${\mathcal{Q}}_{c}$-matrices which anticommute with real symmetric normalized
${\mathcal{M}}_{c}$-matrices except the unit matrix $\varUpsilon^{0}$.

The matrices $\varUpsilon^{\mathsf{A}}$ and $\tilde{\varUpsilon}^{p}$ satisfy
the following Clifford algebra relations:

$\displaystyle\\{\varUpsilon^{\mathsf{A}},\varUpsilon^{\mathsf{B}}\\}=2\delta^{\mathsf{A}\mathsf{B}},\quad\\{\varUpsilon^{0},\varUpsilon^{\mathsf{A}}\\}=2\varUpsilon^{\mathsf{A}},\quad\\{\tilde{\varUpsilon}^{p},\tilde{\varUpsilon}^{q}\\}=2\delta^{pq},$
$\displaystyle\\{\tilde{\varUpsilon}^{p},\varUpsilon^{\mathsf{A}}\\}=0,\quad\mathsf{A},\mathsf{B}=1,\cdots,11,\quad
p,q=1,2,$ (164)

which implies that the ${\mathcal{Q}}_{c}$-spin charge of hexaqubit-spinor
field $\Psi_{\mathsf{Q}^{6}}(x)$ has the following relation with qubit number
$Q_{N}$:

$\displaystyle\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=Q_{N}-4=2,$ (165)

which is equal to the ${\mathcal{Q}}_{c}$-spin charge of biqubit-spinor field
in four-dimensional Minkowski spacetime.

Actually, four real symmetric normalized ${\mathcal{M}}_{c}$-matrices
($\varUpsilon^{0},\varUpsilon^{9},\varUpsilon^{10},\varUpsilon^{11}$) and two
imaginary antisymmetric normalized ${\mathcal{M}}_{c}$-matrices
($\tilde{\varUpsilon}^{1},\tilde{\varUpsilon}^{2}$) are analogous to those for
biqubit-spinor field as shown in Eqs.(IV.2) and (68). The remaining eight real
symmetric normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\bar{A}}$
($\bar{A}=1,\cdots,8$) characterize extra eight spatial dimensions.

In an analogous way, the action of self-conjugated hexaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)$ as hyperqubit-spinor field can be
written as follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{6}}$ $\displaystyle=$
$\displaystyle\int
d^{12}x\,\\{\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)-\frac{1}{2}\lambda_{2}\phi_{p}(x)\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\tilde{\varGamma}^{p}\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)\\},$
(166)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,12$. The $64\times 64$
$\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{\mathsf{A}}},\varGamma^{10},\varGamma^{11},\varGamma^{12})$
($\tilde{\mathsf{A}}=1,2,3,5,\cdots,9$) and $\tilde{\varGamma}^{p}$ ($p=1,2$)
can be constructed as follows:

$\displaystyle\varGamma^{0}=\sigma_{2}\otimes\sigma_{3}\otimes\Gamma^{0},\quad\varGamma^{\tilde{\mathsf{A}}}=\sigma_{2}\otimes\sigma_{3}\otimes\Gamma^{\tilde{A}},$
$\displaystyle\varGamma^{10}=-i\sigma_{2}\otimes\sigma_{2}\otimes\Gamma^{10},\quad\varGamma^{11}=i\sigma_{2}\otimes\sigma_{1}\otimes
I_{16},$ $\displaystyle\varGamma^{12}=-i\sigma_{1}\otimes\sigma_{0}\otimes
I_{16},$
$\displaystyle\tilde{\varGamma}^{1}=i\sigma_{3}\otimes\sigma_{0}\otimes
I_{16},\quad\tilde{\varGamma}^{2}=\sigma_{0}\otimes\sigma_{0}\otimes I_{16},$
(167)

with ($\Gamma^{0},\Gamma^{\tilde{A}},\Gamma^{10}$) presented in Eq.(V.1).
Where four $\varGamma$-matrices
($\varGamma^{0},\varGamma^{9},\varGamma^{10},\varGamma^{11}$) and two
$\tilde{\varGamma}$-matrices ($\tilde{\varGamma}^{1},\tilde{\varGamma}^{2}$)
are similar to those for self-conjugated biqubit-spinor field in four-
dimensional Minkowski spacetime as shown in Eq.(IV.5).

The self-conjugated hexaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)$ as hyperqubit-spinor field has the
following spinor structuce:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)=\binom{\psi_{\mathsf{Q}_{\mathsf{C}}^{5}}}{\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{5}}}=\begin{pmatrix}\psi_{\mathsf{Q}^{4}_{\mathsf{C}+}}(x)\\\
\psi_{\mathsf{Q}^{4}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}+}}(x)\end{pmatrix}=\begin{pmatrix}\psi_{\mathsf{Q}^{3}_{\mathsf{C}++}}(x)\\\
\psi_{\mathsf{Q}^{3}_{\mathsf{C}+-}}(x)\\\
\psi_{\mathsf{Q}^{3}_{\mathsf{C}-+}}(x)\\\
\psi_{\mathsf{Q}^{3}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{3}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{3}_{\mathsf{C}-+}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{3}_{\mathsf{C}+-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{3}_{\mathsf{C}++}}(x)\end{pmatrix},$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{5}}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\psi_{\mathsf{Q}_{\mathsf{C}}^{5}}^{\ast},$
(168)

which satisfies the self-conjugate condition,

$\displaystyle\Psi^{c}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)=C_{\mathsf{Q}_{\mathsf{S}}^{6}}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{S}}^{6}}(x)=\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{S}}^{6}}=\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}.$
(169)

The actions presented in Eq.(162) and (166) get the following associated
symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,11}\ltimes SO(1,11)\,\widetilde{\Join}\,SP(1,11)\times SG(1)\times SP(2)$
(170) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,11)\,\widetilde{\Join}\,SP(1,11)\times SG(1)\times SP(2),$

where SP(1,11) denotes maximal ${\mathcal{M}}_{c}$-spin symmetry as hyperspin
symmetry. SP(2)$\cong$U(1) is the ${\mathcal{Q}}_{c}$-spin symmetry with
respect to ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=2$. PO(1,11) = P${}^{1,11}\times$
SO(1,11) represents Poincaré-type/inhomogeneous Lorentz-type group symmetry
with SO(1,11) the Lorentz-type group symmetry and P1,11 the translation group
symmetry in 12-dimensional Minkowski hyper-spacetime.

###  VI.3 Local coherent state of seven qubits with ${\mathcal{Q}}_{c}$-spin
charge $\mathcal{C}_{{\mathcal{Q}}_{c}}$=3 and the action of heptaqubit-spinor
field as hyperqubit-spinor field in 14D hyper-spacetime

We now come to consider a local coherent state which is form by 7-product
qubit-basis states with qubit number $Q_{N}=7$:

$\displaystyle\Psi_{\mathsf{Q}^{7}}(x)$ $\displaystyle=$
$\displaystyle\sum_{s_{1},\cdots,s_{7}}\psi_{s_{1}\cdots
s_{7}}(x)\varsigma_{s_{1}\cdots
s_{7}}\equiv\sum_{s_{1},\cdots,s_{7}}\psi_{s_{1}\cdots
s_{7}}(x)\,\varsigma_{s_{1}}\otimes\cdots\otimes\varsigma_{s_{7}},$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)$ $\displaystyle=$
$\displaystyle\sum_{s_{1},\cdots,s_{7}}\psi_{s_{1}\cdots
s_{7}}(x)\zeta_{s_{1}\cdots
s_{7}}\equiv\sum_{s_{1},\cdots,s_{7}}\psi_{s_{1}\cdots
s_{7}}(x)\,\zeta_{s_{1}}\otimes\cdots\otimes\zeta_{s_{7}},$ (171)
$\displaystyle\equiv$
$\displaystyle\sum_{s_{1},\cdots,s_{7}}\tilde{\psi}_{s_{1}\cdots
s_{7}}(x)\varsigma_{s_{1}\cdots
s_{7}}\equiv\sum_{s_{1},\cdots,s_{7}}\tilde{\psi}_{s_{1}\cdots
s_{7}}(x)\,\varsigma_{s_{1}}\otimes\cdots\otimes\varsigma_{s_{7}},$

we may refer to hyperqubit-spinor fields $\Psi_{\mathsf{Q}^{7}}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)$ as heptaqubit-spinor field and self-
conjugated heptaqubit-spinor field, respectively. They involve
${\mathcal{D}}_{H}=2^{Q_{N}}=128$ local distribution amplitudes
$\psi_{s_{1}\cdots s_{7}}(x)$ which span 128-dimensional Hilbert space.

The maximally correlated motion of heptaqubit-spinor field
$\Psi_{\mathsf{Q}^{7}}(x)$ brings on the following hermitian action in
14-dimensional Minkowski spacetime:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{7}}$ $\displaystyle=$
$\displaystyle\int
d^{14}x\frac{1}{2}\Psi_{\mathsf{Q}^{7}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}^{7}}(x),$
(172)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,13$. Where $\varUpsilon^{\mathsf{A}}$
are $128\times 128$ real symmetric normalized ${\mathcal{M}}_{c}$-matrices,
which can be expressed as follows with the notation
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}},\varUpsilon^{9},\varUpsilon^{10},\varUpsilon^{11},\varUpsilon^{12},\varUpsilon^{13})$
($\bar{\mathsf{A}}=1,\cdots,8$):

$\displaystyle\varUpsilon^{0}=\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0},\quad\varUpsilon^{\bar{\mathsf{A}}}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{A},$
$\displaystyle\varUpsilon^{9}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{9},\quad\varUpsilon^{10}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\Upsilon^{0},$
$\displaystyle\varUpsilon^{11}=\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\Upsilon^{0},\quad\varUpsilon^{12}=\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0},$
$\displaystyle\varUpsilon^{13}=\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\Upsilon^{0},$
(173)

with $\Upsilon^{0},\Upsilon^{\bar{A}},\Upsilon^{9}$ the $16\times 16$ real
symmetric normalized ${\mathcal{M}}_{c}$-matrices given in Eq.(V.1). The three
imaginary antisymmetric normalized ${\mathcal{Q}}_{c}$-matrices
$\tilde{\varUpsilon}^{p}$ ($p=1,2,3$) are found to be:

$\displaystyle\tilde{\varUpsilon}^{1}=\sigma_{3}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\Upsilon^{0},$
$\displaystyle\tilde{\varUpsilon}^{2}=\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\Upsilon^{0},$
$\displaystyle\tilde{\varUpsilon}^{3}=\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0},$
(174)

which are anti-commuting with real symmetric normalized
${\mathcal{M}}_{c}$-matrices except two matrices $\varUpsilon^{0}$ and
$\varUpsilon^{13}$. The matrices $\varUpsilon^{\mathsf{A}}$ and
$\tilde{\varUpsilon}^{p}$ satisfy the following Clifford algebra relations:

$\displaystyle\\{\varUpsilon^{\mathsf{A}},\varUpsilon^{\mathsf{B}}\\}=2\delta^{\mathsf{A}\mathsf{B}},\quad\\{\varUpsilon^{0},\varUpsilon^{\mathsf{A}}\\}=2\varUpsilon^{\mathsf{A}},\quad\\{\tilde{\varUpsilon}^{p},\tilde{\varUpsilon}^{q}\\}=2\delta^{pq},$
$\displaystyle\mathsf{A},\mathsf{B}=1,\cdots,13,\quad
p,q=1,2,3;\quad\\{\tilde{\varUpsilon}^{p},\varUpsilon^{\mathsf{A}}\\}=0,\;\mathsf{A}\neq
0,13,$ (175)

where six real symmetric normalized ${\mathcal{M}}_{c}$-matrices
($\varUpsilon^{0},\varUpsilon^{9},\varUpsilon^{10},\varUpsilon^{11},\varUpsilon^{12},\varUpsilon^{13}$)
and three antisymmetric normalized ${\mathcal{Q}}_{c}$-matrices
($\tilde{\varUpsilon}^{1},\tilde{\varUpsilon}^{2},\tilde{\varUpsilon}^{3}$)
are analogous to matrices for triqubit-spinor field in six-dimensional
Minkowski spacetime as shown in Eqs.(IV.2) and (72). The other eight real
symmetric normalized ${\mathcal{M}}_{c}$-matrices
$\varUpsilon^{\bar{\mathsf{A}}}$ ($\bar{\mathsf{A}}=1,\cdots,8$) correlate to
extra eight spatial dimensions. The ${\mathcal{Q}}_{c}$-spin charge of
heptaqubit-spinor field is the same as that of triqubit-spinor field, i.e.,
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=3$.

The action of self-conjugated heptaqubit-spinor field as hyperqubit-spinor
field $\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)$ is easily read:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{7}}$ $\displaystyle=$
$\displaystyle\int
d^{14}x\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x),$
(176)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,14$. The fourteen $128\times 128$
$\Gamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{\mathsf{A}}},\varGamma^{10},\varGamma^{11},\varGamma^{12},\varGamma^{13},\varGamma^{14})$
($\tilde{\mathsf{A}}=1,2,3,5,\cdots,9$) are given by the following forms:

$\displaystyle\varGamma^{0}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\Gamma^{0},\quad\varGamma^{10}=-i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\Gamma^{10},$
$\displaystyle\varGamma^{11}=i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}\otimes
I_{16},\quad\varGamma^{12}=-i\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{0}\otimes
I_{16},$
$\displaystyle\varGamma^{13}=-i\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
I_{16},\quad\varGamma^{14}=\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\otimes
I_{16},$
$\displaystyle\varGamma^{\tilde{\mathsf{A}}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\Gamma^{\tilde{A}},\quad\tilde{\mathsf{A}}=1,2,3,5,\cdots,9,$
(177)

with ($\Gamma^{0},\Gamma^{\tilde{A}},\Gamma^{10}$) presented in Eq.(V.1).
Where six $\varGamma$-matrices
($\varGamma^{0},\varGamma^{10},\varGamma^{11},\varGamma^{12},\varGamma^{13},\varGamma^{14}$)
are analogous to those for self-conjugated triqubit-spinor field in six-
dimensional Minkowski spacetime as shown in Eq.(IV.5).

The self-conjugated heptaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)$ as hyperqubit-spinor field is defined
as follows:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)=\binom{\psi_{\mathsf{Q}_{\mathsf{C}}^{6}}}{\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{6}}}=\begin{pmatrix}\psi_{\mathsf{Q}^{5}_{\mathsf{C}+}}(x)\\\
\psi_{\mathsf{Q}^{5}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}+}}(x)\end{pmatrix}=\begin{pmatrix}\psi_{\mathsf{Q}^{4}_{\mathsf{C}++}}(x)\\\
\psi_{\mathsf{Q}^{4}_{\mathsf{C}+-}}(x)\\\
\psi_{\mathsf{Q}^{4}_{\mathsf{C}-+}}(x)\\\
\psi_{\mathsf{Q}^{4}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}-+}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}+-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{4}_{\mathsf{C}++}}(x)\end{pmatrix},$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{6}}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\psi_{\mathsf{Q}_{\mathsf{C}}^{6}}^{\ast},$
(178)

which satisfies the following self-conjugate condition:

$\displaystyle\Psi^{c}_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)=C_{\mathsf{Q}_{\mathsf{S}}^{7}}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{S}}^{7}}(x)=\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{S}}^{7}}=\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}.$
(179)

The actions given in Eq.(172) and (176) turn out to have the following
associated symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,13}\ltimes SO(1,13)\,\widetilde{\Join}\,SP(1,13)\rtimes SG(1)\times
SP(3)$ (180) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,13)\,\widetilde{\Join}\,SP(1,13)\rtimes SG(1)\times SU(2),$

where SP(1,13) is hyperspin symmetry and PO(1,13) = P${}^{1,13}\ltimes$
SO(1,13) is Poincaré-type/inhomogeneous Lorentz-type group symmetry in
14-dimensional Minkowski hyper-spacetime. SP(3)$\cong$SU(2) represents
${\mathcal{Q}}_{c}$-spin symmetry which has the following group generators:

$\displaystyle\lambda_{1}=\frac{1}{2}\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0}\otimes
I_{16},$
$\displaystyle\lambda_{2}=\frac{1}{2}\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\otimes
I_{16},$
$\displaystyle\lambda_{3}=\frac{1}{2}\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
I_{16}.$

It is obvious that the ${\mathcal{Q}}_{c}$-spin charge is given by,

$\displaystyle\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=Q_{N}-4=3.$ (181)

###  VI.4 Local coherent state of eight qubits and the action of octoqubit-
spinor field as hyperqubit-spinor field in 18D hyper-spacetime in analogous to
tetraqubit-spinor field in 10D hyper-spacetime and zeroqubit-spinor field in
2D spacetime

Let us analyze local coherent states of eight qubits formed from two types of
8-product qubit-basis states. They are hyperqubit-spinor fields
$\Psi_{\mathsf{Q}^{8}}(x)$ and $\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ in real
qubit-basis and complex conjugated qubit-basis, respectively, with the
following definitions:

$\displaystyle\Psi_{\mathsf{Q}^{8}}(x)$ $\displaystyle=$
$\displaystyle\sum_{s_{1},\cdots,s_{8}}\psi_{s_{1}\cdots
s_{8}}(x)\varsigma_{s_{1}\cdots
s_{8}}\equiv\sum_{s_{1},\cdots,s_{8}}\psi_{s_{1}\cdots
s_{8}}(x)\,\varsigma_{s_{1}}\otimes\cdots\otimes\varsigma_{s_{8}},$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ $\displaystyle=$
$\displaystyle\sum_{s_{1},\cdots,s_{8}}\psi_{s_{1}\cdots
s_{8}}(x)\zeta_{s_{1}\cdots
s_{8}}\equiv\sum_{s_{1},\cdots,s_{8}}\psi_{s_{1}\cdots
s_{8}}(x)\,\zeta_{s_{1}}\otimes\cdots\otimes\zeta_{s_{8}},$ (182)
$\displaystyle\equiv$
$\displaystyle\sum_{s_{1},\cdots,s_{8}}\tilde{\psi}_{s_{1}\cdots
s_{8}}(x)\varsigma_{s_{1}\cdots
s_{8}}\equiv\sum_{s_{1},\cdots,s_{8}}\tilde{\psi}_{s_{1}\cdots
s_{8}}(x)\,\varsigma_{s_{1}}\otimes\cdots\otimes\varsigma_{s_{8}},$

where $\Psi_{\mathsf{Q}^{8}}(x)$ is referred to as octoqubit-spinor field and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ as self-conjugated octoqubit-spinor
field. They contain ${\mathcal{D}}_{H}=2^{8}=256$ local distribution
amplitudes $\psi_{s_{1}\cdots s_{8}}(x)$ which span 256-dimensional Hilbert
space.

