11institutetext: School of Physical Science and Technology, Northwestern
Polytechnical University, Xi’an 710072, China22institutetext: Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China33institutetext: School of Physical Sciences, University of Chinese
Academy of Sciences, 19A Yuquan Road, Beijing 100049, China44institutetext:
CCAST (World Laboratory), P.O. Box 8730, Beijing 100190, China55institutetext:
Department of Physics, Hebei University, Baoding 071002, China66institutetext:
School of Physics, Harbin Institute of Technology, Harbin 150001, China

# Mass spectra and wave functions of $T_{QQ\bar{Q}\bar{Q}}$ tetraquarks

Qiang Li 2,3,4 Chao-Hsi Chang 5 Guo-Li Wang 6 Tianhong Wang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The compact tetraquark states with fully heavy quark contents
$QQ\bar{Q}\bar{Q}$ are studied as the bound states of the diquark-antidiquark
within the Bethe-Salpeter framework. The (anti)diquark masses and form factors
used are the same as we calculated the doubly heavy baryons in a previous
work. Under the instantaneous approximation, the three-dimensional
(Bethe-)Salpeter equation of the tetraquarks is derived and solved numerically
to obtain the corresponding mass spectra and wave functions of the tetraquarks
with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$. Our results show that the three
ground states of $cc\bar{c}\bar{c}$ locate in the mass range of $6.4\sim
6.5\,$\mathrm{G}\mathrm{e}\mathrm{V}$$, and the $bb\bar{b}\bar{b}$ states in
mass range of $19.2\sim 19.3\,$\mathrm{G}\mathrm{e}\mathrm{V}$$. The obtained
relativistic wave functions naturally include the mixing effects from the
possible $D$ (or $G$) partial waves, and then can be further used to do
precise calculations of the tetraquark decays. Based on the obtained results,
the LHCb’s observation $X(6900)$ is less likely to be the ground states of
compact $cc\bar{c}\bar{c}$ tetraquarks but might be the first or second
excited states. In addition, a widely used propagator-like form factor is also
investigated and discussed.

##  1 Introduction

The quantum chromodynamics and the quark model allows not only the well-known
traditional hadrons, such as the $q\bar{q}$-type mesons and $qqq$-type
baryons, but also the exotic tetraquark states and the pentaquark baryons
GellMann1964 ; Zweig1964 . About 50 years after the predictions of these
exotic states, the LHCb Collaboration first detected the pentaquark baryons
LHCb2015-Pc ; LHCb2019-Pc , and then in 2020 reported a new narrow structure
labeled as $X(6900)$ which covers the predicted masses of states composed of
four charm quarks LHCb2020-X6900 .

Although not the first hint of the tetraquark states, $X(6900)$ causes great
attention in hadron physics for its fully heavy quark contents. Inspired by
this observation, the four-charm states around 6.9 GeV have been investigated
in several models or approaches ZhaoJX2020 ; Gordillo2020 ; Mutuk2021 ;
Giron2020 ; Lundhammar2020 ; DengCR2021 ; WengXZ2021 ; WangZG2020 ; LuQF2020 ;
Faustov2020 ; Bedolla2020 ; KeHW2021 ; ZhuRL2021 ; ChaoKT2020 ; Karliner2020 ,
such as, solving the two- or many-body time-independent Schrödinger equation
ZhaoJX2020 ; Gordillo2020 ; Mutuk2021 ; Giron2020 ; Lundhammar2020 , the
chromomagnetic interaction models DengCR2021 ; WengXZ2021 , the QCD sum rules
combined with the Regge trajectories WangZG2020 , the extended Godfrey and
Isgur (GI) quark model LuQF2020 , the relativistic quark model based on the
quasipotential approach or effective Hamiltonian Faustov2020 ; Bedolla2020 ,
and also the Bethe-Salpeter framework with different interaction kernels and
approximation methods KeHW2021 ; ZhuRL2021 . Also notice by using the heavy
diquark limit $M_{i}\to\infty$, Ref. ZhuRL2021  is in fact dealing with a
Schrödinger equation combined with the Regge trajectories to obtain the mass
spectra. Notice most of these previous studies are based on the
nonrelativistic Schrödinger equation or effective Hamiltonian methods while
the relativistic effects and the possible $S$-$D$ or $D$-$G$ mixing effects
are not included properly. In these previous studies, the ground states of the
$cc\bar{c}\bar{c}$ structures are predicted to be $6.1\sim 6.5$ GeV, and the
$X(6900)$ are usually tentatively identified as the radial or orbital excited
states of tetraquark $cc\bar{c}\bar{c}$ in these recent studies, or
interpreted as the coupled-channel or rescattering effects of two (or more)
charmonia GuoZH2021 ; DongXK2021 ; WangJZ2021 . Other interpretations, such as
the gluonic tetracharm WanBD2020  or light Higgs-like boson ZhuJW2020  are
also proposed. So far, there is still no strong evidences whether these
observed exotic hadrons are the genuine multiquark bound states or just the
loosely bound molecules of the traditional mesons and baryons.

The fully heavy tetraquark states have several advantages in both experimental
and theoretical researches. On the one hand, the mass of fully heavy
tetraquarks locates far away from those of the traditional mesons and doubly
heavy tetraquark $Q\bar{Q}q\bar{q}$ and they can be clearly identified from
the the known hadron spectra. On the other hand, the molecular states of two
charmonia can not be bounded by the light boson exchange which makes the
molecule configuration much more difficult to produce such states. Hence, the
exotic states consisting of four heavy quarks are more likely to be the real
compact tetraquarks.

In this work, we will try to deal with the mass spectra and wave functions of
the fully heavy tetraquark states $T_{QQ\bar{Q}\bar{Q}}$, namely,
$cc\bar{c}\bar{c}$ or $bb\bar{b}\bar{b}$, within the framework of the Bethe-
Salpeter equation. The fully heavy tetraquarks are assumed to be formed by the
diquark $QQ$ and the antidiquark $\bar{Q}\bar{Q}$. The diquark $QQ$ is further
assumed to be in the color $\bar{3}$ configuration in order to produce the
attractive force, and then similarly, the antidiquark $\bar{Q}\bar{Q}$ is in
the color $3$ state. This compact color-$\bar{3}$ diquark picture has already
been used in a previous work to study the doubly heavy baryons LiQ2020 , where
the obtained mass of $\Xi_{cc}^{++}$ is just 20 MeV lower than the
experimental measurements, and other predictions are also consistent with the
recent theoretical researches especially the Lattice QCD results Brown2014 .
We will use the previously calculated mass spectra and form factors of the
$J^{P}=1^{+}$ $cc$ and $bb$ diquarks in this work.  In addition, a propagator-
like form factor is investigated and the corresponding cutoff dependence is
also studied. The color $\bar{3}$ $QQ$ diquark and the color $3$
$\bar{Q}\bar{Q}$ antidiquark can finally form a compact tetraquark in color
singlet by the one-gluon-exchange interaction. Based on the above analysis,
the four-body tetraquark problem can be first reduced into two two-body bound
problems of fermion (antifermon) system, which can be solved by calculating
the original Bethe-Salpeter equation (BSE)SB1951 ; Salpeter1952 . Then we need
to deal with a two-body problem of boson system which will be the main focus
of this work.

The Bethe-Salpeter framework has great advantages in dealing with the two-body
bound states for the relativistic interaction kernel and corresponding wave
functions. The constructed relativistic wave functions are based on the good
quantum number $J^{P(C)}$ rather than the nonrelativistic characteristics spin
and orbital angular momentum ${}^{2S+1}L_{J}$. The BSE framework has been
successfully used in the mass spectra of mesons Chang2005A ; Chang2010 ;
LiQ2019A , traditional baryons and pentaquarks LiQ2020 ; XuH2020 , hadronic
transitions, electro-weak decays, and etc Chang2005 ; WangZ2012A ; WangT2013 ;
WangT2013A ; LiQ2016 ; LiQ2017 ; LiQ2017A . In this work we would try to push
the BSE framework further to study the fully heavy $QQ\bar{Q}\bar{Q}$ system
and develop a precise and systematic approach to describe the compact
tetraquark states.

This manuscript is organized as: in section 2 the (Bethe-)Salpeter equation is
derived in the perspective of the tetraquarks taken as the axial-vector
diquark-antidiquark bound states; in section 3 the Salpeter wave functions of
the tetraquarks with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ are constructed;
then in section 4 the obtained mass spectra and wave functions are given and
discussed; finally a brief summary is presented.

##  2 Tetraquarks as the bound states of diquark and antidiquark

Considering the exclusive principle, the diquark (antidiquark), consisting of
two charm quarks (antiquarks) in the color $\bar{3}$ (3) configuration, could
only be in the $J^{P}=1^{+}$ spin configuration in orbital ground states,
since the flavor wave function is naturally symmetric and spatial wave
function is also symmetric in ground states. In current work, we do not
consider the excitation of the diquarks or antidiquarks. Now we try to deal
with the BSE of the tetraquark states consisting of a $1^{+}$ diquark and a
$1^{+}$ antidiquark, which could form three ground states with spin-parity
configuration $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ respectively.

###  2.1 Bethe-Salpeter equation of two vector bosons

Figure 1: Bethe-Salpeter equation of the tetraquark states in the diquark-
antidiquark picture. The Greeks (red) are used for the Lorentz indices.
$P,~{}p_{1}(k_{1}),~{}p_{2}(k_{2})$ denote the momenta of the tetraquark
state, constituent diquark, and the constituent antidiquark respectively;
$K_{\alpha\alpha^{\prime};\beta^{\prime}\beta}$ denotes the effective
interaction kernel between the diquark and the antidiquark based on the one-
gluon exchange.

The Bethe-Salpeter equations of the bound states consisting of two vector (or
axialvactor) constituents are schematically depicted in Fig. 1. The
corresponding BSE are expressed as the four-dimensional integral of the inner
relative momentum $k$,

$\displaystyle\Gamma^{\alpha\beta}(P,q,\xi)=\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}(-i)K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)[D_{\alpha^{\prime}\mu}(k_{2})\Gamma^{\mu\nu}(P,k,\xi)D_{\nu\beta^{\prime}}(k_{1})],$
(1)

where $\Gamma^{\alpha\beta}(P,q,\xi)$ denotes the vertex of the two
axialvector constituents; symbol $P$ is used to denote the momentum of the
tetraquark state, and we have $P^{2}=M^{2}$ with $M$ representing the
tetraquark mass; symbol $\xi$ indicates the polarization state, with $\xi=\pm
2,0$ for the $J^{P}=2^{+}$ tetraquark states, $\xi=\pm 1,0$ for the $1^{+}$
ones, and $\xi=0$ for the scalar ones;
$K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)$ is the interaction
kernel of the diquark pair based on the one-gluon exchange. The constituent
mass of the diquark (antidiquark) is represented by $M_{1(2)}$. The
constituent masses of the $J^{P}=1^{+}$ $cc$ and $bb$ diquarks are obtained by
solving the corresponding BSE LiQ2020 . The inner relative momenta $q$ and $k$
are defined as $q=\alpha_{2}p_{1}-\alpha_{1}p_{2}$ and
$k=\alpha_{2}k_{1}-\alpha_{1}k_{2}$ respectively, with
$\alpha_{i}\equiv\frac{M_{i}}{M_{1}+M_{2}}$. The effective propagator of the
$1^{+}$ diquark reads,

$\displaystyle
D^{\alpha\beta}(k_{1})=D(k_{1})d^{\alpha\beta}(k_{1\perp}),~{}~{}~{}d^{\alpha\beta}(k_{1\perp})=-g^{\alpha\beta}+\frac{k_{1\perp}^{\alpha}k_{1\perp}^{\beta}}{M_{1}^{2}},$
(2)

where $k_{i\perp}\equiv k_{i}-\frac{k_{i}\cdot P}{M}$, and
$D(k_{1})=i\frac{1}{k_{1}^{2}-M_{1}^{2}+i\epsilon}$ is the usual scalar
propagator. As usual, the Bethe-Salpeter wave function describing the
tetraquark states can be defined as

$\displaystyle
T_{\alpha\beta}(P,q,\xi)=D_{\alpha\mu}(p_{2})\Gamma^{\mu\nu}(P,q,\xi)D_{\beta\nu}(p_{1}).$
(3)

The symbols $P$ and $\xi$ in the BS wave function $T_{\alpha\beta}(P,q,\xi)$
and vertex $\Gamma^{\mu\nu}(P,q,\xi)$ will be omitted unless it is necessary
to specify them. By using the definition of BS wave function, the BSE in Eq.
(1) can also be rewritten as the integral of $T_{\alpha\beta}$,

$\displaystyle
D^{-1}_{\alpha\alpha^{\prime}}(p_{1})T_{\alpha^{\prime}\beta^{\prime}}(q)D^{-1}_{\beta^{\prime}\beta}(p_{2})=\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}(-i)K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)T_{\alpha^{\prime}\beta^{\prime}}(k),$
(4)

where the inverse of the vector propagator reads,

$\displaystyle
D^{-1}_{\alpha\beta}(p_{i})=\vartheta_{\alpha\beta}(p_{i})D^{-1}(p_{i}),~{}~{}~{}\vartheta^{\alpha\beta}(p_{i})=-g^{\alpha\beta}-\frac{p^{\alpha}_{i\perp}p^{\beta}_{i\perp}}{w_{i}^{2}}.$
(5)

It can be easily checked that $\vartheta^{\alpha\beta}(p_{i})$ fulfills the
condition
$\vartheta^{\alpha\beta}(p_{i})d_{\beta\gamma}(p_{i})=\delta^{\alpha}_{\gamma}$,
where $w_{i}\equiv(M_{i}^{2}-p_{i\perp}^{2})^{\frac{1}{2}}$ denotes the
kinetic energy of the $i$th constituent.

