# On unital qubit channels

Man-Duen Choi and Chi-Kwong Li The research of Li was supported by the Simons
Foundation Grant 851334.

###### Abstract

A canonical form for unital qubit channels under local unitary transforms is
obtained. In particular, it is shown that the eigenvalues of the Choi matrix
of a unital quantum channel are complete invariant of the canonical form. It
follows immediately that every unital qubit channel is the average of four
unitary channels. More generally, a unital qubit channel can be expressed as
the convex combination of unitary channels with convex coefficients
$p_{1},\dots,p_{m}$ as long as $(p_{1},\dots,p_{m})$ is majorized by the
vector of eigenvalues of the Choi matrix of the channel. By the fact that
every unital map $\Phi$ on $2\times 2$ matrices preserving trace and Hermitian
matrices corresponds to a linear map $\phi$ on ${\mathbb{R}}^{3}$ that will
transform the Bloch (unit) sphere in ${\mathbb{R}}^{3}$ to an ellipsoid ${\cal
E}$, it is shown that $\Phi$ is a unital qubit channel if and only if $\phi$
is a convex combination of special orthogonal matrices.

In memory of Professor Chandler Davis.

AMS Classification. 81Q10, 47B65, 15B57.

Keywords. Unital qubit channel, completely positive linear map, Choi matrix.

##  1 Introduction

In the mathematical framework of quantum mechanics, quantum states are density
matrices, i.e., positive semidefinite matrices with trace one, and quantum
channels (operations) are trace preserving completely positive linear maps
$\Phi:M_{n}\rightarrow M_{m}$, where $M_{k}$ denotes the set of $k\times k$
complex matrices, admitting the operator sum representation

$\Phi(A)=\sum_{j=1}^{r}F_{j}AF_{j}^{*}\qquad\hbox{ for all }A\in M_{n}$ (1.1)

for some $m\times n$ matrices $F_{1},\dots,F_{r}$ satisfying
$\sum_{j=1}^{r}F_{j}^{*}F_{j}=I_{n}$.

For a linear map $\Phi:M_{n}\rightarrow M_{m}$, its Choi matrix is defined as

$C(\Phi)=(\Phi(E_{ij}))_{1\leq i,j\leq
n}=\sum_{i,j}E_{ij}\otimes\Phi(E_{ij})\in M_{n}(M_{m})\equiv M_{n}\otimes
M_{m},$

where $\\{E_{ij}:1\leq i,j\leq n\\}$ is the standard basis for $M_{n}$. By a
result in [1], a linear map $\Phi:M_{n}\rightarrow M_{m}$ is a quantum channel
if and only if $C(\Phi)$ is positive semidefinite and ${\rm
tr}\,\Phi(E_{ij})=\delta_{ij}$, the Kronecker delta.

We say that two quantum channels $\Phi,\Psi:M_{n}\rightarrow M_{m}$ are
unitarily equivalent if there are unitary matrices $U\in M_{n}$ and $V\in
M_{m}$ such that $\Phi(A)=V\Psi(UAU^{*})V^{*}$ for all $A\in M_{n}$. It turns
out that this happens if and only if $C(\Phi)=(U^{t}\otimes
V)C(\Psi)(U^{t}\otimes V)^{*}$, where $X^{t}$ denotes the transpose of the
matrix $X$. This follows readily from the following lemma.

######  Lemma 1.1.

Let $\Phi:M_{n}\rightarrow M_{k}$ be a linear map. Then for any $R,S\in
M_{n}$,

$\sum_{i,j}E_{ij}\otimes\Phi(RE_{ij}S)=\sum_{i,j}R^{t}E_{ij}S^{t}\otimes\Phi(E_{ij}).$

Proof. Let $R=(r_{ij})=\sum r_{ij}E_{ij}$ and $S=(s_{ij})=\sum s_{ij}E_{ij}$.
Then

$\displaystyle\sum_{i,j}E_{i,j}\otimes\Phi(RE_{ij}S)=\sum_{i,j,p,q}E_{ij}\otimes\Phi(r_{pi}s_{jq}E_{pq})$
$\displaystyle=$
$\displaystyle\sum_{i,j,p,q}r_{pi}s_{jq}E_{ij}\otimes\Phi(E_{p,q})=\sum_{p,q}R^{t}E_{pq}S^{t}\otimes\Phi(E_{p,q}),$

which is the same as $\sum_{i,j}R^{t}E_{ij}S^{t}\otimes\Phi(E_{ij})$.

By Lemma 1.1 that if $\Phi(A)=V\Psi(UAU^{*})V^{*}$ for all $A\in M_{n}$, then

$\displaystyle C(\Phi)$ $\displaystyle=$ $\displaystyle(I\otimes
V)(\sum_{ij}E_{ij}\otimes\Psi(UE_{ij}U^{*}))(I\otimes V)^{*}$ $\displaystyle=$
$\displaystyle(I_{n}\otimes V)(U^{t}\otimes I_{m})C(\Psi)(U^{t}\otimes
I_{m})^{*}(I_{n}\otimes V)^{*}$ $\displaystyle=$ $\displaystyle(U^{t}\otimes
V)C(\Psi)(U^{t}\otimes V)^{*}.$

In this paper, we make use of the above observation to show that if
$\Phi:M_{2}\rightarrow M_{2}$ is a unital qubit channel such that $C(\Phi)$
has eigenvalues $\lambda_{1}\geq\cdots\geq\lambda_{4}\geq 0$ summing up to
two, then there are unitary matrices $U,V\in M_{2}$ such that the map
$A\mapsto V^{*}\Phi(U^{*}AU)V$ has the form

$A\mapsto\frac{1}{2}(\lambda_{1}A+\lambda_{2}XAX+\lambda_{3}YAY+\lambda_{4}ZAZ),$

where

$X=\begin{pmatrix}0&1\cr 1&0\cr\end{pmatrix},\quad Y=\begin{pmatrix}0&-i\cr
i&0\cr\end{pmatrix},\quad\hbox{ and }\quad Z=\begin{pmatrix}1&0\cr
0&-1\cr\end{pmatrix}$ (1.2)

are the Pauli matrices. We then deduce that $\Phi$ can be written as the
average of four unitary channels, i.e., maps of the form $A\mapsto UAU^{*}$
for some unitary $U$. More generally, suppose $C(\Phi)$ has rank $k$, i.e.,
$k$ positive eigenvalues $\lambda_{1},\dots,\lambda_{k}$, and
$p_{1},\dots,p_{m}$ are positive numbers summing up to 1. Then there are
unitary $V_{1},\dots,V_{m}$ such that $\Phi$ has the form
$A\mapsto\sum_{j=1}^{m}p_{j}V_{j}AV_{j}^{*}$ if and only if $m\geq k$ and
$(p_{1},\dots,p_{m})$ is majorized by
$\frac{1}{2}(\lambda_{1},\dots,\lambda_{k},\underbrace{0,\dots,0}_{m-k})$; see
Section 3 for the definition of majorization.

Every unital map $\Phi$ on $M_{2}$ preserving trace and Hermitian matrices
corresponds to a linear map $\phi$ on ${\mathbb{R}}^{3}$ defined by
$\phi(x,y,z)=(\hat{x},\hat{y},\hat{z})$ if

$\Phi(xX+yY+zZ)=\hat{x}X+\hat{y}Y+\hat{z}Z.$

The structure of $\phi$ is completely determined by its effect on the Bloch
(unit) sphere

${\cal B}=\\{(x,y,z):x,y,z\in{\mathbb{R}},\
x^{2}+y^{2}+z^{2}=1\\}\subseteq{\mathbb{R}}^{3}.$

In our discussion, both ${\mathbb{R}}^{1\times 3}$ and ${\mathbb{R}}^{3\times
1}$ will be referred to as ${\mathbb{R}}^{3}$. Researchers have used such
correspondence to study qubit channels; see [3, 4, 7] and their references.
Clearly, a linear map on ${\mathbb{R}}^{3}$ will transform the Bloch (unit)
sphere in ${\mathbb{R}}^{3}$ to an ellipsoid ${\cal E}$. We will show that a
unital linear map $\Phi$ on $M_{2}$ is a qubit channel if and only if the
corresponding linear map $\phi$ on ${\mathbb{R}}^{3}$ is a convex combination
of elements in $SO(3)$, the group of special orthogonal matrices. One can use
basic information of the ellipsoid ${\cal E}=\phi({\cal B})$ to determine
whether $\phi$ lies in the convex hull of $SO(3)$ so that $\Phi$ is a unital
qubit channel. Geometrical features of the convex hull of $SO(3)$ can be
deduced from those of the set unital qubit channels, and vice versa.

##  2 A canonical form of unital qubit channels

We begin with the following simple lemma.

######  Lemma 2.1.

Let $\Phi:M_{2}\rightarrow M_{2}$ be a unital linear map that preserve trace
and Hermitian matrices. Then there are unitary $U,V\in M_{2}$ such that the
map $A\mapsto V^{*}\Phi(UAU^{*})V$ has Choi matrix of the form

$\begin{pmatrix}a&0&0&b\cr 0&1-a&c&0\cr 0&c&1-a&0\cr b&0&0&a\cr\end{pmatrix}\
\hbox{ with }\quad b,c\geq 0\ \hbox{ and }\ 2a\geq 1+b+c.$ (2.1)

Proof. Let $P_{0}\in M_{2}$ be a rank one orthogonal projection such that

$\|\Phi(P_{0})\|=\max\\{\|\Phi(P)\|:P\hbox{ is a rank one orthogonal
projection}\\}.$

Then there are unitary $U,V\in M_{2}$ such that $P_{0}=UE_{11}U^{*}$ and
$\Phi(P_{0})=V{\rm diag}\,(a,1-a)V^{*}$ with $\|\Phi(P_{0})\|=|a|$. So,
$|a|\geq|1-a|$ implies that $a\geq 1/2$. Note that the map
$\hat{\Phi}:A\mapsto V^{*}\Phi(UAU^{*})V$ also preserves traces and Hermitian
matrices; its Choi matrix $C(\hat{\Phi})$ has the form

$\begin{pmatrix}a&0&d&b\mu_{1}\cr
0&1-a&c\mu_{2}&-d\cr\bar{d}&c\bar{\mu}_{1}&1-a&0\cr
b\bar{\mu}_{2}&-\bar{d}&0&a\cr\end{pmatrix}\quad\hbox{ with
}d,\mu_{1},\mu_{2}\in{\mathbb{C}},\ |\mu_{1}|=|\mu_{2}|=1,\hbox{ and }b,c\geq
0.$

We claim that $d=0$. If not, we may let $P=Uww^{*}U^{*}$ with
$w=(\cos\theta,\sin\theta|d|/d)^{t}$. Then the $(1,1)$ entry of $\Phi(P)$
equals

$w^{*}\begin{pmatrix}a&d\cr\bar{d}&1-a\cr\end{pmatrix}w=a\cos^{2}\theta+2\sin\theta\cos\theta|d|+\sin^{2}\theta>a$

if $\cos\theta>(2a-1)\sin\theta/|d|>0$, which is a contraction. Thus, $d=0$ as
asserted.

