Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

Abstract

In this paper the geometry of two- and three-qubit states under local unitary groups is discussed. We first review the one-qubit geometry and its relation with Riemannian sphere under the action of group SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group $SU(2)\otimes SU(2)$ and quaternionic Möbius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group Sp(2), generalizing the familiar Bloch sphere in two-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Möbius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.

Appendices

Appendix 1: Quaternion and octonion

The quaternion skew-field ${\mathbb {Q}}$ is an associative algebra of rank 4 on real number ${\mathbb {R}}$ whose every element can be written as

$$\begin{aligned} q=\sum \limits _{i = 0}^3 {{x_i}{e_i}} ,\quad \quad {x_i} \in {\mathbb {R}}, \end{aligned}$$

(5.1)

where $e_{0}=1$ and $e_{1},e_{2},e_{3}$ satisfy

$$\begin{aligned}&e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=-1, \quad e_{1}e_{2}=-e_{2}e_{1}=e_{3}, \quad e_{2}e_{3}=-e_{3}e_{2}=e_{1},\nonumber \\&\quad e_{3}e_{1}=-e_{1}e_{3}=e_{2}. \end{aligned}$$

(5.2)

It can also be defined equivalently, using the complex numbers $z_{1}=x_{0}+x_{1}e_{1}$ and $z_{2}=x_{2}+x_{3}e_{1}$ in the form $q=z_{1}+z_{2}e_{2}$ endowed with an involutory anti-automorphism (conjugation) such as

$$\begin{aligned} q=z_{1}+z_{2}e_{2} \in {{\mathbb {C}}}\oplus {{\mathbb {C}}}e_{2}\ \longrightarrow \ \bar{q}=x_{0}-\sum \limits _{i = 1}^3 {{x_i}{e_i}}=\bar{z}_{1}-z_{2}e_{2}. \end{aligned}$$

(5.3)

Note that quaternion multiplication is non-commutative so that $\overline{q_{1}q_{2}}={\overline{q_{2}}}\ {\overline{q_{1}}}$ and $e_{2}z=\bar{z}e_{2}$. On the other hand, a two-dimensional quaternionic vector space V defines a four-dimensional complex vector space ${{\mathbb {C}}}V$ by forgetting scalar multiplication by non-complex quaternions (i.e., those involving $e_{2}$). Roughly speaking, if V has quaternionic dimension 2, with basis $|0\rangle _{q},|1\rangle _{q}$, then ${{\mathbb {C}}}V$ has complex dimension 4, with basis $\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$.

The octonionic skew-field ${\mathbb {O}}$ is neither commutative nor associative algebra of rank 8 over ${\mathbb {R}}$ whose every element can be written as

$$\begin{aligned} o=\sum \limits _{i = 0}^7 {{x_i}{e_i}}, \quad {x_i} \in {\mathbb {R}}, \quad {e_0} = 1\quad \mathrm {and} \quad e_i^2 = - 1\; (i = 1,\ldots ,7). \end{aligned}$$

(5.4)

The multiplication of octonion is given by

$$\begin{aligned} {e_i}{e_j} = - {\delta _{ij}}{e_0} + \sum \limits _{k = 1}^7 {{f_{ijk}}{e_k}} \quad \mathrm {for} \; i,j = 1,2,\ldots ,7, \end{aligned}$$

(5.5)

where the ${f_{ijk}}$ are totally anti-symmetric in i, j, and k with values $1, 0,-1$ just as Levi-Civita symbol. Moreover, $f_{ijk} = +1$ for $ijk= 123, 145, 246, 347,617, 725,536$. We can depict this fact graphically as in Fig. 3 where the multiplication role can be read from orientation of arrows, for example:

$$\begin{aligned} e_{1}e_{2}=e_{3} , \quad \quad e_{2}e_{4}=e_{6} , \quad \quad e_{7}e_{2}=e_{5}. \end{aligned}$$

(5.6)

Fig. 3

Multiplication rule for octonion

The complex conjugate of a octonion is given by

$$\begin{aligned} {\bar{o}}=x_{0}-\sum \limits _{i = 1}^7 {{x_i}{e_i}}. \end{aligned}$$

(5.7)