Based on the local coherent-qubits motion postulate with following along the
maximum coherence motion principle, we obtain the following action for
octoqubit-spinor field $\Psi_{\mathsf{Q}^{8}}(x)$:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{8}}$ $\displaystyle=$
$\displaystyle\int
d^{18}x\frac{1}{2}\Psi_{\mathsf{Q}^{8}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}^{8}}(x),$
(183)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,17$. Where the eighteen $2^{8}\times
2^{8}$ real symmetric normalized ${\mathcal{M}}_{c}$-matrices
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}},\varUpsilon^{\bar{\mathsf{A}}+8},\varUpsilon^{9+8})$
($\bar{\mathsf{A}}=1,\cdots,8$) have the following structures:

$\displaystyle\varUpsilon^{0}=\Upsilon^{0}\otimes\Upsilon^{0},\;\;\varUpsilon^{9+8}=\Upsilon^{9}\otimes\Upsilon^{9},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}}=\Upsilon^{0}\otimes\Upsilon^{\bar{A}},\quad\varUpsilon^{\bar{\mathsf{A}}+8}=\Upsilon^{\bar{A}}\otimes\Upsilon^{9},\;\;\bar{\mathsf{A}}=1,\cdots,8,$
(184)

with $\Upsilon^{0},\Upsilon^{\bar{A}},\Upsilon^{9}$ the $16\times 16$ real
symmetric normalized ${\mathcal{M}}_{c}$-matrices defined in Eq.(V.1). The
matrices $\varUpsilon^{\mathsf{A}}$ satisfy the following Clifford algebra:

$\displaystyle\\{\varUpsilon^{\mathsf{A}},\varUpsilon^{\mathsf{B}}\\}=2\delta^{\mathsf{A}\mathsf{B}},\quad\\{\varUpsilon^{0},\varUpsilon^{\mathsf{A}}\\}=2\varUpsilon^{\mathsf{A}},\quad\mathsf{A},\mathsf{B}=1,\cdots,17.$
(185)

As there exists no antisymmetric normalized ${\mathcal{Q}}_{c}$-matrix which
anticommutes with real symmetric normalized ${\mathcal{M}}_{c}$-matrices, the
octoqubit-spinor field $\Psi_{\mathsf{Q}^{8}}(x)$ as hyperqubit-spinor field
has zero ${\mathcal{Q}}_{c}$-spin charge. Such a ${\mathcal{Q}}_{c}$-spin
charge is given by a relation with qubit number as follows:

$\displaystyle\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=Q_{N}-8=0,$ (186)

which repeats the cases for tetraqubit-spinor field in ten-dimensional
Minkowski hyper-spacetime and zeroqubit-spinor field in two-dimensional
Minkowski spacetime. Where the ten real symmetric normalized
${\mathcal{M}}_{c}$-matrices
$\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}},\varUpsilon^{9+8}$ are similar
to those of tetraqubit-spinor field in 10-dimensional Minkowski hyper-
spacetime as shown in Eq.(V.1), and the remaining eight real symmetric
normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\bar{\mathsf{A}}+8}$
($\bar{\mathsf{A}}=1,\cdots,8$) involve additional eight spatial dimensions.
Alternatively, the two real symmetric normalized ${\mathcal{M}}_{c}$-matrices
$\varUpsilon^{0},\varUpsilon^{9+8}$ are similar to the ones for self-
conjugated chiral uniqubit-spinor field or zeroqubit-spinor field in two-
dimensional Minkowski spacetime as shown in Eq.(41), while the other sixteen
real symmetric normalized ${\mathcal{M}}_{c}$-matrices
$\varUpsilon^{\bar{\mathsf{A}}}$ and $\varUpsilon^{\bar{\mathsf{A}}+8}$
($\bar{\mathsf{A}}=1,\cdots,8$) concern extra sixteen spatial dimensions.

The action for self-conjugated octoqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ as hyperqubit-spinor field can be
expressed as follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{8}}$ $\displaystyle=$
$\displaystyle\int
d^{18}x\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x),$
(187)

with $\mathsf{A},\mathsf{M}=0,1,2,3,5,\cdots,18$. Where $2^{8}\times 2^{8}$
$\Gamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{\mathsf{A}}},\varGamma^{\bar{\mathsf{A}}+9},\varGamma^{10+8})$
($\tilde{\mathsf{A}}=1,2,3,5,\cdots,9$, $\bar{\mathsf{A}}=1,\cdots,8$) have
the following forms:

$\displaystyle\varGamma^{0}=\Gamma^{0}\otimes\Gamma^{0},\quad\varGamma^{\tilde{\mathsf{A}}}=\Gamma^{0}\otimes\Gamma^{\tilde{A}},$
$\displaystyle\varGamma^{\bar{\mathsf{A}}+9}=\Gamma^{\tilde{A}}\otimes\Gamma^{10},\quad\varGamma^{10+8}=\Gamma^{10}\otimes\Gamma^{10},$
(188)

with $\Gamma^{0}$, $\Gamma^{\tilde{A}}$ and $\Gamma^{10}$ the $16\times 16$
${\mathcal{M}}_{c}$-matrices given in Eq.(V.1).

It is noticed that the hyper-spacetime in association with
$\varGamma$-matrices $\varGamma^{0}$, $\varGamma^{\tilde{\mathsf{A}}}$ and
$\varGamma^{10+8}$ describes a free motion in ten-dimensional spacetime out of
eighteen-dimensional Minkowski hyper-spacetime, which is in analogous to the
case for self-conjugated tetraqubit-spinor field shown in Eq.(128). The
remaining eight ${\mathcal{M}}_{c}$-matrices $\varGamma^{\bar{\mathsf{A}}+9}$
characterize maximally correlated motion in extra eight spatial dimensions. On
the other hand, we may regard two-dimensional spacetime in association with
${\mathcal{M}}_{c}$-matrices $\varGamma^{0}$ and $\varGamma^{10+8}$ as basic
spacetime for describing a longitudinal free motion, which is in analogous to
a free motion of self-conjugated chiral uniqubit-spinor field or zeroqubit-
spinor field shown in Eq.(141), whereas the other sixteen
${\mathcal{M}}_{c}$-matrices $\varGamma^{\tilde{\mathsf{A}}}$ and
$\varGamma^{\bar{\mathsf{A}}+9}$ reflect the maximally correlated transverse
motion in sixteen spatial dimensions.

The self-conjugated octoqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ as hyperqubit-spinor field is
explicitly defined as follows:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)=\binom{\psi_{\mathsf{Q}_{\mathsf{C}}^{7}}}{\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{7}}}=\begin{pmatrix}\psi_{\mathsf{Q}^{6}_{\mathsf{C}+}}(x)\\\
\psi_{\mathsf{Q}^{6}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{Q}^{6}_{\mathsf{C}-}}(x)\\\
\psi_{\hat{Q}^{6}_{\mathsf{C}+}}(x)\end{pmatrix}=\begin{pmatrix}\psi_{\mathsf{Q}^{5}_{\mathsf{C}++}}(x)\\\
\psi_{\mathsf{Q}^{5}_{\mathsf{C}+-}}(x)\\\
\psi_{\mathsf{Q}^{5}_{\mathsf{C}-+}}(x)\\\
\psi_{\mathsf{Q}^{5}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}--}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}-+}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}+-}}(x)\\\
\psi_{\hat{\mathsf{Q}}^{5}_{\mathsf{C}++}}(x)\end{pmatrix},$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{C}}^{7}}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}\psi_{\mathsf{Q}_{\mathsf{C}}^{7}}^{\ast},$
(189)

which satisfies the following self-conjugate condition:

$\displaystyle\Psi^{c}_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)=C_{\mathsf{Q}_{\mathsf{S}}^{8}}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)=\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{S}}^{8}}=\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}.$
(190)

The maximal associated symmetry of the actions in Eq.(183) and (187) is found
to be,

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,17}\ltimes SO(1,17)\,\widetilde{\Join}\,SP(1,17)\rtimes SG(1)\times
SP(q_{c}=0)$ (191) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,17)\,\widetilde{\Join}\,SP(1,17)\rtimes SG(1),$

where SP(1,17) is maximal ${\mathcal{M}}_{c}$-spin symmetry as hyperspin
symmetry. The semidirect group PO(1,17) = P${}^{1,17}\ltimes$ SO(1,17)
characterizes Poinceré-type/inhomogeneous Lorentz-type group symmetry with
$P^{1,17}$ denoting the translation group symmetry in 18-dimensional Minkowski
hyper-spacetime of coordinates. SP($q_{c}=0$) denotes in formal
${\mathcal{Q}}_{c}$-spin symmetry to classify hyperqubit-spinor field, which
indicates that there exist a class of hyperqubit-spinor field with zero
${\mathcal{Q}}_{c}$-spin charge that gets no scalar coupling.

###  VI.5 Classifications of arbitrary qubit number and spacetime dimension
with four categories of ${\mathcal{Q}}_{c}$-spin charge and the genesis of
spacetime dimension from categoric qubit number

When looking back to ${\mathcal{Q}}_{c}$-spin charge with a periodic behavior
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2,3,0$ in correspondence to basic
qubit-spinor fields, i.e., zeroqubit-spinor, uniqubit-spinor, biqubit-spinor,
triqubit-spinor and tetraqubit-spinor fields, with respective to basic qubit
numbers, i.e., $q_{c}=Q_{N}=0,1,2,3,4$, we observe that local coherent states
of qubits with high qubit number $Q_{N}\geq 4$ bring on a periodic
${\mathcal{Q}}_{c}$-spin charge given by a simple relation with qubit number,
i.e., $\mathcal{C}_{{\mathcal{Q}}_{c}}=Q_{N}-4=q_{c}=0,1,2,3$, which
characterize other four kinds of qubit-spinor fields: tetraqubit-spinor,
pentaqubit-spinor, hexaqubit-spinor and heptaqubit-spinor fields with
respective to four qubit numbers $Q_{N}=4,5,6,7$.

To be manifest, let us summarize the main properties of hyperqubit-spinor
fields and hyper-spacetime dimensions in table 2.

Table 2: The properties of local coherent state of qubits as hyperqubit-spinor field Qubits $Q_{N}$ | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---
Hyperqubit-spinor | $\Psi_{\mathsf{Q}^{4}}$ | $\Psi_{\mathsf{Q}^{5}}$ | $\Psi_{\mathsf{Q}^{6}}$ | $\Psi_{\mathsf{Q}^{7}}$ | $\Psi_{\mathsf{Q}^{8}}$
Qubit-spinor | tetraqubit- | pentaqubit- | hexaqubit- | heptaqubit- | octoqubit-
Hilbert space ${\mathcal{D}}_{H}$ | $16=2^{4}$ | $32=2^{5}$ | $64=2^{6}$ | $128=2^{7}$ | $256=2^{8}$
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}$ |  $Q_{N}$-4=0 |  $Q_{N}$-4=1 |  $Q_{N}$-4=2 |  $Q_{N}$-4=3 |  $Q_{N}$-4$\times$2=0
${\mathcal{Q}}_{c}$-spin sym. | SP(0) | SP(1)$\cong$O(1) | SP(2)$\cong$U(1) | SP(3)$\cong$SU(2) | SP(0)
${\mathcal{M}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{M}}_{c}}$ |  $2Q_{N}$+2=10 |  $2Q_{N}$+1=11 |  $2Q_{N}$=12 |  $2Q_{N}$=14 |  $2Q_{N}$+2=18
Hyper-spacetime $D_{h}$ | 2+$2^{3}$=2+8 | 3+8 | 4+8 | 6+8 | 10+8=2+16
Spinor structure | $\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}}$ | $\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}}$ | $\Psi_{\mathsf{Q}_{\mathsf{S}}^{6}},\Psi_{\mathsf{Q}_{\mathsf{C}}^{5}}$ | $\Psi_{\mathsf{Q}_{\mathsf{S}}^{7}},\Psi_{\mathsf{Q}_{\mathsf{C}}^{6}}$ | $\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}$
Hyperspin sym. | SP(1,9) | SP(1,10) | SP(1,11) | SP(1,13) | SP(1,17)
Lorentz-type sym. | SO(1,9) | SO(1,10) | SO(1,11) | SO(1,13) | SO(1,17)
Poincaré-type sym. | PO(1,9) | PO(1,10) | PO(1,11) | PO(1,13) | PO(1,17)

From the properties shown in table 1 and table 2 for basic qubit-spinor fields
and hyperqubit-spinor fields, the ${\mathcal{Q}}_{c}$-spin charge is resulted
as the mod 4 qubit number, i.e.:

$\displaystyle\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2,3=Q_{N}(\mbox{mod
4}),$ (192)

which enables us to deduce a general theorem for the classification of qubit
number.

Qubit number classification theorem: any qubit number $Q_{N}$ used to form
product qubit-basis states and build local coherent state of qubits can be
classified into four categories as categoric qubit number $Q_{N}^{(q_{c},k)}$,
which is characterized by four ${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2,3$ and integer number
$k=0,1,\cdots$.

As four categories are in correspondence to four basic qubit numbers, i.e.,
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=Q_{N}=0,1,2,3$, we are able to express
any qubit number $Q_{N}$ in terms of ${\mathcal{Q}}_{c}$-spin charge with the
following general relation:

$\displaystyle Q_{N}^{(q_{c},k)}\equiv q_{c}+4k,$ $\displaystyle
q_{c}=0,1,2,3,\quad k=0,1,2,\cdots,$ (193)

which indicates that ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}$ plays an essential role as qubit
category characteristic number. The periodic feature of
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}$ for any
qubit number $Q_{N}^{(q_{c},k)}=q_{c}+4k$ can be expressed as follows:

$\displaystyle\mathcal{C}_{{\mathcal{Q}}_{c}}\equiv\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c})}=q_{c},$
$\displaystyle q_{c}=0,1,2,3,\quad k=0,1,2,\cdots,$ (194)

where the integer number $k$ denotes as cycle number of period for
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}$. Therefore,
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}$ is regarded as categoric
${\mathcal{Q}}_{c}$-spin charge in category-$q_{c}$ and $k$-th period and
$Q_{N}^{(q_{c},k)}$ is considered as categoric qubit number in
category-$q_{c}$ and $k$-th period.

Following along the qubit number classification theorem and local coherent-
qubits motion postulate, we are able to further deduce a general theorem for
the classification of spacetime dimension.

Spacetime dimension classification theorem: any spacetime dimension determined
by the maximally correlated motion of qubit-spinor field with arbitrary
categoric qubit number can be classified into four categories as categoric
spacetime dimension $D_{h}^{(q_{c},k)}$, which is characterized by four
${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2,3$ and periodic number
$k=0,1,\cdots$.

As the basic spacetime dimensions generating from maximally correlated free
motions of basic qubit-spinor fields with respective to four basic qubit
numbers $Q_{N}=0,1,2,3$ are determined to be $D_{q_{c}}=2,3,4,6$ which are
characterized by four ${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=Q_{N}=0,1,2,3$, we can express any
spacetime dimension $D_{h}$ in terms of categoric basic dimension $D_{q_{c}}$
with the following general relation:

$\displaystyle D_{h}^{(q_{c},k)}\equiv D_{q_{c}}+8k,\quad k=0,1,2,\cdots,$
$\displaystyle D_{q_{c}}=2,3,4,6,\quad q_{c}=0,1,2,3,$ (195)

which is referred to as category-$q_{c}$ spacetime dimension in $k$-th period.
$D_{q_{c}}$ represents basic spacetime dimension in category-$q_{c}$, which is
completely determined by ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}$ with the following identity:

$\displaystyle D_{q_{c}}=2+2^{q_{c}-1}\theta(q_{c}-1),$ (196)

which brings on four basic spacetime dimensions $D_{q_{c}}=2,3,4,6$ with
respective to four ${\mathcal{Q}}_{c}$-spin charges $q_{c}=0,1,2,3$. Where
$\theta(z)$ is the $\theta$-function with $\theta(z)=1$ for $z\geq 0$ and
$\theta(z)=0$ for $z<0$.

Therefore, we are able to obtain the following general relation between
categoric spacetime dimension $D_{h}^{(q_{c},k)}$ and categoric qubit number
$Q_{N}^{(q_{c},k)}$,

$\displaystyle
D_{h}^{(q_{c},k)}=2Q_{N}^{(q_{c},k)}-2(q_{c}-1)+2^{q_{c}-1}\theta(q_{c}-1),$
(197)

which implies how the genesis of spacetime dimension is correlated to
categoric qubit number and ${\mathcal{Q}}_{c}$-spin charge.

###  VI.6  Categorization of hyperqubit-spinor field and hyper-spacetime with
four categories of ${\mathcal{Q}}_{c}$-spin charge and the action of
hyperqubit-spinor field in category-$q_{c}$ with hyperspin symmetry

In analogous to qubit number classification theorem and spacetime dimension
classification theorem, it is straightforward to deduce a general
categorization theorem for qubit-spinor field.

Qubit-spinor categorization theorem: any qubit-spinor field as local coherent
state of qubits with arbitrary qubit number can be categorized into four
categories as categoric qubit-spinor field $\Psi_{Q_{N}^{(q_{c},k)}}(x)$,
which is characterized by four ${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0,1,2,3$ and periodic number
$k=0,1,\cdots$.

To make the categorizations for both hyperqubit-spinor field and hyper-
spacetime through four ${\mathcal{Q}}_{c}$-spin charges with respective to
four basic qubit numbers and four basic spacetime dimensions, it is
appropriate to build the action of hyperqubit-spinor field as local coherent
state of qubits with categoric qubit number $Q_{N}^{(q_{c},k)}=q_{c}+4k$ in
$k$-th period. Such a hyperqubit-spinor field is denoted by
$\Psi_{Q_{N}^{(q_{c},k)}}(x)$ and referred to as category-$q_{c}$ hyperqubit-
spinor field in $k$-th period, which spans
${\mathcal{D}}_{H}=2^{Q_{N}^{(q_{c},k)}}$ dimensional Hilbert space defined on
$D_{h}^{(q_{c},k)}$ dimensional Minkowski hyper-spacetime in category-$q_{c}$
and $k$-th period.