Let $D_{1}=I_{2}\otimes{\rm diag}\,(e^{i\theta},e^{-i\theta})$ and $D_{2}={\rm
diag}\,(e^{i\phi},1)\otimes I_{2}$ such that $\mu_{1}e^{2i\theta}$ and
$\mu_{2}e^{-2i\theta}$ have the same argument $-\phi$. We may further replace
$\hat{\Phi}$ by the map $\tilde{\Phi}:T\mapsto
D_{2}(\hat{\Phi}(D_{1}TD_{1}^{*}))D_{2}^{*},$ whose Choi matrix
$C(\tilde{\Phi})$ has the asserted form (2.1) with $b,c\geq 0$.

To prove that $2a\geq 1+b+c$, consider $P=\frac{1}{2}\begin{pmatrix}1&1\cr
1&1\cr\end{pmatrix}$. Then $\tilde{\Phi}(P)=\frac{1}{2}\begin{pmatrix}1&b+c\cr
b+c&1\cr\end{pmatrix}$ so that

$a=\|\Phi(P_{0})\|\geq\|\tilde{\Phi}(P)\|=\frac{1}{2}(1+b+c).$

Hence, $a-b\geq 1-a+c$.

Applying the above lemma to unital qubit channels, we have the following.

######  Theorem 2.2.

Let $\Phi:M_{2}\rightarrow M_{2}$ be a unital linear map that preserve trace
and Hermitian matrices. Then the Choi matrix of $\Phi$ has real eigenvalues
summing up to two. If the eigenvalues are
$\lambda_{1}\geq\cdots\geq\lambda_{4}$, then there are unitary $U,V\in M_{2}$
such that the map $A\mapsto V^{*}\Phi(UAU^{*})V$ has the form

$A\mapsto\frac{1}{2}(\lambda_{1}A+\lambda_{2}ZAZ+\lambda_{3}XAX+\lambda_{4}YAY),$
(2.2)

where $X,Y,Z$ are the Pauli matrices described in (1.2), and its Choi matrix
has the form

$\frac{1}{2}{\small\begin{pmatrix}\lambda_{1}+\lambda_{2}&0&0&\lambda_{1}-\lambda_{2}\cr
0&\lambda_{3}+\lambda_{4}&\lambda_{3}-\lambda_{4}&0\cr
0&\lambda_{3}-\lambda_{4}&\lambda_{3}+\lambda_{4}&0\cr\lambda_{1}-\lambda_{2}&0&0&\lambda_{1}+\lambda_{2}\cr\end{pmatrix}}.$
(2.3)

In particular, $\Phi$ is a unital qubit channel if and only if
$\lambda_{4}\geq 0$.

Proof. By Lemma 2.1, there are unitary $U,V\in M_{2}$ such that the Choi
matrix of the map $A\mapsto V^{*}\Phi(U^{*}AU)V$ has the form (2.1). Since
$b,c\geq 0$ and $2a+1\geq 1+b+c$, we see that

$(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(a+b,a-b,1-a+c,1-a-c).$

Thus, the map $A\mapsto V^{*}\Phi(U^{*}AU)V$ has the form (2.2) with Choi
matrix (2.3).

The last assertion follows from the fact that $\Phi$ is a unital qubit channel
if and only if its Choi matrix is positive semidefinite.

By Theorem 2.2, we see that the four eigenvalues of the Choi matrix of a
unital qubit channel form the complete invariant under unitary equivalence.

######  Corollary 2.3.

Let $\Phi,\Psi:M_{2}\rightarrow M_{2}$ be unital linear maps that preserve
trace and Hermitian matrices. Then $\Phi$ and $\Psi$ are unitarily equivalent
if and only if $C(\Phi)$ and $C(\Psi)$ have same eigenvalues (counting
multiplicity). In particular, two unital qubit channels are unitarily
equivalent if and only if their Choi matrices have the same eigenvalues.

Proof. The if part follows from Theorem 2.2. The only if part follows from
Lemma 1.1.

Several remarks are in order. First, up to unitary equivalence the Choi matrix
always has the form (2.3), and the operator sum representation has the form
(2.2).

Second, the roles of $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ in
(2.2) are symmetric because a unital qubit channel with Choi matrix

$\frac{1}{2}{\small\begin{pmatrix}\lambda_{j_{1}}+\lambda_{j_{2}}&0&0&\lambda_{j_{1}}-\lambda_{j_{2}}\cr
0&\lambda_{j_{3}}+\lambda_{j_{4}}&\lambda_{j_{3}}-\lambda_{j_{4}}&0\cr
0&\lambda_{j_{3}}-\lambda_{j_{4}}&\lambda_{j_{3}}+\lambda_{j_{4}}&0\cr\lambda_{j_{1}}-\lambda_{j_{2}}&0&0&\lambda_{j_{1}}+\lambda_{j_{2}}\cr\end{pmatrix}}$

for any permutation $(j_{1},j_{2},j_{3},j_{4})$, the Choi matrix has
eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$. By
Theorem 2.2, it is equivalent to the channel with Choi matrix (2.3). In the
following, we will often consider linear map on $M_{2}$ of the form
$A\mapsto\mu_{1}A+\mu_{2}XAX+\mu_{3}YAY+\mu_{4}ZAZ$; its Choi matrix will have
eigenvalues $2\mu_{1},\dots,2\mu_{4}$.

Third, under unitary similarity, the eigenvalues are the complete invariant of
Hermitian matrices. In general, two Hermitian matrices in $M_{n}(M_{m})$ with
the same eigenvalues may not be similar under a untiary matrix of the form
$U\otimes V$. It is interesting that the Choi matrices of two unital qubit
channels are unitarily similar if and only if they are unitarily similar via a
matrix of the form $U\otimes V$.

######  Example 2.4.

Define $\Lambda:M_{2}\rightarrow M_{2}$ by $\Lambda(A)=\frac{1}{2}({\rm
tr}\,A)I$ for all $A\in M_{2}$. Then $\Lambda$ is the best known unital quibit
channel satisfying $U\Lambda(VAV^{*})U^{*}=\Lambda(A)$ for all unitary $U,V\in
M_{2}$. Moreover, $\Lambda(A)=\frac{1}{4}(A+XAX+YAY+ZAZ)$, and the Choi matrix
$C(\Lambda)=\frac{1}{2}I_{4}$ with four equal eigenvalues.

Using this example and the previous remark, we can prove the following.

######  Corollary 2.5.

Let $\Phi,\Psi:M_{2}\rightarrow M_{2}$ be linear maps of the form

$A\mapsto\mu_{1}A+\mu_{2}XAX+\mu_{3}YAY+\mu_{4}ZAZ\ \hbox{ and }\
A\mapsto\nu_{1}A+\nu_{2}XAX+\nu_{3}YAY+\nu_{4}ZAZ,$

respectively, where $\mu_{j},\nu_{j}\in{\mathbb{R}}$ for $j=1,\dots,4$, and
$X,Y,Z$ are the Pauli matrices defined as in (1.2). Then $\Phi$ and $\Psi$ are
unitarily equivalent if and only if $(\mu_{1},\dots,\mu_{4})$ is a permutation
of $(\nu_{1},\dots,\nu_{4})$.

Proof. Suppose $\Phi$ and $\Psi$ are unitarily equivalent. By Lemma 1.1 and
the remark after it, there are unitary $U,V\in M_{2}$ such that
$C(\Phi)=(U^{t}\otimes V)C(\Psi)(U^{t}\otimes V)^{*}$. So, $C(\Phi)$ and
$C(\Psi)$ have the same eigenvalues, which are $2\mu_{1},\dots,2\mu_{4}$, and
$2\nu_{1},\dots,2\nu_{4}$, respectively. Thus, $(\mu_{1},\dots,\mu_{4})$ is a
permutation of $(\nu_{1},\dots,\nu_{4})$.

Conversely, assume $(\mu_{1},\dots,\mu_{4})$ is a permutation of
$(\nu_{1},\dots,\nu_{4})$. Then
$\sum_{j=1}^{4}\mu_{j}=\sum_{j=1}^{4}\nu_{j}=a$. If $\Lambda$ is defined as in
Example 2.4, then $\hat{\Phi}=\Phi+(1-a)\Lambda$ and
$\hat{\Psi}=\Psi+(1-a)\Lambda$ are unital linear maps preserving trace and
Hermitian matrices. Let $\xi=(1-a)/2$. Then

$C(\hat{\Phi})=C(\Phi)+\xi I\quad\hbox{ and }\quad C(\hat{\Psi})=C(\Psi)+\xi
I$

have the same eigenvalues $2\mu_{1}+\xi,\dots,2\mu_{4}+\xi$. By Corollary 2.3,
$\hat{\Phi}$ and $\hat{\Psi}$ are unitarily equivalent. So, there are unitary
$U,V\in M_{2}$ such that $C(\hat{\Phi})=(U^{t}\otimes
V)C(\hat{\Psi})(U^{t}\otimes V)^{*}.$ Clearly, we also have
$C(\Phi)=(U^{t}\otimes V)C(\Psi)(U^{t}\otimes V)^{*}.$ Hence,
$\Phi(A)=V\Psi(UAU^{*})V^{*}$ for all $A$. So, $\Phi$ and $\Psi$ are unitarily
equivalent.