Any octonion number can be represented in terms of four complex number $z_{0}=x_{0}+x_{1}e_{1}$, $z_{1}=x_{2}+x_{3}e_{1}$, $z_{2}=x_{4}+x_{5}e_{1}$ and $z_{3}=x_{6}+x_{7}e_{1}$ as

$$\begin{aligned} o=z_{0}+z_{1}e_{2}+(z_{3}+z_{3}e_{2})e_{4}, \end{aligned}$$

(5.8)

and it’s complex conjugate is given by

$$\begin{aligned} {\bar{o}}={\bar{z}}_{0}-z_{1}e_{2}-(z_{3}+z_{3}e_{2})e_{4}, \end{aligned}$$

(5.9)

or equivalently, in terms of quaternion numbers $q_{1}=z_{0}+z_{1}e_{2}$ and $q_{2}=z_{2}+x_{3}e_{2}$, we can write

$$\begin{aligned} o&=q_{1}+q_{2}e_{4}, \nonumber \\ {\bar{o}}&=\bar{q}_{1}-q_{2}e_{4}. \end{aligned}$$

(5.10)

The multiplication of two octonions

$$\begin{aligned} o_{1}&=z_{0}+z_{1}e_{2}+(z_{3}+z_{3}e_{2})e_{4},\nonumber \\ o_{2}&=p_{0}+p_{1}e_{2}+(p_{3}+p_{3}e_{2})e_{4}, \end{aligned}$$

(5.11)

using the multiplication rule of $e_{i}$, is an octonion

$$\begin{aligned} o_{3}=o_{1}o_{2}=s_{0}+s_{1}e_{2}+(s_{3}+s_{3}e_{2})e_{4} , \end{aligned}$$

(5.12)

where $s_{i}, i=0,1,2,3$ are complex numbers and are given by

$$\begin{aligned} s_{0}&=z_{0}p_{0}-z_{1}\bar{p}_{1}-z_{2}\bar{p}_{2}-{\bar{z}}_{3}p_{3}, \nonumber \\ s_{1}&=z_{0}p_{1}+z_{1}\bar{p}_{0}+{\bar{z}}_{2} p_{3}- z_{3}\bar{p}_{2}, \nonumber \\ s_{2}&=z_{0}p_{2}-{\bar{z}}_{1} p_{3}+z_{2} \bar{p}_{0}+ z_{3}\bar{p}_{1}, \nonumber \\ s_{3}&={\bar{z}}_{0} p_{3}+ z_{1} p_{2}-z_{2} \bar{p}_{1}+ z_{3}\bar{p}_{0}. \end{aligned}$$

(5.13)

Every nonzero ($p\in {\mathbb {Q}}$ or ${\mathbb {O}}$) is invertible, and the unique inverse is given by $p^{-1}=\frac{1}{|p|^2}\bar{p}$ where the quaternion or octonionic norm |p| is defined by $|p|^2=p\bar{p}.$ The norm of two quaternions or octonions $p_{1}$ and $p_{2}$ satisfies $|p_{1}p_{2}| = |p_{2}p_{1}| = |p_{1}||p_{2}|.$ Note that octonion multiplication is non-commutative and non-associative that is $(o_{1}o_{2})o_{3}\ne o_{1}(o_{2}o_{3})$. On the other hand, a two-dimensional octonionic vector space V defines a four-dimensional quaternionic vector space ${\mathbb {Q}}V$ by forgetting scalar multiplication by octonion (i.e., those involving $e_{4}$). Roughly speaking if V has octonionic dimension 2, with basis $\{|0\rangle _{o},|1\rangle _{0}\}$, then ${\mathbb {Q}}V$ has quaternion dimension 4, with basis $\{|00\rangle _{q},|01\rangle _{q},|10\rangle _{q},|11\rangle _{q}\}$ and ${{\mathbb {C}}}V$ has complex dimension 8, with basis $\{|000\rangle ,|001\rangle ,|010\rangle ,|011\rangle ,|100\rangle ,|101\rangle ,|110\rangle ,|111\rangle \}$.

Appendix 2: Calculating

In this appendix, we calculate the terms appearing in the main commutative diagram related to three-qubit pure state.