###  2.2 Diquark form factors and tetraquark interaction kernel

For the tetraquark states consisting of the $1^{+}$ diquark and antidiquark
constituents, each vector constituent has internal structure and usually can
not be regarded as the pointlike particle. The diquark (antidiquark) is
usually described by the corresponding form factors. Generally speaking, the
form factors have great effects on the energy splittings of the tetraquark
bound states. Then the potential $V(s)$ between the two constituents will be
smeared by the two form factors. With these two form factors, the interaction
kernels of tetraquarks can be expressed as,

$\displaystyle
K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)=f_{\mathrm{1V}}f_{\mathrm{2V}}g^{\alpha\alpha^{\prime}}g^{\beta^{\prime}\beta}(k_{1}+p_{1})^{\nu}(k_{2}+p_{2})_{\nu}V(s),$
(6)

where $f_{\mathrm{1V}}$ and $f_{\mathrm{2V}}$ denote the vector form factor of
the diquark and antidiquark, and in the fully heavy $cc\bar{c}\bar{c}$ or
$bb\bar{b}\bar{b}$ system, we have
$f_{1\mathrm{V}}=f_{\mathrm{2V}}=f_{\mathrm{V}}$; $V(s)$ denotes the one-gluon
exchange potential with $s\equiv(k-q)$ being the momentum of the exchanged
gluon. Compared with the case in baryon problem LiQ2020 , there are two form
factors to describe the nonpointlike structures, which make it much more
complicated to deal with the tetraquark system.

Under the instantaneous approximation, the one-gluon-exchange potential is
assumed to be static, namely, $V(s)\sim V(s_{\perp})$ with
$s_{\perp}=s-\frac{P\cdot s}{M}$, which reads in the Coulomb gauge as Chao1992
; DingYB1993 ; DingYB1995 ; Kim2004

$V(\vec{s}\,)=V_{\mathrm{Coul}}+V_{\mathrm{Conf}}=-\frac{4}{3}\frac{4\pi\alpha_{s}(\vec{s}\,)}{\vec{s}\,^{2}+a_{1}^{2}}+\left[(2\pi)^{3}\delta^{3}(\vec{s}\,)\left(\frac{\lambda}{a_{2}}+V_{0}\right)-\frac{8\pi\lambda}{(\vec{s}\,^{2}+a_{2}^{2})^{2}}\right],$
(7)

where $\frac{4}{3}$ is the color factor in the color singlet; $a_{1(2)}$ is
introduced to avoid the divergence in small momentum transfer zone; the
potential $V_{\mathrm{Conf}}(\vec{s}\,)$ describing the confinement effects is
introduced phenomenologically, which is characterized by the the string
constant $\lambda$ and the factor $a_{2}$. The potential used here is based on
the famous Cornell potential Eichten1978 ; Eichten1980 , which behaves as the
one-gluon exchange Coulomb-type potential at short distance and a linear
growth confinement one at long distance, and then modified as the
aforementioned one to incorporate the color screening effects Laermann1986 ;
Born1989  in the linear confinement potential. $V_{0}$ is a constant fixed by
fitting to the meson data. The strong coupling constant $\alpha_{s}$ has the
following form,

$\alpha_{s}(\vec{s}\,)=\frac{12\pi}{(33-2N_{f})}\frac{1}{\ln\left(a+{\vec{s}\,^{2}}/{\Lambda^{2}_{\mathrm{QCD}}}\right)},$

where $\Lambda_{\mathrm{QCD}}$ is the scale of the strong interaction; $N_{f}$
is the active flavor number which is 3 for the $cc$ interaction while 4 for
the $bb$ interaction; $a=e$ is a regulator constant. For later convenience, we
further split $V(\vec{s}\,)$ into two parts as

$V(\vec{s}\,)=(2\pi)^{3}\delta^{3}(\vec{s}\,)V_{1}+V_{2}(\vec{s}\,).$ (8)

Namely, all the dependence on $\vec{s}$ is incorporated into
$V_{2}(\vec{s}\,)$, while $V_{1}$ is just the constant item.

Also since the spatial parts are suppressed by a factor of
$(\frac{v}{c})^{2}$, we only consider the dominate time component ($\nu=0$) in
the one-gluon-exchange Lorentz structure, namely,

$\displaystyle(k_{1}+p_{1})^{\nu}(k_{2}+p_{2})_{\nu}\simeq\varrho_{1}\varrho_{2},$
(9)

where we used the abbreviations $\varrho_{i}\equiv 2(\alpha_{i}M+q_{P})$ with
definition $q_{P}\equiv\frac{P\cdot q}{M}$. Using the notations and
approximations introduced above, the tetraquark interaction kernel behaves as

$\displaystyle
K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)\simeq\varrho_{1}\varrho_{2}g^{\alpha\alpha^{\prime}}g^{\beta^{\prime}\beta}\varkappa\left(s_{\perp}\right),$
(10)

where $\varkappa\equiv f_{\mathrm{V}}^{2}V(\vec{s}\,)$; the factor
$\varrho_{1}\varrho_{2}$ has been stripped off for later convenience. Notice
the kernel $\varkappa(s_{\perp})$ here has no dependence on the time component
of momentum transfer $s$.

The diquark form factors $f_{\mathrm{V}}(s^{2})$ can be calculated by using
the BS wave functions of the diquark. The attractive diquark is in the color
antitriplet with the corresponding color factor $(-\frac{2}{3})$, which makes
the bound interaction be half of the corresponding meson. Namely, in the
rainbow-ladder truncation the effective interaction for the diquarks is
reduced by a factor of 2 compared to that in the meson channel. Then after a
charge conjugate transformation, a diquark fulfills the same Bethe-Salpeter
equation with a meson system with only the interaction kernel halved and the
parity flipped Cahill1987 ; Maris2002 ; Roberts2011 . Then by solving the BSE
of the $J^{P}=1^{-}$ $Q\bar{Q}$ meson system, we can obtain the corresponding
mass spectra and wave functions for the doubly heavy diquarks $cc$ and $bb$
(see Ref.LiQ2020  for detailed calculations). The corresponding form factors
$f_{\mathrm{V}}$ describing the interaction between the diquark and a gluon
have also been obtained in the previous work LiQ2020 , which are showed
graphically in Fig. 2(a) labeled as method I. On the other hand, a colored
diquark is never observed in nature alone and the confinement in diquark is an
open problem in quark model. Therefore, we also calculated the corresponding
form factors without considering the confinement item $V_{\mathrm{Conf}}(s)$
in the Cornell potential of the diquark interaction kernel, and the obtained
results are showed in Fig. 2(b) labeled as method II. Both form factors will
be used in the calculation of the tetraquarks. The form factors fall faster
with $s^{2}$ in method II than in method I.  Also notice the form factors used
here are calculated from the relevant BS wave functions combined with the
Mandelstam formalism, which makes this work a self-contained framework and is
different from the parameterized treatment in Ref. KeHW2021 .

(a) Method I

(b) Method II

Figure 2: Form factors of the $1^{+}$ diquarks $cc$ and $bb$. Method I denotes
the results obtained by containing $V_{\mathrm{Conf}}$ in the diquark
interaction kernel while method II not.

Also notice the form factors in method I and II are calculated based on the
on-shell diquark bound state, and then are simply generalized to the off-shell
diquark propagator to describe the structure’s non-pointlike effects. In order
to investigate the influences of different form factors and make comparisons,
we will also use another phenomenological form factors in this work, namely,

$\displaystyle
f_{\mathrm{V}}(s^{2})=\frac{\Lambda_{c}^{2}}{-s^{2}+\Lambda^{2}_{c}},$ (11)

where $\Lambda_{c}$ is a introduced regulator parameter. We notice that this
propagator-like form factor is widely used in literature. Generally speaking,
the parameter $\Lambda_{c}$ should be determined by fitting to the data.
However, we found that this propagator-like function can not describe the
obtained form factors well in a large range of the momentum transfer. Hence,
in this work, we only fit in the small zone of $s^{2}$, namely, about
$1(3)\sim 0$ GeV2 for $cc\,(bb)$ diquark, and the determined $\Lambda_{c}$ is
$1.7$ and $2.8$ GeV for the $cc$ and $bb$ diquarks respectively. Also we will
let $\Lambda_{c}$ change from $0.1$ GeV to $20$ GeV to investigate the
corresponding dependence.

###  2.3 BSE under the instantaneous approximation

Generally speaking, solving the four dimensional BSE 1 is not an easy
computational problem. However, under the instantaneous approximation
introduced above, following Salpeter’s procedures to cope with the fermion-
antifermion system Salpeter1952 , we can reduce the four-dimensional BSE of
the tetraquarks into the three-dimensional integral equation. Inserting the
instantaneous kernel 10 into the BSE 1, the tetraquark vertex can now be
further expressed as
$\Gamma^{\alpha\beta}(q)=\varrho_{1}\varrho_{2}\Theta^{\alpha\beta}(q_{\perp})$,
where the three-dimensional vertex $\Theta^{\alpha\beta}$ is defined as

$\displaystyle\Theta^{\alpha\beta}(q_{\perp})$
$\displaystyle=\int\frac{\mathrm{d}^{3}k_{\perp}}{(2\pi)^{3}}\varkappa\left(k_{\perp}-q_{\perp}\right)\varphi^{\alpha\beta}(k_{\perp}),$
(12)

where the Salpeter wave function is defined by absorbing the integration over
the time component as usual,

$\displaystyle\varphi^{\alpha\beta}(k_{\perp})\equiv-i\int\frac{\mathrm{d}k_{P}}{2\pi}T^{\alpha\beta}(k).$
(13)

The Salpeter wave function $\varphi^{\alpha\beta}(k_{\perp})$ is only
explicitly dependent on the three-dimensional momentum $k_{\perp}$.

Following the standard procedure, performing the contour integral over $q_{P}$
on both sides of Eq. (3), we obtain the three-dimensional Salpeter equation
(SE),

$\displaystyle\varphi_{\alpha\beta}(q_{\perp})$
$\displaystyle=d_{\alpha\mu}(p_{1\perp})\Theta^{\mu\nu}(q_{\perp})d_{\nu\beta}(p_{2\perp})\left(\frac{1}{M-w_{1}-w_{2}}-\frac{1}{M+w_{1}+w_{2}}\right).$
(14)

Also we can define the positive and negative energy wave functions as

$\displaystyle\varphi^{\pm}_{\alpha\beta}(P,q_{\perp})\equiv
d_{\alpha\mu}(p_{1\perp})\Theta^{\mu\nu}(q_{\perp})d_{\nu\beta}(p_{2\perp})\frac{1}{\pm
M-(w_{1}+w_{2})},$ (15)

and we have
$\varphi_{\alpha\beta}=\varphi_{\alpha\beta}^{+}+\varphi_{\alpha\beta}^{-}$.
In the weak binding condition $M\sim(w_{1}+w_{2})$,
$\varphi_{\alpha\beta}^{+}\gg\varphi_{\alpha\beta}^{-}$, and the positive
energy wave function $\varphi_{\alpha\beta}^{+}(q_{\perp})$ dominates. The SE
can be further rewritten as the following simple Shrödinger type

$\displaystyle M^{2}\varphi_{\alpha\beta}$
$\displaystyle=(w_{1}+w_{2})^{2}\varphi_{\alpha\beta}+2(w_{1}+w_{2})d_{\alpha\mu}(p_{1\perp})\Theta^{\mu\nu}(q_{\perp})d_{\nu\beta}(p_{2\perp}).$
(16)

The obtained three-dimensional BSE, namely, Eq. (16), indicates that the mass
of the tetraquark state consists of two parts, the kinetic energy and the
potential energy. Also we notice Eq. (16) is in fact the integral equation of
the Salpeter wave function $\varphi_{\alpha\beta}(q_{\perp})$, and $M^{2}$
behaves as the eigenvalue of the corresponding Salpeter wave function. By
solving this eigenvalue equation, we can obtain the mass spectra and wave
functions of the corresponding tetraquarks.