The hyperqubit-spinor field $\Psi_{\mathsf{Q}^{n}}(x)$ with arbitrary qubit
number $Q_{N}=n$ is defined by real $n$-product qubit-basis
$\\{\varsigma_{s_{1}\cdots s_{n}}\\}$ as follows:

$\displaystyle\Psi_{\mathsf{Q}^{n}}(x)=\sum_{s_{1},\cdots,s_{n}}\psi_{s_{1}\cdots
s_{n}}(x)\varsigma_{s_{1}\cdots
s_{n}}\equiv\sum_{s_{1},\cdots,s_{n}}\psi_{s_{1}\cdots
s_{n}}(x)\,\varsigma_{s_{1}}\otimes\cdots\otimes\varsigma_{s_{n}},$

with $\psi_{s_{1}\cdots s_{n}}(x)$ regarded as evolving local distribution
amplitudes. Any hyperqubit-spinor field $\Psi_{\mathsf{Q}^{n}}(x)$ can always
be classified into four categories
$\Psi_{\mathsf{Q}^{n}}(x)\equiv\Psi_{Q_{N}^{(q_{c},k)}}(x)$ with respective to
${\mathcal{Q}}_{c}$-spin charges $q_{c}=0,1,2,3$ in $k$-th period.

For any qubit number in Category-0, i.e., $Q_{N}^{(0,k)}=n=4k$, the action for
Category-0 hyperqubit-spinor field $\Psi_{Q_{N}^{(0,k)}}(x)$ with zero
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0$ in
$k$-th period can generally be expressed as follows:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(0)}}$ $\displaystyle=$
$\displaystyle\int[dx]\frac{1}{2}\Psi_{Q_{N}^{(0,k)}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(0,k)}}(x),$
(198)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,D_{h}^{(0,k)}-1$. The maximally
correlated motion of Category-0 hyperqubit-spinor field leads to genesis of
Category-0 hyper-spacetime with dimension $D_{h}^{(0,k)}=2+8k$. For the case
$k=0$, it provides the basic action for zeroqubit-spinor field or bit-spinor
field $\psi_{\mathsf{Q}^{0}_{\pm}}(x)$ in two-dimensional spacetime as shown
in Eq.(141). When $k\geq 1$, the $2^{4k}\times 2^{4k}$ real symmetric
normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\mathsf{A}}$ can
generally be expressed as
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}+8j},\varUpsilon^{9+8(k-1)})$
($\bar{\mathsf{A}}=1,\cdots,8$; $j=0,1,\cdots,k-1$). For the cases $k=1,2$,
the ${\mathcal{M}}_{c}$-matrices are presented in Eqs.(V.1) and (VI.4),
respectively. In the general case, we can express the
${\mathcal{M}}_{c}$-matrices by the following explicit forms:

$\displaystyle\varUpsilon^{0}$ $\displaystyle=$
$\displaystyle\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8\times 0}$ $\displaystyle=$
$\displaystyle\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{\bar{A}}_{(k)},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8j}$ $\displaystyle=$
$\displaystyle\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{\bar{A}}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle j=1,\cdots,k-2,$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8(k-1)}$ $\displaystyle=$
$\displaystyle\Upsilon^{\bar{A}}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle\varUpsilon^{9+8(k-1)}$ $\displaystyle=$
$\displaystyle\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
(199)

where the matrices $\Upsilon^{0},\Upsilon^{\bar{A}},\Upsilon^{9}$ are
$16\times 16$ real symmetric normalized ${\mathcal{M}}_{c}$-matrices given in
Eq.(V.1). The subscript of ${\mathcal{M}}_{c}$-matrices $\Upsilon^{A}_{(i)}$
($i=1,2\cdots,k$) labels the column order in the product tensor.

For any qubit number in Category-1, i.e., $Q_{N}^{(1,k)}=1+4k$, the action for
Category-1 hyperqubit-spinor field $\Psi_{Q_{N}^{(1,k)}}(x)$ with
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=1$ in
$k$-th period is generally obtained as follows:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(1)}}=\int[dx]\\{\frac{1}{2}\Psi_{Q_{N}^{(1,k)}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(1,k)}}(x)-\frac{1}{2}\lambda_{1}\phi_{1}(x)\Psi_{Q_{N}^{(1,k)}}^{\dagger}(x)\tilde{\varUpsilon}\Psi_{Q_{N}^{(1,k)}}(x)\\},$
(200)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,D_{h}^{(1,k)}-1$. The Category-1 hyper-
spacetime gets dimensions $D_{h}^{(1,k)}=3+8k$ ($k=0,1\cdots$) with three-
dimensional spacetime as a basic one. In the simplest case with $k=0$, the
action for uniqubit-spinor field $\psi_{\mathsf{Q}^{1}}(x)$ in three-
dimensional spacetime is presented in Eq.(44). The
${\mathcal{M}}_{c}$-matrices and ${\mathcal{Q}}_{c}$-matrix for the case $k=1$
are given in Eq.(VI.1).

In the general case $k\geq 1$, the $2^{4k+1}\times 2^{4k+1}$ real symmetric
normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\mathsf{A}}$ are
expressed as
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}+8j},\varUpsilon^{9+8(k-1)},\varUpsilon^{10+8(k-1)})$
($\bar{\mathsf{A}}=1,\cdots,8$; $j=0,1,\cdots,k-1$), which can be obtained by
extending straightforwardly the case $k=1$ to the general case $k\geq 1$ with
the following explicit structures:

$\displaystyle\varUpsilon^{0}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8j}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{\bar{A}}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle j=0,1,\cdots,k-1,$ $\displaystyle\varUpsilon^{9+8(k-1)}$
$\displaystyle=$
$\displaystyle\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle\varUpsilon^{10+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\tilde{\varUpsilon}$ $\displaystyle=$
$\displaystyle\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)}.$
(201)

Similarly, the action for Category-2 hyperqubit-spinor field
$\Psi_{Q_{N}^{(2,k)}}(x)$ with categoric qubit number $Q_{N}^{(2)}=2+4k$ in
the $k$-th period has the following general form:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(2)}}=\int[dx]\\{\frac{1}{2}\Psi_{Q_{N}^{(2,k)}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(2,k)}}(x)-\frac{1}{2}\lambda_{2}\phi_{p}(x)\Psi_{Q_{N}^{(2,k)}}^{\dagger}(x)\tilde{\varUpsilon}^{p}\Psi_{Q_{N}^{(2,k)}}(x)\\},$
(202)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,D_{h}^{(2,k)}-1$. The dimension of
Category-2 hyper-spacetime is given by $D_{h}^{(2,k)}=4+8k$. The simplest case
with $k=0$ corresponds to the action of biqubit-spinor field
$\psi_{\mathsf{Q}^{2}}(x)$ in four-dimensional spacetime as shown in Eq.(66).
For the case $k=1$, the ${\mathcal{M}}_{c}$-matrices and
${\mathcal{Q}}_{c}$-matrices are presented in Eq.(VI.2).

For the general case $k\geq 1$, the $2^{4k+2}\times 2^{4k+2}$ real symmetric
normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\mathsf{A}}$ can be
written as
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}+8j},\varUpsilon^{9+8(k-1)},\varUpsilon^{10+8(k-1)},\varUpsilon^{11+8(k-1)})$
($\bar{\mathsf{A}}=1,\cdots,8$; $j=0,1,\cdots,k-1$). The explicit structures
of ${\mathcal{M}}_{c}$-matrices and ${\mathcal{Q}}_{c}$-matrices in Category-2
can be expressed as follows:

$\displaystyle\varUpsilon^{0}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8j}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{\bar{A}}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle j=0,1,\cdots,k-1,$ $\displaystyle\varUpsilon^{9+8(k-1)}$
$\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle\varUpsilon^{10+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{1}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{11+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{2}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\tilde{\varUpsilon}^{1}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\tilde{\varUpsilon}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)}.$
(203)

In analogous, the action for Category-3 hyperqubit-spinor field
$\Psi_{Q_{N}^{(3,k)}}(x)$ with categoric qubit number $Q_{N}^{(3,k)}=3+4k$ in
the $k$-th period can be written in the following general form:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(3)}}=\int[dx]\,\frac{1}{2}\Psi_{Q_{N}^{(3,k)}}^{\dagger}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varUpsilon^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(3,k)}}(x),$
(204)

with $\mathsf{A},\mathsf{M}=0,1,\cdots,D_{h}^{(3,k)}-1$. The spacetime
dimension of Category-3 hyper-spacetime is given by $D_{h}^{(2,k)}=6+8k$. The
action with $k=0$ is presented in Eq.(70) for triqubit-spinor field
$\psi_{\mathsf{Q}^{3}}(x)$ in six-dimensional spacetime. For the case $k=1$,
the ${\mathcal{M}}_{c}$-matrices and ${\mathcal{Q}}_{c}$-matrices are shown in
Eqs.(VI.3) and (VI.3), respectively.

For the general case $k\geq 1$, the $2^{4k+3}\times 2^{4k+3}$ real symmetric
normalized ${\mathcal{M}}_{c}$-matrices $\varUpsilon^{\mathsf{A}}$ can be
written as
$\varUpsilon^{\mathsf{A}}=(\varUpsilon^{0},\varUpsilon^{\bar{\mathsf{A}}+8j},\varUpsilon^{9+8(k-1)},\varUpsilon^{10+8(k-1)},\varUpsilon^{11+8(k-1)},\varUpsilon^{12+8(k-1)},\varUpsilon^{13+8(k-1)})$
($\bar{\mathsf{A}}=1,\cdots,8$; $j=0,1,\cdots,k-1$). The
${\mathcal{M}}_{c}$-matrices in Category-3 can be expressed as follows:

$\displaystyle\varUpsilon^{0}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{\bar{\mathsf{A}}+8j}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{\bar{A}}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle j=0,1,\cdots,k-1,$ $\displaystyle\varUpsilon^{9+8(k-1)}$
$\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\Upsilon^{9}_{(1)}\otimes\cdots\Upsilon^{9}_{(k-j)}\cdots\otimes\Upsilon^{9}_{(k)},$
$\displaystyle\varUpsilon^{10+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{11+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{12+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\varUpsilon^{13+8(k-1)}$ $\displaystyle=$
$\displaystyle\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)}.$
(205)

The three imaginary antisymmetric normalized ${\mathcal{Q}}_{c}$-matrices
$\tilde{\varUpsilon}^{p}$ ($p=1,2,3$) in Category-3 are given by,

$\displaystyle\tilde{\varUpsilon}^{1}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\tilde{\varUpsilon}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)},$
$\displaystyle\tilde{\varUpsilon}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\Upsilon^{0}_{(1)}\otimes\cdots\Upsilon^{0}_{(k-j)}\cdots\otimes\Upsilon^{0}_{(k)}.$
(206)

The matrices $\varUpsilon^{\mathsf{A}}$ and $\tilde{\varUpsilon}^{p}$ satisfy
the following Clifford algebra relations:

$\displaystyle\\{\varUpsilon^{\mathsf{A}},\varUpsilon^{\mathsf{B}}\\}=2\delta^{\mathsf{A}\mathsf{B}},\quad\\{\varUpsilon^{0},\varUpsilon^{\mathsf{A}}\\}=2\varUpsilon^{\mathsf{A}},\quad\\{\tilde{\varUpsilon}^{p},\tilde{\varUpsilon}^{q}\\}=2\delta^{pq},$
$\displaystyle\mathsf{A},\mathsf{B}=1,\cdots,D_{h}^{(3,k)}-1,\quad p,q=1,2,3,$
$\displaystyle\\{\tilde{\varUpsilon}^{p},\varUpsilon^{\mathsf{A}}\\}=0,\quad\mathsf{A}\neq\varUpsilon^{0},\varUpsilon^{13+8(k-1)}.$
(207)

In general, the action for category-$q_{c}$ hyperqubit-spinor field
$\Psi_{Q_{N}^{(q_{c},k)}}(x)$ in $k$-th period possesses the following
associated symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,D_{h}^{(q_{c},k)}-1}\ltimes
SO(1,D_{h}^{(q_{c},k)}-1)\,\widetilde{\Join}\,SP(1,D_{h}^{(q_{c},k)}-1)\rtimes
SG(1)\times SP(q_{c})$ (208) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,D_{h}^{(q_{c},k)}-1)\,\widetilde{\Join}\,SO(1,D_{h}^{(q_{c},k)}-1)\rtimes
SG(1)\times SP(q_{c}).$

##  VII  String-like qubit-spinor field with free motion in two-dimensional
spacetime and the self-conjugated chiral qubit-spinor field with full anti-
commuting $\Gamma$-matrices

The ${\mathcal{Q}}_{c}$-spin charge as qubit category characteristic number
enables us to categorize all hyperqubit-spinor fields by utilizing categoric
qubit number $Q_{N}^{(q_{c},k)}=q_{c}+4k$ in the $k$-th period
($k=0,1,\cdots$) and meanwhile to construct category-$q_{c}$ hyperqubit-spinor
field $\Psi_{Q_{N}^{(q_{c},k)}}(x)$ in hyper-spacetime based on the periodic
feature of ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c})}=q_{c}$.
The dimension of hyper-spacetime is also classified into categoric spacetime
dimension $D_{h}^{(q_{c},k)}=D_{q_{c}}+8k$ with $D_{q_{c}}$ representing four
basic spacetime dimensions $D_{q_{c}}=2,3,4,6$ in correspondence to four
${\mathcal{Q}}_{c}$-spin charges $Q_{c}=0,1,2,3$. In this section, we are
going to investigate self-conjugated chiral qubit-spinor structure. On one
hand, a self-conjugated chiral-like qubit-spinor structure allows us to
construct string-like qubit-spinor field with free motion in two-dimensional
spacetime from self-conjugated qubit-spinor field defined in
$D_{h}$-dimensional spacetime. On the other hand, a self-conjugate qubit-
spinor field with arbitrary qubit number $Q_{N}$ in category-$q_{c}$ can
always be described by a self-conjugated chiral qubit-spinor field with one
additional qubit number $Q_{N}+1$ but with the same category-$q_{c}$ in large
Hilbert space. As a specific example, a massless Dirac fermion as category-3
self-conjugated triqubit-spinor field in six-dimensional spacetime is shown to
be described by self-conjugated chiral tetraqubit-spinor field in category-3.
In particular, we will demonstrate that in light of self-conjugated chiral
qubit-spinor field, all $\Gamma$-matrices represented in large Hilbert space
become anti-commuting.

###  VII.1  Self-conjugated chiral-like spinor structure and string-like
qubit-spinor field

Based on the periodic feature of ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}=\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c})}=q_{c}$,
all qubit-spinor fields $\Psi_{Q_{N}^{(q_{c},k)}}(x)$ in the same
category-$q_{c}$ but with different period ($k=1,2,\cdots$) should possess the
same ${\mathcal{Q}}_{c}$-spin symmetry, and the ${\mathcal{M}}_{c}$-matrices
should have a similar matrix structure in correlation to four basic spacetime
dimensions $D_{q_{c}}=2,3,4,6$.

Let us begin with examining the zeroqubit-spinor field
$\psi_{\mathsf{Q}^{0}}(x)$, tetraqubit-spinor field $\Psi_{\mathsf{Q}^{4}}(x)$
and octoqubit-spinor field $\Psi_{\mathsf{Q}^{8}}(x)$ in category-0. As the
zeroqubit-spinor field can be regarded as a self-conjugated chiral uniqubit-
spinor field in two-dimensional Minkowski spacetime shown in Eq.(141), the
relevant two ${\mathcal{M}}_{c}$-matrices are in analogous to two
${\mathcal{M}}_{c}$-matrices $\Upsilon^{0}$ and $\Upsilon^{9}$ for tetraqubit-
spinor field $\Psi_{\mathsf{Q}^{4}}(x)$ or $\Gamma$-matrices $\Gamma^{0}$ and
$\Gamma^{10}$ for self-conjugated tetraqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}}(x)$, such two ${\mathcal{M}}_{c}$-matrices
characterize longitudinal free motion in a relevant two-dimensional Minkowski
spacetime.

To make an explicit illustration, it is useful to express the tetraqubit-
spinor field $\Psi_{\mathsf{Q}^{4}}(x)$ as the sum of two self-conjugated
chiral-like tetraqubit-spinor fields $\Psi_{\mathsf{Q}^{4}\pm}(x)$, i.e.:

$\displaystyle\Psi_{\mathsf{Q}^{4}}(x)=\Psi_{\mathsf{Q}^{4}-}(x)+\Psi_{\mathsf{Q}^{4}+}(x),$
$\displaystyle\Psi_{\mathsf{Q}^{4}\pm}(x)=\Upsilon_{\pm}\Psi_{\mathsf{Q}^{4}}(x),\quad\Upsilon_{\pm}=\frac{1}{2}(1\pm\Upsilon^{9}).$
(209)

In a similar way, we make the following decomposition for
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}}(x)$:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}}(x)=\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)+\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x),$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)=\Gamma_{\pm}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}}(x),\quad\Gamma_{\pm}=\frac{1}{2}(1\pm\Gamma^{0}),$
(210)

where the project operators $\Upsilon_{\pm}$ and $\Gamma_{\pm}$ define a self-
conjugated chiral-like structure for tetraqubit-spinor field, which allows us
to rewrite the actions in Eqs.(128) and (131) into the following forms in ten-
dimensional hyper-spacetime:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{4}}$ $\displaystyle=$
$\displaystyle\int
d^{10}x\frac{1}{2}\\{\Psi_{\mathsf{Q}^{4}+}^{\dagger}(x)\delta_{a}^{\;\mu}\Upsilon^{a}i\partial_{\mu}\Psi_{\mathsf{Q}^{4}+}(x)+\Psi_{\mathsf{Q}^{4}-}^{\dagger}(x)\delta_{a}^{\;\mu}\Upsilon^{a}i\partial_{\mu}\Psi_{\mathsf{Q}^{4}-}(x)$
(211) $\displaystyle+$
$\displaystyle\Psi_{\mathsf{Q}^{4}+}^{\dagger}(x)\delta_{\bar{A}}^{\;\bar{M}}\Upsilon^{\bar{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}^{4}-}(x)+\Psi_{\mathsf{Q}^{4}-}^{\dagger}(x)\delta_{\bar{A}}^{\;\mathsf{M}}\Upsilon^{\bar{A}}i\partial_{\bar{M}}\Psi_{\mathsf{Q}^{4}+}(x)\\},$

with $a,\mu=0,9$ and $\bar{A},\bar{M}=1,2,\cdots,8$, and

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{4}}$ $\displaystyle=$
$\displaystyle\int
d^{10}x\frac{1}{2}\\{\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)+\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)$
(212) $\displaystyle+$
$\displaystyle\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)\delta_{\tilde{A}}^{\;\tilde{M}}\Gamma^{\tilde{A}}i\partial_{\tilde{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)+\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)\delta_{\tilde{A}}^{\;\tilde{M}}\Gamma^{\tilde{A}}i\partial_{\tilde{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)\\},$

with $a,\mu=0,10$ and $\tilde{A},\tilde{M}=1,2,3,5,\cdots,9$.