Calculating ${\mathcal {P}}{\mathcal {O}}{\mathcal {Q}}B|\psi \rangle$

It is convenient to start with the first statement in Eq. (4.21),

$$\begin{aligned} {\mathcal {P}}{\mathcal {O}}{\mathcal {Q}}B|\psi \rangle ={\mathcal {P}}{\mathcal {O}}{\mathcal {Q}}|\psi \rangle '={\mathcal {P}}{\mathcal {O}}|\psi \rangle '_{q}={\mathcal {P}}|\psi \rangle '_{o} , \end{aligned}$$

(5.14)

where $|\psi \rangle '_{o}=o'_{1}|0\rangle _{o}+o'_{2}|1\rangle _{o}$ and $o'_{1}$ and $o'_{2}$ are results of the action of local unitary transformation $B=A_{1}\otimes A_{2}\otimes A_{3}$ on the general three-qubit pure state (4.1) followed by the map ${\mathcal {O}}{\mathcal {Q}}$ as

$$\begin{aligned} { o}'_{1}= & {} t_{0}'+t_{1}'e_{2}+(t_{2}'+t_{3}'e_{2})e_{4}\nonumber \\= & {} (t_{0}{a_{1}}{a_{2}}{a_{3}}+t_{1}{a_{1}}{a_{2}}{b_{3}} +t_{2}{a_{1}}{b_{2}}\bar{a_{3}}+t_{3}{a_{1}}{b_{2}}{b_{3}} +t_{4}{b_{1}}{a_{2}}{a_{3}}\nonumber \\&+\,t_{5}{b_{1}}{a_{2}}{b_{3}} +t_{6}{b_{1}}{b_{2}}{a_{3}}+t_{7}{b_{1}}{b_{2}}{b_{3}})\nonumber \\&+\,(-t_{0}{a_{1}}\bar{a_{2}}\bar{b_{3}}+t_{1}{a_{1}}\bar{a_{2}}\bar{a_{3}} -t_{2}{a_{1}}{b_{2}}\bar{b_{3}}+t_{3}{a_{1}}{b_{2}}\bar{a_{3}} -t_{4}{b_{1}}\bar{a_{2}}\bar{b_{3}}\nonumber \\&+\,t_{5}{b_{1}}{a_{2}}{a_{3}} -t_{6}{b_{1}}{b_{2}}\bar{b_{3}}+t_{7}{b_{1}}{b_{2}}\bar{a_{3}})e_{2}\nonumber \\&+\,(-t_{0}{a_{1}}\bar{b_{2}}{a_{3}}-t_{1}{a_{1}}\bar{b_{2}}{b_{3}} +t_{2}{a_{1}}\bar{a_{2}}{a_{3}}+t_{3}{a_{1}}\bar{a_{2}}{b_{3}} -t_{4}{b_{1}}\bar{b_{2}}{a_{3}}\nonumber \\&-\,t_{5}\bar{b_{1}}\bar{b_{2}}\bar{b_{3}} +t_{6}{b_{1}}\bar{a_{2}}{a_{3}}+t_{7}{b_{1}}\bar{a_{2}}{b_{3}})e_{4}\nonumber \\&+\,([t_{0}{a_{1}}{b_{2}}\bar{b_{3}}-t_{1}{a_{1}}\bar{b_{2}}\bar{a_{3}} -t_{2}{a_{1}}\bar{a_{2}}\bar{b_{3}}+t_{3}{a_{1}}\bar{a_{2}}\bar{a_{3}} +t_{4}{b_{1}}{b_{2}}\bar{b_{3}}\nonumber \\&-\,t_{5}{b_{1}}\bar{b_{2}}\bar{a_{3}} -t_{6}{b_{1}}\bar{a_{2}}\bar{b_{3}} +t_{7}{b_{1}}\bar{a_{2}}\bar{a_{3}}]e_{2})e_{4},\end{aligned}$$

(5.15)