Instead of using the unitary equivalence via the Choi matrices, we can show
that a map on $M_{2}$ of the form
$A\mapsto\mu_{1}A+\mu_{2}XAX+\mu_{3}YAY+\mu_{4}ZAZ$ is unitarily equivalent to
a map on $M_{2}$ of the form
$A\mapsto\mu_{j_{1}}A+\mu_{j_{2}}XAX+\mu_{j_{3}}YAY+\mu_{j_{4}}ZAZ$ for any
permutation $(j_{1},j_{2},j_{3},j_{4})$ of $(1,2,3,4)$ as follows. Let
$H=(X+Z)/\sqrt{2}$ and $H_{1}=(X+Y)/\sqrt{2}$. Using the anti-commuting
relations $XY=-YX=iZ,YZ=-ZY=iX,ZX=-XZ=iY$, we have

$\displaystyle A$ $\displaystyle\mapsto$
$\displaystyle\Psi_{1}(A)=H\Psi(HAH)H=\mu_{1}A+\mu_{3}ZAZ+\mu_{2}XAX+\mu_{4}YAY,$
$\displaystyle A$ $\displaystyle\mapsto$
$\displaystyle\Psi_{2}(A)=H_{1}\Psi(H_{1}AH_{1})H_{1}=\mu_{1}A+\mu_{2}ZAZ+\mu_{4}XAX+\mu_{3}YAY,$
$\displaystyle A$ $\displaystyle\mapsto$
$\displaystyle\Psi_{3}(A)=Z\Psi(A)Z=\mu_{2}A+\mu_{1}ZAZ+\mu_{4}XAX+\mu_{3}YAY.$

The composition of $\Psi_{2}$ and $\Psi_{3}$ yields

$A\mapsto\Psi_{4}(A)=ZH_{1}\Psi(H_{1}AH_{1})H_{1}Z=\mu_{2}A+\mu_{1}ZAZ+\mu_{3}XAX+\mu_{4}YAY.$

It is known that the group of permutations of $\\{1,2,3,4\\}$ is generated by
the transpositions (2-cycles) $(1,2),(2,3),(3,4)$. Thus, we may get any
permutation $(\mu_{i_{1}},\dots,\mu_{i_{4}})$ of $(\mu_{1},\dots,\mu_{4})$ by
a composition of $\Psi_{1},\Psi_{2},\Psi_{4}$.

##  3 A unital qubit channel as combination of unitary channels

######  Theorem 3.1.

Let $\Phi$ be a unital qubit channel. Then $\Phi$ is the average of four
unitary channels.

Proof. Note that the Choi matrix of the unitary channel $A\mapsto VAV^{*}$
equals $vv^{*}$ with $v=(\alpha,-\beta,\bar{\beta},\bar{\alpha})^{t}$ if
$V=\begin{pmatrix}\alpha&\bar{\beta}\cr-\beta&\bar{\alpha}\end{pmatrix}$. As a
result, if $\alpha,\beta\in{\mathbb{C}}$ satisfy $|\alpha|^{2}+|\beta|^{2}=1$,
then for

$V_{1}=\begin{pmatrix}\alpha&\bar{\beta}\cr-\beta&\bar{\alpha}\end{pmatrix},\quad
V_{2}=\begin{pmatrix}\alpha&-\bar{\beta}\cr\beta&\bar{\alpha}\end{pmatrix},\quad
V_{3}=\begin{pmatrix}\bar{\alpha}&\beta\cr-\bar{\beta}&\alpha\end{pmatrix},\quad
V_{4}=\begin{pmatrix}\bar{\alpha}&-\beta\cr\bar{\beta}&\alpha\end{pmatrix},$

we can let

$v_{1}=(\alpha,-\beta,\bar{\beta},\bar{\alpha})^{t},\quad
v_{2}=(\alpha,\beta,-\bar{\beta},\bar{\alpha})^{t},\quad
v_{3}=(\bar{\alpha},-\bar{\beta},\beta,\alpha)^{t},\quad
v_{4}=(\bar{\alpha},\bar{\beta},-\beta,\alpha)^{t},$

so that the Choi matrix of the unital channel
$A\mapsto\frac{1}{4}\sum_{j=1}^{4}V_{j}AV_{j}^{*}$ equals

$\frac{1}{4}\sum_{j=1}^{4}v_{j}v_{j}^{*}=\begin{pmatrix}|\alpha|^{2}&0&0&\frac{1}{2}(\alpha^{2}+\bar{\alpha}^{2})\cr
0&|\beta|^{2}&\frac{-1}{2}(\beta^{2}+\bar{\beta}^{2})&0\cr
0&\frac{-1}{2}(\beta^{2}+\bar{\beta}^{2})&|\beta|^{2}&0\cr\frac{1}{2}(\alpha^{2}+\bar{\alpha}^{2})&0&0&|\alpha|^{2}\cr\end{pmatrix}.$

Now, suppose $\Phi$ is a unital channel. By Theorem 2.2, up to a local unitary
similarity transform, a unital qubit channel has Choi matrix

$\frac{1}{2}\begin{pmatrix}\lambda_{1}+\lambda_{2}&0&0&\lambda_{1}-\lambda_{2}\cr
0&\lambda_{3}+\lambda_{4}&\lambda_{3}-\lambda_{4}&0\cr
0&\lambda_{3}-\lambda_{4}&\lambda_{3}+\lambda_{4}&0\cr\lambda_{1}-\lambda_{2}&0&0&\lambda_{1}+\lambda_{2}\cr\end{pmatrix}$

with $\lambda_{1},\dots,\lambda_{4}\geq 0$ summing up to two. Let
$\alpha=\frac{1}{\sqrt{2}}(\sqrt{\lambda_{1}}+i\sqrt{\lambda_{2}}),\beta=\frac{1}{\sqrt{2}}(\sqrt{\lambda_{4}}+i\sqrt{\lambda_{3}})$.
Then

$|\alpha|^{2}=\frac{1}{2}(\lambda_{1}+\lambda_{2}),\quad\alpha^{2}+\bar{\alpha}^{2}=\lambda_{1}-\lambda_{2},\quad|\beta|^{2}=\frac{1}{2}(\lambda_{3}+\lambda_{4}),\quad\beta^{2}+\bar{\beta}^{2}=\lambda_{4}-\lambda_{3}.$

So, up to a unitary similarity transform, $\Phi$ has the form
$A\mapsto\frac{1}{4}\sum_{j=1}^{4}V_{j}AV_{j}^{*}$ defined above.

The set ${\cal S}$ of unital qubit channels is linearly isomorphic to the
compact convex set of Choi matrices of unital qubit channels, that are
$4\times 4$ positive semidefinite matrices in block form $(C_{ij})_{1\leq
i,j\leq 2}$ with $C_{11},C_{12},C_{21},C_{22}\in M_{2}$ satisfying
$C_{11}+C_{22}=I_{2}$, ${\rm tr}\,C_{11}={\rm tr}\,C_{22}=1$ and ${\rm
tr}\,C_{12}={\rm tr}\,C_{21}=0$. Thus, the set of Choi matrices has real
affine dimension 9, and it is known that the Choi matrices of unitary channels
are extreme points of the convex set. By the Caratheodory theorem, every
element in ${\cal S}$ can be written as a convex combination of no more than
10 extreme points. It is remarkable that our result ensures that every element
$\Phi\in{\cal S}$ is a convex combination of $\ell$ extreme points, where
$\ell\leq 4$ is the rank of $C(\Phi)$, and $\Phi$ is the average of four
extreme points.

In the following, we will determine all the possible convex coefficients
$p_{1},\dots,p_{m}$ so that $\Phi=\sum_{j=1}^{m}p_{j}\Psi_{j}$ for some
unitary channels $\Psi_{1},\dots,\Psi_{m}$. To achieve this, we need the
concept of majorization.

Recall that a vector $u\in{\mathbb{R}}^{m}$ is majorized by another vector
$v\in{\mathbb{R}}^{m}$, denoted by $u\prec v$, if the sum of the $k$ largest
entries of $u$ is not larger than that of $v$ for $k=1,\dots,m$, with equality
holding for $k=m$. We can extend the notion of majorization to vectors of
different sizes as follows. Suppose the sum of entries of
$u\in{\mathbb{R}}^{m}$ and $v\in{\mathbb{R}}^{n}$ are the same. We can change
$u,v$ to $\tilde{u},\tilde{v}$ by adding zero entries to the vector with lower
dimension. Then we say that $u\prec v$ if $\tilde{u}\prec\tilde{v}$.

Our main theorem is the following.

######  Theorem 3.2.

Let $\Phi:M_{2}\rightarrow M_{2}$ be a qubit quantum channel such that the
Choi matrix of $\Phi$ has $k$ positive eigenvalues
$\lambda_{1}\geq\cdots\geq\lambda_{k}$. Suppose $p_{1},\dots,p_{m}$ are
positive real numbers summing up to one. Then there are unitary matrices
$V_{1},\dots,V_{m}\in M_{2}$ such that

$\Phi(A)=\mu_{1}V_{1}AV_{1}^{*}+\cdots+\mu_{m}V_{m}AV_{m}^{*}\qquad\hbox{ for
all }A\in M_{2}$ (3.1)

if and only if
$(\mu_{1},\dots,\mu_{m})\prec\frac{1}{2}(\lambda_{1},\dots,\lambda_{k}).$

To prove the theorem, we first obtain some auxiliary results, which are of
independent interest. Note that if $\Psi:M_{n}\rightarrow M_{m}$ is (unital or
/ and trace preserving) completely positive linear map, then $\Psi$ admits a
representation $A\mapsto\sum_{j=1}^{\ell}V_{j}AV_{j}^{*}$ for some
$V_{1},\dots,V_{\ell}\in M_{m,n}$ such that
$\sum_{j=1}^{\ell}V_{j}^{*}V_{j}=I_{n}$
($\sum_{j=1}^{\ell}V_{j}^{*}V_{j}=I_{m}$ or / and
$\sum_{j=1}^{\ell}V_{j}V_{j}^{*}=I_{n}$, respectively); see [1].