It is noticed from the above actions that each self-conjugated chiral-like
tetraqubit-spinor field $\Psi_{\mathsf{Q}^{4}+}(x)$
($\Psi_{\mathsf{Q}^{4}-}(x)$) or $\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)$
($\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)$) gets a free motion only in two-
dimensional Minkowski spacetime. The free motion in other eight spatial
dimensions only occurs when two self-conjugated chiral-like tetraqubit-spinor
fields with opposite chirality $\Psi_{\mathsf{Q}^{4}+}(x)$ and
$\Psi_{\mathsf{Q}^{4}-}(x)$ or $\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)$ present simultaneously and obey the
maximum coherence motion principle. Namely, their interferential terms for the
derivative operators in association with eight ${\mathcal{M}}_{c}$-matrices
$\Upsilon^{\bar{A}}$ ($\bar{A}=1,2,\cdots,8$) or eight $\Gamma$-matrices
$\Gamma^{\tilde{A}}$ ($\tilde{A}=1,2,3,5,\cdots,9$), describe the maximally
correlated motion in eight spatial dimensions. Therefore, the action for each
self-conjugated chiral-like tetraqubit-spinor field can be expressed as
follows in two-dimensional Minkowski spacetime:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{4}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\Psi_{\mathsf{Q}^{4}\pm}^{\dagger}(x)\delta_{a}^{\;\mu}\Upsilon^{a}i\partial_{\mu}\Psi_{\mathsf{Q}^{4}\pm}(x),$
$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x),$
(213)

with $a,\mu=0,9$ for $\Psi_{\mathsf{Q}^{4}\pm}(x)$ and $a,\mu=0,10$ for
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)$.

Similarly, it can be verified that the self-conjugated chiral-like octoqubit-
spinor fields $\Psi_{\mathsf{Q}^{8}\pm}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)$ get a free motion in two-
dimensional Minkowski spacetime. Their actions can be presented as follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{8}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\Psi_{\mathsf{Q}^{8}\pm}^{\dagger}(x)\delta_{a}^{\;\mu}\varUpsilon^{a}i\partial_{\mu}\Psi_{\mathsf{Q}^{8}\pm}(x),$
$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)\delta_{a}^{\;\mu}\varGamma^{a}i\partial_{\mu}\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x),$
(214)

with $a,\mu=0,17$ for $\Psi_{\mathsf{Q}^{8}\pm}(x)$ and $a,\mu=0,18$ for
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)$. Where we have introduced the
following definitions:

$\displaystyle\Psi_{\mathsf{Q}^{8}\pm}(x)=\varUpsilon_{\pm}\Psi_{\mathsf{Q}^{8}}(x),\quad\varUpsilon_{\pm}=\frac{1}{2}(1\pm\varUpsilon^{9+8}),$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)=\varGamma_{\pm}\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x),\quad\varGamma_{\pm}=\frac{1}{2}(1\pm\varGamma^{0}).$
(215)

with $\varUpsilon^{9+8}\equiv\varUpsilon^{17}$ given in Eq.(VI.4) and
$\varGamma^{0}$ presented in Eq.(VI.4). Where $\Psi_{\mathsf{Q}^{8}}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}}(x)$ are octoqubit-spinor field and self-
conjugated octoqubit-spinor field defined in Eq.(VI.4), respectively, which
are shown in Eqs.(183) and (187) to have a free motion in 18-dimensional
hyper-spacetime.

It is easy to check that the above actions possess ${\mathcal{M}}_{c}$-spin
symmetry SP(1,1) and global scaling symmetry SG(1) in association with
Lorentz-type group symmetry SO(1,1) and translation group symmetry $P^{1,1}$
as well as conformal scaling symmetry SC(1). Meanwhile, the
${\mathcal{M}}_{c}$-spin symmetry concerning $D_{h}-2$ spatial dimensions
becomes an intrinsic hyperspin symmetry SP($D_{h}$-2).

For the self-conjugated chiral-like tetraqubit-spinor field
$\Psi_{\mathsf{Q}^{4}\pm}(x)$ ($\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)$),
the associated symmetry is given by,

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,1}\ltimes SO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times SP(8)$
(216) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,1)\,\widetilde{\Join}\,SP(1,1)\times SG(1)\times SP(8),$

where SP(8)$\cong$SO(8) provides a maximal internal hyperspin symmetry.

For the self-conjugated chiral-like octoqubit-spinor field
$\Psi_{\mathsf{Q}^{8}\pm}(x)$ ($\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)$),
the associated symmetry reads as follows:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,1}\ltimes SO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times SP(16)$
(217) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times SP(16),$

which brings on the maximal internal hyperspin symmetry SP(16)$\cong$SO(16).

It is straightforward to generalize the above analyses to arbitrary
hyperqubit-spinor field in category-0 with self-conjugated chiral-like
hyperqubit-spinor structure. A self-conjugated chiral-like hyperqubit-spinor
field in category-0 can generally be defined as follows:

$\displaystyle\Psi_{Q_{N}^{(0,k)}\pm}(x)=\varUpsilon_{\pm}\Psi_{Q_{N}^{(0,k)}}(x),\quad\varUpsilon_{\pm}=\frac{1}{2}(1\pm\Upsilon^{9+8(k-1)}),$
(218)

with $\varUpsilon_{\pm}$ the project operators and $k$ the periodic number
($k=0,1,\cdots$). For the cases with $k=0,1,2$, they are already analyzed
above. For a general case $k\geq 3$, the action for self-conjugated chiral-
like hyperqubit-spinor field $\Psi_{Q_{N}^{(0,k)}\pm}(x)$ in category-0 gets
the following general form:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(0)}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\Psi_{Q_{N}^{(0,k)}\pm}^{\dagger}(x)\delta_{a}^{\;\mu}\varUpsilon^{a}i\partial_{\mu}\Psi_{Q_{N}^{(0,k)}\pm}(x),$
(219)

which has a free motion only in two-dimensional Minkowski spacetime with
$a,\mu=0,(1+8k)$. Such an action possesses the following associated symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,1}\ltimes SO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times SP(8k)$
(220) $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times SP(8k),$

where SP(8k) represents internal hyperspin symmetry.

As a self-conjugated chiral-like qubit-spinor field has a free motion only in
two-dimensional spacetime, we are naturally inspired to associate it with
string theoryST1 ; ST2 . So, the self-conjugated chiral-like tetraqubit-spinor
field $\Psi_{\mathsf{Q}^{4}\pm}(x)$ or
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)$ and self-conjugated chiral-like
octoqubit-spinor field $\Psi_{\mathsf{Q}^{8}\pm}(x)$ or
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{8}\pm}(x)$ may be regarded as a string-like
tetraqubit-spinor field and string-like octoqubit-spinor field, respectively.
In general, a self-conjugated chiral-like hyperqubit-spinor field in
category-0 $\Psi_{Q_{N}^{(0,k)}\pm}(x)$ with a free motion in two-dimensional
spacetime is viewed as a string-like hyperqubit-spinor field in category-0,
which is in analogous to the zeroqubit-spinor field shown in Eq.(141).

The above discussions can be extended to any hyperqubit-spinor field in
category-$q_{c}$ with self-conjugated chiral-like hyperqubit-spinor structure,
i.e., $\Psi_{Q_{N}^{(q_{c},k)}\pm}(x)$, its action can be written into the
following form:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(q_{c},k)}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\Psi_{Q_{N}^{(q_{c},k)}\pm}^{\dagger}(x)\delta_{a}^{\;\mu}\varUpsilon^{a}i\partial_{\mu}\Psi_{Q_{N}^{(q_{c},k)}\pm}(x),$
(221)

with $a,\mu=0,(1+8k)$ for the $k$-th period of ${\mathcal{Q}}_{c}$-spin charge
and the definition,

$\displaystyle\Psi_{Q_{N}^{(q_{c},k)}\pm}(x)=\varUpsilon_{\pm}\Psi_{Q_{N}^{(q_{c},k)}}(x),\quad\varUpsilon_{\pm}=\frac{1}{2}(1\pm\varUpsilon^{9+8(k-1)}).$
(222)

To be more general, we can define a string-like hyperqubit-spinor field in
category-$q_{c}$ with any self-conjugated chiral-like hyperqubit-spinor
structure, i.e.:

$\displaystyle\Psi_{Q_{N}^{(q_{c},k)}\pm}^{(j)}(x)=\varUpsilon_{\pm}^{j}\Psi_{Q_{N}^{(q_{c},k)}}^{(j)}(x),\quad\varUpsilon_{\pm}^{j}=\frac{1}{2}(1\pm\varUpsilon^{j}),$
(223)

where $\varUpsilon^{j}$ can be any $\varUpsilon$-matrix with
$j=1,2,\cdots,1+8k$ for the $k$-th period of ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}^{(q_{c},k)}$. The action for such a general
string-like hyperqubit-spinor field is presented as follows:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(q_{c},k)}\pm}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\frac{1}{2}\Psi_{Q_{N}^{(q_{c},k)}\pm}^{(j)\dagger}(x)\delta_{a}^{\;\mu}\varUpsilon^{a}i\partial_{\mu}\Psi_{Q_{N}^{(q_{c},k)}\pm}^{(j)}(x),$
(224)

with $a,\mu=0,j$.

The action for a general string-like hyperqubit-spinor field possesses the
following associated symmetry:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,1}\ltimes SO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times
SP(D_{q_{c}}-2+8k)\times SP(q_{c})$ (225) $\displaystyle=$ $\displaystyle
SC(1)\ltimes PO(1,1)\,\widetilde{\Join}\,SP(1,1)\rtimes SG(1)\times
SP(D_{q_{c}}-2+8k)\times SP(q_{c}),$

with $D_{q_{c}}=3,4,6$ for $q_{c}=1,2,3$.
SP$(D_{q_{c}}-2+8k)\times$SP($q_{c}$) represents the maximal internal symmetry
for self-conjugated chiral-like hyperqubit-spinor field in category-$q_{c}$
and $k$-th period.

Therefore, we can always construct a self-conjugated chiral-like qubit-spinor
field $\Psi_{Q_{N}^{(q_{c},k)}\pm}(x)$ from a general qubit-spinor field in
category-$q_{c}$ with categoric qubit number $Q_{N}^{(q_{c},k)}$, so that it
moves freely only in two-dimensional spacetime. Such a self-conjugated chiral-
like qubit-spinor field $\Psi_{Q_{N}^{(q_{c},k)}\pm}(x)$ is referred to as
string-like qubit-spinor field.

###  VII.2  Self-conjugated chiral hyperqubit-spinor field and full anti-
commuting $\Gamma$-matrices in an extended Hilbert space

When the dimension ${\mathcal{D}}_{H}$ of Hilbert space spanned by product
qubit basis states is larger than the dimension $D_{h}$ of Minkowski spacetime
determined from the maximally correlated motion of qubit-spinor field, the
motion correlation $\varGamma$-matrices are not all satisfying anti-commuting
relations, which indicates that such a Hilbert space formed by product qubit
basis states is not large enough. To make all $\varGamma$-matrices becoming
anti-commuting, it is appropriate to consider self-conjugated chiral qubit-
spinor structure for qubit-spinor field represented in an enlarged Hilbert
space. A self-conjugated chiral qubit-spinor field can generally be
constructed from self-conjugated qubit-spinor field with one additional qubit
number in chiral qubit-spinor representation so as to keep the same
independent degrees of freedom with equal ${\mathcal{M}}_{c}$-spin charge and
${\mathcal{Q}}_{c}$-spin charge.

Let us begin with examining the tetraqubit-spinor field. As shown in Eq.(V.2),
not all $\Gamma$-matrices $\Gamma^{\mathsf{A}}$ defined in Eq.(V.1) are anti-
commuting. The tetraqubit-spinor field in 16-dimensional Hilbert space belongs
to a qubit-spinor representation in vector space of eight dimensions, i.e.,
${\mathcal{D}}_{H}=2^{D/2}=16$ with $D=8$. Whereas its maximally correlated
motion brings on ten-dimensional Minkowski hyper-spacetime $D_{h}=10$. It
naturally inspires us to represent such a tetraqubit-spinor field with 16
independent degrees of freedom in a qubit-spinor representation of vector
space with the same dimension as that of Minkowski hyper-spacetime, i.e.,
$D=D_{h}$.

It is straightforward to construct a self-conjugated chiral hyperqubit-spinor
field $\Psi_{\mathsf{Q}^{5}-}(x)$ in terms of self-conjugated chiral-like
hyperqubit-spinor fields $\Psi_{\mathsf{Q}^{4}\pm}(x)$ defined in Eq.(VII.1),
its explicit form is given as follows:

$\displaystyle\Psi_{\mathsf{Q}^{5}-}(x)=\binom{\Psi_{\mathsf{Q}^{4}-}(x)}{\Psi_{\mathsf{Q}^{4}+}(x)}\equiv\varGamma_{-}\Psi_{\mathsf{Q}^{5}}(x),$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\gamma_{\mathsf{Q}^{5}}),\quad\gamma_{\mathsf{Q}^{5}}=\sigma_{3}\otimes\Upsilon^{9},$
(226)

where the 16 independent degrees of freedom of tetraqubit-spinor field are
represented as a self-conjugated chiral hyperqubit-spinor field with chiral
qubit-spinor representation defined in 10 dimensions $D=10$. The action in
Eq.(211) can be rewritten into the following form:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{5}-}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{S}}_{\mathsf{Q}^{4}}=\int
d^{10}x\,\frac{1}{2}\bar{\Psi}_{\mathsf{Q}^{5}-}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}^{5}-}(x),$
(227)

with
$\bar{\Psi}_{\mathsf{Q}^{5}-}(x)=\Psi^{\dagger}_{\mathsf{Q}^{5}-}\varGamma^{0}$.
The $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{A}},\varGamma^{10})$
($\tilde{A}=1,2,3,5,\cdots,9$) are explicitly given as follows:

$\displaystyle\varGamma^{0}=\;\sigma_{2}\otimes\Upsilon^{0},\qquad\varGamma^{10}=-i\sigma_{1}\otimes\Upsilon^{0},$
$\displaystyle\varGamma^{\tilde{A}}=-i\sigma_{3}\otimes\Upsilon^{\bar{A}},\quad\tilde{A}=1,2,3=\bar{A},$
$\displaystyle\varGamma^{\tilde{A}}=-i\sigma_{3}\otimes\Upsilon^{\bar{A}+1},\;\;\tilde{A}=5,\cdots,9=\bar{A}+1,$
(228)

where $\Upsilon^{0}$ and $\Upsilon^{\bar{A}}$ ($\bar{A}=1,\cdots,8$) are
presented in Eq.(V.1).

Similarly, the action in Eq.(212) can be expressed into the following form:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{5}-}$
$\displaystyle\equiv$
$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{4}}=\int
d^{10}x\,\frac{1}{2}\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x),$
(229)

with
$\bar{\Psi}_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)=\Psi^{\dagger}_{\mathsf{Q}_{\mathsf{S}}^{5}-}\varGamma^{0}$.
The $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{10},\varGamma^{\tilde{A}})$
($\tilde{A}=1,2,3,5,\cdots,9$) get the following structure:

$\displaystyle\varGamma^{0}=\sigma_{1}\otimes\Gamma^{0},\quad\varGamma^{10}=-i\sigma_{2}\otimes\Gamma^{0},$
$\displaystyle\varGamma^{\tilde{A}}=\sigma_{0}\otimes\Gamma^{\tilde{A}},\quad\tilde{A}=1,2,3,5,\cdots,9,$
(230)

where $\Gamma^{0}$ and $\Gamma^{\tilde{A}}$ ($\tilde{A}=1,2,3,5,\cdots,9$) are
defined in Eq.(V.1). $\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)$ is prsented as a
self-conjugated chiral hyperqubit-spinor field with the following explicit
form:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)=\binom{\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}-}(x)}{\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}+}(x)}\equiv\varGamma_{-}\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}}(x),$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\gamma_{\mathsf{Q}_{\mathsf{S}}^{5}}),\quad\gamma_{\mathsf{Q}_{\mathsf{S}}^{5}}=\sigma_{3}\otimes\Gamma^{0},$
(231)

where $\Psi_{\mathsf{Q}_{\mathsf{S}}^{4}\pm}(x)$ are the self-conjugated
chiral-like hyperqubit-spinor fields defined in Eq.(VII.1).

It can be verified that both $\Psi_{\mathsf{Q}^{5}-}(x)$ and
$\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)$ satisfy the following self-conjugated
chiral conditions:

$\displaystyle\Psi^{c}_{\mathsf{Q}^{5}-}(x)=C_{\mathsf{Q}^{5}-}\bar{\Psi}^{T}_{\mathsf{Q}^{5}-}(x)=\Psi_{\mathsf{Q}^{5}-}(x),\quad\gamma_{\mathsf{Q}^{5}}\Psi_{\mathsf{Q}^{5}-}(x)=-\Psi_{\mathsf{Q}^{5}-}(x),$
$\displaystyle\Psi^{c}_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)=C_{\mathsf{Q}_{\mathsf{S}}^{5}-}\bar{\Psi}^{T}_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)=\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x),\quad\gamma_{\mathsf{Q}_{\mathsf{S}}^{5}}\Psi_{\mathsf{Q}^{5}-}(x)=-\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x),$
$\displaystyle C_{\mathsf{Q}^{5}-}=\varGamma^{0},\quad
C_{\mathsf{Q}_{\mathsf{S}}^{5}-}=\sigma_{1}\otimes
C_{\mathsf{Q}_{\mathsf{S}}^{4}}=\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2},$
(232)

with $C_{\mathsf{Q}_{\mathsf{S}}^{4}}$ defined in Eq.(V.2).