The normalization condition of the Bethe-Salpeter wave function for the
tetraquark state is generally expressed as,

$\displaystyle
i\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\bar{T}_{\alpha\beta}(q,\xi^{\prime})\frac{\partial}{\partial
P^{0}}\left[\mathrm{I}^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)\right]T_{\alpha^{\prime}\beta^{\prime}}(k,\xi)=(2M)\delta_{\xi\xi^{\prime}},$

where the conjugate wave function is defined as
$\bar{T}_{\alpha\beta}\equiv\gamma^{0}T_{\alpha\beta}^{\dagger}\gamma_{0}$;
the integral kernel reads,

$\displaystyle\mathrm{I}^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q)=(2\pi)^{4}\delta^{4}(k-q)[D_{\alpha\alpha^{\prime}}(p_{1})]^{-1}[D_{\beta^{\prime}\beta}(p_{2})]^{-1}+iK^{\alpha\alpha^{\prime};\beta^{\prime}\beta}(P,k,q).$

Both the inverses of the propagators and the interaction kernel $K$ are
dependent on $P^{0}$ and $q_{P}$. Inserting the inverses of the propagators,
the integration involving the propagators behaves as,

$\displaystyle\mathrm{N}_{\mathrm{propagator}}$
$\displaystyle=\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}\bar{\varphi}^{\alpha\beta}\Theta_{\alpha\beta}\frac{8Mw_{1}w_{2}}{M^{2}-(w_{1}+w_{2})^{2}}.$
(17)

While the partial differential of the kernel gives

$\displaystyle\frac{\partial
K^{\alpha\alpha^{\prime};\beta^{\prime}\beta}}{\partial
P_{0}}=2(\alpha_{1}\varrho_{2}+\alpha_{2}\varrho_{2})\varkappa
g^{\alpha\alpha^{\prime}}g^{\beta^{\prime}\beta},$ (18)

and then we obtain normalization related to the interaction kernel as,

$\displaystyle\mathrm{N}_{\mathrm{kernel}}$
$\displaystyle=\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}\bar{\varphi}^{\alpha\beta}\Theta_{\alpha\beta}\frac{4Mw_{q}}{(w_{1}+w_{2})},$
(19)

where we have used the Salpter equation (14);
$w_{q}\equiv\alpha_{2}w_{1}+\alpha_{1}w_{2}$. On the other hand, the vertex
$\Theta_{\alpha\beta}$ can also be expressed by the wave function as,

$\displaystyle\Theta_{\alpha\beta}=\frac{M^{2}-(w_{1}+w_{2})^{2}}{2(w_{1}+w_{2})}\vartheta_{\alpha\mu}(p_{1\perp})\varphi^{\mu\nu}\vartheta_{\nu\beta}(p_{2\perp}).$
(20)

Putting the two part together and inserting the equation above, we obtain the
normalization of the Salpeter wave function,

$\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}N_{0}\,\bar{\varphi}^{\alpha\beta}_{(\xi^{\prime})}\vartheta_{\alpha\mu}(p_{1\perp})\vartheta_{\nu\beta}(p_{2\perp})\varphi^{\mu\nu}_{(\xi)}=\delta_{\xi\xi^{\prime}},$
(21)

where the abbreviation
$N_{0}\equiv\left[\frac{2w_{1}w_{2}}{(w_{1}+w_{2})}+\frac{w_{q}M^{2}}{(w_{1}+w_{2})^{2}}-w_{q}\right]$
is used. By the dimensional analysis, we can conclude that the dimension of
the Salpeter wave function $\varphi_{\alpha\beta}$ is $(-2)$ in units of mass.

It should be noted that, the obtained three-dimensional (Bethe-)Salpeter
equation 16 and the corresponding normalization condition 21 are universal to
any fully heavy tetraquark states consisting of two (axial)vector
constituents, and do not depend on the specific properties of the total
angular momentum $J$ or parity. These obtained equations are applicable to the
Salpeter wave functions with $J^{PC}=0^{++}$, $1^{+-}$, $2^{++}$ or any other
possible spin-parity. This is different from the approaches adopted in Ref.
KeHW2021 , where the Salpeter equations are coupled with three certain wave
functions.The obtained equations and relevant results can also be further
extended to deal with the molecular states consisting of two vector mesons.

##  3 BS wave functions of the tetraquark states

A $1^{+}$ diquark and a $1^{+}$ antidiquark can form a boson with
$J^{P}=0^{+}$, $1^{+}$, or $2^{+}$ in the ground state, namely,
$\bm{1}\times\bm{1}=\bm{0}+\bm{1}+\bm{2}$. For the tetraquark state, the
spatial parity is expressed as $P=(-1)^{l}$, with $l$ stands for the quantum
number of the angular momentum. A tetraquark state containing two $c$ quarks
and two $\bar{c}$ quarks also occupies the definite $C$-parity. The charge
conjugate parity ($C$-parity) is expressed as $C=(-1)^{l+s}$ with $s$ stands
for the spin of the particle. In order to simplify the expressions, from now
on, we will always use the abbreviation
$x_{\alpha}\equiv\frac{q_{\perp\alpha}}{|\vec{q}\,|}$. Since throughout this
manuscript we work in the momentum space, this abbreviation is supposed not to
cause confusion.

According to Lorentz condition, spin-parity and also considering the
relativistic covariance, the Salpeter wave function of tetraquark states with
$J^{P}=0^{+}$ can be generally expressed as,

$\displaystyle\varphi_{\alpha\beta}(q_{\perp})$
$\displaystyle=g_{1}\left(\hat{P}_{\alpha}\hat{P}_{\beta}-g_{\alpha\beta}\right)+g_{2}x_{\alpha}x_{\beta},$
(22)

where $\hat{P}_{\alpha}=\frac{P_{\alpha}}{M}$; the radial wave function
$g_{i}(|\vec{q}\,|)~{}(i=1,\,2)$ just depends on $|\vec{q}\,|$ explicitly. It
is clear to see that $g_{1}$ corresponds to the $S$-wave component, and
$g_{2}$ contributes to both the $S$ and $D$ partial waves (see Ref. LiQ2020
for a detailed expression in terms of the spherical harmonics $Y_{l}^{m}$). By
inserting the two wave functions into Eq. (21), the normalization condition of
the $0^{++}$ Salpeter wave function is obtained,

$\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}N_{0}\left[n_{0}(g_{1}+g_{2})^{2}+2g_{1}^{2}\right]=1,$
(23)

where $n_{0}=\left(\frac{M_{1}M_{2}}{w_{1}w_{2}}\right)^{2}$.

Similarly, the $J^{PC}=1^{+-}$ Salpeter wave function can be constructed by
using the the antisymmetric Levi-Civita tensor $\epsilon_{\alpha\beta\mu\nu}$
as

$\displaystyle\varphi_{\alpha\beta}(q_{\perp})$
$\displaystyle=h_{1}\epsilon_{\alpha\beta e\hat{P}}+h_{2}\epsilon_{\alpha\beta
x\hat{P}}e\cdot x,$ (24)

where $\epsilon_{\alpha\beta
e\hat{P}}=\epsilon_{\alpha\beta\mu\nu}e^{\mu}\hat{P}^{\nu}$; $e_{\alpha}(\xi)$
is the polarization vector with $\xi\,(\xi=0,~{}\pm 1)$ denoting the possible
polarization states, and fulfills the following conditions

$\displaystyle P_{\alpha}e^{\alpha}=0,$ (25)
$\displaystyle\sum_{\xi}e_{\alpha}(\xi)e_{\beta}(\xi)={G}_{\alpha\beta}\equiv\frac{P_{\alpha}P_{\beta}}{M^{2}}-g_{\alpha\beta}.$
(26)

It is clear to see that $h_{1}$ and $h_{2}$ parts in Eq. (24) represent the
$S$ and $D$-wave components, respectively.  . The $J^{PC}=1^{+-}$ tetraquark
wave function is antisymmetric under the interchange of the two free Lorentz
index, which is different from the $0^{++}$ case. The $1^{+-}$ normalization
is finally expressed as,

$\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}\frac{2}{3}N_{0}\left[n_{1}h^{2}_{1}+(h_{2}-h_{1})^{2}\right]=1,$
(27)

where $n_{1}={M_{1}^{2}}/{w_{1}^{2}}+{M_{2}^{2}}/{w_{2}^{2}}$.  Notice in a
similar work KeHW2021  only part of the $S$-wave components in the wave
functions of $J^{PC}=0^{++}$ and $1^{+-}$ are included, while our results show
that the $D$-wave components, namely, the $g_{2}$ and $h_{2}$ items, and the
the possible $S$-$D$ mixing effects also play important roles especially in
the excited states (see the obtained wave functions in Fig. 5).

The $J^{PC}=2^{++}$ Salpeter wave function of the tetraquarks can be
constructed as

$\displaystyle\varphi_{\alpha\beta}(q_{\perp})$
$\displaystyle=i_{1}e_{\alpha\beta}+i_{2}\left(\hat{P}_{\alpha}\hat{P}_{\beta}-g_{\alpha\beta}\right)e_{xx}+i_{3}\left(e_{x\alpha}x_{\beta}+e_{x\beta}x_{\alpha}\right)+i_{4}\left(x_{\alpha}x_{\beta}\right)e_{xx},$
(28)

where $e_{xx}\equiv e_{\alpha\beta}x^{\alpha}x^{\beta}$;
$e_{\alpha\beta}(\xi)$ is the symmetric polarization tensor with $\xi\,(=\pm
2,\pm 1,0)$ denoting the possible polarization states. The symmetric
$e_{\alpha\beta}(\xi)$ is traceless, and also fulfills the Lorentz condition
and the completeness relationship

$\displaystyle e_{\alpha\beta}=e_{\beta\alpha},$ (29) $\displaystyle
e_{\alpha\beta}g^{\alpha\beta}=0,$ (30) $\displaystyle
P^{\alpha}e_{\alpha\beta}=0,$ (31)
$\displaystyle\sum_{\xi}e_{\alpha\beta}(\xi)e_{\alpha_{1}\beta_{1}}(\xi)=\frac{1}{2}\left(G_{\alpha\alpha_{1}}G_{\beta\beta_{1}}+G_{\alpha\beta_{1}}G_{\beta\alpha_{1}}\right)-\frac{1}{3}G_{\alpha\beta}G_{\alpha_{1}\beta_{1}}.$
(32)

It is clear to see that $i_{1}$, $i_{2(3)}$, and $i_{4}$ parts represent the
$S$ , $D$, and $G$-wave components, respectively. In Ref. KeHW2021  only the
$i_{1}$ item representing the dominant $S$-wave component is considered, while
both the $D$ and the possible $G$-wave components are ignored, which would
have effects on the mass splittings and damage the completeness of the wave
functions. The normalization for $2^{++}$ Salpeter wave function is expressed
as,

$\displaystyle\int\frac{\mathrm{d}^{3}q_{\perp}}{(2\pi)^{3}}$
$\displaystyle\frac{N_{0}}{15}\sum_{m=1,n\geqslant m}^{4}c_{mn}i_{m}i_{n}=1$
(33)

where the coefficients $c_{ij}$s are defined as

$\displaystyle c_{11}$ $\displaystyle=5n_{0}-3c_{0}+10,$ $\displaystyle
c_{12}$ $\displaystyle=4n_{0}-4,$ (34) $\displaystyle c_{13}$
$\displaystyle=-14n_{0}+6c_{0}-6,~{}~{}$ $\displaystyle c_{14}$
$\displaystyle=4n_{0},$ $\displaystyle c_{22}$ $\displaystyle=2n_{0}+4,$
$\displaystyle c_{23}$ $\displaystyle=-8n_{0},$ $\displaystyle c_{24}$
$\displaystyle=4n_{0},$ $\displaystyle c_{33}$
$\displaystyle=11n_{0}-3c_{0}+3,$ $\displaystyle c_{34}$
$\displaystyle=-8n_{0},$ $\displaystyle c_{44}$ $\displaystyle=2n_{0},$

with $c_{0}=(\frac{q^{2}}{w_{1}w_{2}})^{2}$.