Let ${\cal S}$ is the set of unital or / and trace preserving completely
positive maps. It is usually impossible to express $\Psi\in{\cal S}$ as
$A\mapsto\sum_{j=1}^{\ell}V_{j}AV_{j}^{*}$ such that each of the map $A\mapsto
V_{j}AV_{j}^{*}$ is a multiple of an element in ${\cal S}$. By Theorem 3.2,
the convex set of unital qubit channels, enjoys this nice property.

######  Lemma 3.3.

Let $\theta,\eta_{1},\eta_{2},\nu_{1},\nu_{2}$ be nonnegative numbers such
that $\eta_{1}\geq\nu_{1}\geq\nu_{2}\geq\eta_{2}$ and
$\eta_{1}+\eta_{2}=\nu_{1}+\nu_{2}=d$. Then there are
$\theta_{1},\theta_{2}\in[0,2\pi)$ with
$\nu_{1}e^{i\theta_{1}}+\nu_{2}e^{i\theta_{2}}=\eta_{1}+\eta_{2}e^{i\theta}$.

Proof. By the given conditions,

$\nu_{1}-\nu_{2}\leq\eta_{1}-\eta_{2}\leq|\eta_{1}+\eta_{2}e^{i\theta}|\leq\eta_{1}+\eta_{2}=\nu_{1}+\nu_{2}.$

Thus, there is $\phi\in[0,2\pi)$ such that
$|\nu_{1}+e^{i\phi}\nu_{2}|=|\eta_{1}+\eta_{2}e^{i\theta}|$. Hence, there is
$\theta_{1}\in{\mathbb{R}}$ such that
$e^{i\theta_{1}}(\nu_{1}+e^{i\phi}\nu_{2})=\eta_{1}+\eta_{2}e^{i\theta}$. Let
$\theta_{2}\equiv\phi+\theta_{1}$ (mod $2\pi$). The result follows.

######  Corollary 3.4.

Suppose $\Psi(A)=\eta_{1}V_{1}AV_{1}^{*}+\eta_{2}V_{2}AV_{2}^{*}$ for all
$A\in M_{n}$, where $\eta_{1}\geq\eta_{2}\geq 0$, $V_{1},V_{2}\in M_{2}$ are
unitary matrices. Then for any $\nu_{1},\nu_{2}$ such that
$\eta_{1}\geq\nu_{1}\geq\nu_{2}\geq\eta_{2}$ with
$\eta_{1}+\eta_{2}=\nu_{1}+\nu_{2}$, there are unitary matrices
$U_{1},U_{2}\in M_{2}$ such that
$\Psi(A)=\nu_{1}U_{1}AU_{1}^{*}+\nu_{2}U_{2}AU_{2}^{*}$ for all $A\in M_{2}$.

Proof. Consider the special case when $V_{1}=I_{2}$ and $V_{2}=D={\rm
diag}\,(1,e^{i\theta})$. Define $\Psi(A)=\eta_{1}A+\eta_{2}DAD^{*}$. It
follows that

$C(\Psi)=\eta_{1}\begin{pmatrix}1&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr
1&0&0&1\cr\end{pmatrix}+\eta_{2}\begin{pmatrix}1&0&0&e^{-i\theta}\cr
0&0&0&0\cr 0&0&0&0\cr e^{i\theta}&0&0&1\cr\end{pmatrix}.$

By Lemma 3.3, there are $\theta_{1},\theta_{2}\in[0,2\pi)$ such that
$\eta_{1}+\eta_{2}e^{i\theta}=\nu_{1}e^{i\theta_{1}}+\nu_{2}e^{i\theta_{2}}$.
Let $\tilde{U}_{1}={\rm diag}\,(1,e^{i\theta_{1}}),\tilde{U}_{2}={\rm
diag}\,(1,e^{i\theta_{2}})$. Then
$\Psi(A)=\nu_{1}\tilde{U}_{1}A\tilde{U}_{1}^{*}+\nu_{2}\tilde{U}_{2}A\tilde{U}_{2}^{*}$.

Next, consider general unitary $V_{1},V_{2}\in M_{2}$. Then
$V_{1}^{*}V_{2}=\alpha WDW^{*}$ with a complex unit $\alpha$, a unitary $W\in
M_{2}$ and $D={\rm diag}\,(1,e^{i\theta})$. Then

$\displaystyle\Psi(A)$ $\displaystyle=$
$\displaystyle\eta_{1}V_{1}AV_{1}^{*}+\eta_{2}V_{2}AV_{2}^{*}=\eta_{1}V_{1}AV_{1}^{*}+\eta_{2}V_{1}(V_{1}^{*}V_{2}A_{2}V_{2}^{*}V_{1})V_{1}^{*}$
$\displaystyle=$ $\displaystyle\eta_{1}V_{1}AV_{1}^{*}+\eta_{2}V_{1}(\alpha
WDW^{*}A\bar{\alpha}WD^{*}W^{*})V_{1}^{*}=V_{1}W\Phi(W^{*}AW)W^{*}V_{1}^{*}$

with $\Phi(X)=\eta_{1}X+\eta_{2}DXD^{*}$ for $X\in M_{2}$. By the special
case, there are diagonal unitary matrices $\tilde{U}_{1},\tilde{U}_{2}\in
M_{2}$ such that

$\Phi(W^{*}AW)=\eta_{1}(W^{*}AW)+\eta_{2}D(W^{*}AW)D^{*}=\nu_{1}\tilde{U}_{1}(W^{*}AW)\tilde{U}_{1}^{*}+\nu_{2}\tilde{U}_{2}(W^{*}AW)\tilde{U}_{2}^{*}.$

Let $U_{j}=V_{1}W\tilde{U}_{j}W^{*}$ for $j=1,2$. Then
$\Psi(A)=\nu_{1}U_{1}AU_{1}^{*}+\nu_{2}U_{2}AU_{2}^{*}$ for all $A\in M_{2}$.

######  Lemma 3.5.

Let $m\geq 2$ and ${\bf u}=(u_{1},\dots,u_{m}),{\bf
v}=(v_{1},\dots,v_{m})\in{\mathbb{R}}^{m}$ with entries arranged in descending
order, and ${\bf u}\prec{\bf v}$. Then one of the following holds.

  * (1)

There is $i\in\\{1,\dots,m\\}$ such that $u_{i}=v_{i}$.

  * (2)

There is $j\in\\{1,\dots,m-1\\}$ with $v_{j}>u_{j}\geq u_{j+1}>v_{j+1}$. If
$\delta=\min\\{v_{j}-u_{j},u_{j+1}-v_{j+1}\\}$, then the vector $\tilde{\bf
v}$ obtained from ${\bf v}$ by replacing the pair of entries $(v_{j},v_{j+1})$
to $(v_{j}-\delta,v_{j+1}+\delta)$ will satisfy ${\bf u}\prec\tilde{\bf
v}\prec{\bf v}$, and ${\bf u}$ and $\tilde{\bf v}$ will have the same value at
the $j$th or $(j+1)$st position.

Proof. Direct verification.

In general, suppose ${\bf y}=(y_{1},\dots,y_{m})$ with $y_{1}\geq\cdots\geq
y_{m}$, and ${\bf x}=(x_{1},\dots,x_{m})$ is obtained from ${\bf y}$ by
changing a pair of entries $(y_{p},y_{q})$ to
$(x_{p},x_{q})=(y_{p}-\delta,y_{q}+\delta)$ with
$\delta\in(0,(y_{p}-y_{q})/2)$ for some $p<q$ and $y_{q}-y_{p}>0$. We say that
${\bf x}$ is obtained from ${\bf y}$ by a pinching. By Lemma 3.5, if ${\bf
x},{\bf y}\in{\mathbb{R}}^{m}$ satisfy ${\bf x}\prec{\bf y}$, we may remove
the entries $x_{\ell},y_{\ell}$ whenever $x_{\ell}=y_{\ell}$, and apply (2) to
get a vectors $\tilde{\bf y}$ such that the pair of vectors ${\bf x}$ and
$\tilde{\bf y}$ have one more common entry than the pair of vectors ${\bf x}$
and ${\bf y}$. Thus, one can change ${\bf y}$ to ${\bf x}$ by at most $m-1$
pinchings.

######  Corollary 3.6.

Suppose $\Phi$ is a qubit channel having the form

$A\mapsto\sum_{j=1}^{k}\nu_{j}U_{j}AU_{j}^{*}$ (3.2)

for some nonnegative real numbers $\nu_{1},\dots,\nu_{k}$ summing up to 1, and
unitary $U_{1},\dots,U_{k}\in M_{2}$. If
$(\mu_{1},\dots,\mu_{m})\prec(\nu_{1},\dots,\nu_{k})$, then $\Phi$ admits a
representation of the form $A\mapsto\sum_{j=1}^{m}\mu_{j}V_{j}AV_{j}^{*}$ for
some unitary $V_{1},\dots,V_{m}\in M_{2}$.

Proof. We may assume that $m=k$ by adding zeros to the vector with lower
dimension. Assume $(\mu_{1},\dots,\mu_{m})\neq(\nu_{1},\dots,\nu_{m})$ to
avoid trivial consideration. Furthermore, we may assume that the entries of
the two vectors are arranged in descending order. Suppose
$(\mu_{1},\dots,\mu_{m})$ is obtained from $(\nu_{1},\dots,\nu_{m})$ by a
pinching, say, $(\mu_{p},\mu_{q})=(\nu_{p}-\delta,\nu_{q}+\delta)$ with
$\delta\in(0,(\nu_{q}-\nu_{p})/2)$ for some $p<q$. By Lemma 3.4, we can
replace the expression $\nu_{p}U_{p}AU_{p}^{*}+\nu_{q}U_{q}AU_{q}^{*}$ in
(3.2) by $\mu_{p}V_{p}AV_{p}^{*}+\mu_{q}V_{q}AV_{q}^{*}$ for some unitary
matrices $V_{p},V_{q}\in M_{2}$. By the remark after Lemma 3.5, we can reduce
$(\nu_{1},\dots,\nu_{m})$ by at most $m-1$ pinchings. Thus, we can repeat the
above argument for at most $m-1$ times to get the conclusion.

Now we are ready to present the following.