The self-conjugated chiral hyperqubit-spinor fields
$\Psi_{\mathsf{Q}^{5}-}(x)$ and $\Psi_{\mathsf{Q}_{\mathsf{S}}^{5}-}(x)$ are
referred to as self-conjugated chiral pentaqubit-spinor fields. In such a
chiral qubit-spinor representation, all $\varGamma$-matrices
$\varGamma^{\mathsf{A}}$ defined in Eqs.(VII.2) and (VII.2) become anti-
commuting,

$\displaystyle\\{\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}\\}=2\eta^{\mathsf{A}\mathsf{B}},$
(233)

with $\eta^{\mathsf{A}\mathsf{B}}=\mathop{\rm diag}.(1,-1,\cdots,-1)$. Thus
the group generators of hyperspin symmetry SP(1,9) can simply be given by the
commutators of $\varGamma$-matrices,

$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],$
(234)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,10$.

The above analysis can be generalized to build the action for any hyperqubit-
spinor field in category-$q_{c}$. For the general hyperqubit-spinor field in
category-$0$, the action can be rewritten into the following form:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(1)}-}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{S}}_{Q_{N}^{(0)}}=\int
d^{D_{h}^{(0,k)}}x\,\frac{1}{2}\bar{\Psi}_{Q_{N}^{(1,k)}-}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(1,k)}-}(x),$
(235)

with
$\bar{\Psi}_{Q_{N}^{(1,k)}-}(x)=\Psi^{\dagger}_{Q_{N}^{(1,k)}-}\varGamma^{0}$.
Where $\Psi_{Q_{N}^{(1,k)}-}(x)$ defines a self-conjugated chiral hyperqubit-
spinor field with the following form:

$\displaystyle\Psi_{Q_{N}^{(1,k)}-}(x)=\binom{\Psi_{Q_{N}^{(0,k)}-}(x)}{\Psi_{Q_{N}^{(0,k)}+}(x)}\equiv\varGamma_{-}\Psi_{Q_{N}^{(1,k)}}(x),$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\gamma_{D_{h}^{(0,k)}+1}),\quad\gamma_{D_{h}^{(0,k)}+1}=\sigma_{3}\otimes\varUpsilon^{9+8(k-1)},$
(236)

where $\Psi_{Q_{N}^{(0,k)}\pm}(x)$ are the so-called self-conjugated chiral-
like hyperqubit-spinor field in category-0 defined in Eq.(218), and
$\Psi_{Q_{N}^{(1,k)}}(x)$ is a hyperqubit-spinor field in category-1. The
$\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{A}+8j},\varGamma^{10+8(k-1)})$
($\tilde{A}=1,2,3,5,\cdots,9$; $j=1,\cdots,k-1$; $k=1,2,\cdots,$) have the
following explicit structure:

$\displaystyle\varGamma^{0}=\;\sigma_{2}\otimes\varUpsilon^{0},$
$\displaystyle\varGamma^{\tilde{A}+8j}=-i\sigma_{3}\otimes\varUpsilon^{\bar{A}+8j},$
$\displaystyle\varGamma^{10+8(k-1)}=-i\sigma_{1}\otimes\varUpsilon^{0},$ (237)

where $\varUpsilon^{0}$, $\varUpsilon^{9+8(k-1)}$ and
$\varUpsilon^{\bar{A}+8j}$ ($\bar{A}=1,\cdots,8$) are presented in Eq.(VI.6).

Analogously, for the general hyperqubit-spinor field in category-$1$, its
action can be expressed as follows:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(2)}-}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{S}}_{Q_{N}^{(1)}}=\int
d^{D_{h}^{(1,k)}}x\,\\{\frac{1}{2}\bar{\Psi}_{Q_{N}^{(2,k)}-}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(2,k)}-}(x)$
(238) $\displaystyle-$
$\displaystyle\frac{1}{2}\lambda_{1}\phi_{1}(x)\bar{\Psi}_{Q_{N}^{(2,k)}-}(x)\tilde{\varGamma}\Psi_{Q_{N}^{(2,k)}-}(x)\\},$

with
$\bar{\Psi}_{Q_{N}^{(2,k)}-}(x)=\Psi^{\dagger}_{Q_{N}^{(2,k)}-}\varGamma^{0}$.
$\Psi_{Q_{N}^{(2,k)}-}(x)$ is defined as a self-conjugated chiral hyperqubit-
spinor field,

$\displaystyle\Psi_{Q_{N}^{(2,k)}-}(x)=\binom{0}{\Psi_{Q_{N}^{(1,k)}}(x)}\equiv\varGamma_{-}\Psi_{Q_{N}^{(2,k)}}(x),$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\gamma_{D_{h}^{(1,k)}+1}),\quad\gamma_{D_{h}^{(1,k)}+1}=\sigma_{3}\otimes\sigma_{0}\otimes\varUpsilon^{0},$
(239)

where $\Psi_{Q_{N}^{(2,k)}}(x)$ is a hyperqubit-spinor field in category-2.
The $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{A}+8j},\varGamma^{10+8(k-1)},\varGamma^{11+8(k-1)},\tilde{\varGamma})$
($\tilde{A}=1,2,3,5,\cdots,9$; $j=1,\cdots,k-1$; $k=1,\cdots,$) get the
following structure:

$\displaystyle\varGamma^{0}=\;\sigma_{1}\otimes\sigma_{2}\otimes\varUpsilon^{0},$
(240)
$\displaystyle\varGamma^{\tilde{A}+8j}=i\sigma_{1}\otimes\sigma_{1}\otimes\varUpsilon^{\bar{A}+8j},$
$\displaystyle\varGamma^{10+8(k-1)}=i\sigma_{1}\otimes\sigma_{1}\otimes\varUpsilon^{9+8(k-1)},$
$\displaystyle\varGamma^{11+8(k-1)}=-i\sigma_{1}\otimes\sigma_{3}\otimes\varUpsilon^{0},$
$\displaystyle\tilde{\varGamma}=-i\sigma_{2}\otimes\sigma_{0}\otimes\varUpsilon^{0}.$

Similarly, we are able to rewrite the action for the general hyperqubit-spinor
field in category-$2$ as follows:

$\displaystyle{\mathcal{S}}_{Q_{N}^{(3)}-}$ $\displaystyle\equiv$
$\displaystyle{\mathcal{S}}_{Q_{N}^{(2)}}=\int
d^{D_{h}^{(2,k)}}x\,\\{\frac{1}{2}\bar{\Psi}_{Q_{N}^{(3,k)}-}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\varGamma^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{Q_{N}^{(3,k)}-}(x)$
(241) $\displaystyle-$
$\displaystyle\frac{1}{2}\lambda_{2}\tilde{\phi}_{p}(x)\bar{\Psi}_{Q_{N}^{(3,k)}-}(x)\tilde{\varGamma}^{p}\Psi_{Q_{N}^{(3,k)}-}(x)\\},$

with
$\bar{\Psi}_{Q_{N}^{(3,k)}-}(x)=\Psi^{\dagger}_{Q_{N}^{(3,k)}-}\varGamma^{0}$.
$\Psi_{Q_{N}^{(3,k)}-}(x)$ is defined as a self-conjugated chiral hyperqubit-
spinor field,

$\displaystyle\Psi_{Q_{N}^{(3,k)}-}(x)=\binom{0}{\Psi_{Q_{N}^{(2,k)}}(x)}\equiv\varGamma_{-}\Psi_{Q_{N}^{(3,k)}}(x),$
$\displaystyle\varGamma_{-}=\frac{1}{2}(1-\gamma_{D_{h}^{(2,k)}+1}),\quad\gamma_{D_{h}^{(2,k)}+1}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\varUpsilon^{0},$
(242)

where $\Psi_{Q_{N}^{(3,k)}}(x)$ is a hyperqubit-spinor field in category-3.
The $\varGamma$-matrices
$\varGamma^{\mathsf{A}}=(\varGamma^{0},\varGamma^{\tilde{A}+8j},\varGamma^{10+8(k-1)},\varGamma^{11+8(k-1)},\varGamma^{12+8(k-1)}$)
($\tilde{A}=1,2,3,5,\cdots,9$; $j=1,\cdots,k-1$; $k=1,2,\cdots,$) and
$\tilde{\varGamma}_{p}$ ($p=1,2$) have the following form:

$\displaystyle\varGamma^{0}=\;\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\varUpsilon^{0},$
$\displaystyle\varGamma^{\tilde{A}+8j}=i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\varUpsilon^{\bar{A}+8j},$
$\displaystyle\varGamma^{10+8(k-1)}=i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\varUpsilon^{9+8(k-1)},$
$\displaystyle\varGamma^{11+8(k-1)}=i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\varUpsilon^{0},$
$\displaystyle\varGamma^{12+8(k-1)}=i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}\otimes\varUpsilon^{0},$
$\displaystyle\tilde{\varGamma}^{1}=i\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\varUpsilon^{0},$
$\displaystyle\tilde{\varGamma}^{2}=i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\varUpsilon^{0}.$
(243)

It can be verified that $\Psi_{Q_{N}^{(q_{c},k)}-}(x)$ ($q_{c}=1,2,3$)
satisfies the following self-conjugated chiral condition:

$\displaystyle\Psi^{c}_{Q_{N}^{(q_{c},k)}-}(x)=C_{Q_{N}^{(q_{c},k)}}\bar{\Psi}^{T}_{Q_{N}^{(q_{c},k)}-}(x)=\Psi_{Q_{N}^{(q_{c},k)}-}(x),\quad
C_{Q_{N}^{(q_{c},k)}}=\varGamma^{0},$
$\displaystyle\gamma_{D_{h}^{(q_{c}-1,k)}+1}\Psi_{Q_{N}^{(q_{c},k)}-}(x)=-\Psi_{Q_{N}^{(q_{c},k)}-}(x)=\gamma_{D_{h}^{(q_{c}-1,k)}+1}\Psi^{c}_{Q_{N}^{(q_{c},k)}-}(x),$
(244)

where $\Psi_{Q_{N}^{(q_{c},k)}-}(x)$ ($q_{c}=1,2,3$) is referred to as self-
conjugated chiral hyperqubit-spinor field in category-($q_{c}$-1) and $k$-th
period. In such a self-conjugated chiral qubit-spinor representation, all
$\varGamma$-matrices become anti-commuting,

$\displaystyle\\{\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}\\}=2\eta^{\mathsf{A}\mathsf{B}},\quad\\{\varGamma^{\mathsf{A}},\tilde{\varGamma}^{p}\\}=0,\quad\\{\tilde{\varGamma}^{p},\tilde{\varGamma}^{q}\\}=2\eta^{pq},$
(245)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,D_{h}^{(q_{c}-1,k)}$ and
$p,q=1,\cdots,q_{c}-1$. The group generators of hyperspin symmetry SP(1,
$D_{h}^{(q_{c}-1,k)}$-1) and ${\mathcal{Q}}_{c}$-spin symmetry SP($q_{c}-1$)
are directly defined by the commutators of $\varGamma$-matrices as follows:

$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\varGamma^{\mathsf{A}},\varGamma^{\mathsf{B}}],\quad\tilde{\Sigma}^{pq}=\frac{i}{4}[\tilde{\varGamma}^{p},\tilde{\varGamma}^{p}].$
(246)

Therefore, when the categoric hyperqubit-spinor field
$\Psi_{Q_{N}^{(q_{c},k)}}(x)$ with categoric qubit number $Q_{N}^{(q_{c},k)}$
in category-$q_{c}$ and $k$-th period is represented by a self-conjugated
chiral hyperqubit-spinor field $\Psi_{Q_{N}^{(q_{c}+1,k)}-}(x)$ constructed
from the hyperqubit-spinor field $\Psi_{Q_{N}^{(q_{c}+1,k)}}(x)$ in
category-($q_{c}$+1) with one additional qubit number
$Q_{N}^{(q_{c}+1,k)}=Q_{N}^{(q_{c},k)}+1$, all $\varGamma$-matrices that
reflect both ${\mathcal{M}}_{c}$-spin charge of hyperspin symmetry and
${\mathcal{Q}}_{c}$-spin charge of ${\mathcal{Q}}_{c}$-spin symmetry become
full anti-commuting. The categoric hyperqubit-spinor field
$\Psi_{Q_{N}^{(q_{c},k)}}(x)$ and self-conjugated chiral hyperqubit-spinor
field $\Psi_{Q_{N}^{(q_{c}+1,k)}-}(x)$ are defined in Hilbert space with
dimensions ${\mathcal{D}}_{H}=2^{Q_{N}^{(q_{c},k)}}$ and
${\mathcal{D}}_{H}=2^{Q_{N}^{(q_{c},k)}+1}$, respectively, but they contain
the same independent degrees of freedom with equal ${\mathcal{M}}_{c}$-spin
charge and ${\mathcal{Q}}_{c}$-spin charge.

###  VII.3  Massless Dirac fermion as self-conjugated chiral tetraqubit-spinor
field in category-$3$ with full anti-commuting ${\mathcal{M}}_{c}$-matrices
and ${\mathcal{Q}}_{c}$-matrices in 6D spacetime

The actions in Eqs.(98) and (110) indicate that a massless Dirac fermion can
be described by a self-conjugated triqubit-spinor field
$\psi_{\mathsf{Q}_{\mathsf{S}}^{3}}(x)$ given in Eq.(IV.5) or equivalently by
a self-conjugated triqubit-spinor field
$\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)$ presented in Eq.(IV.7) via a unitary
transformation. Nevertheless, in such qubit-spinor representations, the
$\Gamma$-matrices $\Gamma^{\hat{a}}$ presented in Eq.(IV.5) or Eq.(IV.7) are
not all anti-commuting, which motivates us to consider an extended qubit-
spinor representation with self-conjugated chiral spinor structure. In fact,
leptons and quarks in the electroweak interaction of SM behave as Weyl
fermions with a chiral spinor structure in four-dimensional spacetime.

The self-conjugated triqubit-spinor field
$\psi_{\mathsf{Q}_{\mathsf{S}}^{3}}(x)$ can generally be rewritten into the
sum of two self-conjugated chiral triqubit-spinor fields as follows:

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{S}}^{3}}(x)=\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)+\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x),$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}(x)=\hat{\Gamma}_{\pm}\psi_{\mathsf{Q}_{\mathsf{S}}^{3}}(x),$
$\displaystyle\hat{\Gamma}_{\pm}=\frac{1}{2}(1\pm\hat{\gamma}_{7}),\quad\hat{\gamma}_{7}=\sigma_{3}\otimes\gamma_{5},$
(247)

with the definitions,

$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)\equiv\binom{\psi_{\mathsf{Q}_{\mathsf{C}}^{2}-}(x)}{\psi_{\mathsf{Q}_{\mathsf{C}}^{2}-}^{\;\bar{c}}(x)}\equiv\binom{\psi_{\mathsf{Q}_{\mathsf{W}-}^{2}}(x)}{\psi_{\mathsf{Q}_{\mathsf{W}-}^{2}}^{\;\bar{c}}(x)},$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)\equiv\binom{\psi_{\mathsf{Q}_{\mathsf{C}}^{2}+(x)}}{\psi_{\mathsf{Q}_{\mathsf{C}}^{2}+}^{\;\bar{c}}(x)}\equiv\binom{\psi_{\mathsf{Q}_{\mathsf{W}+}^{2}}(x)}{\psi_{\mathsf{Q}_{\mathsf{W}+}^{2}}^{\;\bar{c}}(x)},$
(248) $\displaystyle\psi_{\mathsf{Q}_{\mathsf{C}}^{2}\pm}^{\;\bar{c}}\equiv
C_{D}\bar{\psi}_{\mathsf{Q}_{\mathsf{C}}^{2}\pm}^{T},\quad
C_{D}=-i\sigma_{3}\otimes\sigma_{2},$

where $\psi_{\mathsf{Q}_{\mathsf{C}}^{2}-}(x)$ and
$\psi_{\mathsf{Q}_{\mathsf{C}}^{2}+}(x)$ are chiral complex biqubit-spinor
fields,

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{C}}^{2}\pm}\equiv\gamma_{\pm}\psi_{\mathsf{Q}_{\mathsf{C}}^{2}}\equiv\psi_{\mathsf{Q}_{\mathsf{W}\pm}^{2}}\equiv\gamma_{\pm}\psi_{D},$
$\displaystyle\gamma_{\pm}=\frac{1}{2}(1\pm\gamma_{5}),\quad\gamma_{5}=\sigma_{3}\otimes\sigma_{0},$
(249)

which are regarded as the left-handed and right-handed Weyl fermions defined
from Dirac fermion in the spinor representation of four dimensions,

In light of self-conjugated chiral triqubit-spinor fields
$\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}(x)$ which satisfy the following
self-conjugated chiral conditions:

$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}^{c}(x)=C_{\mathsf{Q}_{\mathsf{S}}^{3}}\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}^{T}(x)=\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}(x),$
$\displaystyle\hat{\gamma}_{7}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}(x)=\hat{\gamma}_{7}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}^{c}(x)=\pm\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}\pm}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{S}}^{3}}=-i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{2},$
(250)

the action in Eq.(87) can be rewritten into the following form in six-
dimensional spacetime:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{3}}$ $\displaystyle=$
$\displaystyle\int
d^{6}x\,\frac{1}{2}[\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)+\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)$
(251) $\displaystyle+$
$\displaystyle\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)\delta_{k}^{\;m}\Gamma^{k}i\partial_{m}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)+\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)\delta_{k}^{\;m}\Gamma^{k}i\partial_{m}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)],$

with $a,\mu=0,1,2,3$ and $k,m=5,6$. Where the correlation motions in the fifth
and sixth spatial dimensions can only appear between two self-conjugated
chiral triqubit-spinor fields $\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)$
and $\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)$.