$$\begin{aligned} { o}'_{2}= & {} t_{4}'+t_{5}'e_{2}+(t_{6}'+t_{7}'e_{2})e_{4} \nonumber \\= & {} (-t_{0}\bar{b_{1}}{a_{2}}{a_{3}}-t_{1}\bar{b_{1}}{a_{2}}{b_{3}} -t_{2}\bar{b_{1}}{b_{2}}{a_{3}}-t_{3}\bar{b_{1}}{b_{2}}{b_{3}}+t_{4}\bar{a_{1}}{a_{2}}{a_{3}}\nonumber \\&+\,t_{5}\bar{a_{1}}{a_{2}}{b_{3}}+t_{6}\bar{a_{1}}{b_{2}}{a_{3}}+t_{7}\bar{a_{1}}{b_{2}}{b_{3}})\nonumber \\&+\,(t_{0}\bar{b_{1}}\bar{a_{2}}\bar{b_{3}}-t_{1}\bar{b_{1}}\bar{a_{2}}\bar{a_{3}}+t_{2}\bar{b_{1}}{b_{2}}\bar{b_{3}} -t_{3}\bar{b_{1}}r{b_{2}}\bar{a_{3}}-t_{4}\bar{a_{1}}\bar{a_{2}}\bar{b_{3}}\nonumber \\&+\,t_{5}\bar{a_{1}}\bar{a_{2}}\bar{a_{3}}-t_{6}\bar{a_{1}}{b_{2}}\bar{b_{3}}+t_{7}\bar{a_{1}}{b_{2}}\bar{a_{3}})e_{2}\nonumber \\&+\,(t_{0}\bar{b_{1}}\bar{b_{2}}{a_{3}}+t_{1}\bar{b_{1}}\bar{b_{2}}{b_{3}}-t_{2}\bar{b_{1}}\bar{a_{2}}{a_{3}}-t_{3}\bar{b_{1}}\bar{a_{2}}{b_{3}}-t_{4}\bar{a_{1}}\bar{b_{2}}{a_{3}}\nonumber \\&-\,t_{5}\bar{a_{1}}\bar{b_{2}}{b_{3}}+t_{6}\bar{a_{1}}\bar{a_{2}}{a_{3}}+t_{7}\bar{a_{1}}\bar{a_{2}}{b_{3}})e_{4}\nonumber \\&+\,([t_{0}\bar{b_{1}}\bar{b_{2}}\bar{b_{3}}+t_{1}\bar{b_{1}}\bar{b_{2}}\bar{a_{3}}+t_{2}\bar{b_{1}}\bar{a_{2}}\bar{b_{3}}-t_{3}\bar{b_{1}}\bar{a_{2}}\bar{a_{3}} +t_{4}\bar{a_{1}}\bar{b_{2}}\bar{b_{3}}\nonumber \\&-\,t_{5}\bar{a_{1}}\bar{b_{2}}\bar{a_{3}}-t_{6}\bar{a_{1}}\bar{a_{2}}\bar{b_{3}}+t_{7}\bar{a_{1}}\bar{a_{2}}\bar{a_{3}}]e_{2})e_{4}, \end{aligned}$$

(5.16)

According to the relation ${\mathcal {O}}|\psi \rangle '_{o}=({\mathcal {O}}B){\mathcal {O}}|\psi \rangle _{o}$, the $o'_{1}$ and $o'_{2}$ can be factorized as

$$\begin{aligned} o'_{1}&=a_{1}\left[ \{o_{1}{\mathcal {A}}_{3}^{(q)}\}{\mathcal {A}}_{2}^{(o)}\right] +b_{1}\left[ \left\{ o_{2}{\mathcal {A}}_{3}^{(q)}\right\} {\mathcal {A}}_{2}^{(o)}\right] \ ,\nonumber \\ o'_{2}&=-\bar{b}_{1}\left[ \left\{ o_{1}{\mathcal {A}}_{3}^{(q)}\right\} {\mathcal {A}}_{2}^{(o)}\right] +\bar{a}_{1} \left[ \left\{ o_{2}{\mathcal {A}}_{3}^{(q)}\right\} {\mathcal {A}}_{2}^{(o)}\right] , \end{aligned}$$

(5.17)

where ${\mathcal {A}}_{3}^{(q)}$ and ${\mathcal {A}}_{2}^{(o)}$ are quaternion and octonionic form of unitary transformation $A_{3}$ and $A_{2}$, introduced in (3.9) and (4.18), respectively.