Proof of Theorem 3.2. Suppose $\Psi(A)=\sum_{j=1}^{\ell}V_{j}AV_{j}^{*}$ for
all $A\in M_{n}$. Then $C(\Psi)=\sum_{j=1}^{\ell}v_{j}v_{j}^{*}$, where for
$j=1,\dots,\ell$, $v_{j}\in{\mathbb{C}}^{mn}$ has the first $m$ entries equal
to the first column of $V_{j}$, the next $m$ entries equal to the second
column of $V_{j}$, etc. Suppose $R=[v_{1}|\cdots|v_{\ell}]$. Then
$C(\Psi)=RR^{*}$ has the same nonzero eigenvalues as $R^{*}R\in M_{\ell}$,
which is a Hermitian matrix with diagonal entries

$(v_{1}^{*}v_{1},\dots,v_{\ell}^{*}v_{\ell})=({\rm
tr}\,(V_{1}^{*}V_{1}),\dots,{\rm tr}\,(V_{\ell}^{*}V_{\ell})).$

Then $\ell\geq k$ and the majorization relation holds by the result in [5].

For the sufficiency, we see that $\Phi$ can be written in the form
$A\mapsto\frac{1}{2}\sum_{j=1}^{k}\lambda_{j}U_{j}AU_{j}^{*}$ by Theorem 2.2.
By Corollary 3.6, If
$(\mu_{1},\dots,\mu_{m})\prec\frac{1}{2}(\lambda_{1},\dots,\lambda_{k}),$
there are unitary $V_{1},\dots,V_{m}\in M_{n}$ such that $\Phi$ can be written
in the form (3.1).

From Theorem 3.2, one easily deduces the following.

######  Corollary 3.7.

Suppose $\Phi$ is a unital qubit channel such that $C(\Phi)$ has rank
$k\in\\{1,2,3,4\\}$. Then $\Phi$ can be written as the average of $m$ unital
channels for any positive integer $m$ satisfying $k\leq m$.

Proof. If $C(\Phi)$ has eigenvalues $\lambda_{1},\dots,\lambda_{k}$, then
$(1/m,\dots,1/m)\prec\frac{1}{2}(\lambda_{1},\dots,\lambda_{k})$. So,
$C(\Phi)$ is the average of $m$ unitary channels by Theorem 3.2.

We conclude this section with some comments on the unusual convexity features
of the set ${\cal S}$ of unital qubit channels, whose extreme points are
unitary channels.

By Theorem 3.2, if $\Phi\in{\cal S}$ such that $C(\Phi)$ has rank $k$ with
positive eigenvalues $\lambda_{1}\geq\cdots\geq\lambda_{k}$, then $\Phi$ can
be written as a convex combination of unitary channels
$A\mapsto\sum_{j=1}^{m}\mu_{j}U_{j}AU_{j}^{*}$ as long as
$(\mu_{1},\dots,\mu_{m})\prec\frac{1}{2}(\lambda_{1},\dots,\lambda_{k})$.
Consequently, the four eigenvalues of $C(\Phi)$ can be characterized as the
supremum of set of four numbers $p_{1}\geq\cdots\geq p_{4}$ such that $\Phi$
admits a representation of the form
$A\mapsto\frac{1}{2}(p_{1}V_{1}AV_{1}^{*}+\cdots+p_{4}V_{4}AV_{4}^{*})$ for
some unitary matrices $V_{1},\dots,V_{4}\in M_{2}$. Of course,
$(\lambda_{1},\cdots,\lambda_{4})$ can also be characterized as $({\rm
tr}\,V_{1}^{*}V_{1},\dots,{\rm tr}\,V_{4}^{*}V_{4})$ such that $\Phi$ admits a
representation of the form $A\mapsto V_{1}AV_{1}^{*}+\cdots+V_{4}AV_{4}^{*}$
for some matrices $V_{1},\dots,V_{4}\in M_{2}$ satisfying ${\rm
tr}\,V_{i}^{*}V_{j}=0$ for all $1\leq i,j\leq 4$. If $\Phi$ is a unitary
channel so that $\Phi$ is a unital qubit channel so that $C(\Phi)$ has
eigenvalues $1,0,0,0$, then $\Phi$ can be written as
$\sum_{j=1}^{m}\mu_{j}U_{j}AU_{j}^{*}$ for any choice of convex coefficients
$\mu_{1},\dots,\mu_{m}$. In some sense, the eigenvalues of $C(\Phi)$ can be
viewed as a measure on how close $\Phi$ is to a unitary channel, an extreme
point of ${\cal S}$.

##  4 Unital qubit channels and the Bloch sphere

Researchers have studied qubit channels using the correspondence between
linear maps on the Bloch (unit) ball in ${\mathbb{R}}^{3}$, and linear maps on
$M_{2}$ that preserve trace and Hermitian matrices; e.g., see [3, 4, 7] and
their references. In this section, we present our results in Sections 2 and 3
in terms of the corresponding linear maps on the Bloch sphere.

Identify the standard basis $\\{e_{1},e_{2},e_{3}\\}$ for ${\mathbb{R}}^{3}$
with the orthnormal basis $\\{X/\sqrt{2},Y/\sqrt{2},Z/\sqrt{2}\\}$ for the
real linear space for trace zero Hermitian matrices in $M_{2}$, then a linear
map $\phi$ on ${\mathbb{R}}^{3}$ of the form
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ with $d_{1},d_{2},d_{3}\in{\mathbb{R}}$
induces a linear map $R_{\phi}$ on $M_{2}$ sending $(I_{2},X,Y,Z)$ to
$(I_{2},d_{1}X,d_{2}Y,d_{3}Z)$. Clearly, $R_{\phi}$ is a unital map preserving
trace and Hermitian matrices. We have the following.

######  Theorem 4.1.

Let $\phi$ be the linear map on ${\mathbb{R}}^{3}$ of the form
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$, and $R_{\phi}$ be the corresponding
unital map on $M_{2}$. $C(R_{\phi})$ has the form

$\frac{1}{2}\begin{pmatrix}1+d_{3}&0&0&d_{1}+d_{2}\cr
0&1-d_{3}&d_{1}-d_{2}&0\cr 0&d_{1}-d_{2}&1-d_{3}&0\cr
d_{1}+d_{2}&0&0&1+d_{3}\cr\end{pmatrix},$ (4.1)

which has eigenvalues
$\frac{1}{2}(1+d_{1}+d_{2}+d_{3}),\frac{1}{2}(1+d_{3}-d_{1}-d_{2}),\frac{1}{2}(1-d_{3}+d_{1}-d_{2}),\frac{1}{2}(1-d_{3}-d_{1}+d_{2})$.

By Theorem 4.1 and Theorem 2.2, we see that $R_{\phi}$ can be written in the
form

$A\mapsto\mu_{1}A+\mu_{2}XAX+\mu_{3}YAY+\mu_{4}ZAZ,$ (4.2)

which is a unital map on $M_{2}$ preserving trace and Hermitian matrices. In
fact, there is a one-one correspondence between the maps $\phi$ on
${\mathbb{R}}^{3}$ of the form $(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ and the
maps in the form (4.2). The effect of $\phi$ is completely determined in terms
of its action on the Bloch (unit) sphere in ${\mathbb{R}}^{3}$, namely, the
Bloch sphere is transformed to the ellipsoid with scaling factors
$d_{1},d_{2},d_{3}$. Here, if $d_{i}<0$, then the effect is a scaling of
$|d_{i}|$ followed by a reflection. It is interesting that such maps on
${\mathbb{R}}^{3}$ correspond to the maps in the form (4.2) that play a
crucial role in the study of qubit channels.

By the above discussion, one can connect the results in Sections 2 and 3 to
real linear maps on ${\mathbb{R}}^{3}$ of the form
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$. For example, by Corollary 2.5, we have
the following.

######  Theorem 4.2.

Let $\phi,\psi$ be the real linear maps

$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ and
$(x,y,z)\mapsto(\tilde{d}_{1}x,\tilde{d}_{2}y,\tilde{d}_{3}z)$.

Suppose $R_{\phi}$ and $R_{\psi}$ are the corresponding unital maps on $M_{2}$
that preserve trace and Hermitian matrices. The following conditions are
equivalent.

(a) $R_{\phi}$ and $R_{\psi}$ are unitarily equivalent.

(b)
$(\tilde{d}_{1}+\tilde{d}_{2}+\tilde{d}_{3},\tilde{d}_{1}-\tilde{d}_{2}-\tilde{d}_{2},-\tilde{d}_{1}+\tilde{d}_{2}-\tilde{d}_{3},-\tilde{d}_{1}-\tilde{d}_{2}+\tilde{d}_{3})$
is a permutation of

$(d_{1}+d_{2}+d_{3},d_{1}-d_{2}-d_{3},-d_{1}+d_{2}-d_{3},-d_{1}-d_{2}+d_{3})$.

(c) $(|\tilde{d}_{1}|,|\tilde{d}_{2}|,|\tilde{d}_{3}|)$ is a permutation of
$(|d_{1}|,|d_{2}|,|d_{3}|)$, and
$\tilde{d}_{1}\tilde{d}_{2}\tilde{d}_{3}=d_{1}d_{2}d_{3}$.

Proof. By Corollary 2.3, (a) holds if and only if $C(R_{\phi})$ and
$C(R_{\psi})$ have the same eigenvalues, equivalently, condition (b) holds by
(4.1).

To prove the equivalence of (b) and (c), let
$T=\frac{1}{2}\begin{pmatrix}1&1&-1&-1\cr 1&-1&1&-1\cr
1&-1&-1&1\cr\end{pmatrix}$. Then $TT^{*}=I_{3}$ and $T^{*}T=I_{4}-J_{4}/4$,
where $J_{4}\in M_{4}$ has all entries equal to 1. Moreover, for any
permutation matrix $P\in M_{4}$, $PT^{*}T=T^{*}TP$, and $T^{*}I_{4}T=I_{3}$.
Consequently, the map $X\mapsto TXT^{*}$ defines a group homomorphism from the
group of permutation matrices in $M_{4}$ to the group of the 24 matrices
$Q_{1},\dots,Q_{24}\in M_{3}$, where each $Q_{j}$ is the product of a
permutation matrix in $M_{3}$ and a diagonal orthogonal matrix with
determinant 1.