Notice in the construction of the Salpeter wave functions, we work based on
the good quantum number $J^{PC}$ while not the nonrelativistic
characteristics, such as spin $S$ or orbital angular momentum $L$. Then the
contributions from the $D$ or $G$-wave components are naturally included in
the $0^{++}$, $1^{+-}$, and $2^{++}$ Salpeter wave functions, and are
determined by the dynamics of the Bethe-Salpeter equation while not the man-
made mixing effects. This is one of the advantages of the relativistic Bethe-
Salpeter methods. The relativistic effects are naturally included in both the
adoption of the BSE and the construction of the wave functions though the
instantaneous approximation partly destroyed the covariance. Inserting these
constructed Salpeter wave functions into the three-dimensional Salpeter
equation 16 and solving this eigenvalue problem numerically, we can obtain the
corresponding mass spectra and wave functions, which are presented in
following section.

##  4 Numerical results and discussions

Before giving the numerical results of the mass spectra and wave functions, we
specify the numerical values of the parameters first. The model parameters
used in this work are kept the same with that we applied in previous meson and
baryon calculations Chang2010 ; WangT2013 ; WangT2013A ; LiQ2016 ; LiQ2017 ;
LiQ2017A ; LiQ2020 ,

$a=e=2.7183,~{}~{}\lambda=0.21~{}$\mathrm{G}\mathrm{e}\mathrm{V}$^{2},~{}~{}a_{1}=a_{2}=0.06~{}$\mathrm{G}\mathrm{e}\mathrm{V}$;$

the strong interaction scale
$\Lambda_{\text{QCD}}=0.20\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ for $bb$
interaction while takes $0.27\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ for other
cases; and the constituent quark masses used are
$m_{c}=1.62~{}$\mathrm{G}\mathrm{e}\mathrm{V}$,~{}m_{b}=4.96~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$.
The obtained constitute masses of the diquarks are
$M_{cc}=3.303\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$M_{bb}=9.816\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ LiQ2020  within method I, and
$M_{cc}=3.135\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$M_{bb}=9.732\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ within the method II.

The free parameter $V_{0}$ plays a role in shifting the mass spectra and is
determined by the spin-weighted average methods. By using the Clebsch–Gordan
coefficients, the $J^{P}=0^{+}$ tetraquark formed by the two $1^{+}$ diquark
and antidiquarks can be decomposed as,

$\displaystyle\textstyle\ket{(12)_{1}(34)_{1}}_{0}=\frac{1}{2}\ket{(14)_{1}(23)_{1}}_{0}+\frac{\sqrt{3}}{2}\ket{(14)_{0}(23)_{0}}_{0},$
(35)

where label 1, 2 denote the quarks, and 3, 4 denote the antiquarks;
$\ket{(12)_{1}(34)_{1}}$ means quark-1 and quark-2 are in the spin-1 state,
while the antiquark-3 and antiquark-4 are also in the spin-1 state; then other
notations are also implied. Then the parameter $V_{0}$ for $0^{+}$
${cc\bar{c}\bar{c}}$ will be expressed as
$V_{0({cc\bar{c}\bar{c}})}=\frac{1}{4}V_{0(J/\psi)}+\frac{3}{4}V_{0(\eta_{c})}$.
By similar analysis, the $J^{P}=1^{+}$ tetraquark is decomposed as,

$\displaystyle\textstyle\ket{(12)_{1}(34)_{1}}_{1}=\frac{1}{\sqrt{2}}\ket{(14)_{0}(23)_{1}}_{1}+\frac{1}{\sqrt{2}}\ket{(14)_{1}(23)_{0}}_{1},$
(36)

and then the corresponding parameter is determined as
$V_{0({cc\bar{c}\bar{c}})}=\frac{1}{2}\left[V_{0(J/\psi)}+V_{0(\eta_{c})}\right]$.
The $V_{0}$ for $2^{+}$ ${cc\bar{c}\bar{c}}$ is totally decided by the
$V_{0(J/\psi)}$ since any two quarks (anitquarks) inside are in the spin-1
state. Finally, the obtained parameters $V_{0}$ for $0^{+}$, $1^{+}$, and
$2^{+}$ $(cc\bar{c}\bar{c})$ are $-0.304$, $-0.276$, and $-0.221$ GeV
respectively; for $bb\bar{b}\bar{b}$ are $-0.207$, $-0.192$, and $-0.162$ GeV,
respectively.

(a) $(cc\bar{c}\bar{c})$ in method I

(b) $(cc\bar{c}\bar{c})$ in method II

(c) $(cc\bar{c}\bar{c})$ in method III

(d) $(bb\bar{b}\bar{b})$ in method I

(e) $(bb\bar{b}\bar{b})$ in method II

(f) $(bb\bar{b}\bar{b})$ in method III

Figure 3: Mass spectra of the tetraquark states $(cc\bar{c}\bar{c})$ and
$(bb\bar{b}\bar{b})$ with all the diquarks (antiquarks) in the ground states.
In method I (II) the diquark masses and corresponding form factors are
calculated with (without) considering the confinement item $V_{\mathrm{Conf}}$
in the interaction kernel;  in method III the diquark form factors are assumed
to be $\frac{\Lambda_{c}^{2}}{-s^{2}+\Lambda_{c}^{2}}$ with
$\Lambda_{c}=1.7\,(2.8)$ GeV for $cc\,(bb)$ diquark.

The obtained mass spectra of the tetraquarks $cc\bar{c}\bar{c}$ and
$bb\bar{b}\bar{b}$ are showed in Fig. 3, where method I denotes the results
with considering the confinement item $V_{\mathrm{Conf}}$ in calculating the
diquark masses and the corresponding form factors, while method II not. The
results labeled as method III represents the ones where the diquark form
factors are assumed to be propagator-like type in Eq. (11) with
$\Lambda_{c}=1.7$ GeV for $cc$ diquark and $2.8\,$GeV for $bb$ diquark
respectively. In order to see the dependence on the regulator $\Lambda_{c}$ in
method III, we show the variation of the mass spectra along with $\Lambda_{c}$
in Fig. 4 by changing the values of $\Lambda_{c}$ from 0.1 GeV to 20 GeV. With
the increases of $\Lambda_{c}$, $f_{\mathrm{V}}$ becomes more and more flat,
which means the diquark is more and more similar to a pointlike particle. Also
it should be pointed out that the regulator parameter $\Lambda_{c}$ can not be
taken too large which may cause the instability of the instantaneous BSE in
high momentum zone. The plots reveal that the mass spectra in ground states
are sensitive to the diquark form factors.

(a) $(cc\bar{c}\bar{c})$

(b) $(bb\bar{b}\bar{b})$

Figure 4:  Dependence on the regulator parameter $\Lambda_{c}$ of the tetraquark mass spectra in method III, where the diquark form factors are assumed to be $f_{\mathrm{V}}(s^{2})=\frac{\Lambda_{c}^{2}}{-s^{2}+\Lambda_{c}^{2}}$. Table I: Comparison of the predicted $cc\bar{c}\bar{c}$ masses in ground states with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ in units of GeV.  $J^{PC}$ | This | DengCR2021  | LiuMS2019  | WangGJ2019  | LuQF2020  | ChenW2017  | ZhaoJX2020  | Gordillo2020  | Faustov2020  | WengXZ2021
---|---|---|---|---|---|---|---|---|---|---
$0^{++}$ | 6.419 | 6.407, 6.491 | 6.470 | 6.371 | 6.435 | 6.44 | 6.346 | 6.351 | 6.190 | 6.045
$1^{+-}$ | 6.456 | 6.463, 6.580 | 6.512 | 6.450 | 6.515 | 6.37 | 6.441 | 6.441 | 6.271 | 6.231
$2^{++}$ | 6.516 | 6.486, 6.607 | 6.534 | 6.534 | 6.543 | 6.51 | 6.475 | 6.471 | 6.367 | 6.287

The obtained mass spectra show that the ground $cc\bar{c}\bar{c}$ tetraquarks
locate in the range $6.4\sim 6.5\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ when
considering $V_{\mathrm{Conf}}$ in diquarks. The mass splittings for the
$1^{+-}$ and $2^{++}$ to $0^{++}$ states are about
$35\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ and
$100\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ respectively. When the confinement
item is not included in the diquarks, the corresponding masses then locate
about $40\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ lower. The first excited states
are always about $400\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ higher than their
corresponding ground states. A comparison of our predictions with recent
researches is listed in Tab. I. Our obtained results for the three ground
$cc\bar{c}\bar{c}$ states are roughly consistent with other studies.  The mean
values in Tab. I are 6.36, 6.43, and 6.48 GeV for the $0^{++}$, $1^{+-}$, and
$2^{++}$ $cc\bar{c}\bar{c}$ in ground states respectively; and the
corresponding standard derivations are 0.13, 0.10 and 0.09 GeV respectively.
All the results listed in Tab. I are above the threshold of the lowest
quarkonium pair $\eta_{c}\eta_{c}$. Hence these three ground states are
expected to be broad, since all of them can decay to a pair of quarkonia
$\eta_{c}\eta_{c}$ or $J/\psi J/\psi$ through the (anti)quark rearrangements.
These kind of decays are favored both dynamically and kinematically. From Tab.
I we can conclude that the obtained masses of the ground $cc\bar{c}\bar{c}$
states are usually much lower (about $400\sim 500$ MeV) than the $X(6900)$
observed by the LHCb collaboration. The observed $X(6900)$ is less likely to
be the ground state of the compact tetraquark $cc\bar{c}\bar{c}$ states, but
might be the first or second radial excited states. However, more detailed
information is needed to investigate the inner structure of $X(6900)$.  We
also notice that the masses of the $cc\bar{c}\bar{c}$ in ground states is near
to the $X(6900)$ with $\Lambda_{c}\sim 1$ GeV in method III.

The obtained $bb\bar{b}\bar{b}$ masses in ground states are in the range
$19.2\sim 19.3$ GeV, which are higher than the $\Upsilon\Upsilon$ threshold
$1.892$ GeV and $\eta_{b}\eta_{b}$ threshold $1.880$ GeV but lower than the
$\chi_{b0}\chi_{b0}$ threshold PDG2014 . Also we notice that $M_{bb}$ in
method II is about 80 MeV lower than that in method I, while the obtained mass
spectra of $bb\bar{b}\bar{b}$ are about 20 MeV lower in ground states. The
comparison of our predictions with other researches is collected in Tab. II,
from which we can see that the theoretical masses of $bb\bar{b}\bar{b}$ in
ground states locate in a large range of $18.75\sim
19.35\,$\mathrm{G}\mathrm{e}\mathrm{V}$$ in researches.  The mean values of
the ground $bb\bar{b}\bar{b}$ tetraquarks in Tab. II are about 19.14, 19.20,
and 19.22 GeV for the $0^{++}$, $1^{+-}$, and $2^{++}$, respectively, where
the corresponding standard derivations are 0.20, 0.18 and 0.16 GeV
respectively. Except the 3 results predicted in Ref. Bedolla2020  and the
$0^{++}$ state in Ref. WengXZ2021 , all other results listed in Tab. II are
higher than the $\Upsilon\Upsilon$ threshold. Therefore, the three states can
decay to a pair of quarkonia $\eta_{b}\eta_{b}$ or $\Upsilon\Upsilon$ through
the quark rearrangements, and hence are expected to be broad.

Table II: Comparison of the $bb\bar{b}\bar{b}$ masses in ground states with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ in units of GeV.  $J^{PC}$ | This | LuQF2020  | ZhaoJX2020  | Faustov2020  | LiuMS2019  | WengXZ2021  | Gordillo2020  | WangGJ2019  | Bedolla2020
---|---|---|---|---|---|---|---|---|---
$0^{++}$ | 19.205 | 19.201 | 19.154 | 19.314 | 19.322 | 18.836 | 19.199 | 19.243 | 18.748
$1^{+-}$ | 19.221 | 19.251 | 19.214 | 19.320 | 19.329 | 18.969 | 19.276 | 19.329 | 18.828
$2^{++}$ | 19.253 | 19.262 | 19.232 | 19.330 | 19.341 | 19.000 | 19.289 | 19.325 | 18.900

Fig. 5 shows the obtained Salpeter wave functions for the first four
$(cc\bar{c}\bar{c})$ states obtained by the method I. Notice that the $D$
partial wave gives important contribution in the third and the forth states,
and the possible $D$ or $G$-wave mixing are included naturally. The obtained
wave functions show the rich information of the inner structure, and can be
further used to do precise calculations on the decays, magnetic moments, or
other properties of the tetraquarks.