By introducing an alternative qubit-spinor field
$\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}$ with satisfying the following
self-conjugated chiral conditions:

$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}=\binom{\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}}{\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}},$
$\displaystyle\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}^{c}(x)=C_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}}\bar{\psi}^{T}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}(x)=\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-},$
$\displaystyle\hat{\gamma}_{9}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}=\hat{\gamma}_{9}\psi^{c}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}=-\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-},\quad\hat{\gamma}_{9}=\sigma_{3}\otimes\hat{\gamma}_{7},$
$\displaystyle C_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}}=\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{S}}^{3}}=-i\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{2},$
(252)

we can express the above action into the following simple form:

$\displaystyle{\mathcal{S}}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}\equiv{\mathcal{S}}_{\mathsf{Q}_{\mathsf{S}}^{3}}$
$\displaystyle=$ $\displaystyle\int
d^{6}x\,\frac{1}{2}\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}(x)\delta_{\hat{a}}^{\;\hat{\mu}}\varGamma^{\hat{a}}i\partial_{\hat{\mu}}\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}(x),$
(253)

with
$\bar{\psi}_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}(x)=\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}^{\dagger}(x)\varGamma^{0}$.
We may refer to $\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{4}-}(x)$ as self-
conjugated chiral tetraqubit-spinor field. In such a chiral spinor
representation, the $\varGamma$-matrices
$\varGamma^{\hat{a}}\equiv(\varGamma^{a},\varGamma^{k})$ ($a=0,1,2,3$;
$k=5,6$) get following explicit form:

$\displaystyle\varGamma^{a}=\sigma_{0}\otimes\sigma_{0}\otimes\gamma^{a},$
$\displaystyle\varGamma^{5}=i\sigma_{2}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle\varGamma^{6}=i\sigma_{1}\otimes\sigma_{3}\otimes\gamma_{5},$
(254)

with $\gamma^{a}$ defined in Eq.(IV.5). The ${\mathcal{Q}}_{c}$-matrices
$\tilde{\Gamma}^{p}=\varGamma^{6+p}$($p=1,2,3$) are found to be,

$\displaystyle\varGamma^{7}=i\sigma_{1}\otimes\sigma_{1}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{1},$
$\displaystyle\varGamma^{8}=i\sigma_{1}\otimes\sigma_{2}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{2},$
$\displaystyle\varGamma^{9}=i\sigma_{3}\otimes\sigma_{0}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{3}.$
(255)

It can be verified that all $\Gamma$-matrices become anti-commuting,

$\displaystyle\\{\Gamma^{\hat{a}},\Gamma^{\hat{b}}\\}=2\eta^{\hat{a}\hat{b}},\quad\\{\tilde{\Gamma}^{p},\tilde{\Gamma}^{q}\\}=2\eta^{pq},\quad\\{\Gamma^{\hat{a}},\tilde{\Gamma}^{p}\\}=0.$
(256)

The group generators of ${\mathcal{M}}_{c}$-spin symmetry SP(1,5) and
${\mathcal{Q}}_{c}$-spin symmetry SP(3) can simply be given by the commutators
of $\varGamma$-matrices,

$\displaystyle\Sigma^{\hat{a}\hat{b}}=\frac{i}{4}[\varGamma^{\hat{a}},\varGamma^{\hat{b}}],\quad\tilde{\Sigma}^{pq}=\frac{i}{4}[\tilde{\varGamma}^{p},\tilde{\varGamma}^{q}],$
(257)

where ${\mathcal{Q}}_{c}$-spin symmetry SP(3)$\cong$SU(2) reflects an U(1)
charge-spin between the chiral complex biqubit-spinor field
$\psi_{\mathsf{Q}_{\mathsf{C}}^{2}\pm}(x)$ and its charge conjugated one
$\psi_{\mathsf{Q}_{\mathsf{C}}^{2}\pm}^{\;\bar{c}}$ as well as a rotational-
spin property between two self-conjugated chiral triqubit-spinor fields
$\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}-}(x)$ and
$\psi_{\hat{\mathsf{Q}}_{\mathsf{S}}^{3}+}(x)$.

##  VIII  Entangled hyperqubit-spinor field from locally entangled-qubits
motion postulate and inhomogeneous hyperspin symmetry as basic symmetry of
lepton-quark states

It is shown that in light of self-conjugated chiral qubit-spinor structure,
all $\Gamma$-matrices become anti-commuting and the group generators of
${\mathcal{M}}_{c}$-spin and ${\mathcal{Q}}_{c}$-spin symmetries are directly
presented by the commutators of $\Gamma$-matrices, which provides an
appropriate qubit-spinor representation in Hilbert space for building the
action of qubit-spinor field. In this section, we are going to investigate
further the intrinsic property of self-conjugated chiral qubit-spinor field,
which allows us to extend the motion-correlation ${\mathcal{M}}_{c}$-spin
symmetry to be an inhomogeneous ${\mathcal{M}}_{c}$-spin symmetry in
association with inhomogeneous Lorentz-type/Poincaré-type group symmetry. In
particular, we are led to make a locally entangled-qubits motion postulate by
which a locally entangled state of qubits is proposed to be as an entangled
qubit-spinor field, which enables us to show how inhomogeneous hyperspin
symmetry of entangled hyperqubit-spinor field reflect basic symmetries of
lepton-quark states.

###  VIII.1 Entangled state in quantum theory and locally entangled-qubits
motion postulate

It is known in quantum physics that multiple qubits can exhibit quantum
entanglement, which is regarded as the distinguishing feature between a qubit
and a bit. For instance, the entanglement of quantum state of two qubits can
be denoted either by Dirac “bra-ket” notation or by spin-like “column-row”
notation as follows:

$\displaystyle|\Psi_{E}\rangle=\alpha_{00}|0\rangle\otimes|0\rangle+\alpha_{11}|1\rangle\otimes|1\rangle\equiv\alpha_{00}|00\rangle+\alpha_{11}|11\rangle,$
$\displaystyle\Psi_{E}=\alpha_{++}\binom{1}{0}\otimes\binom{1}{0}+\alpha_{--}\binom{0}{1}\otimes\binom{0}{1}=\begin{pmatrix}\alpha_{++}\\\
0\\\ 0\\\ \alpha_{--}\end{pmatrix},$
$\displaystyle|\alpha_{00}|^{2}+|\alpha_{11}|^{2}=1,\quad|\alpha_{++}|^{2}+|\alpha_{--}|^{2}=1,$
(258)

where $|\alpha_{00}|^{2}$ or $|\alpha_{++}|^{2}$ and $|\alpha_{11}|^{2}$ or
$|\alpha_{--}|^{2}$ are probabilities of measuring $|0\rangle\otimes|0\rangle$
or $\binom{1}{0}\otimes\binom{1}{0}$ and $|1\rangle\otimes|1\rangle$ or
$\binom{0}{1}\otimes\binom{0}{1}$, respectively. For
$\alpha_{00}=\alpha_{11}=1/{\sqrt{2}}$ or
$\alpha_{++}=\alpha_{--}=1/{\sqrt{2}}$, it is called equal superposition with
equal probability $|1/{\sqrt{2}}|^{2}=1/2$.

In general, an entangled system is considered to be the system in which its
quantum state cannot be factored as a product of pure states.

For a more formal analysis, let us consider two noninteracting systems $S$ and
$S^{\prime}$ with Hilbert spaces HS and H${}_{S^{\prime}}$, respectively. The
Hilbert space of their composite system is defined by the tensor product
H${}_{S}\otimes$H${}_{S^{\prime}}$. If two systems are in states $\Psi_{S}$
and $\Psi_{S^{\prime}}$, the state of the composite system is given by the
tensor product $\Psi_{S}\otimes\Psi_{S^{\prime}}$. If states of the composite
system can be represented in such a form in the tensor product, they are
called product states or separable states. Whereas not all states can be
expressed as product states or separable states.

Given a basis $\\{\xi_{i}\\}$ for HS and a basis $\\{\xi^{\prime}_{i}\\}$ for
H${}_{S^{\prime}}$, a general state in H${}_{S}\otimes$H${}_{S^{\prime}}$ can
be expressed as follows:

$\displaystyle\Psi=\sum_{i,j}\psi_{ij}\,\xi_{i}\otimes\xi^{\prime}_{j},$

where $\Psi$ is called separable if

$\displaystyle\psi_{ij}=\psi_{i}\,\psi^{\prime}_{j},\quad\Psi=\psi\otimes\psi^{\prime},$
$\displaystyle\psi=\sum_{i}\psi_{i}\,\xi_{i},\quad\psi^{\prime}=\sum_{j}\psi^{\prime}_{j}\,\xi^{\prime}_{j},$

with $\psi$ and $\psi^{\prime}$ representing two vectors.

When $\Psi$ becomes inseparable if

$\displaystyle\psi_{ij}\neq\psi_{i}\,\psi^{\prime}_{j},\quad\Psi\neq\psi\otimes\psi^{\prime},$

hold for all vectors $\psi_{i}$ and $\psi^{\prime}_{j}$.

If a state is inseparable, it is referred to as entangled state. As an
example, let us consider the simplest case with two basis states
$\varsigma_{s}=\\{|0\rangle,|1\rangle\\}$ or
$\varsigma_{s}=\\{\binom{1}{0},\binom{0}{1}\\}$ of Hilbert space HS and two
basis states $\varsigma_{s^{\prime}}=\\{|0\rangle,|1\rangle\\}$ or
$\varsigma_{s^{\prime}}=\\{\binom{1}{0},\binom{0}{1}\\}$ of Hilbert space
H${}_{S^{\prime}}$, an entangled state can be constructed as

$\displaystyle|\psi_{E_{-}}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle\otimes|1\rangle+|1\rangle\otimes|0\rangle\right)\equiv\frac{1}{\sqrt{2}}\left(|01\rangle+|10\rangle\right)$

with the Dirac “bra-ket” notation, which can also be expressed equivalently by
using the spin-like “column-row” notation as follows:

$\displaystyle\psi_{E_{-}}=\frac{1}{\sqrt{2}}\left(\binom{1}{0}\otimes\binom{0}{1}+\binom{0}{1}\otimes\binom{1}{0}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
1\\\ 1\\\ 0\end{pmatrix},$ (259)

which is actually one of four Bell states. In general, Bell states are
maximally entangled pure states in Hilbert space
H${}_{S}\otimes$H${}_{S^{\prime}}$, but they cannot be separated into pure
states of each HS and H${}_{S^{\prime}}$.

Let us make a detailed discussion on the maximally entangled pure states in
H${}_{S}\otimes$H${}_{S^{\prime}}$ based on the real basic qubit-basis
$\\{\varsigma_{s}\\}$ defined in Eq.(35). Applying for 2-product qubit-basis
states $\varsigma_{s_{1}s_{2}}$ defined in Eq.(64), we can express a biqubit-
spinor state into the following general form:

$\displaystyle\Psi=\sum_{s_{1}s_{2}}\,\psi_{s_{1}s_{2}}\varsigma_{s_{1}s_{2}}\equiv\sum_{s_{1}s_{2}}\psi_{s_{1}s_{2}}\,\varsigma_{s_{1}}\otimes\varsigma_{s_{2}}=\begin{pmatrix}\psi_{++}\\\
\psi_{+-}\\\ \psi_{-+}\\\ \psi_{--}\end{pmatrix},$ (260)

with $s_{1},s_{2}=\pm$. where $\psi_{s_{1}s_{2}}$ denote four constant
amplitudes. The four Bell states as maximally entangled pure states in such a
spinor representation can be read from the above biqubit-spinor state with the
following conventions:

$\displaystyle\psi_{E_{\pm}}=\hat{\gamma}_{\pm}\Psi,\quad\psi_{E^{\prime}_{\pm}}=\gamma_{5}\hat{\gamma}_{\pm}\Psi,\quad\psi_{s_{1}s_{2}}=\frac{1}{\sqrt{2}},$
$\displaystyle\hat{\gamma}_{\pm}=\frac{1}{2}(1\pm\hat{\gamma}_{5}),\quad\hat{\gamma}_{5}=\sigma_{3}\otimes\sigma_{3},\quad\gamma_{5}=\sigma_{3}\otimes\sigma_{0},$
(261)

where $\psi_{E_{\pm}}$ and $\psi_{E^{\prime}_{\pm}}$ are regarded as chiral
biqubit-spinor states.

To be more explicit, let us write down four Bell states in the spinor
representation as follows:

$\displaystyle\psi_{E_{+}}=\psi_{++}\varsigma_{+}\otimes\varsigma_{+}+\psi_{--}\varsigma_{-}\otimes\varsigma_{-}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
0\\\ 0\\\ 1\end{pmatrix},$
$\displaystyle\psi_{E_{-}}=\psi_{-+}\varsigma_{-}\otimes\varsigma_{+}+\psi_{+-}\varsigma_{+}\otimes\varsigma_{-}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
1\\\ 1\\\ 0\end{pmatrix},$
$\displaystyle\psi_{E^{\prime}_{+}}=\psi_{++}\varsigma_{+}\otimes\varsigma_{+}-\psi_{--}\varsigma_{-}\otimes\varsigma_{-}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
0\\\ 0\\\ -1\end{pmatrix},$
$\displaystyle\psi_{E^{\prime}_{-}}=\psi_{-+}\varsigma_{-}\otimes\varsigma_{+}-\psi_{+-}\varsigma_{+}\otimes\varsigma_{-}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
1\\\ -1\\\ 0\end{pmatrix}.$ (262)

So that the general entangled states can be represented as follows:

$\displaystyle\Psi_{E_{+}}=\hat{\gamma}_{+}\Psi=\frac{1}{\sqrt{2}}(\psi_{++}+\psi_{--})\,\psi_{E_{+}}+\frac{1}{\sqrt{2}}(\psi_{++}-\psi_{--})\,\psi_{E^{\prime}_{+}}=\begin{pmatrix}\psi_{++}\\\
0\\\ 0\\\ \psi_{--}\end{pmatrix},$
$\displaystyle\Psi_{E_{-}}=\hat{\gamma}_{-}\Psi=\frac{1}{\sqrt{2}}(\psi_{+-}+\psi_{-+})\,\psi_{E_{-}}+\frac{1}{\sqrt{2}}(\psi_{+-}+\psi_{-+})\,\psi_{E^{\prime}_{-}}=\begin{pmatrix}0\\\
\psi_{+-}\\\ \psi_{-+}\\\ 0\end{pmatrix},$
$\displaystyle|\psi_{++}|^{2}+|\psi_{--}|^{2}=1,\quad|\psi_{+-}|^{2}+|\psi_{-+}|^{2}=1,$
(263)

which indicates that a general entangled state for two qubits can be expressed
as a superposition of Bell states and meanwhile constructed by a chiral qubit-
spinor state with global amplitudes constrained by the total probability.

From the above analyses, we come to the statement that by adopting qubit basis
states in the spinor representation, an entangled state in quantum physics can
formally be expressed into a chiral qubit-spinor state. Therefore, a general
entangled state in quantum physics can be regarded as an entangled state of
qubits with the probability amplitudes satisfying the normalization condition
constrained from the total probability.

It is noticed that all above analyses and discussions about entangled state as
an inseparable state in the tensor product Hilbert space are actually
applicable to local states of qubits rather than only global states. Namely,
they should hold for both global and local coherent states of qubits.

Let us now turn to investigate local coherent state of qubits as qubit-spinor
field. It is demonstrated in the previous section that a self-conjugated
chiral qubit-spinor field becomes more appropriate to build the action, so
that all $\Gamma$-matrices satisfy the anticommutative relations. Meanwhile,
the group algebra of ${\mathcal{M}}_{c}$-spin and ${\mathcal{Q}}_{c}$-spin
symmetries can simply be formed from the commutators of $\Gamma$-matrices. In
analogous to the above analyses and discussions on entangled states in quantum
physics as global coherent states of qubits, we come to a similar statement
that a self-conjugated chiral qubit-spinor field is conceptually regarded as
an entangled qubit-spinor field. For that, we are led to make the following
postulate.

Locally entangled-qubits motion postulate: a locally entangled state of qubits
as a column vector field of local distribution amplitudes obeying the maximum
coherence motion principle brings on an entangled qubit-spinor field which is
postulated to be as fundamental building block of nature. Such an entangled
qubit-spinor field can generally be expressed as a self-conjugated chiral
qubit-spinor field.

###  VIII.2  Majorana-Weyl type fermions as entangled qubit-spinor fields in
category-2 $\And$ category-3 and the inhomogeneous ${\mathcal{M}}_{c}$-spin
symmetries associating with inhomogeneous Lorentz group symmetries in four-
and six-dimensional Minkowski spacetimes

The actions in Eqs.(110) and (117) indicate that the self-conjugated triqubit-
spinor field $\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)$ in category-3 defined in
Eq.(IV.7) characterizes a massless Dirac fermion $\psi_{D}(x)$ in six-
dimensional spacetime. Whereas in such a triqubit-spinor representation, the
eight-dimensional Hilbert space spanned by 3-product qubit basis states is not
large enough to make all $\Gamma$-matrices anti-commuting. As shown in
Eq.(253), it is more appropriate to extend eight-dimensional Hilbert space to
16-dimensional Hilbert space spanned by 4-product qubit basis states with a
chiral structure so as to keep the same independent degrees of freedom.

Let us begin with introducing entangled triqubit-spinor fields
$\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x)$, which are defined by the following
self-conjugated chiral triqubit-spinor fields
$\psi_{\mathbf{Q}_{\mathsf{S}}^{3}\pm}(x)$:

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)\equiv\psi_{\mathbf{Q}_{\mathsf{S}}^{3}-}(x)\equiv\Gamma_{-}\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)=\binom{\psi_{L}(x)}{\psi_{L}^{\;\bar{c}}(x)},$
$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)\equiv\psi_{\mathbf{Q}_{\mathsf{S}}^{3}+}(x)\equiv\Gamma_{+}\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)=\binom{\psi_{R}(x)}{\psi_{R}^{\;\bar{c}}(x)},$
$\displaystyle\Gamma_{\pm}=\frac{1}{2}(1\pm\hat{\gamma}_{7}),\quad\hat{\gamma}_{7}=\sigma_{3}\otimes\gamma_{5},$
(264)

where $\psi_{L}$ and $\psi_{R}$ are the usual left-handed and right-handed
Weyl fermions defined by the Dirac fermion $\psi_{D}(x)$ as follows:

$\displaystyle\psi_{L,R}=\gamma_{\mp}\psi_{D},\quad\gamma_{\pm}=\frac{1}{2}(1\pm\gamma_{5}),$
$\displaystyle\psi_{L,R}^{\;\bar{c}}=C_{D}\bar{\psi}_{L,R}^{T},\quad
C_{D}=-i\sigma_{3}\otimes\sigma_{2},$ (265)

with $\psi_{L,R}^{\;\bar{c}}$ being complex charge-conjugated ones. In
general, $\psi_{D}$ can be regarded as complex biqubit-spinor field
$\psi_{D}\equiv\psi_{\mathsf{Q}_{\mathsf{C}}^{2}}$, and $\psi_{L,R}$ are
viewed as chiral complex biqubit-spinor fields
$\psi_{L,R}\equiv\psi_{\mathbf{Q}_{\mathsf{W}\pm}^{2}}$.