Note that
$(d_{1}+d_{2}+d_{3},d_{1}-d_{2}-d_{3},-d_{1}+d_{2}-d_{3},-d_{1}-d_{2}+d_{3})=2(d_{1},d_{2},d_{3})T.$
Thus, if (b) holds, then there is a permutation matrix $P\in M_{4}$ such that
$(\tilde{d}_{1},\tilde{d}_{2},\tilde{d}_{3})T=(d_{1},d_{2},d_{3})TP$. Hence,
$(\tilde{d}_{1},\tilde{d}_{2},\tilde{d}_{3})=(d_{1},d_{2},d_{3})TPT^{*}=(d_{1},d_{2},d_{3})Q_{j}$
for some $j=1,\dots,24$, i.e., condition (c) holds.

If (c) holds, then
$(\tilde{d}_{1},\tilde{d}_{2},\tilde{d}_{3})=(d_{1},d_{2},d_{3})Q$ for some
$Q\in\\{Q_{j}:1\leq j\leq 24\\}$ so that
$(\tilde{d}_{1},\tilde{d}_{2},\tilde{d}_{3})T=(d_{1},d_{2},d_{3})(TPT^{*})T=(d_{1},d_{2},d_{3})T(T^{*}T)P=(d_{1},d_{2},d_{3})TP$
for some permutation $P\in M_{4}$. Thus, condition (b) holds.

For any unital map $\Psi$ on $M_{2}$ that preserves trace and Hermitian
matrices, one can use the corresponding real linear map $\psi$ on
${\mathbb{R}}^{3}$ to study its property. For instance, the set of positive
semidefinite matrices in $M_{2}$ that have the form
$\frac{1}{2}\begin{pmatrix}1+z&x-iy\cr x+iy&1-z\end{pmatrix}$ with
$x^{2}+y^{2}+z^{2}\leq 1$ corresponding to the unit ball in
${\mathbb{R}}^{3}$. Thus, a unital tracing preserving and Hermitian preserving
map $\Psi$ on $M_{2}$ is a positive map, i.e., $\Psi$ sends the set positive
semidefinite matrices to itself, if and only if the corresponding linear map
$\psi$ on ${\mathbb{R}}^{3}$ is a contraction. However, not every contractive
map $\phi$ on ${\mathbb{R}}^{3}$ corresponds to a unital qubit channel $\Phi$.
For example, the map $\phi:(x,y,z)\mapsto(x,y,0)$ corresponding to

$\Phi:\frac{1}{2}\begin{pmatrix}1+z&x-iy\cr
x+iy&1-z\cr\end{pmatrix}\mapsto\frac{1}{2}\begin{pmatrix}1&x-iy\cr
x+iy&1\cr\end{pmatrix}\quad\hbox{ with }\quad
C(\Phi)=\frac{1}{2}\begin{pmatrix}1&0&0&2\cr 0&1&0&0\cr 0&0&1&0\cr
2&0&0&1\cr\end{pmatrix}$ (4.3)

is not a quantum channel because $C(\Phi)$ is not positive semidefinite.

We have shown that up to unitarily equivalence, every unit qubit channel can
be written as a convex combination of the following four maps

$A\mapsto A,\qquad A\mapsto XAX,\qquad A\mapsto YAY,\qquad A\mapsto ZAZ.$

The corresponding linear maps on ${\mathbb{R}}^{3}$ are

$(x,y,z)\mapsto(x,y,z),\ (x,y,z)\mapsto(x,-y,-z),\ (x,y,z)\mapsto(-x,y,-z),\
(x,y,z)\mapsto(-x-y,z),$

which are the identity map, the rotation of $\pi$ of the $(y,z)$-plane,
$(x,z)$-plane, and $(x,y)$-plane, respectively. Equivalently, $\phi={\rm
diag}\,(d_{1},d_{2},d_{3})$ is a convex combination of matrices

${\rm diag}\,(1,1,1),\ {\rm diag}\,(1,-1,-1),\ {\rm diag}\,(-1,1,-1),\ {\rm
diag}\,(-1,-1,1).$

In terms of the $(d_{1},d_{2},d_{3})$, the map $\Phi$ is a unital qubit
channel if and only if the eigenvalues of $C(\Phi)$ are nonnegative, i.e.,

$1+d_{1}+d_{2}+d_{3},1+d_{3}-d_{1}-d_{2},1-d_{3}+d_{1}-d_{2},1-d_{3}-d_{1}+d_{2}\in[0,\infty).$
(4.4)

One can determine the set of extreme points of the compact convex set
$(d_{1},d_{2},d_{3})\in{\mathbb{R}}^{3}$ satisfying (4.4). Every extreme point
of the set must attain equality for at least three of the linear inequality
constraints. A direct computation shows that there are four extreme points,
namely,

$(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1),$

and the set of real vectors $(d_{1},d_{2},d_{3})$ satisfying (4.4) is the
convex hull of these four points, which is a regular tetrahedron.

######  Theorem 4.3.

Suppose $\phi$ is a linear map on ${\mathbb{R}}^{3}$ defined by
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ for some
$d_{1},d_{2},d_{3}\in{\mathbb{R}}$, and $R_{\phi}$ is the corresponding linear
map on $M_{2}$ so that $C(R_{\phi})$ has the form (4.1).

  * (a)

The map $\phi$ determines a unital trace preserving positive map $R_{\phi}$ if
and only if $(d_{1},d_{2},d_{3})$ lies in the cube ${\rm
conv}\\{(\xi_{1},\xi_{2},\xi_{3}):\xi_{j}\in\\{1,-1\\}\hbox{ for }j=1,2,3\\}.$

  * (b)

The map $\phi$ determines a unital qubit channel $R_{\phi}$ if and only if any
one of the following equivalent conditions hold.

(b.1) $(d_{1},d_{2},d_{3})$ belongs to the regular tetrahedron with vertices

$(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1).$

(b.2) The vector
$(1+d_{1}+d_{2}+d_{3},1+d_{1}-d_{2}-d_{3},1-d_{1}+d_{2}-d_{3},1-d_{1}-d_{2}+d_{3})$
is

nonnegative.

(b.3) The diagonal matrix ${\rm diag}\,(d_{1},d_{2},d_{3})$ is a convex
combination of matrices in $SO(3)$.

In Theorem 4.3, (b.1) and (b.2) establish a correspondence between the regular
tetrahedron in ${\mathbb{R}}^{3}$ with vertices
$(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$, and the convex set

$\left\\{(\lambda_{1},\dots,\lambda_{4}):\lambda_{1},\dots,\lambda_{4}\geq
0,\sum_{j=1}^{4}\lambda_{j}=2\right\\}.$

Mathematically, it is simple to verify the correspondence. Nevertheless, it is
physically astonishing that the unital qubit channels serve as the natural
link between these two convex sets. Namely, each quantum channel $\Phi$
determines four eigenvalues $\lambda_{1},\dots,\lambda_{4}$ of $C(\Phi)$ as
well as the shrinking effects $(d_{1},d_{2},d_{3})$ of the Block sphere in
${\mathbb{R}}^{3}$. For example, if $(d_{1},d_{2},d_{3})=(1,1,0)$, i.e., the
Bloch sphere is transformed to the unit disk on the $(x,y)$-plane, then it
does not correspond to a unital qubit channel. Also, if
$(d_{1},d_{2},d_{3})=(1,1,-1)$, i.e., the Bloch sphere is transformed to
itself by a reflection about the $(x,y)$-plane, then it does not correspond to
a unital qubit channel.

In fact, it not hard to show that for a nonngative number $\gamma$ the map
$(x,y,z)\mapsto\gamma(x,y,-z)$ corresponds to a unital qubit channel if and
only if $\gamma\in[0,1/3]$. It turns out that this and other limitations on
admissible scalings of the Bloch sphere corresponding to unital qubit maps can
be deduced from the next theorem, where the assumption $d_{1}\geq
d_{2}\geq|d_{3}|$ is imposed without loss of generality by virtue of Theorem
4.2 (c).

######  Theorem 4.4.

Suppose $\phi$ is a linear map on ${\mathbb{R}}^{3}$ defined by
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ with $d_{1}\geq d_{2}\geq|d_{3}|$, and
$R_{\phi}$ is the corresponding linear map on $M_{2}$. Then $R_{\phi}$ is a
unital qubit channel if and only if $1+d_{3}\geq d_{1}+d_{2}$, equivalently,
$(d_{1},d_{2},d_{3})\in{\rm conv}\\{(0,0,0),(1,1,1),(1,0,0),(1,1,-1)/3\\}.$

Proof. By Theorem 4.3, under the assumption that $d_{1}\geq d_{2}\geq|d_{3}|$,
condition (b.2) reduces to $1+d_{3}\geq d_{1}+d_{2}$. The convex set
$S=\\{(d_{1},d_{2},d_{3})\in{\mathbb{R}}^{3}:d_{1}\geq
d_{2}\geq|d_{3}|,1+d_{3}\geq d_{1}+d_{2}\\}$ is defined by the four
inequalities $d_{1}\geq d_{2},d_{2}\geq d_{3},d_{2}\geq-d_{3}$, and
$1+d_{3}\geq d_{1}+d_{2}$; an extreme point of the set $S$ must attain at
least three of these inequalities. Thus the extreme points of $S$ are
$(0,0,0),(1,1,1),(1,0,0),(1,1,-1)/3$, and $S={\rm
conv}\\{(0,0,0),(1,1,1),(1,0,0),(1,1,-1)/3\\}$.

By Theorem 4.4, $R_{\phi}$ is a unital qubit channel if and only if $\phi$ is
a convex combination of the maps $\psi_{1}:(x,y,z)\mapsto(0,0,0)$,
$\psi_{2}:(x,y,z)\mapsto(x,y,z)$, $\psi_{3}:(x,y,z)\mapsto(x,0,0)$,
$\psi_{4}:(x,y,z)\mapsto(0,0,-z).$ These maps on the Bloch sphere have the
following effects, respectively:

  * •

shrinking the sphere to the origin,

  * •

leaving the sphere invariant,

  * •

shrinking the Bloch sphere to the line segment joining $(-1,0,0)$ to
$(1,0,0)$,

  * •

shrinking the Bloch sphere by a factor of $1/3$ followed by a refection over
the $(x,y)$-plane.

