sqrt{\lambda _{C}}\right) \log _{2}\left( 1+\frac{1}{3}\sqrt{\lambda _{C}}\right) \right. \nonumber \\&+\left. \left( 1-\frac{1}{3}\sqrt{\lambda _{C}}\right) \log _{2}\left( 1-\frac{1}{3}\sqrt{\lambda _{C}}\right) \right] \nonumber \\&= 1-\frac{1}{2}\varepsilon \left( \frac{1}{3}\sqrt{\lambda _{C}}\right) , \end{aligned}$$

(61)

where $\varepsilon (x)$ is given by (26). The function $S(x)=1-\frac{1}{2}\varepsilon (x)$ is known in the literature as an estimator of correlations and relative entropies in bipartite systems [24], and its expression holds only for $-1\le x\le 1$ (in our case it is always $x>0$). Due to its symmetry properties, $S(x)$ has its maximum value for $x=0$ and decreases monotonically as $x$ approaches $1$ (or $-1$). Therefore, we conclude that:

$$\begin{aligned} S(x_{1})<S(x_{2})\Longleftrightarrow x_{1}>x_{2}\quad \forall x_{1},x_{2}\ge 0 \end{aligned}$$

(62)

If $\phi _{2}=0$, then $\lambda _{C}$ takes the following value:

$$\begin{aligned} \lambda _{C}=\frac{9}{16}\, f(0,0)=\frac{9}{16}(3c_{2}+c_{1})^{2}, \end{aligned}$$

(63)

and the corresponding expression for $S_{rel}$ is:

$$\begin{aligned} S_{2}=S_{rel}({\textstyle \frac{\pi }{4}},{\textstyle \frac{\pi }{4}},0,0)=1-\frac{1}{2}\varepsilon \left( \frac{|3c_{2}+c_{1}|}{4}\right) =1-\frac{1}{2}\varepsilon \left( \frac{3c_{2}+c_{1}}{4}\right) \end{aligned}$$

(64)

The $\lambda _{C}$ expression for $\phi _{2}=\bar{\phi }_{2}=\arccos \left( -\frac{c_{1}+c_{2}}{2c_{1}}\right)$ is the following one:

$$\begin{aligned} \lambda _{C}=\frac{9}{16}\, f(0,\bar{\phi }_{2})=\frac{9}{16}\frac{\left( c_{1}-c_{2}\right) ^{3}}{c_{1}}, \end{aligned}$$

(65)

which appears under a square root (for a real eigenvalue), and therefore is acceptable only if:

$$\begin{aligned} \frac{\left( c_{1}-c_{2}\right) }{c_{1}}\ge 0\quad \Rightarrow \quad {\left\{ \begin{array}{ll} c_{1}\ge c_{2} &{} \,\text {if}\,\; c_{1}>0\\ c_{1}\le c_{2}. &{} \,\text {if}\,\; c_{1}<0 \end{array}\right. }. \end{aligned}$$

(66)

The corresponding expression for $S_{rel}$ becomes:

$$\begin{aligned} S_{3}=S_{rel}({\textstyle \frac{\pi }{4}},{\textstyle \frac{\pi }{4}},0,\bar{\phi }_{2})=1-\frac{1}{2}\varepsilon \left( {\textstyle \frac{1}{4}\sqrt{\frac{\left( c_{1}-c_{2}\right) ^{3}}{c_{1}}}}\right) \end{aligned}$$

(67)

When both $S_{2}$ and $S_{3}$ expressions are well defined (see Eq. (24)), then the absolute minimum of $S_{rel}$ can occur only in the lowest of these two values, and according to Eq. (62) we conclude that

$$\begin{aligned} S_{3}<S_{2}\Longleftrightarrow \frac{\left( c_{1}-c_{2}\right) ^{3}}{c_{1}}>(3c_{2}+c_{1})^{2}\Longleftrightarrow c_{1}\cdot c_{2}<0. \end{aligned}$$

(68)

Now, collecting together the last Eq. (68) and the existence conditions for $S_{3}$ (Eqs. (60) and (66)), we conclude that:

$$\begin{aligned} S_{3}<S_{2}\Longleftrightarrow c_{1}\cdot c_{2}<0\quad \text {and}\quad |c_{2}|\le |3c_{1}|. \end{aligned}$$

(69)