Calculating ${\mathcal {P}}{\mathcal {O}}({\mathbb {Q}}B){\mathcal {Q}}|\psi \rangle$

We can proceed to another approach to understand more about three-qubit entangled pure state. Unlike in definition of ${\mathbb {Q}}B$, in order to correctly represent the complex transformation on the three-qubit pure state, the transformation B on the two-quaterbit should be represented by left action of the $4 \times 4$ matrix $A_{1} \otimes A_{2}$ and right multiplication with the quaternion ${\mathcal {A}}_{3}^{(q)}$ as in Eq. (4.17), hence applying the ${\mathbb {Q}}B$ on a two-quaterbit $|\psi \rangle _{q}$ one can get

$$\begin{aligned} A_{1} \otimes A_{2}\left[ |\psi \rangle {\mathcal {A}}_{3}^{(q)}\right] =\left( {\begin{array}{*{20}{c}} {t{'_0} + t{'_1}{e_2}} \\ {t{'_2} + t{'_3}{e_2}} \\ {t{'_4} + t{'_5}{e_2}} \\ {t{'_6} + t{'_7}{e_2}} \end{array}} \right) , \end{aligned}$$

(5.18)

where $t'_{0},\ldots ,t'_{7}$ are the same forms as in Eq. (5.15). It is clear that the operation ${\mathcal {O}}$ leads to Eq. (5.17), and hence the relevant part of the crucial diagram (the first quadrangle) is commutative, i.e.,

$$\begin{aligned} {\mathcal {O}}{\mathcal {Q}}B|\psi \rangle ={\mathcal {O}}({\mathbb {Q}}B){\mathcal {Q}}|\psi \rangle , \end{aligned}$$

(5.19)

and it is clear that applying the octonionic stereographic projection on the both side of the above equation lead to the first equality in Eq. (4.21).

Calculating ${\mathcal {P}}({\mathbb {O}}B){\mathcal {O}}{\mathcal {Q}}|\psi \rangle$

We mentioned that the octonionic form of local unitary transformation should be represented by left action of the $2 \times 2$ matrix $(A_{1}\in SU(2))$ and right multiplication with the octonion ${\mathcal {A}}_{3}^{(q)}, {\mathcal {A}}_{2}^{(o)}$ keeping in mind that the ordering is as Eq. (5.17). Now, applying the $({\mathbb {O}}B)$ on a octabit one can get

$$\begin{aligned} {A_1}\left\{ \left[ | \psi \rangle _{o}{\mathcal {A}}_{3}^{(q)}\right] {\mathcal {A}}_{2}^{(o)}\right\} = \left( {\begin{array}{*{20}{c}} {{a_1}}&{}{b_{1}} \\ {-\bar{b_{1}}}&{}{{a_1}} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {\{ {o_1}{\mathcal {A}}_{3}^{(q)}\}{\mathcal {A}}_{2}^{(o)}} \\ {\{ {o_2}{\mathcal {A}}_{3}^{(q)}\} {\mathcal {A}}_{2}^{(o)}} \end{array}} \right) =\left( {\begin{array}{*{20}{c}} {o{'_1}} \\ {o{'_2}} \end{array}} \right) , \end{aligned}$$

(5.20)

where $o{'_1}$ and $o{'_2}$ have the same forms as in Eq. (5.17). Altogether, we obtain

$$\begin{aligned} {\mathcal {O}}{\mathcal {Q}}({{\mathbb {C}}}B)|\psi \rangle ={\mathcal {O}}({\mathbb {Q}}B){\mathcal {Q}}|\psi \rangle =({\mathbb {O}}B){\mathcal {O}}{\mathcal {Q}}|\psi \rangle , \end{aligned}$$

(5.21)

Now, applying the octonionic stereographic projection ${\mathcal {P}}$ on above equation lead to the second equality in (4.21), or more precisely we have

$$\begin{aligned} {\mathcal {P}}(|\psi \rangle '_{o})=\frac{1}{|o'_{2}|^{2}}\left[ a_{1}b_{1}({\tilde{o}}_{2}\bar{{\tilde{o}}}_{2}-o_{1}{\bar{o}}_{1})+\left( a_{1}\frac{o_{1}\bar{{\tilde{o}}}_{2}}{|o_{2}|^{2}}a_{1}-b_{1}\frac{{{\tilde{o}}_{2}}{\bar{o}}_{1}}{|o_{2}|^{2}}b_{1}\right) \right] , \end{aligned}$$