(a) $0^{++}(n=1)$

(b) $1^{+-}(n=1)$

(c) $2^{++}(n=1)$

(d) $0^{++}(n=2)$

(e) $1^{+-}(n=2)$

(f) $2^{++}(n=2)$

(g) $0^{++}(n=3)$

(h) $1^{+-}(n=3)$

(i) $2^{++}(n=3)$

(j) $0^{++}(n=4)$

(k) $1^{+-}(n=4)$

(l) $2^{++}(n=4)$

Figure 5: Salpeter wave functions of the $cc\bar{c}\bar{c}$ tetraquarks with
$J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$.

##  5 Summary

In this work, we study the compact tetraquark states with fully heavy quark
contents $QQ\bar{Q}\bar{Q}$ based on the Bethe-Salpeter equation. The compact
tetraquark is taken as the bound state of the diquark-antidiquark where the
(anti)diquark form factors are calculated with and without considering the
confinement item respectively.  In addition, a propagator-like form factor is
also used to calculate the tetrequark states and the corresponding parameter
dependence is also investigated. Under the instantaneous approximation, the
three-dimensional (Bethe-)Salpeter equation of the tetraquarks are derived,
and the Salpeter wave functions with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$
are then constructed and solved numerically to obtain the corresponding mass
spectra of the tetraquarks.  The compact tetraquarks as the bound states of
the diquarks and antidiquarks follows a natural step of our previous work to
deal with doubly heavy baryons. Therefore some results and approaches, such as
the relevant interaction kernel, diquark constituent masses and form factors,
the reduction of BSE, etc., can be directly adopted or extended to deal with
compact tetraquarks, which allows us to take a systematic, relativistic, and
unified framework to treat the heavy hadron systems, including the mesons,
diquarks, baryons, and tetraquarks. Also notice all the parameters used to
deal with the tetraquarks have already been fixed by the meson data and we do
not introduce any free parameters. The well behaviors of the obtained wave
functions allow to do the further precise calculations on the decays or other
properties of the tetraquarks.

Our results show that the three ground states of $cc\bar{c}\bar{c}$ locate in
the mass range $6.4\sim 6.5\,$\mathrm{G}\mathrm{e}\mathrm{V}$$, and the
$bb\bar{b}\bar{b}$ states in mass range $19.2\sim
19.3\,$\mathrm{G}\mathrm{e}\mathrm{V}$$. Based on the obtained results, the
LHCb’s observation $X(6900)$ is less likely to be the ground states of compact
$cc\bar{c}\bar{c}$ tetraquarks but might be the first or second radially
excited states. The obtained relativistic wave functions are based on the good
quantum numbers and can naturally include the mixing effects from the possible
$D$(or $G$)-wave components.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) under Grant Nos. 12005169, 12075073, 12075301, 11821505 and 11947302.
It is also supported by the Natural Science Basic Research Program of Shaanxi
(Program No. 2021JQ-074), and the Fundamental Research Funds for the Central
Universities.

## References

  * (1) M. Gell-Mann, Phys. Lett. 8 (1964) 214–215.  DOI:10.1016/S0031-9163(64)92001-3.
  * (2) G. Zweig, Report No. CERN-TH-401 (1964).
  * (3) R. Aaij, et al., Phys. Rev. Lett. 115 (2015) 072001.  arXiv:1507.03414, DOI:10.1103/PhysRevLett.115.072001.
  * (4) R. Aaij, et al., Phys. Rev. Lett. 122 (22) (2019) 222001.  arXiv:1904.03947, DOI:10.1103/PhysRevLett.122.222001.
  * (5) R. Aaij, et al., Sci. Bull. 65 (23) (2020) 1983–1993.  arXiv:2006.16957, DOI:10.1016/j.scib.2020.08.032.
  * (6) J. Zhao, S. Shi, P. Zhuang, Phys. Rev. D 102 (11) (2020) 114001.  arXiv:2009.10319, DOI:10.1103/PhysRevD.102.114001.
  * (7) M. C. Gordillo, F. De Soto, J. Segovia, Phys. Rev. D 102 (11) (2020) 114007.  arXiv:2009.11889, DOI:10.1103/PhysRevD.102.114007.
  * (8) H. Mutuk, Eur. Phys. J. C 81 (4) (2021) 367.  arXiv:2104.11823, DOI:10.1140/epjc/s10052-021-09176-8.
  * (9) J. F. Giron, R. F. Lebed, Phys. Rev. D 102 (7) (2020) 074003.  arXiv:2008.01631, DOI:10.1103/PhysRevD.102.074003.
  * (10) P. Lundhammar, T. Ohlsson, Phys. Rev. D 102 (5) (2020) 054018.  arXiv:2006.09393, DOI:10.1103/PhysRevD.102.054018.
  * (11) C. Deng, H. Chen, J. Ping, Phys. Rev. D 103 (1) (2021) 014001.  arXiv:2003.05154, DOI:10.1103/PhysRevD.103.014001.
  * (12) X.-Z. Weng, X.-L. Chen, W.-Z. Deng, S.-L. Zhu, Phys. Rev. D 103 (3) (2021) 034001\.  arXiv:2010.05163, DOI:10.1103/PhysRevD.103.034001.
  * (13) Z.-G. Wang, Chin. Phys. C 44 (11) (2020) 113106.  arXiv:2006.13028, DOI:10.1088/1674-1137/abb080.
  * (14) Q.-F. Lü, D.-Y. Chen, Y.-B. Dong, Eur. Phys. J. C 80 (9) (2020) 871.  arXiv:2006.14445, DOI:10.1140/epjc/s10052-020-08454-1.
  * (15) R. N. Faustov, V. O. Galkin, E. M. Savchenko, Phys. Rev. D 102 (11) (2020) 114030\.  arXiv:2009.13237, DOI:10.1103/PhysRevD.102.114030.
  * (16) M. A. Bedolla, J. Ferretti, C. D. Roberts, E. Santopinto, Eur. Phys. J. C 80 (11) (2020) 1004.  arXiv:1911.00960, DOI:10.1140/epjc/s10052-020-08579-3.
  * (17) H.-W. Ke, X. Han, X.-H. Liu, Y.-L. Shi, Eur. Phys. J. C 81 (5) (2021) 427.  arXiv:2103.13140, DOI:10.1140/epjc/s10052-021-09229-y.
  * (18) R. Zhu, Nucl. Phys. B 966 (2021) 115393.  arXiv:2010.09082, DOI:10.1016/j.nuclphysb.2021.115393.
  * (19) K.-T. Chao, S.-L. Zhu, Sci. Bull. 65 (23) (2020) 1952–1953.  arXiv:2008.07670, DOI:10.1016/j.scib.2020.08.031.
  * (20) M. Karliner, J. L. Rosner, Phys. Rev. D 102 (11) (2020) 114039.  arXiv:2009.04429, DOI:10.1103/PhysRevD.102.114039.
  * (21) Z.-H. Guo, J. A. Oller, Phys. Rev. D 103 (3) (2021) 034024.  arXiv:2011.00978, DOI:10.1103/PhysRevD.103.034024.
  * (22) X.-K. Dong, V. Baru, F.-K. Guo, C. Hanhart, A. Nefediev, Phys. Rev. Lett. 126 (13) (2021) 132001.  arXiv:2009.07795, DOI:10.1103/PhysRevLett.126.132001.
  * (23) J.-Z. Wang, D.-Y. Chen, X. Liu, T. Matsuki, Phys. Rev. D 103 (7) (2021) L071503.  arXiv:2008.07430, DOI:10.1103/PhysRevD.103.L071503.
  * (24) B.-D. Wan, C.-F. Qiao (12 2020).  arXiv:2012.00454.
  * (25) J.-W. Zhu, X.-D. Guo, R.-Y. Zhang, W.-G. Ma, X.-Q. Li (11 2020).  arXiv:2011.07799.
  * (26) Q. Li, C.-H. Chang, S.-X. Qin, G.-L. Wang, Chin. Phys. C 44 (2020) 013102.  arXiv:1903.02282, DOI:10.1088/1674-1137/44/1/013102.
  * (27) Z. S. Brown, W. Detmold, S. Meinel, K. Orginos, Phys. Rev. D 90 (9) (2014) 094507\.  arXiv:1409.0497, DOI:10.1103/PhysRevD.90.094507.
  * (28) E. E. Salpeter, H. A. Bethe, Phys. Rev. 84 (1951) 1232–1242.  DOI:10.1103/PhysRev.84.1232.
  * (29) E. E. Salpeter, Phys. Rev. 87 (1952) 328–343.  DOI:10.1103/PhysRev.87.328.
  * (30) C.-H. Chang, J.-K. Chen, X.-Q. Li, G.-L. Wang, Commun. Theor. Phys. 43 (2005) 113–118.  arXiv:hep-ph/0406050, DOI:10.1088/0253-6102/43/1/023.
  * (31) C. H. Chang, G. L. Wang, Sci. China Phys. Mech. Astron. 53 (2010) 2005–2018.  arXiv:1003.3827, DOI:10.1007/s11433-010-4156-1.
  * (32) Q. Li, T. Wang, Y. Jiang, G.-L. Wang, C.-H. Chang, Phys. Rev. D 100 (7) (2019) 076020\.  arXiv:1802.06351, DOI:10.1103/PhysRevD.100.076020.
  * (33) H. Xu, Q. Li, C.-H. Chang, G.-L. Wang, Phys. Rev. D 101 (5) (2020) 054037.  arXiv:2001.02980, DOI:10.1103/PhysRevD.101.054037.
  * (34) C.-H. Chang, C. Kim, G.-L. Wang, Phys. Lett. B 623 (2005) 218–226.  DOI:10.1016/j.physletb.2005.07.059.
  * (35) Z.-H. Wang, G.-L. Wang, C.-H. Chang, J. Phys. G: Nucl. Part. Phys. 39 (2012) 015009\.  arXiv:1107.0474, DOI:10.1088/0954-3899/39/1/015009.
  * (36) T. Wang, G.-L. Wang, H.-F. Fu, W.-L. Ju, JHEP 07 (2013) 120.  arXiv:1305.1067, DOI:10.1007/JHEP07(2013)120.
  * (37) T. Wang, G.-L. Wang, W.-L. Ju, Y. Jiang, JHEP 03 (2013) 110.  arXiv:1303.1563, DOI:10.1007/JHEP03(2013)110.
  * (38) Q. Li, T. Wang, Y. Jiang, H. Yuan, G.-L. Wang, Eur. Phys. J. C 76 (8) (2016) 454\.  DOI:10.1140/epjc/s10052-016-4306-3.
  * (39) Q. Li, T. Wang, Y. Jiang, H. Yuan, T. Zhou, G.-L. Wang, Eur. Phys. J. C 77 (1) (2017) 12.  DOI:10.1140/epjc/s10052-016-4588-5.
  * (40) Q. Li, Y. Jiang, T. Wang, H. Yuan, G.-L. Wang, C.-H. Chang, Eur. Phys. J. C 77 (5) (2017) 297.  arXiv:1701.03252, DOI:10.1140/epjc/s10052-017-4865-y.
  * (41) K.-T. Chao, Y.-B. Ding, D.-H. Qin, Commun. Theor. Phys. 18 (1992) 321–326.
  * (42) Y.-B. Ding, K.-T. Chao, D.-H. Qin, Chin. Phys. Lett. 10 (1993) 460–463.  DOI:10.1088/0256-307X/10/8/004.
  * (43) Y.-B. Ding, K.-T. Chao, D.-H. Qin, Phys. Rev. D 51 (1995) 5064–5068.  arXiv:hep-ph/9502409, DOI:10.1103/PhysRevD.51.5064.
  * (44) C. S. Kim, G.-L. Wang, Phys. Lett. B 584 (2004) 285–293.  arXiv:hep-ph/0309162, DOI:10.1016/j.physletb.2004.01.058.
  * (45) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T.-M. Yan, Phys. Rev. D 17 (1978) 3090.  DOI:10.1103/PhysRevD.17.3090,10.1103/physrevd.21.313.2.
  * (46) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T.-M. Yan, Phys. Rev. D 21 (1980) 203.  DOI:10.1103/PhysRevD.21.203.
  * (47) E. Laermann, F. Langhammer, I. Schmitt, P. M. Zerwas, Phys. Lett. B 173 (1986) 437–442.  DOI:10.1016/0370-2693(86)90411-9.
  * (48) K. D. Born, E. Laermann, N. Pirch, T. F. Walsh, P. M. Zerwas, Phys. Rev. D 40 (1989) 1653–1663.  DOI:10.1103/PhysRevD.40.1653.
  * (49) R. T. Cahill, C. D. Roberts, J. Praschifka, Phys. Rev. D 36 (1987) 2804.  DOI:10.1103/PhysRevD.36.2804.
  * (50) P. Maris, Few Body Syst. 32 (2002) 41–52.  arXiv:nucl-th/0204020, DOI:10.1007/s00601-002-0111-7.
  * (51) H. L. L. Roberts, L. Chang, I. C. Cloet, C. D. Roberts, Few Body Syst. 51 (2011) 1–25.  arXiv:1101.4244, DOI:10.1007/s00601-011-0225-x.
  * (52) M.-S. Liu, Q.-F. Lü, X.-H. Zhong, Q. Zhao, Phys. Rev. D 100 (1) (2019) 016006\.  arXiv:1901.02564, DOI:10.1103/PhysRevD.100.016006.
  * (53) G.-J. Wang, L. Meng, S.-L. Zhu, Phys. Rev. D 100 (9) (2019) 096013.  arXiv:1907.05177, DOI:10.1103/PhysRevD.100.096013.
  * (54) W. Chen, H.-X. Chen, X. Liu, T. G. Steele, S.-L. Zhu, Phys. Lett. B 773 (2017) 247–251.  arXiv:1605.01647, DOI:10.1016/j.physletb.2017.08.034.
  * (55) K. A. Olive, et al., Chin. Phys. C 38 (2014) 090001.  DOI:10.1088/1674-1137/38/9/090001.