In terms of $\Phi$, it means that every unital qubit channel can be written as
a convex combination of $\Psi_{1},\dots,\Psi_{4}$ defined, respectively, by

$A\mapsto\frac{1}{2}({\rm tr}\,A)I_{2},\quad A\mapsto A,\quad
A\mapsto\frac{1}{2}(A+XAX),\quad\hbox{ and }\quad
A\mapsto\frac{1}{3}(A+XAX+YAY).$

Using Theorem 4.2 (c), one may impose other assumptions on the scaling factors
$d_{1},d_{2},d_{3}$ of the Bloch sphere, and deduce additional results on
unital qubit channels. For example, we may assume that $d_{1}\geq d_{2}\geq
d_{3}$ and $d_{2}\geq 0$. In such a case,
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$ will correspond to a unital qubit
channel if and only if $d_{1}\geq d_{2}\geq d_{3}$, $d_{2}\geq 0$, and
$1+d_{3}\geq d_{1}+d_{2}$, i.e., $(d_{1},d_{2},d_{3})\in{\rm
conv}\\{(0,0,0),(1,1,1),(1,0,0),(0,0,-1)\\}$. Thus, a unital qubit channel can
be written as a convex combination of the maps

$A\mapsto\frac{1}{2}({\rm tr}\,A)I_{2},\quad A\mapsto A,\quad
A\mapsto\frac{1}{2}(A+XAX),\quad\hbox{ and }\ A\mapsto\frac{1}{2}(XAX+YAY).$

A unital map on $M_{2}$ that preserves trace and Hermitian matrices
corresponds to a real linear map on trace zero Hermitian matrices. With the
identification of the orthonormal basis
$\\{X/\sqrt{2},Y/\sqrt{2},Z/\sqrt{2}\\}$ for trace zero hermitian matrices and
the standard basis $\\{e_{1},e_{2},e_{3}\\}$ for ${\mathbb{R}}^{3}$, a real
linear map on trace zero Hermitian matrices can be represented by a real
matrix in $M_{3}$. In particular, it is well known and easy to check that a
map on trace zero matrices in $M_{2}$ of the form $A\mapsto UAU^{*}$ for a
unitary $U\in M_{2}$ corresponds to a matrix $P$ in the special orthogonal
group $SO(3)$, i.e., the set of real orthogonal matrices in $M_{3}$ with
positive determinant. In Section 2, it is shown that every unital map $\Phi$
on $M_{2}$ that preserves trace and Hermitian matrices is unitarily equivalent
to a map of the form (4.2). In terms of the corresponding linear map $\phi$ on
${\mathbb{R}}^{3}$, realized as a real $3\times 3$ matrix, it simply means
that there are $P,Q\in SO(3)$ such that $P\phi Q$ is the diagonal matrix ${\rm
diag}\,(d_{1},d_{2},d_{3})$ so that the map has the form
$(x,y,z)\mapsto(d_{1}x,d_{2}y,d_{3}z)$.

From our results, one sees that every qubit channel corresponds to a real
matrix in $M_{3}$ that can be written as the combination of matrices in
$SO(3)$, and every matrix in ${\rm conv}SO(3)$ corresponds to a qubit channel.
Thus, the set of qubit channels can be identified with ${\rm conv}SO(3)$, one
may use results on ${\rm conv}SO(3)$ to give alternative proofs for the
results on unital qubit channels. For example, the result in [6] can be used
to give a different proof of Theorem 3.1. Also, we can give a quick proof that
every $\phi$ satisfying the condition in Theorem 4.4 is the average of four
matrices in $SO(3)$.

######  Theorem 4.5.

Let $D={\rm diag}\,(d_{1},d_{2},d_{3})$ satisfy $d_{1}\geq d_{2}\geq|d_{3}|$
such that $1+d_{3}\geq d_{1}+d_{2}$. Then there is $A\in SO(3)$ such that
$D=\frac{1}{4}\sum_{j=1}^{4}D_{j}AD_{j}$, where

$D_{1}=I_{3},\ D_{2}={\rm diag}\,(1,-1,-1),\ D_{3}={\rm diag}\,(-1,1,-1),\
D_{4}={\rm diag}\,(-1,-1,1).$

Proof. Assume $d_{1}\geq d_{2}\geq|d_{3}|$ such that $1+d_{3}\geq
d_{1}+d_{2}$. Let $c=(d_{1}+d_{2})/(1+d_{3})\in[0,1]$ and $s=\sqrt{1-c^{2}}$.
Then $cd_{3}\leq|d_{3}|\leq d_{2}$ and $c=c(1+d_{3})-cd_{3}\geq
d_{1}+d_{2}-d_{2}=d_{1}$. Set

$B(t)=\begin{pmatrix}\cos t&\sin t\cr-\sin t&\cos
t\cr\end{pmatrix}\begin{pmatrix}c&s\cr-s&c\cr\end{pmatrix}\begin{pmatrix}1&0\cr
0&d_{3}\cr\end{pmatrix}\begin{pmatrix}\cos t&-\sin t\cr\sin t&\cos
t\cr\end{pmatrix},\qquad t\in[0,\pi/2].$

Then ${\rm tr}\,B(t)=c(1+d_{3})=d_{1}+d_{2}$ for all $t\in[0,\pi/2]$, the
$(1,1)$ entries of $B(0)$ and $B(\pi/2)$ are $c$ and $cd_{3}$, respectively.
Since $d_{1}\in[cd_{1},c]$, by continuity there is $t\in[0,\pi/2]$ such that
the $(1,1)$ entry of $B(t)$ equals $d_{1}$, and hence the $(2,2)$ entry of
$B(t)$ equals ${\rm tr}\,(B(t))-d_{2}=d_{2}$. Consequently, for
$P_{1}=\begin{pmatrix}\cos t&\sin t\cr-\sin t&\cos
t\cr\end{pmatrix}\begin{pmatrix}c&s\cr-s&c\cr\end{pmatrix}$ and
$P_{2}=\begin{pmatrix}\cos t&-\sin t\cr\sin t&\cos t\cr\end{pmatrix}$, the
matrix

$A=\begin{pmatrix}P_{1}&\cr&1\cr\end{pmatrix}\begin{pmatrix}1&0&0\cr
0&d_{3}&\sqrt{1-d_{3}^{2}}\cr
0&-\sqrt{1-d_{3}^{2}}&d_{3}\cr\end{pmatrix}\begin{pmatrix}P_{2}&\cr&&1\cr\end{pmatrix}=\begin{pmatrix}d_{1}&*&*\cr*&d_{2}&*\cr*&*&d_{3}\cr\end{pmatrix}$

satisfies the assertion.

There are other interesting implications of the results on unital qubit
channels to matrices in ${\rm conv}SO(3)$. For instance, we can translate
Theorem 3.2 to the following result showing some interesting geometric
properties of the set ${\rm conv}SO(3)$.

######  Theorem 4.6.

Let $A\in{\rm conv}SO(3)$ with singular values $d_{1}\geq d_{2}\geq\varepsilon
d_{3}$, where $\varepsilon=1$ or $-1$ depending on $\det(A)>0$ or $\det(A)\leq
0$. Suppose $p_{1},\dots,p_{m}$ are positive numbers summing up to one. Then
there are $Q_{1},\dots,Q_{m}\in SO(3)$ such that $A=\sum_{j=1}^{m}p_{j}Q_{j}$
if and only if $m\geq k$ and

$(p_{1},\dots,p_{m})\prec\frac{1}{2}(d_{1}+d_{2}+d_{3},d_{1}-d_{2}-d_{3},-d_{1}+d_{2}-d_{3},-d_{1}-d_{2}+d_{3}).$

By Theorem 4.6, we see that there are different ways to represent an element
in ${\rm conv}SO(3)$ as convex combinations of elements in $SO(3)$. Among the
different representations, there is one whose convex coefficients
$(\mu_{1},\mu_{2},\mu_{3},\mu_{3})$ which will dominate the others under the
majorization ordering. In particular, these coefficients are attained if
$A=\mu_{1}Q_{1}+\dots+\mu_{4}Q_{4}$ for some $Q_{1},\dots,Q_{4}\in SO(3)$ such
that $Q_{1}^{t}Q_{2},Q_{1}^{t}Q_{3},Q_{1}^{t}Q_{4}$ commute.

## References

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  * [3] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2004.
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(Choi) Department of Mathematics, University of Toronto, Toronto, Ontario,
Canada M5S 3G3.

Email<EMAIL_ADDRESS>
(Li) Department of Mathematics, College of William & Mary, Williamsburg, VA
23187, USA.

Email<EMAIL_ADDRESS>


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# Barrow Entropy and AdS Black Holes in RPS Thermodynamics

Yahya Ladghami<EMAIL_ADDRESS>Laboratory of Physics of Matter and
Radiation, Mohammed I University, BP 717, Oujda, Morocco  Astrophysical and
Cosmological Center, BP 717, Oujda, Morocco  Brahim Asfour
<EMAIL_ADDRESS>Laboratory of Physics of Matter and Radiation,
Mohammed I University, BP 717, Oujda, Morocco  Astrophysical and Cosmological
Center, BP 717, Oujda, Morocco  Amine Bouali<EMAIL_ADDRESS>Laboratory
of Physics of Matter and Radiation, Mohammed I University, BP 717, Oujda,
Morocco  Astrophysical and Cosmological Center, BP 717, Oujda, Morocco  Ahmed
Errahmani<EMAIL_ADDRESS>Laboratory of Physics of Matter and
Radiation, Mohammed I University, BP 717, Oujda, Morocco  Astrophysical and
Cosmological Center, BP 717, Oujda, Morocco  Taoufik Ouali<EMAIL_ADDRESS>Laboratory of Physics of Matter and Radiation, Mohammed I University, BP 717,
Oujda, Morocco  Astrophysical and Cosmological Center, BP 717, Oujda, Morocco

###### Abstract

In this paper, we examine the restricted phase space (RPS) thermodynamics for
charged AdS black holes by considering the impact of quantum gravity on the
event horizon area. The primary aim of this work is to elucidate the influence
of quantum gravitational effects on thermodynamic behaviors, critical
phenomena, phase transitions, and the stability of black holes. We observe
that charged AdS black holes exhibit thermodynamic behavior similar to that of
Van der Waals fluids when influenced by quantum gravity. Furthermore, we
introduce a novel black hole thermodynamic phenomenon, which we term
“resistance of phase transitions”. Our study uncovers a violation of the
homogeneity property of the Smarr relation in RPS thermodynamics due to the
effects of quantum gravity.