(5.22)

where we used the relations ${\tilde{o}}_{2}\bar{{\tilde{o}}}_{2}-o_{1}{\bar{o}}_{1}=|o_{2}|^{2}(1-o{\bar{o}})$) and $o={\mathcal {P}}(|{\psi }\rangle _{o})=\frac{{\tilde{o}}_{1}\bar{{\tilde{o}}}_{2}}{|o_{2}|^{2}}$ to get

$$\begin{aligned} {\mathcal {P}}(o'_{1}|0\rangle _{o}+o'_{2}|1\rangle _{o})&=\frac{|o_{2}|^{2}}{|o'_{2}|^{2}}(a_{1}b_{1}(1-o{\bar{o}})+a_{1}oa_{1}-b_{1}{\bar{o}} b_{1}) \nonumber \\&=\frac{|o_{2}|^{2}}{|o'_{2}|^{2}}[S'_{0}+S'_{1}e_{2}+(S'_{2}+S'_{3}e_{2})e_{4}], \end{aligned}$$

(5.23)

where

$$\begin{aligned} S'_{0}&=[a_{1}b_{1}(1-|S_{0}|^2-|S_{1}|^2-|S_{2}|^2-|S_{3}|^2)+(a_{1}^{2}S_{0}-b_{1}^{2}\bar{S_{0}})+S_{1}e_{2},\nonumber \\ S'_{1}&=S_{1}\nonumber \\ S'_{2}&=S_{2}\nonumber \\ S'_{3}&=S_{3} \end{aligned}$$

(5.24)

and $S_{0}, S_{1}, S_{2}$ and $S_{3}$ are complex number that we introduce in (4.9).

Calculating ${\mathcal {F}}_{_{B}}{\mathcal {P}}{\mathcal {O}}{\mathcal {Q}}|\psi \rangle$

So far we have shown that the first two equalities in Eq. (4.21) holds. We will henceforth focus on the role of octonionic Möbius transformation. Using the linear map ${\mathcal {Q}}$ and ${\mathcal {O}}$ together with octonionic stereographic projection ${\mathcal {P}}$ on a three-qubit pure state in Eq. (4.1) yields

$$\begin{aligned} {\mathcal {P}}{\mathcal {O}}{\mathcal {Q}}|\psi \rangle ={\mathcal {P}}{\mathcal {O}}|\psi \rangle _{q}={\mathcal {P}}|\psi \rangle _{o}={\tilde{o}}_{1}{\tilde{o}}^{-1}_{2}=\frac{1}{|o_{2}|^2 }(S_{0}+S_{1}e_{2}+(S_{2}+S_{3}e_{2})e_{4}). \end{aligned}$$

(5.25)

Furthermore, this point is mapped under the action of the octonionic Möbius transformation in Eq. (4.20) as follows

$$\begin{aligned}&{\mathcal {F}}_{B}\left( \frac{1}{|o_{2}|^2}(S_{0}+S_{1}e_{2}+(S_{2}+S_{3}e_{2})e_{4}\right) \nonumber \\&\quad = \frac{1}{|o_{2}|^2}\left[ a_{1}b_{1}(1-|S_{0}|^2-|S_{1}|^2-|S_{2}|^2 -|S_{3}|^2)+(a_{1}^{2}S_{0}-b_{1}^{2}\bar{S_{0}})\right. \nonumber \\&\left. \qquad +\,S_{1}e_{2}+(S_{2}e_{4}+S_{3}e_{2})e_{4})\right] \nonumber \\&\quad =\frac{1}{|o_{2}|^2}[S'_{0}+S'_{1}e_{2}+(S'_{2}e_{4}+S'_{3}e_{2})e_{4}], \end{aligned}$$

(5.26)

in which the term $S_{1}e_{2}+S_{2}e_{4}+(S_{3}e_{2})e_{4}$ is invariant. Remembering that the entanglement between the first qubit and the last two qubits of the state (4.1) is given by concurrence measure (4.14), we have $S_{1}=S_{2}=S_{3}=0$ for separable three-qubit states. For completeness we mention that the first qubit lives on the base space of the Hopf fibration while the other two-qubit lives on the fiber space of the Hopf map. On the other hand, since there is, in general, a symmetry in choosing the partitions of a three-qubit state, we could have started from $|{\psi }\rangle _{2,(13)}$ or $|{\psi }\rangle _{3,(12)}$ (instead of $|{\psi }\rangle _{1,(23)}$), and hence there are three kind of octonionification and subsequently three stereographic projections. Two others are

$$\begin{aligned}&|{\psi }\rangle _{o'}=m_{1}|{ 0 }\rangle _{o'}+m_{2}|{ 1 }\rangle _{o'}\nonumber \\&|{\psi }\rangle _{o''}=n_{1}|{ 0 }\rangle _{o''}+n_{2}|{ 1 }\rangle _{o''}, \end{aligned}$$