revtex4-2Repair the float

# Designing Stochastic Channels

Matthew A. Graydon<EMAIL_ADDRESS>Institute for Quantum Computing,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Department of
Physics & Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Joshua Skanes-Norman  Institute for Quantum Computing, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada  Department of Applied Mathematics,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Keysight
Technologies Canada, Kanata, ON K2K 2W5, Canada  Joel J. Wallman  Institute
for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada  Department of Applied Mathematics, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada  Keysight Technologies Canada, Kanata, ON K2K 2W5,
Canada

###### Abstract

Stochastic channels are ubiquitous in the field of quantum information because
they are simple and easy to analyze. In particular, Pauli channels and
depolarizing channels are widely studied because they can be efficiently
simulated in many relevant quantum circuits. Despite their wide use, the
properties of general stochastic channels have received little attention. In
this paper, we prove that the diamond distance of a general stochastic channel
from the identity coincides with its process infidelity to the identity. We
demonstrate with an explicit example that there exist multi-qubit stochastic
channels that are not unital. We then discuss the relationship between unitary
1-designs and stochastic channels. We prove that the twirl of an arbitrary
quantum channel by a unitary 1-design is always a stochastic channel. However,
unlike with unitary 2-designs, the twirled channel depends upon the choice of
unitary 1-design. Moreover, we prove by example that there exist stochastic
channels that cannot be obtained by twirling a quantum channel by a unitary
1-design.

##  I Introduction

Quantum computers hold tremendous promise, but noise severely limits the
reliability of quantum devices in the noisy intermediate-scale quantum (NISQ)
era [1]. Quantum error correction and noise mitigation methods hinge on
assumptions about the noise in question [2, 3]. A standard assumption is that
the noise is a Pauli channel, because such channels are simple and easy to
analyze. In this paper, we consider a more general class of stochastic error
processes. A stochastic channel is a completely positive and trace-preserving
(CPTP) map admitting a set of Hilbert-Schmidt orthogonal Kraus operators
containing a strictly positive multiple of the identity [4]. Operationally, a
stochastic channel corresponds to an error occurring with some probability
$p\in[0,1)$. Some nice examples are depolarizing channels, decoherence
channels, multi-qubit Pauli channels and qudit Weyl-Heisenberg channels [5].
In this paper, we identify significant operational properties common to all
stochastic channels.

The diamond distance to the identity [6] (hereafter, the diamond distance) is
a metric for quantifying error rates in quantum circuits. The diamond distance
can be expressed as a semidefinite program [7], but no efficient protocols
have been found to measure it directly for arbitrary channels. However, we
prove that the diamond distance of a stochastic channel is its process
infidelity to the identity (theorem 1); moreover, these measures are simply
the probability that an error occurs. The process infidelity to the identity
(hereafter, the process infidelity) can be efficiently estimated via
randomized benchmarking and generalizations thereof [8, 9, 10]. Stochastic
channels are thus a class of channels wherein the difficult problem of
calculating the diamond distance becomes quite easy.

Stochastic channels can be created with low overhead. Twirling over unitary
$t$-designs [11, 12, 13, 14, 15, 16, 17, 18, 19] is an industry standard
technique in quantum information science. A unitary $t$-design is a normalized
measure on the unitary group mimicking its $m{{}^{\text{th}}}$ Haar moments
for all $m\leq t$. The twirl of a quantum channel over a unitary 2-design is a
depolarizing channel with the same process infidelity. In this paper, we
exhibit a similar mechanism with $t=1$. We point out that channel twirling
commutes with the Choi-Jamiołkowski isomorphism (lemma 2). We prove that the
twirl of any quantum channel over any unitary $1$-design is a stochastic
channel with $p$ the process fidelity of the input channel (theorem 3). We
emphasize that noise can be tailored as such without resorting to some
potentially higher unitary $t$-designs, e.g. Clifford groups [20]. The twirl
of a quantum channel over a unitary 1-design, however, depends on the chosen
unitary 1-design. Many unitary 1-designs exist in all finite dimensions [21].

A natural question is whether or not the class of stochastic channels extends
beyond the class of mixed unitary channels implemented by unitary operator
bases containing the identity. Indeed, the twirl of a quantum channel over a
unitary 1-design is unital (theorem 4). We nevertheless establish a positive
answer. We define a noisy rank-2 state preparation channel for a system of two
qubits that is stochastic but not unital. We also prove that all stochastic
qubit channels are unital (theorem 5).

The balance of this paper is structured as follows. In section II, we recall
preliminary notions from quantum information theory in order to establish
three equivalent definitions of stochastic channels. We then define via Eq. 17
a stochastic channel that is not unital. In section III, we prove that the
diamond distance and process infidelity are equal for arbitrary stochastic
channels. In section IV, we discuss connections between stochastic channels
and unitary 1-designs and we prove lemma 2, theorem 3, and theorem 4. Finally,
in section V we conclude and discuss directions for future research.

##  II Stochastic Channels

We begin by defining stochastic channels, which are informally a noiseless
process probabilistically mixed with orthogonal errors. A formal definition of
a stochastic channel (in particular, the notion of orthogonal channels) is
stated most naturally in terms of the Choi state, which we now review. Let
$\mathpzc{H}$ be a finite-dimensional complex Hilbert space and
$\mathfrak{B}(\mathpzc{H})$ be the set of bounded complex-linear maps from
$\mathpzc{H}$ to itself equipped with the Hilbert-Schmidt inner product,

$\displaystyle\langle\\!\langle
A|B\rangle\\!\rangle\equiv\operatorname{Tr}A^{\dagger}B\quad\forall
A,B\in\mathfrak{B}(\mathpzc{H}).$ (1)

Then $\mathfrak{B}(\mathpzc{H})$ is itself a finite-dimensional complex
Hilbert space and so $\mathfrak{B}(\mathfrak{B}(\mathpzc{H}))$ is well-
defined. Canonically, $\mathpzc{H}$ is the set of complex vectors, that is,
$\mathpzc{H}=\mathbb{C}^{d}$ for some finite positive integer $d$, and
$\mathfrak{B}(\mathpzc{H})$ are $d\times d$ complex matrices. We can represent
the elements of $\mathfrak{B}(\mathbb{C}^{d})$ as vectors using the column-
stacking vectorization map col, and then elements $\Phi$ of
$\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ by their Choi-Jamiołkowski state
(hereafter their Choi state),

$\mathpzc{J}(\Phi)\equiv\frac{1}{d}\Phi\otimes\text{id}_{d}\left(\text{col}(\mathds{1}_{d})\text{col}(\mathds{1}_{d})^{\dagger}\right),$
(2)

where $\mathds{1}_{d}\in\mathfrak{B}(\mathbb{C}^{d})$ is the identity operator
and $\text{id}_{d}$ is the identity element of
$\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$. Applying a singular value
decomposition to $\mathpzc{J}(\Phi)$ and using the fact that the vectorization
map is invertible, we find that there exist suitable vectors
$l_{k},r_{k}\in\mathbb{C}^{d^{2}}$ and matrices
$L_{k},R_{k}\in\mathfrak{B}(\mathbb{C}^{d})$ so that

$\displaystyle\mathpzc{J}(\Phi)=\frac{1}{d}\sum_{k=1}^{d^{2}}l_{k}r_{k}^{\dagger}\equiv\frac{1}{d}\sum_{k=1}^{d^{2}}\text{col}(L_{k}^{\text{T}})\text{col}(R_{k}^{\text{T}})^{\dagger}\text{.}$
(3)

Using the vectorization identity

$\text{col}(ABC)=(C^{\text{T}}\\!\otimes\\!A)\text{col}(B)\quad\forall
A,B,C\in\mathfrak{B}(\mathbb{C}^{d}),$ (4)

we thus see that we can write $\Phi$ in a (non-unique) Kraus form as

$\displaystyle\Phi(X)=\sum_{k=1}^{d^{2}}L_{k}XR_{k}^{\dagger}.$ (5)

Using the vectorization identity

$\operatorname{Tr}A^{\dagger}B=\text{col}(A)^{\dagger}\text{col}(B),$ (6)

the orthogonality of the vectors in the singular value decomposition implies
that for all $j\neq k$, we have

$\displaystyle\operatorname{Tr}L_{j}^{\dagger}L_{k}=\operatorname{Tr}R_{j}^{\dagger}R_{k}=0.$
(7)

For any CPTP map, the Choi state is a positive matrix, so that $L_{k}=R_{k}$
for all $k$ and we can write

$\displaystyle\Phi=\sum_{k=1}^{d^{2}}\text{Ad}_{L_{k}}$ (8)

where for any matrix $A\in\mathfrak{B}(\mathbb{C}^{d})$ we define

$\text{Ad}_{A}:\mathfrak{B}(\mathbb{C}^{d})\longrightarrow\mathfrak{B}(\mathbb{C}^{d})::x\longmapsto
AxA^{\dagger}\text{.}$ (9)

Using the cyclic invariance of the trace, the trace preservation condition on
a CPTP map is

$\sum_{k=1}^{d^{2}}L_{k}^{\dagger}L_{k}=\mathds{1}_{d}\text{.}$ (10)

A stochastic channel is a CPTP map
$\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ such that for some
$0<\lambda\leq 1$

$\mathpzc{J}(\Phi)\text{col}(\mathds{1}_{d})=\lambda\text{col}(\mathds{1}_{d})\text{.}$
(11)

Put otherwise, a stochastic channel is a CPTP map whose set of canonical Kraus
operators contains a strictly positive multiple of the identity. A stochastic
channel is therefore a map

$\Phi:\mathfrak{B}(\mathbb{C}^{d})\longrightarrow\mathfrak{B}(\mathbb{C}^{d})::x\longmapsto\lambda
x+(1-\lambda)\Phi^{\perp}(x)\text{,}$ (12)

where $\Phi^{\perp}\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ is an error
channel that is orthogonal to the identity, that is

$\mathpzc{J}(\Phi^{\perp})\text{col}(\mathds{1}_{d})=0\text{.}$ (13)

The set of stochastic channels is manifestly convex via Eq. 12. The class of
stochastic channels contains channels of known types. A mixed unitary channel
is a quantum channel such that its Kraus decomposition may be taken in terms
of a set of appropriately subnormalized unitary matrices. A mixed unitary
channel is therefore always unital, which is to say that it leaves the
maximally mixed state invariant. Indeed, some quantum channels might not
appear to be mixed unitary at first sight. It turns out that question is
actually NP-Hard [22]. We proceed now by way of an example that our definition
of stochastic channels encompasses channels outside the mixed unitary class.
Define the following $4\times 4$ matrices with $\lambda\in(0,1]$

$\displaystyle A_{1}$
$\displaystyle\equiv\sqrt{\lambda}\mathds{1}_{2}\otimes\mathds{1}_{2}\text{,}$
(14) $\displaystyle A_{2}$
$\displaystyle\equiv\sqrt{1-\lambda}\sigma_{z}\otimes|0\rangle\\!\langle
0|\text{,}$ (15) $\displaystyle A_{3}$
$\displaystyle\equiv\sqrt{1-\lambda}\sigma_{z}\otimes|0\rangle\\!\langle
1|\text{.}$ (16)

where $\sigma_{z}$ is the usual Pauli Z matrix. Defining the quantum channel

$\displaystyle\Phi=\sum_{k=1}^{3}\text{Ad}_{A_{k}},$ (17)

one can easily verify that $\Phi$ is a stochastic quantum channel. One can
check, furthermore, that
$\Phi(\mathds{1}_{4})=\lambda\mathds{1}_{4}+(1-\lambda)\mathds{1}_{2}\otimes
2|0\rangle\langle 0|\neq\mathds{1}_{4}$. $\Phi$ prepares a rank-2 state with
probability $(1-\lambda)$, and with probability $\lambda$ the stochastic
channel $\Phi$ is noiseless. This construction generalizes to all even
dimensions larger than 2. In appendix A we prove that all stochastic qubit
channels are unital. We leave open the question of whether or not all
stochastic channels are unital for $d=3$.