The self-conjugated triqubit-spinor field
$\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)$ can be expressed as the superposition
of two entangled triqubit-spinor fields, i.e.:

$\displaystyle\psi_{\mathbf{Q}_{\mathsf{S}}^{3}}(x)\equiv\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)+\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)=\binom{\psi_{D}(x)}{\psi_{D}^{\bar{c}}(x)},$
(266)

where $\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x)$ satisfy the following self-
conjugated chiral conditions:

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{3}}\bar{\psi}_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}^{T}(x)=\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x),\quad
C_{\mathsf{Q}_{\mathsf{E}}^{3}}=\sigma_{1}\otimes C_{D},$
$\displaystyle\hat{\gamma}_{7}\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x)=\pm\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x),\quad\hat{\gamma}_{7}\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}^{c}(x)=\pm\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}^{c}(x).$
(267)

Such defined entangled triqubit-spinor fields
$\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x)$ are also considered as Majorana-
Weyl type fermions. To be distinguishable with ordinary left-handed and right-
handed Weyl fermions defined in four-dimensional spacetime via the project
operators $\gamma_{\mp}=\frac{1}{2}(1\mp\gamma_{5})\equiv\gamma_{L,R}$, the
entangled qubit-spinor fields defined with the project operators
$\Gamma_{\mp}=\frac{1}{2}(1\mp\hat{\gamma}_{D+1})$ in high dimensional
spacetime are referred to as westward qubit-spinor field and eastward qubit-
spinor field, respectively. So that, $\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)$
is called as westward entangled triqubit-spinor field and
$\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)$ as eastward entangled triqubit-spinor
field.

In terms of the westward and eastward entangled triqubit-spinor fields
$\psi_{\mathsf{Q}_{\mathsf{E}\pm}^{3}}(x)$, the actions in Eqs.(117) and (113)
in category-3 can be rewritten into the following form:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{4}}\equiv{\mathcal{S}}_{\mathbf{Q}_{\mathsf{S}}^{3}}$
$\displaystyle=$ $\displaystyle\int
d^{6}x\,\frac{1}{2}[\bar{\psi}_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)+\bar{\psi}_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)\delta_{a}^{\;\mu}\Gamma^{a}i\partial_{\mu}\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)]$
(268) $\displaystyle+$
$\displaystyle\bar{\psi}_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)\delta_{k}^{\;m}\Gamma^{k}i\partial_{m}\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)+\bar{\psi}_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)\delta_{k}^{\;m}\Gamma^{k}i\partial_{m}\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)],$
$\displaystyle\equiv$ $\displaystyle\int
d^{6}x\,\bar{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)\delta_{\hat{a}}^{\;\hat{\mu}}\Sigma_{-}^{\hat{a}}i\partial_{\hat{\mu}}\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x),$

with $a,\mu=0,1,2,3$, $\,k,m=5,6$ and $\hat{a},\hat{\mu}=0,1,2,3,5,6$ labeling
six-dimensional spacetime. We have introduced in the second equality the
matrices $\Sigma_{-}^{\hat{a}}$ defined by $\Gamma$-matrices as follows:

$\displaystyle\Sigma_{-}^{\hat{a}}=\frac{1}{2}\Gamma^{\hat{a}}\Gamma_{-},\quad\Gamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{9}),$
$\displaystyle\hat{\gamma}_{9}=\sigma_{3}\otimes\hat{\gamma}_{7}=\sigma_{3}\otimes\sigma_{3}\otimes\gamma_{5}.$
(269)

Where $\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}$ represents a westward entangled
qubit-spinor field satisfying the following self-conjugated chiral conditions:

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}=\binom{\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}}{\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}}=\begin{pmatrix}\psi_{L}(x)\\\
\psi_{L}^{\;\bar{c}}(x)\\\ \psi_{R}(x)\\\
\psi_{R}^{\;\bar{c}}(x)\end{pmatrix}\equiv\Gamma_{-}\psi_{\mathbf{Q}_{\mathsf{S}}^{4}},$
$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{4}}\bar{\psi}^{T}_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)=\psi_{\mathsf{Q}_{\mathsf{E}}^{4}},$
$\displaystyle\hat{\gamma}_{9}\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}=-\psi_{\mathsf{Q}_{\mathsf{E}}^{4}},\quad\hat{\gamma}_{9}\psi^{c}_{\mathsf{Q}_{\mathsf{E}}^{4}}=-\psi^{c}_{\mathsf{Q}_{\mathsf{E}}^{4}},$
$\displaystyle C_{\mathsf{Q}_{\mathsf{E}}^{4}}=\sigma_{0}\otimes
C_{\mathsf{Q}_{\mathsf{E}}^{3}}=-i\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{2}.$
(270)

For convenience, $\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}$ is referred to as
entangled tetraqubit-spinor field, which is also regarded as Majorana-Weyl
type fermion.

In such a westward spinor representation, both $\Gamma$-matrices
$\Gamma^{\hat{a}}\equiv(\Gamma^{a},\Gamma^{k})$ ($a=0,1,2,3$; $k=5,6$) and
${\mathcal{Q}}_{c}$-matrices $\tilde{\Gamma}^{p}$ ($p=1,2,3$) can be expressed
as the following forms:

$\displaystyle\Gamma^{a}=\sigma_{0}\otimes\sigma_{0}\otimes\gamma^{a},$
$\displaystyle\Gamma^{5}=i\sigma_{1}\otimes\sigma_{3}\otimes\gamma_{5},$
$\displaystyle\Gamma^{6}=i\sigma_{2}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle\Gamma^{7}=i\sigma_{1}\otimes\sigma_{1}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{1},$
$\displaystyle\Gamma^{8}=i\sigma_{1}\otimes\sigma_{2}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{2},$
$\displaystyle\Gamma^{9}=i\sigma_{3}\otimes\sigma_{0}\otimes\gamma_{5}\equiv\tilde{\Gamma}^{3},$
(271)

with $\gamma^{a}$ ($a=0,1,2,3$) defined in Eq.(IV.7). In terms of the
entangeled tetraqubit-spinor field $\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}$, it is
easy to check that there exists no scalar coupling term in associating to
$\tilde{\Gamma}^{p}$ matrices. This is because the matrices $\Gamma^{\hat{a}}$
and $\tilde{\Gamma}^{p}$ have different commutation relations with
$\hat{\gamma}_{9}$, i.e.:

$\displaystyle[\tilde{\Gamma}^{p},\hat{\gamma}_{9}]=0,\quad\\{\Gamma^{\hat{a}},\hat{\gamma}_{9}\\}=0.$
(272)

It is noticed from the action in Eq.(268) that the correlation motion in the
fifth and sixth spatial dimensions can occur only between two entangled
triqubit-spinor fields $\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)$ and
$\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)$, which indicates that each entangled
triqubit-spinor field $\psi_{\mathsf{Q}_{\mathsf{E}+}^{3}}(x)$ (or
$\psi_{\mathsf{Q}_{\mathsf{E}-}^{3}}(x)$) has a free motion only in four-
dimensional spacetime. So that the action of entangled triqubit-spinor field
$\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)$ can present an equivalent action for
category-2 biqubit-spinor field in four-dimensional spacetime, i.e.:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{3}}$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}x\,\\{\bar{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)\delta_{a}^{\;\mu}\Sigma_{-}^{a}i\partial_{\mu}\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)-\lambda_{2}\phi_{p}(x)\bar{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)\tilde{\Sigma}_{-}^{p}\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)$
(273) $\displaystyle+$
$\displaystyle\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi_{p}(x)\partial_{\nu}\phi^{p}(x)-\frac{1}{4}\lambda_{Q}(\phi_{p}(x)\phi^{p}(x))^{2}\\},$

with $a,\mu=0,1,2,3$ and $p=1,2$. Where $\Sigma_{-}^{a}$ and
$\tilde{\Sigma}_{-}^{p}$ matrices are given by the $\Gamma$-matrices
$\Gamma^{\hat{a}}$ through the following explicit forms:

$\displaystyle\Sigma_{-}^{a}=\frac{1}{2}\Gamma^{a}\Gamma_{-},\quad\tilde{\Sigma}_{-}^{p}=\frac{1}{2}\tilde{\Gamma}^{p}\Gamma_{-},$
$\displaystyle\Gamma^{a}=\sigma_{0}\otimes\gamma^{a},$
$\displaystyle\Gamma^{5}=i\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0}\equiv\tilde{\Gamma}^{1},$
$\displaystyle\Gamma^{6}=i\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0}\equiv\tilde{\Gamma}^{2},$
$\displaystyle\Gamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{7}),$ (274)

with $\gamma^{a}$ ($a=0,1,2,3$) given in Eq.(IV.7).

In the entangled qubit-spinor representation, all $\Gamma$-matrices given in
Eqs.(VIII.2) and (VIII.2) become anti-commuting,

$\displaystyle\\{\Gamma^{\hat{a}},\Gamma^{\hat{b}}\\}=2\eta^{\hat{a}\hat{b}},\quad\\{\Gamma^{\hat{a}},\tilde{\Gamma}^{p}\\}=0,\quad\\{\tilde{\Gamma}^{p},\tilde{\Gamma}^{q}\\}=2\eta^{pq},$
(275)

with $\hat{a},\hat{b}=0,1,2,3,5,6$, $p,q=1,2,3$, and

$\displaystyle\\{\Gamma^{a},\Gamma^{b}\\}=2\eta^{ab},\quad\\{\Gamma^{a},\tilde{\Gamma}^{p}\\}=0,\quad\\{\tilde{\Gamma}^{p},\tilde{\Gamma}^{q}\\}=2\eta^{pq},$
(276)

with $a,b=0,1,2,3$, $p,q=1,2$.

The commutators of $\Gamma$-matrices provide directly the group generators of
extended ${\mathcal{M}}_{c}$-spin symmetry, i.e.:

$\displaystyle\Sigma^{\hat{a}\hat{b}}=\frac{i}{4}[\Gamma^{\hat{a}},\Gamma^{\hat{b}}],\quad\Sigma_{-}^{\hat{a}}=\frac{1}{2}\Gamma^{\hat{a}}\Gamma_{-},\quad\Gamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{9}),$
(277)

which satisfy the following group algebra:

$\displaystyle[\Sigma^{\hat{a}\hat{b}},\Sigma^{\hat{c}\hat{d}}]=i(\Sigma^{\hat{a}\hat{d}}\eta^{\hat{b}\hat{c}}-\Sigma^{\hat{b}\hat{d}}\eta^{\hat{a}\hat{c}}-\Sigma^{\hat{a}\hat{c}}\eta^{\hat{b}\hat{d}}+\Sigma^{\hat{b}\hat{c}}\eta^{\hat{a}\hat{d}}),$
$\displaystyle[\Sigma^{\hat{a}\hat{b}},\Sigma_{-}^{\hat{c}}]=i(\Sigma_{-}^{\hat{a}}\eta^{\hat{b}\hat{c}}-\Sigma_{-}^{\hat{b}}\eta^{\hat{a}\hat{c}}),\quad[\Sigma_{-}^{\hat{a}},\Sigma_{-}^{\hat{b}}]=0.$
(278)

A similar group algebra holds for $\Sigma^{ab}$ and $\Sigma_{-}^{a}$,

$\displaystyle\Sigma^{ab}=\frac{i}{4}[\Gamma^{a},\Gamma^{b}],\quad\Sigma_{-}^{a}=\frac{1}{2}\Gamma^{a}\Gamma_{-},\quad\Gamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{7}).$
(279)

The group algebra for the group generators $\Sigma^{ab}$ or
$\Sigma^{\hat{a}\hat{b}}$ characterize the motion-correlation
${\mathcal{M}}_{c}$-spin symmetry SP(1,$D_{h}$-1) for $D_{h}=4$ or $D_{h}=6$.
The commutating group algebra for the group generators $\Sigma_{-}^{a}$ or
$\Sigma_{-}^{\hat{a}}$ brings on an Abelian-type group symmetry for the action
of entangled qubit-spinor field.

Let us study explicitly such an Abelian-type group symmetry for both westward
entangled triqubit-spinor field and westward entangled tetraqubit-spinor
field. They transform under the group operation as follows:

$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)\to\psi^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)=S(\varpi)\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x),\quad
S(\varpi)=e^{i\varpi_{a}\Sigma_{-}^{a}/2},$
$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)\to\psi^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)=S(\varpi)\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x),\quad
S(\varpi)=e^{i\varpi_{\hat{a}}\Sigma_{-}^{\hat{a}}/2},$ (280)

with the explicit forms,

$\displaystyle\psi^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)=\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(w)}(x)+\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(e)}(x),\quad
S(\varpi)=1+i\varpi_{a}\Sigma_{-}^{a}/2,$
$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(w)}(x)\equiv\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x),\quad\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(e)}(x)\equiv
i\frac{1}{2}\varpi_{a}\Sigma_{-}^{a}\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x),$
(281)

and

$\displaystyle\psi^{\prime}_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)=\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(w)}(x)+\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(e)}(x),\quad
S(\varpi)=1+i\varpi_{\hat{a}}\Sigma_{-}^{\hat{a}}/2,$
$\displaystyle\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(w)}(x)\equiv\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x),\quad\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(e)}(x)\equiv
i\frac{1}{2}\varpi_{\hat{a}}\Sigma_{-}^{\hat{a}}\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x),$
(282)

where $\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(e)}(x)$ and
$\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(e)}(x)$ represent the shifted
new parts of entangled qubit-spinor fields after such Abelian-type group
transformations. The superscripts ‘$w$’ and ‘$e$’ on the entangled qubit-
spinor fields label their westward and eastward chirality properties,
respectively,

$\displaystyle\hat{\gamma}_{7}\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(w)}(x)=-\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(w)}(x),\quad\hat{\gamma}_{7}\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(e)}(x)=+\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{3}}^{(e)}(x),$
$\displaystyle\hat{\gamma}_{9}\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(w)}(x)=-\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(w)}(x),\quad\hat{\gamma}_{9}\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(e)}(x)=+\tilde{\psi}_{\mathsf{Q}_{\mathsf{E}}^{4}}^{(e)}(x).$
(283)

Such an Abelian-type group symmetry reflects a translation-like spin group
symmetry of entangled qubit-spinor field in Hilbert space. It is noted that
the shifted new part of entangled qubit-spinor field gets a sign flip in
chirality as shown in Eq.(VIII.2). Namely, when the entangled qubit-spinor
field is westward, its shifted new part after the transformation becomes
eastward, which holds the other way round.

This enlarged translation-like Abelian spin group symmetry is analogous to the
translation group symmetry P${}^{1,D_{h}-1}$ of coordinates in Minkowski
spacetime. For convenience, we may use the notation W${}^{1,D_{h}-1}$ to
denote such a translation-like Abelian spin group symmetry since it brings on
a sign flip in chirality for the shifted new part after such a group
transformation. As such a translation-like Abelian spin group symmetry
W${}^{1,D_{h}-1}$ arises initially from the locally entangled-qubits motion
postulate, it is regarded as an entanglement-correlated Abelian-type spin
group symmetry in the entangled qubit-spinor representation of Hilbert space.

As such an entanglement-correlated translation-like spin group symmetry
W${}^{1,D_{h}-1}$ is associated with the sign flip in chirality of westward
and eastward entangled qubit-spinor fields for the initial and shifted new
parts, we may refer to W${}^{1,D_{h}-1}$ as ${\mathcal{W}}_{e}$-spin symmetry
for short. The extended ${\mathcal{M}}_{c}$-spin symmetry is referred to as
inhomogeneous ${\mathcal{M}}_{c}$-spin symmetry which is denoted as follows:

$\displaystyle WS(1,D_{h}-1)\equiv SP(1,D_{h}-1)\rtimes W^{1,D_{h}-1},$ (284)

where the symbol ‘$\rtimes$’ is used to indicate that the inhomogeneous
${\mathcal{M}}_{c}$-spin symmetry WS(1,$D_{h}$-1) is a semidirect product
group with ${\mathcal{M}}_{c}$-spin symmetry group SP(1, $D_{h}$-1) and
${\mathcal{W}}_{e}$-spin symmetry group W${}^{1,D_{h}-1}$.

The group generators of ${\mathcal{Q}}_{c}$-spin symmetry is simply given by

$\displaystyle\Sigma^{pq}=\frac{i}{4}[\Gamma^{D_{h}+p},\Gamma^{D_{h}+q}]=\frac{i}{4}[\tilde{\Gamma}^{p},\tilde{\Gamma}^{q}]\equiv\tilde{\Sigma}^{pq},$
(285)

with ${\mathcal{Q}}_{c}$-spin charges
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=2$ and
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=3$ in correspondence to the entangled
qubit-spinor fields $\psi_{\mathsf{Q}_{\mathsf{E}}^{3}}(x)$ and
$\psi_{\mathsf{Q}_{\mathsf{E}}^{4}}(x)$.

The actions in Eqs.(273) and (268) possess the following inhomogeneous
${\mathcal{M}}_{c}$-spin symmetry and ${\mathcal{Q}}_{c}$-spin symmetry:

$\displaystyle G_{S}=WS(1,D_{h}-1)\times SP(q_{c})\equiv SP(1,D_{h}-1)\rtimes
W^{1,D_{h}-1}\times SP(q_{c}),$ (286)

with $D_{h}=4$ and $D_{h}=6$ for $q_{c}=2$ and $q_{c}=3$, respectively. Such
an inhomogeneous ${\mathcal{M}}_{c}$-spin symmetry is in association with
inhomogeneous Lorentz-type/Poincaré-type group symmetry in Minkowski
spacetime,

$\displaystyle PO(1,D_{h}-1)=P^{1,D_{h}-1}\ltimes SO(1,D_{h}-1),$

which is a semidirect product group symmetry with P${}^{1,D_{h}-1}$
representing the translation group symmetry of coordinates in Minkowski
spacetime.