(5.27)

where $m_{1}, m_{2}, n_{1}, n_{2}$ are octonion numbers

$$\begin{aligned}&m_{1}=t_{0}+t_{1}e_{2}+(t_{4}+ t_{5}e_{2})e_{4}, \nonumber \\&m_{2}=t_{2}+t_{3}e_{2}+(t_{6}+ t_{7}e_{2})e_{4}, \nonumber \\&n_{1}=t_{0}+t_{2}e_{2}+(t_{4}+ t_{6}e_{2})e_{4}, \nonumber \\&n_{2}=t_{1}+t_{3}e_{2}+(t_{5}+ t_{7}e_{2})e_{4}. \end{aligned}$$

(5.28)

Acting the stereographic projection ${\mathcal {P}}$ on two states $|{\psi }\rangle _{o'}$ and $|{\psi }\rangle _{o''}$ yields

$$\begin{aligned} M&=( t_{0}\bar{t_{2}}+ t_{1}\bar{t_{3}}+ t_{4}\bar{t_{6}}+ t_{3}\bar{t_{7}})+( t_{1}t_{2} - t_{0}t_{3}+\bar{t_{5}}\bar{t_{6}}-\bar{t_{4}}\bar{t_{7}})e_{2} \nonumber \\&\quad +\,[( t_{4}t_{2} - t_{0}t_{6}+\bar{t_{1}}\bar{t_{7}}-\bar{t_{5}}\bar{t_{3}})+( t_{4}t_{3} - t_{1}t_{6}+\bar{t_{5}}\bar{t_{2}}-\bar{t_{0}}\bar{t_{7}})e_{2}]e_{4} \nonumber \\ N&=( t_{0}\bar{t_{1 }}+ t_{2 }\bar{t_{3 }}+ t_{4 }\bar{t_{5 }}+ t_{6 }\bar{t_{7}})+( t_{2 }t_{1 } - t_{0}t_{3 }+\bar{t_{6 }}\bar{t_{5 }}-\bar{t_{4 }}\bar{t_{7}})e_{2} \nonumber \\&\quad +\,[( t_{4 }t_{1 } - t_{0}t_{5 }+\bar{t_{2 }}\bar{t_{7}}-\bar{t_{6 }}\bar{t_{3 }})+( t_{4 }t_{3 } - t_{2 }t_{5 }+\bar{t_{6 }}\bar{t_{1 }}-\bar{t_{0}}\bar{t_{7}})e_{2}]e_{4} \end{aligned}$$

(5.29)

respectively. Corresponding to each octonification there are concurrence measures which relate the entanglement between one-qubit with other two-qubit states, i.e.,

$$\begin{aligned} C^{2(13)}= & {} 2\left( |t_{0}t_{3}-t_{1}t_{2}|^{2}+|t_{0}t_{6}-t_{2}t_{4}|^{2}+|t_{0}t_{7}-t_{2}t_{5}|^{2}\right. \nonumber \\&\left. +\,|t_{1}t_{6}-t_{3}t_{4}|^{2}+|t_{1}t_{7}-t_{3}t_{5}|^{2}+|t_{4}t_{7}-t_{5}t_{6}|^{2}\right) ^{\frac{1}{2}},\nonumber \\ C^{3(12)}= & {} 2\left( |t_{0}t_{3}-t_{1}t_{2}|^{2}+|t_{0}t_{5}-t_{1}t_{4}|^{2}+|t_{0}t_{7}-t_{1}t_{6}|^{2}\right. \nonumber \\&\left. +\,|t_{2}t_{5}-t_{3}t_{4}|^{2}+|t_{2}t_{7}-t_{3}t_{6}|^{2}+|t_{4}t_{7}-t_{5}t_{6}|^{2}\right) ^{\frac{1}{2}}. \end{aligned}$$

(5.30)

According the above concurrence, the results M and N, which arise from projecting of the states $|{\psi }\rangle _{o'}$ and $|{\psi }\rangle _{o''}$, are also entanglement sensitive.