##  III Faithful Diamonds

In this section, we prove that stochastic channels are such that their diamond
distance equals their process infidelity. We recall that the trace norm is
$\|\cdot\|_{1}:\mathfrak{B}(\mathbb{C}^{d})\longrightarrow\mathbb{R}_{\geq
0}::X\longmapsto\mathrm{Tr}\sqrt{X^{\dagger}X}$. The induced trace norm on
superoperators is defined via

$\displaystyle\|\cdot\|_{1}$ $\displaystyle:$
$\displaystyle\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))\longrightarrow\mathbb{R}_{\geq
0}$ (18) $\displaystyle::$
$\displaystyle\Gamma\longmapsto\text{max}\left\\{\|\Phi(X)\|_{1}\;\big{|}\;\|X\|_{1}\leq
1\right\\}\text{.}$

The completely bounded trace norm [5] is also commonly referred to as the
diamond norm [6] and defined as follows

$\displaystyle\|\cdot\|_{\diamond}$ $\displaystyle:$
$\displaystyle\mathfrak{B}\big{(}\mathfrak{B}(\mathbb{C}^{d})\big{)}\longrightarrow\mathbb{R}_{\geq
0}$ (19) $\displaystyle::$
$\displaystyle\Gamma\longmapsto\|\Gamma\otimes\text{id}_{d}\|_{1}\text{.}$

The diamond norm induces for any quantum channel $\Phi$ its diamond distance
from the identity

$r_{\diamond}(\Phi)\equiv\frac{1}{2}\|\Phi-\text{id}_{d}\|_{\diamond}\text{.}$
(20)

The process fidelity to the identity of $\Phi$ is

$\displaystyle F_{e}(\Phi)$
$\displaystyle=\operatorname{Tr}\left(\mathpzc{J}(\text{id}_{d})^{\dagger}\mathpzc{J}(\Phi)\right)$
(21)
$\displaystyle=\operatorname{Tr}\left(\text{col}(\mathds{1}_{d})\text{col}(\mathds{1}_{d})^{\dagger}\mathpzc{J}(\Phi)\right)$
(22)
$\displaystyle=\text{col}(\mathds{1}_{d})^{\dagger}\mathpzc{J}(\Phi)\text{col}(\mathds{1}_{d}).$
(23)

The process infidelity is [23]

$r(\Phi)\equiv 1-F_{e}(\Phi)\text{.}$ (24)

The following theorem generalizes a similar result for generalized Pauli
channels [24]. Note that, in our more general case of interest, the canonical
Kraus operators for an arbitrary stochastic channel need not be unitary.

######  Theorem 1.

Let $\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ be a stochastic
quantum channel. Then

$r_{\diamond}(\Phi)=r(\Phi)\text{.}$ (25)

###### Proof.

For any CPTP map, we have the bound [25]

$\displaystyle r_{\diamond}(\Phi)\geq r(\Phi).$ (26)

Therefore we need only prove that for a stochastic map $\Phi$,

$\displaystyle r_{\diamond}(\Phi)\leq r(\Phi).$ (27)

To prove this, recall that by the definition of a stochastic channel we have

$\displaystyle\Phi=\sum_{k}\text{Ad}_{B_{k}}$ (28)

with $B_{1}=\sqrt{\lambda}\mathds{1}_{d}$ for some $\lambda\in(0,1)$. By the
triangle inequality, we therefore have

$\displaystyle\|\Phi-\text{id}_{d}\|_{\diamond}$
$\displaystyle\leq(1-\lambda)\|\text{id}_{d}\|_{\diamond}+\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}}\|_{\diamond}$
$\displaystyle\leq
1-\lambda+\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}}\|_{\diamond}$ (29)

where we have used the fact that $\|\text{id}_{d}\|_{\diamond}=1$. Now from
the definition of the diamond norm and the triangle inequality, we have

$\displaystyle\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}}\|_{\diamond}$
$\displaystyle=\sup_{\rho}\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}\otimes\mathds{1}_{d}}(\rho)\|_{1}$
$\displaystyle=\sup_{\rho}\sum_{k=2}^{d^{2}}\|\text{Ad}_{B_{k}\otimes\mathds{1}_{d}}(\rho)\|_{1}.$
(30)

For any operator $B$ and any positive operator $\rho$, $B\rho B^{\dagger}$ is
positive and so $\|\text{Ad}_{B}(\rho)\|_{1}=\operatorname{Tr}B\rho
B^{\dagger}=\operatorname{Tr}B^{\dagger}B\rho$ by the cyclic invariance of the
trace. Therefore using Eq. 10, we find that for any CPTP map $\Phi$,

$\displaystyle\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}}\|_{\diamond}$
$\displaystyle\leq\sup_{\rho}\sum_{k=2}^{d^{2}}\operatorname{Tr}(B_{k}^{\dagger}B_{k}\otimes\mathds{1}_{d})\rho$
$\displaystyle=\sup_{\rho}\operatorname{Tr}\big{(}(\mathds{1}_{d}-B_{1}^{\dagger}B_{1})\otimes\mathds{1}_{d}\big{)}\rho.$
(31)

For a stochastic channel, $B_{1}=\sqrt{\lambda}\mathds{1}_{d}$ and so

$\displaystyle\|\sum_{k=2}^{d^{2}}\text{Ad}_{B_{k}}\|_{\diamond}$
$\displaystyle\leq 1-\lambda.$ (32)

We thus have

$\displaystyle r_{\diamond}(\Phi)=1-\lambda.$ (33)

∎

##  IV Twirling

We now show that unitary 1-designs can be used to construct stochastic
channels. We also show that there exist stochastic channels that cannot be
constructed using a unitary 1-design. Let $\mathsf{U}(\mathpzc{H})$ be the
unitary group on a Hilbert space $\mathpzc{H}$. Let
$\mu:\Sigma(\mathsf{U}(\mathbb{C}^{d}))\longrightarrow\mathbb{R}_{\geq 0}$ be
a normalized measure on the unitary group $\mathsf{U}(\mathbb{C}^{d})$, where
$\Sigma(\mathsf{U}(\mathbb{C}^{d}))$ denotes the Borel $\sigma$-algebra of
$\mathsf{U}(\mathbb{C}^{d})$ [26]. Then the $\mu$-twirl of a map
$\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ is the map

$\Phi_{\mu}=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)\mathrm{Ad}_{u}\circ\Phi\circ\mathrm{Ad}_{u^{\dagger}}.$
(34)

######  Lemma 2.

Let $\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$. Let $\mu$ be a
measure on the unitary group. Then

$\mathpzc{J}\left(\Phi_{\mu}\right)=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)\mathrm{Ad}_{u\otimes\bar{u}}\left(\mathpzc{J}(\Phi)\right).$
(35)

###### Proof.

Let $L_{k}$ and $R_{k}$ satisfy Eq. 5. For fixed elements
$u,v\in\mathsf{U}(\mathbb{C}^{d})$,

$\displaystyle\Phi^{\prime}=\text{Ad}_{u}\circ\Phi\circ\text{Ad}_{v}$ (36)

can be written in the form

$\displaystyle\Phi^{\prime}(X)=\sum_{k=1}^{d^{2}}uL_{k}vXv^{\dagger}R_{k}^{\dagger}u^{\dagger}.$
(37)

By eqns 3 and 4, we thus have

$\displaystyle\mathpzc{J}(\Phi^{\prime})$
$\displaystyle=\frac{1}{d}\sum_{k=1}^{d^{2}}\text{col}(v^{\text{T}}L_{k}^{\text{T}}u^{\text{T}})\text{col}(\bar{u}R_{k}^{\text{T}}\bar{v})^{\dagger}$
$\displaystyle=\frac{1}{d}u\otimes
v^{\text{T}}\left(\sum_{k=1}^{d^{2}}\text{col}(L_{k}^{\text{T}})\text{col}(R_{k}^{\text{T}})^{\dagger}\right)(v^{\dagger}\otimes\bar{u})^{\dagger}$
$\displaystyle=(u\otimes v^{\text{T}})\mathpzc{J}(\Phi^{\prime})(v\otimes
u^{\text{T}}).$ (38)

Setting $v=u^{\dagger}$ and averaging over $\mu$ completes the proof. ∎

There are many equivalent definitions of unitary 1-designs [12][18]. A unitary
$1$-design (in dimension $d$) is a normalized measure $\mu$ on the unitary
group such that

$\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)\text{Ad}_{u}=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu_{\text{H}}(u)\text{Ad}_{u}\text{,}$
(39)

where $\mu_{\text{H}}$ is the normalized Haar measure on
$\mathsf{U}(\mathbb{C}^{d})$ [27]. We remark that

$\displaystyle\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)uXu^{\dagger}=\frac{\mathrm{Tr}(X)}{d}\mathds{1}_{d}\text{,}$
(40)
$\displaystyle\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)u\otimes\bar{u}=\frac{1}{d}\mathrm{col}(\mathds{1}_{d})\mathrm{col}(\mathds{1}_{d})^{\dagger}\text{,}$
(41)

are then consequences of Schur’s lemma [28] where
$X\in\mathfrak{B}(\mathbb{C}^{d})$.

The close connection between unitary 1-designs and irreducible unitary
representations (in dimension $d$) [21] means that many unitary 1-designs
exist. We especially recommend Bannai et al. [29] for explicit examples. The
GAP system [30] could also be harvested. The Pauli group on a single qubit is
a unitary 1-design. The multipartite Weyl-Heisenberg group [31] is a unitary
1-design for any collection of quantum systems. Another example is always the
Clifford group [20].

We now prove that the twirl of a quantum channel over a unitary 1-design is a
stochastic quantum channel. Our proof reveals that the resulting stochastic
quantum channel is noiseless with probability the process fidelity of the
input channel, which underlies protocols that are designed to estimate the
process fidelity [32] or suppress coherent errors [33].

######  Theorem 3.

Let $\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ be quantum channel
with nonzero process fidelity. Let $\mu$ be a unitary $1$-design. Then
$\Phi_{\mu}$ is a stochastic channel.

###### Proof.

We shall prove that $\text{col}(\mathds{1}_{d})$ is an eigenvector of
$\mathpzc{J}(\Phi_{\mu})$ with a strictly positive eigenvalue, hence the
identifications in section II yield the result. We calculate

$\displaystyle\hskip
4.26773pt\mathpzc{J}(\Phi_{\mu})\text{col}(\mathds{1}_{d})$
$\displaystyle=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)u\otimes\bar{u}\mathpzc{J}(\Phi)u^{\dagger}\otimes
u^{\text{T}}\text{col}(\mathds{1}_{d})$ (by lemma 2)
$\displaystyle=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)u\otimes\bar{u}\mathpzc{J}(\Phi)\text{col}(\mathds{1}_{d})$
(by Eq. 6)
$\displaystyle=\frac{1}{d}\mathrm{col}(\mathds{1}_{d})\mathrm{col}(\mathds{1}_{d})^{\dagger}\mathpzc{J}(\Phi)\text{col}(\mathds{1}_{d})$
(by Eq. 41) $\displaystyle=F_{e}(\Phi)\text{col}(\mathds{1}_{d})\text{,}$

where the final equality follows from Eq. 21. ∎

The example defined via Eqs. (14)-(17) is not a unital stochastic channel. In
light of the foregoing example and the following theorem, we conclude that
some stochastic channels cannot be accessed via the twirl of any unitary
1-design.