In general, the maximal symmetry of the actions in Eqs.(273) and (268) can be
expressed as the following associated symmetry with coincidental
transformations:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
P^{1,D_{h}-1}\ltimes SO(1,D_{h}-1)\,\widetilde{\Join}\,SP(1,D_{h}-1)\rtimes
W^{1,D_{h}-1}\times SG(1)\times SP(q_{c})$ (287) $\displaystyle=$
$\displaystyle SC(1)\ltimes
PO(1,D_{h}-1)\,\widetilde{\Join}\,WS(1,D_{h}-1)\times SG(1)\times SP(q_{c}),$

with $D_{h}=4,6$ for $q_{c}=2,3$, respectively. Where the symbol
“$\,\widetilde{\Join}\,$” is used to indicate the associated symmetry in which
the transformation of ${\mathcal{M}}_{c}$-spin symmetry SP(1,$D_{h}$-1) in
Hilbert space must be coincidental to that of isomorphic Lorentz-type group
symmetry SO(1,$D_{h}$-1) in globally flat Minkowski spacetime. SP($q_{c}$)
represents the ${\mathcal{Q}}_{c}$-spin symmetry and
SC(1)$\,\widetilde{\Join}\,$SG(1) denotes the associated symmetry for global
scaling transformations.

###  VIII.3  Entangled pentaqubit-spinor field as entangled hyperqubit-spinor
field in category-0 and inhomogeneous hyperspin symmetry WS(1,9) as unified
spin-color symmetry of chiral lepton-quark state in 10D hyper-spacetime

A massless Dirac fermion as triqubit-spinor field in category-3 is shown to be
characterized by an entangled tetraqubit-spinor field with inhomogeneous
${\mathcal{M}}_{c}$-spin symmetry WS(1,5) in six-dimensional spacetime. Based
on the locally entangled-qubits motion postulate, let us further examine an
entangled pentaqubit-spinor field constructed from a chiral spinor structure
of tetraqubit-spinor field in Category-0 with zero ${\mathcal{Q}}_{c}$-spin
charge $\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=0$ in ten-dimensional hyper-
spacetime. Such an entangled pentaqubit-spinor field is verified to unify
chiral spin-color lepton-quark states into an entangled hyperqubit-spinor
field.

In light of chiral qubit-spinor structure of tetraqubit-spinor field, we are
able to construct a self-conjugated chiral pentaqubit-spinor field as follows:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)}{\Psi^{\,\bar{c}}_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)},$
(288)

where $\Psi_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)$ is a chiral complex tetraqubit-
spinor filed with $\Psi^{\;\;\bar{c}}_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)$ as the
complex charge-conjugated one. They have the following explicit forms:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)\equiv\begin{pmatrix}\psi_{\mathbf{Q}_{\mathsf{W}++}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{W}+-}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{W}-+}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{W}--}^{2}}\end{pmatrix}\equiv\begin{pmatrix}Q^{r}_{L}(x)\\\
Q^{b}_{L}(x)\\\ Q^{g}_{L}(x)\\\ Q^{w}_{L}(x)\end{pmatrix},$
$\displaystyle\Psi^{\,\bar{c}}_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)\equiv\begin{pmatrix}\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{W}++}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{W}+-}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{W}-+}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{W}--}^{2}}\end{pmatrix}\equiv\begin{pmatrix}Q^{r\,\bar{c}}_{L}(x)\\\
Q^{b\,\bar{c}}_{L}(x)\\\ Q^{g\,\bar{c}}_{L}(x)\\\
Q^{w\,\bar{c}}_{L}(x)\end{pmatrix},$ (289)

with the definitions,

$\displaystyle\Psi^{\,\bar{c}}_{\mathbf{Q}_{\mathsf{W}}^{4}}(x)=C_{\mathbf{Q}_{\mathsf{W}}^{4}}\bar{\Psi}^{\,T}_{\mathbf{Q}_{\mathsf{W}}^{4}}(x),\quad
C_{\mathbf{Q}_{\mathsf{W}}^{4}}=\sigma_{0}\otimes\sigma_{0}\otimes C_{D},$
$\displaystyle\psi_{\mathbf{Q}_{\mathsf{W}s_{1}s_{2}}^{2}}\equiv
Q^{\alpha}_{L}(x)\equiv\gamma_{-}Q^{\alpha}(x),\quad\gamma_{-}=\frac{1}{2}(1-\gamma_{5}),$
$\displaystyle\psi_{\mathbf{Q}_{\mathsf{W}s_{1}s_{2}}^{2}}^{\bar{c}}\equiv
Q^{\alpha\,\bar{c}}_{L}(x)=C_{D}\bar{Q}^{\alpha\;T}_{L}(x),\quad
C_{D}=-i\sigma_{3}\otimes\sigma_{2},$ $\displaystyle
s_{1},s_{2}=\pm,\quad\alpha=\\{s_{1}s_{2}\\}\equiv(r,b,g,w),$ (290)

where $Q^{\alpha}_{L}(x)$ with $\alpha=r,b,g,w$ represent the left-handed
Dirac fermions with unified four color-spin charges in correspondence to the
‘red’, ‘blue’, ‘green’ and ‘white’. The first three color-spin charges ‘red’,
‘blue’ and ‘green’ characterize three chromo-quarks and the fourth color-spin
charge ‘white’ reflects a relevant lepton. $Q^{\alpha}_{L}(x)$ and
$Q^{\alpha\,\bar{c}}_{L}(x)$ are defined as left-handed lepton-quark state and
its complex charge-conjugated one. Each $Q^{\alpha}_{L}(x)$ is regarded as a
chiral complex biqubit-spinor field
$\psi_{\mathbf{Q}_{\mathsf{W}s_{1}s_{2}}^{2}}$.

Based on the locally entangled-qubits motion postulate, the self-conjugated
chiral pentaqubit-spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)$ is
referred to as entangled pentaqubit-spinor field, its action in ten-
dimensional hyper-spacetime is built to be as follows:

$\displaystyle{\mathcal{S}}_{\mathsf{Q}_{\mathsf{E}}^{5}}$ $\displaystyle=$
$\displaystyle\int
d^{10}x\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)\delta_{\mathsf{A}}^{\;\mathsf{M}}\Sigma_{-}^{\mathsf{A}}i\partial_{\mathsf{M}}\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x),$
(291)

where $\Sigma_{-}^{\mathsf{A}}$ ($\mathsf{A}=0,1,2,3,5,\cdots,10$) matrices
are defined with $\Gamma$-matrices
$\Gamma^{\mathsf{A}}=(\Gamma^{a},\Gamma^{A})$ ($a=0,1,2,3,A=5,\cdots,10$),
which have the following explicit forms:

$\displaystyle\Sigma_{-}^{\mathsf{A}}=\frac{1}{2}\Gamma^{\mathsf{A}}\Gamma_{-},\quad\Gamma_{-}=\frac{1}{2}(1-\hat{\gamma}_{11}),$
$\displaystyle\hat{\gamma}_{11}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
$\displaystyle\Gamma^{0}=\,\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{1}\otimes\sigma_{0},$
$\displaystyle\Gamma^{1}=i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1},$
$\displaystyle\Gamma^{2}=i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{2},$
$\displaystyle\Gamma^{3}=i\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3},$
$\displaystyle\Gamma^{5}=i\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\Gamma^{6}=i\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\Gamma^{7}=i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\Gamma^{8}=i\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{0}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\Gamma^{9}=i\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{3}\otimes\sigma_{0},$
$\displaystyle\Gamma^{10}=i\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{0},$
(292)

where all $\Gamma$-matrices become anti-commuting,

$\displaystyle\\{\Gamma^{\mathsf{A}},\Gamma^{\mathsf{B}}\\}=2\eta^{\mathsf{A}\mathsf{B}}.$
(293)

In general, $\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)$ is also referred to as
entangled hyperqubit-spinor field for chiral lepton-quark state, which
satisfies the following self-conjugated chiral conditions:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}^{c}(x)=C_{\mathsf{Q}_{\mathsf{E}}^{5}}\bar{\Psi}_{\mathsf{Q}_{\mathsf{E}}^{5}}^{T}(x)=\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x),$
$\displaystyle\hat{\gamma}_{11}\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)=\hat{\gamma}_{11}\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}^{c}(x)=-\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x),$
$\displaystyle
C_{\mathsf{Q}_{\mathsf{E}}^{5}}=\sigma_{1}\otimes\sigma_{0}\otimes\sigma_{0}\otimes
C_{D},$ (294)

It can be verified in the entangled pentaqubit-spinor representation that
besides the $\Gamma$-matrices $\Gamma^{\mathsf{A}}$
($\mathsf{A}=0,1,2,3,5,\cdots,10$) and $\hat{\gamma}_{11}$, there exists no
any other anti-commuting $\Gamma$-matrix that can bring on scalar coupling
term with the entangled pentaqubit-spinor field
$\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)$. Therefore, the entangled pentaqubit-
spinor field $\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)$ has zero
${\mathcal{Q}}_{c}$-spin charge $\mathcal{C}_{Q_{c}}=q_{c}=0$ and the action
in Eq.(291) gets no scalar coupling.

The action in Eq.(291) possesses an associated symmetry in which the
inhomogeneous hyperspin symmetry WS(1,9) is in association with inhomogeneous
Lorentz-type/Poincaré-type group symmetry PO(1,9) in ten-dimensional Minkowski
hyper-spacetime. Such an associated symmetry is presented as follows:

$\displaystyle G_{S}$ $\displaystyle=$ $\displaystyle SC(1)\ltimes
PO(1,9)\,\widetilde{\Join}\,WS(1,9)\rtimes SG(1)\times SP(q_{c}=0)$ (295)
$\displaystyle=$ $\displaystyle SC(1)\ltimes P^{1,9}\ltimes
SO(1,9)\,\widetilde{\Join}\,SP(1,9)\rtimes W^{1,9}\rtimes SG(1),$

where we have used the symbol “$\,\widetilde{\Join}\,$” to indicate the
associated symmetry which requires the transformation of hyperspin symmetry
SP(1,9) be coincidental to that of Lorentz-type group symmetry SO(1,9) $\cong$
SP(1,9). The group generators of WS(1,9) can directly be presented by the
commutators of $\Gamma$-matrices:

$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}=\frac{i}{4}[\Gamma^{\mathsf{A}},\Gamma^{\mathsf{B}}],\quad\Sigma_{-}^{\mathsf{A}}=\frac{1}{2}\Gamma^{\mathsf{A}}\Gamma_{-},$
(296)

with $\mathsf{A},\mathsf{B}=0,1,2,3,5,\cdots,10$. Where
$\Sigma^{\mathsf{A}\mathsf{B}}$ represent the group generators of hyperspin
symmetry SP(1,9) and $\Sigma_{-}^{\mathsf{A}}$ the group generators of
${\mathcal{W}}_{e}$-spin symmetry W1,9.

The hyperspin symmetry SP(1,9) reflects a unified spin $\&$ color charge
symmetry for chiral lepton-quark state. To be more explicit, it is natural to
decompose the hyperspin symmetry SP(1,9) into the following subgroups:

$\displaystyle SP(1,9)$ $\displaystyle\supset$ $\displaystyle SP(1,3)\times
SP(6)\cong SO(1,3)\times SO(6)$ (297) $\displaystyle\cong$ $\displaystyle
SO(1,3)\times SU_{C}(4),$

with the corresponding group generators,

$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=0,1,2,3)\in
sp(1,3)\cong so(1,3)$
$\displaystyle\Sigma^{\mathsf{A}\mathsf{B}}(\mathsf{A},\mathsf{B}=5,6,7,8,9,10)\in
sp(6)\cong so(6)\cong su_{C}(4).$ (298)

Where the subgroup symmetry SP(1,3) $\cong$ SO(1,3) in connection with
ordinary four-dimensional Minkowski spacetime represents the usual spin
symmetry, and the subgroup symmetry SP(6)$\cong$SO(6) $\cong$ SUC(4) in
correlation with extra six spacial dimensions characterizes a unified four
color-spin charge symmetry of chiral lepton-quark state.

###  VIII.4 Entangled hexaqubit-spinor field in Category-1 as entangled
hyperqubit-spinor field and inhomogeneous hyperspin symmetry as unified spin-
color symmetry of lepton-quark state in 11D hyper-spacetime

It is known that quarks in the strong interaction of SM behave as vector-like
Dirac fermions, which indicates that the entangled pentaqubit-spinor field as
self-conjugated chiral pentaqubit-spinor field is not an appropriate entangled
hyperqubit-spinor field for characterizing quarks in SM. We should consider an
entangled hyperqubit-spinor field that contains both left-handed and right-
handed lepton-quark states.

Let us introduce an entangled hyperqubit-spinor field with the following
hyperqubit-spinor structure:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}}^{6}}(x)=\binom{\Psi_{\mathsf{Q}_{\mathsf{E}-}^{5}}(x)}{\Psi_{\mathsf{Q}_{\mathsf{E}+}^{5}}(x)},$
$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}-}^{5}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{W}-}^{4}}(x)}{\Psi^{\;\;\bar{c}}_{\mathbf{Q}_{\mathsf{W}-}^{4}}(x)},\quad\Psi_{\mathsf{Q}_{\mathsf{E}+}^{5}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{W}+}^{4}}(x)}{\Psi^{\;\;\bar{c}}_{\mathbf{Q}_{\mathsf{W}+}^{4}}(x)},$
(299)

where $\Psi_{\mathsf{Q}_{\mathsf{E}\pm}^{5}}(x)$ are entangled pentaqubit-
spinor fields defined as follows:

$\displaystyle\Psi_{\mathsf{Q}_{\mathsf{E}\pm}^{5}}(x)\equiv\Gamma_{\pm}\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x),$
$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)=\binom{\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)}{\Psi^{\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)},$
$\displaystyle\Gamma_{\pm}=\frac{1}{2}(1\pm\hat{\gamma}_{11}),\;\;\hat{\gamma}_{11}=\sigma_{3}\otimes\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5}$
(300)

with
$\Psi_{\mathsf{Q}_{\mathsf{E}-}^{5}}(x)\equiv\Psi_{\mathsf{Q}_{\mathsf{E}}^{5}}(x)$
the entangled pentaqubit-spinor field as shown in Eq.(288).
$\Psi_{\mathbf{Q}_{\mathsf{S}}^{5}}(x)$ is a self-conjugated pentaqubit-spinor
field in Category-1 with ${\mathcal{Q}}_{c}$-spin charge
$\mathcal{C}_{{\mathcal{Q}}_{c}}=q_{c}=1$, which is formed from the following
complex tetraqubit-spinor field:

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)=\begin{pmatrix}\psi_{\mathbf{Q}_{\mathsf{C}++}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{C}+-}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{C}-+}^{2}}\\\
\psi_{\mathbf{Q}_{\mathsf{C}--}^{2}}\end{pmatrix}\equiv\begin{pmatrix}Q^{r}(x)\\\
Q^{b}(x)\\\ Q^{g}(x)\\\ Q^{w}(x)\end{pmatrix},$
$\displaystyle\Psi^{\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)=\begin{pmatrix}\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{C}++}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{C}+-}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{C}-+}^{2}}\\\
\psi^{\bar{c}}_{\mathbf{Q}_{\mathsf{C}--}^{2}}\end{pmatrix}\equiv\begin{pmatrix}Q^{r\,\bar{c}}(x)\\\
Q^{b\,\bar{c}}(x)\\\ Q^{g\,\bar{c}}(x)\\\ Q^{w\,\bar{c}}(x)\end{pmatrix},$
$\displaystyle\psi_{\mathbf{Q}_{\mathsf{C}s1s2}^{2}}\equiv
Q^{\alpha}(x),\;\;s_{1},s_{2}=\pm,\;\;\alpha\equiv\\{s_{1}s_{2}\\}\equiv(r,b,g,w),$
$\displaystyle\psi_{\mathbf{Q}_{\mathsf{C}s1s2}^{2}}^{\bar{c}}\equiv
Q^{\alpha\,\bar{c}}(x)=C_{D}\bar{Q}^{\alpha\,T}(x)\equiv\sigma_{2}\otimes\sigma_{2}\psi_{\mathbf{Q}_{\mathsf{C}s1s2}^{2}}^{\ast},$
(301)

where the complex tetraqubit-spinor field
$\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$ is composed of complex biqubit-spinor
fields $\psi_{\mathbf{Q}_{\mathsf{C}s1s2}^{2}}$ that are regarded as Dirac
fermion states $Q^{\alpha}(x)$. Such Dirac fermion states $Q^{\alpha}(x)$
labeled with four color-spin charges ‘red’, ‘blue’, ‘green’ and ‘white’ form a
lepton-quark state. $Q^{\alpha\,\bar{c}}(x)$ is defined as the complex charge
conjugation state of lepton-quark state $Q^{\alpha}(x)$.

The complex tetraqubit-spinor field $\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)$
can generally be written as the sum of chiral complex tetraqubit-spinor fields
$\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{4}}(x)$,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)\equiv\Psi_{\mathbf{Q}_{\mathsf{W}-}^{4}}(x)+\Psi_{\mathbf{Q}_{\mathsf{W}+}^{4}}(x),$
(302)

where $\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{4}}(x)$ and their complex charge-
conjugated ones $\Psi^{\;\bar{c}}_{\mathbf{Q}_{\mathsf{W}\pm}^{4}}(x)$ are
given by,

$\displaystyle\Psi_{\mathbf{Q}_{\mathsf{W}\pm}^{4}}(x)\equiv\varGamma^{5}_{\pm}\Psi_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)=\begin{pmatrix}Q^{r}(x)\\\
Q^{b}(x)\\\ Q^{g}(x)\\\ Q^{w}(x)\end{pmatrix}_{R,L},$
$\displaystyle\Psi^{\;\bar{c}}_{\mathbf{Q}_{\mathsf{W}\pm}^{4}}(x)\equiv\varGamma^{5}_{\mp}\Psi^{\;\bar{c}}_{\mathbf{Q}_{\mathsf{C}}^{4}}(x)=\begin{pmatrix}Q^{r\,\bar{c}}(x)\\\
Q^{b\,\bar{c}}(x)\\\ Q^{g\,\bar{c}}(x)\\\
Q^{w\,\bar{c}}(x)\end{pmatrix}_{L,R},$
$\displaystyle\varGamma^{5}_{\pm}=\frac{1}{2}(1\pm\tilde{\gamma}_{5}),\quad\tilde{\gamma}_{5}=\sigma_{0}\otimes\sigma_{0}\otimes\gamma_{5},$
(303)

which are related to the left-handed and right-handed lepton-quark states
$Q^{\alpha}_{L,R}(x)$.

The action of entangled hyperqubit-spinor field