######  Theorem 4.

Let $\Phi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{d}))$ be quantum channel.
Let $\mu$ be a unitary $1$-design. Then $\Phi_{\mu}$ is unital.

###### Proof.

We calculate

$\displaystyle\Phi_{\mu}(\mathds{1}_{d})$
$\displaystyle=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)\text{Ad}_{u}\circ\Phi\circ\text{Ad}_{u^{\dagger}}(\mathds{1}_{d})$
(by Eq. 34)
$\displaystyle=\int_{\mathsf{U}(\mathbb{C}^{d})}d\mu(u)\text{Ad}_{u}(\Phi(\mathds{1}_{d}))$
(by Eq. 9)
$\displaystyle=\frac{\operatorname{Tr}\Phi(\mathds{1}_{d})}{d}\mathds{1}_{d}$
(by Eq. 40) $\displaystyle=\mathds{1}_{d}\text{,}$

where the final step follows as $\Phi$ is trace preserving. ∎

##  V Conclusion

We have considered properties of the class of stochastic channels. We proved
that the diamond distance of a stochastic channel is its process infidelity
(theorem 1). We illuminated the relationship between stochastic channels and
unitary 1-designs. In general, we pointed out that channel twirling commutes
with the Choi-Jamiołkowski isomorphism (lemma 2). We then demonstrated a way
to access stochastic channels. The twirl of a quantum channel over a unitary
1-design is a stochastic channel (theorem 3). Next, we proved that twirling
over unitary 1-designs results in unital channels (theorem 4.) We provided an
example of a stochastic channel that is not unital. In appendix A, we prove
that all qubit stochastic channels are unital (theorem 5).

There are some potential avenues for future research into designing stochastic
channels. One open question is whether or not a quantum channel $\Phi$ exists
outside the class of stochastic channels but with
$r_{\diamond}(\Phi)=r(\Phi)$. Another open question is whether or not every
qutrit stochastic channel is a mixed unitary channel. Future work might also
explore effects obtained via twirling over different unitary 1-designs. We
observe from our proof of theorem 3 that twirling a quantum channel over any
unitary 1-design results in a stochastic channel whose canonical Kraus
operators can depend on the unitary 1-design in question. For instance,
consider a group unitary operator basis $\mathsf{B}$ and then its conjugation
via any element of the unitary group, $\text{Ad}_{U}(\mathsf{B})$. The twirl
with respect to the latter is a rotated version of the former. The difference
between the two twirled channels is subtle and can be missed, e.g., in [33],
however, since both channels have the same diamond distance and process
fidelity, the difference is typically unimportant.

Finally, the set of stochastic channels is convex. A natural avenue for future
research is to determine the shape of this convex set.

##  Appendix A Stochastic Qubit Channels

In this appendix we prove that all stochastic qubit channels are unital.

######  Theorem 5.

Let $\Xi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{2}))$ be a stochastic quantum
channel. Then $\Xi$ is unital.

###### Proof.

Any completely positive and trace preserving map
$\Xi\in\mathfrak{B}(\mathfrak{B}(\mathbb{C}^{2}))$ can be decomposed as
$\Xi=\text{Ad}_{u}\circ\Phi\circ\text{Ad}_{v}$ with $u,v\in\mathsf{SU}(2)$ and
where $\mathpzc{J}(\Phi)$ is the following matrix

$\small{\begin{pmatrix}1+\eta_{z}+\kappa_{z}&0&\kappa_{x}+i\kappa_{y}&\eta_{x}+\eta_{y}\\\
0&1-\eta_{z}+\kappa_{z}&\eta_{x}-\eta_{y}&\kappa_{x}+i\kappa_{y}\\\
\kappa_{x}-i\kappa_{y}&\eta_{x}-\eta_{y}&1-\eta_{z}-\kappa_{z}&0\\\
\eta_{x}+\eta_{y}&\kappa_{x}-i\kappa_{y}&0&1+\eta_{z}-\kappa_{z}\end{pmatrix}}$
(42)

with parameters $\eta_{x},\eta_{y},\eta_{z}$ characterizing the distortion of
the Bloch ball, and $\kappa_{x},\kappa_{y},\kappa_{z}$ defining the Bloch
sphere coordinates of the action of $\Phi$ on $\mathds{1}_{2}/2$ [34]. In
particular, $\kappa_{x}=\kappa_{y}=\kappa_{z}=0$ if and only if $\Phi$ is
unital. There are constraints on the foregoing parameters in order to ensure
completely positivity and trace preservation. We shall, however, need only the
facts $\kappa_{x},\kappa_{y},\kappa_{z}\in\mathbb{R}$. By section IV, the Choi
state of $\Xi$ is

$\displaystyle\mathpzc{J}(\Xi)=(u\otimes
v^{\text{T}})\mathpzc{J}(\Phi)(v\otimes u^{\text{T}})\text{.}$ (43)

If $\Xi$ is stochastic, then there exists some $\lambda>0$ such that

$\displaystyle\mathpzc{J}(\Xi)\text{col}(\mathds{1}_{2})=\text{col}(\mathds{1}_{2})\lambda,$
(44)

or, multiplying both sides by $(u\otimes v^{\text{T}})^{\dagger}$ and using
Eq. 4 gives

$\displaystyle\mathpzc{J}(\Phi)\text{col}((vu)^{\text{T}})=\lambda\text{col}(\overline{vu}).$
(45)

$\mathsf{SU}(2)$ is a group, so we can write $(vu)^{\text{T}}$ as

$(vu)^{\text{T}}=\small{\begin{pmatrix}\hskip 6.544pt\gamma_{1}&\gamma_{2}\\\
-\overline{\gamma_{2}}&\overline{\gamma_{1}}\end{pmatrix}}\text{,}$ (46)

with $\gamma_{1},\gamma_{2}\in\mathbb{C}$ such that
$|\gamma_{1}|^{2}+|\gamma_{2}|^{2}=1$. We then have

$\displaystyle\mathpzc{J}(\Phi)\begin{pmatrix}\gamma_{1}\\\
-\overline{\gamma_{2}}\\\ \gamma_{2}\\\
\overline{\gamma_{1}}\end{pmatrix}=\lambda\begin{pmatrix}\overline{\gamma_{1}}\\\
\overline{\gamma_{2}}\\\ -\gamma_{2}\\\ \gamma_{1}\end{pmatrix}.$ (47)

Subtracting the conjugate of the fourth equation and using the fact that the
parameters of $\mathpzc{J}(\Phi)$ are real gives

$\displaystyle-\kappa_{z}\gamma_{1}=(\kappa_{x}+i\kappa_{y})\gamma_{2}.$ (48)

Similarly, adding the conjugate of the second equation to the third equation
and using the fact that the parameters of $\mathpzc{J}(\Phi)$ are real gives

$\displaystyle\kappa_{z}\gamma_{2}=(\kappa_{x}-i\kappa_{y})\gamma_{1}.$ (49)

If $\gamma_{1}=0$ then Eq. 48 yields $\kappa_{x}=-i\kappa_{y}$, whence by the
reality of $\kappa_{x}$ and $\kappa_{y}$ we have $\kappa_{x}=\kappa_{y}=0$ and
then by Eq. 49 we conclude that $\kappa_{z}=0$. If instead $\gamma_{2}=0$ then
we reach the same conclusion. If both $\gamma_{1}$ and $\gamma_{2}$ are
nonzero then we multiply Eq. 48 and Eq. 49 to find

$\displaystyle-\kappa_{z}^{2}\gamma_{1}\gamma_{2}$
$\displaystyle=(\kappa_{x}-i\kappa_{y})\gamma_{1}(\kappa_{x}+i\kappa_{y})\gamma_{2}$
$\displaystyle=(\kappa_{x}^{2}+\kappa_{y}^{2})\gamma_{1}\gamma_{2},$ (50)

and so we must have $\kappa_{x}=\kappa_{y}=\kappa_{z}=0$ and we conclude again
that $\Xi$ is unital. ∎

###### Acknowledgements.

We would like to thank Arnaud Carignan-Dugas and Raymond Laflamme for some
illuminating discussions. MG acknowledges support from NSERC DG2018-03968.
This research was undertaken thanks in part to funding from the Canada First
Research Excellence Fund, the Government of Ontario, and the Government of
Canada through NSERC.

## References

  * Preskill [2018] J. Preskill, Quantum 2, 79 (2018).
  * Cao  _et al._ [2021] N. Cao, J. Lin, D. Kribs, Y.-T. Poon, B. Zeng, and R. Laflamme, arXiv preprint arXiv:2111.02345 (2021).
  * Carignan-Dugas [2019] A. Carignan-Dugas,  _A walk through quantum noise: a study of error signatures and characterization methods_ , Ph.D. thesis, University of Waterloo (2019).
  * Wallman [2015] J. J. Wallman, arXiv preprint arXiv:1511.00727 (2015).
  * Watrous [2018] J. Watrous,  _The theory of quantum information_ (Cambridge University Press, 2018).
  * Kitaev [1997] A. Y. Kitaev, Uspekhi Matematicheskikh Nauk 52, 53 (1997).
  * Watrous [2009] J. Watrous, Theory Of Computing 5, 217 (2009).
  * Emerson  _et al._ [2005] J. Emerson, R. Alicki, and K. Życzkowski, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005).
  * Wallman [2018] J. J. Wallman, Quantum 2, 47 (2018).
  * Helsen  _et al._ [2020] J. Helsen, I. Roth, E. Onorati, A. H. Werner, and J. Eisert, arXiv preprint arXiv:2010.07974 (2020).
  * DiVincenzo  _et al._ [2002] D. DiVincenzo, D. Leung, and B. Terhal, IEEE Transactions on Information Theory 48, 580 (2002).
  * Dankert [2005] C. Dankert,  _Efficient simulation of random quantum states and operators_ , Master’s thesis, University of Waterloo, Waterloo, Canada (2005).
  * Dankert  _et al._ [2009] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Physical Review A 80, 012304 (2009).
  * Cleve  _et al._ [2015] R. Cleve, D. Leung, L. Liu, and C. Wang, arXiv preprint arXiv:1501.04592 (2015).
  * Zhu  _et al._ [2016] H. Zhu, R. Kueng, M. Grassl, and D. Gross, arXiv preprint arXiv:1609.08172 (2016).
  * Webb [2016] Z. Webb, Quantum Information and Computation 16, 1379 (2016).
  * Zhu [2017] H. Zhu, Physical Review A 96, 062336 (2017).
  * Haferkamp  _et al._ [2020] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, arXiv preprint arXiv:2002.09524 (2020).
  * Skanes-Norman [2021] J. Skanes-Norman,  _On Realistic Errors in Quantum Computers_ , Master’s thesis, University of Waterloo (2021).
  * Graydon  _et al._ [2021] M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, arXiv preprint arXiv:2108.04200 (2021).
  * Roy and Scott [2009] A. Roy and A. J. Scott, Designs, codes and cryptography 53, 13 (2009).
  * Lee and Watrous [2020] C. D.-Y. Lee and J. Watrous, Quantum 4, 253 (2020).
  * Fuchs and Van De Graaf [1999] C. A. Fuchs and J. Van De Graaf, IEEE Transactions on Information Theory 45, 1216 (1999).
  * Magesan  _et al._ [2012] E. Magesan, J. M. Gambetta, and J. Emerson, Physical Review A 85, 042311 (2012).
  * Wallman and Flammia [2014] J. J. Wallman and S. T. Flammia, New Journal of Physics 16, 103032 (2014).
  * Halmos [2013] P. R. Halmos,  _Measure theory_ , Vol. 18 (Springer, 2013).
  * Knapp [2016] A. W. Knapp,  _Representation theory of semisimple groups_ (Princeton university press, 2016).
  * Kirillov [1976] A. Kirillov,  _Elements of the theory of representations_ (Springer, 1976).
  * Bannai  _et al._ [2019] E. Bannai, M. Nakahara, D. Zhao, and Y. Zhu, Journal of Physics A: Mathematical and Theoretical 52, 495301 (2019).
  * Group [2021] T. G. Group, Gap – groups, algorithms, and programming, version 4.11.1 (2021).
  * Appleby [2005] D. M. Appleby, Journal of Mathematical Physics 46 (2005).
  * Erhard  _et al._ [2019] A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Nature communications 10, 1 (2019).
  * Wallman and Emerson [2016] J. J. Wallman and J. Emerson, Physical Review A 94, 052325 (2016).
  * Bengtsson and Życzkowski [2017] I. Bengtsson and K. Życzkowski, arXiv preprint arXiv:1701.07902 (2017).