##  1 Introduction

Black hole thermodynamics is crucial for comprehending the nature of quantum
gravity. The exploration of black hole thermodynamics has garnered substantial
interest, commencing with the works of Hawking, Bekenstein, Bardeen and
Carter, who formulated four laws for black holes, analogous to those of
classical thermodynamics [1]. They found that black hole emits a radiation,
has a temperature, and possesses entropy. Moreover, the entropy was found to
be proportional to the event horizon area of the black hole [2, 3, 4]. In
addition to explaining the mass of the black hole as internal energy, a
fundamental phenomenon has been discovered for asymptotically anti-de Sitter
(AdS) black holes: the Hawking-Page phase transition [5]. This transition
occurs between the Schwarzschild-AdS black hole and pure thermal AdS space. In
addition to studying the evolution of black hole shadows [6] and the
thermodynamics with negative entropy of the black holes [7]. Later, this
formalism was referred to as “the traditional black hole thermodynamics”
(TBHT).

The inclusion of the negative cosmological constant as a thermodynamic
variable was a fundamental step in constructing a new formalism for black hole
thermodynamics. This formalism, known as “extended phase space thermodynamics”
(EPST) or black hole chemistry, was introduced by Kastor, Ray, and Traschen
[8]. The cosmological constant is related to the pressure through the equation
$P=-\Lambda/8\pi G$. Pressure and its conjugate variable, the thermodynamic
volume, together form a new pair of thermodynamic variables, $(P,V)$. In this
formalism, the black hole mass is reinterpreted as enthalpy. The EPST
formalism plays a important role in black holes thermodynamics because it
reveals various phenomena related to the thermodynamic behavior of black
holes. Among these phenomena are phase transitions between small and large
black holes [9], multiple critical points [10, 11], polymer-type phase
transitions [12], Joule–Thomson expansion [13] and stretched quintessence
phase[14].

Recently, Visser has developed a new version of extended phase space
thermodynamics [15], within the framework of Anti-de Sitter/Conformal Field
Theory (AdS/CFT) correspondence [16]. Based on Visser’s thermodynamics, which
introduced the number of colors and the chemical potential as new variables,
Gao and Zhao [17] proposed a new formalism for black holes thermodynamics by
considering the cosmological constant as fixed while the Newton’s constant is
treated as a thermodynamic variable. This formalism, known as “restricted
phase space (RPS) thermodynamics”, is free of a pair pressure-volume
$(P,\,V)$, and recovers the interpretation of mass as internal energy.
Recently, several studies have utilized this formalism to investigate black
holes, including Kerr-AdS black holes [18], Taub-NUT AdS black holes and the
weak gravity conjecture [19], Kerr-Sen-AdS black holes [20], 4D-EGB black
holes [21], black holes in higher dimensions and higher curvature gravities
[22], and charged AdS black holes in conformal gravity [23].

In the context of quantum gravity and black hole physics, Barrow made
contributions by incorporating the effects of quantum gravity on the structure
of black holes, as detailed in his work [24]. The influence of quantum gravity
leads to a deformation in the event horizon area of black holes. Barrow
proposes that, under such effects, the event horizon area undergoes changes
that can be modeled by a fractal structure. By incorporating fractals into his
statement, Barrow modifies the conventional understanding of a smooth and
simple event horizon when quantum gravity effects are taken into account.
These changes in the event horizon area are what led Barrow to modify the
expression of the entropy of black holes. The effects of quantum gravity,
characterizing the deformation of the event horizon area of black holes, are
quantified by a fractal parameter, $\delta$. Barrow entropy has been
extensively discussed in both the context of black holes and cosmology. Among
these, we cite constraints imposed by the generalized second law of
thermodynamics [25], Barrow holographic dark energy [26, 27, 28, 29, 30] and
applications in the conjecture of “gravity-thermodynamics” [31, 32]. In
addition, it has been used in the traditional thermodynamics of black holes
[33], thermodynamic geometries of BTZ black holes and charged AdS black hole
with a global mono pole [34], the black holes quasinormal modes and fractal
structure [35], Barrow entropy and the equipartition theorem [36].

Our aim is to explore the evaluation of validity of the RPS formalism for
black holes by considering the effect of quantum gravity. Moreover, this paper
seeks to examine how quantum gravity influences the thermodynamic properties
of charged AdS black holes, while also establishing connections between the
fractal parameter, phase transitions, and the stability of these black holes.

This paper is structured as follows: In Section 2, we introduce restricted
phase space thermodynamics incorporating a fractal structure. In Section 3, we
present charged AdS black holes with a fractal structure. Section 4 is
dedicated to defining the thermodynamic variables and equations of state in
RPS thermodynamics for charged AdS black holes. Section 5 explores the impact
of the fractal structure on critical phenomena, phase transitions, and
stability of black holes. Section 6, presents and discusses our most
significant findings related to the current study. In this paper, we adopt the
unit $\hbar=c=k_{B}=1$.

##  2 RPS Thermodynamics

In this section, we develop restricted phase space thermodynamics with respect
to the Barrow statement. To begin, we establish the formalism of RPS, which
was originally formulated by Gao and Zhao [17]. This formalism is grounded in
Visser’s thermodynamics [15] within the framework of AdS/CFT correspondence
[16]. Notably, it takes into consideration a fixed cosmological constant and a
variable Newton constant, wherein the black hole mass is interpreted as the
internal energy.

The AdS/CFT correspondence states that the partition function of AdS
spacetime, $Z_{AdS}$, is equal the CFT dual one, $Z_{CFT}$, [37]. The
partition function of CFT is related to the free energy, $W$, by

$W=-TlnZ_{CFT},$ (1)

where $T$ represents the temperature. Furthermore, we can express the free
energy in terms of the central charge $C$ and the chemical potential, $\mu$,
as follows [15]

$W=\mu\,C.$ (2)

The central charge of CFT is defined as [17, 18]

$C=\frac{l^{d-2}}{G},$ (3)

where $l$ is the AdS radius, $d$ is the dimension of the bulk spacetime, and
$G$ is the Newton constant. The partition function of AdS spacetime is related
to the Euclidean action $I_{E}$ [38, 39] by

$I_{E}=-lnZ_{AdS}.$ (4)

For charged and rotating AdS black holes in a $d$ dimension, the Euclidean
action writes

$I_{\mathrm{E}}=-\frac{1}{16\pi G}\int d^{d}x\sqrt{-g}\left(R-2\Lambda-
F_{\mu\nu}F^{\mu\nu}\right),$ (5)

where $R$ is the Ricci scalar, $\Lambda$ is the cosmological constant, and
$F_{\mu\nu}$ represents the Maxwell strength field. From the formulation of
the AdS/CFT correspondence, as expressed in Eqs. (1)-(4) with the equality
$Z_{AdS}=Z_{CFT}$, one can infer that the free energy is correlated with the
parameters characterizing black holes by [39, 40]

$\mu C=TI_{E}=M-TS-\Phi Q-\Omega J,$ (6)

where $M$ denotes the black hole mass and $S$ corresponds to the Bekenstein-
Hawking entropy. Additionally, $\Phi$ is the electric potential in the event
horizon due to the black hole’s electric charge, denoted by Q. $\Omega$ and
$J$ represent the angular velocity and angular momentum, respectively. The
Smarr relation within the RPS formalism can be written as follows

$M=TS+\tilde{\Phi}\tilde{Q}+\Omega J+\mu C.$ (7)

where we have introduced the rescaled electric potential, $\tilde{\Phi}$, and
the rescaled electric charge, $\tilde{Q}$, as

$\tilde{\Phi}=\frac{\Phi}{\sqrt{C}}\quad\text{and}\quad\tilde{Q}=Q\,\sqrt{C},$
(8)

respectively. The first law in the RPS formalism is expressed as follows [17]

$\mathrm{d}M=T\mathrm{~{}d}S+\Omega\mathrm{d}J+\tilde{\Phi}\mathrm{d}\tilde{Q}+\mu\mathrm{d}C.$
(9)

After obtaining the first law and the Smarr relation, we will reformulate the
restricted phase space thermodynamics by incorporating Barrow’s corrections.
Barrow has demonstrated that the effects of quantum gravity can induce
deformations on the black hole’s surface, which are described by a fractal
structure. This, in turn, leads to a modification in the expression of the
black hole entropy [24], dubbed Barrow entropy, as

$S_{B}=\left(\frac{A}{A_{\text{PI}}}\right)^{1+\frac{\delta}{2}},$ (10)

where $A$ represents the event horizon area of a black hole,
$A_{\text{PI}}=4G$ is the Planck area, and $\delta$ refers to the parameter
quantifying deformations of quantum gravity on the event horizon area, with
$0\leq\delta\leq 1$. When $\delta=0$, the Bekenstein-Hawking entropy is
restored, signifying the absence of a fractal structure. In the case where
$\delta=0$, we recover the normal study of Reissner-Nordström AdS black holes,
i.e., without a fractal structure of the event horizon area[17]. Conversely,
when $\delta=1$, it signifies the most deformed and complex fractal structure
of the black hole’s event horizon area.

In black hole thermodynamics, temperature is the conjugate quantity of
entropy. Consequently, the Barrow temperature $T_{B}$ is defined as

$T_{B}=\frac{\partial M}{\partial S_{B}}.$ (11)

We establish the relationship between the modified quantities
$\left(S_{B},T_{B}\right)$ and the ordinary quantities $\left(S,T\right)$,
using Eqs. (10) and (11), as

$T\,S=\left(1+\frac{\delta}{2}\right)T_{B}\,S_{B}.$ (12)

We substitute Eq. (12) into Eq. (7), and thereby we derive the Smarr relation
in terms of the modified quantities

$M=\left(1+\frac{\delta}{2}\right)T_{B}\,S_{B}+\tilde{\Phi}\tilde{Q}+\Omega
J+\mu\,C.$ (13)

In this Smarr relation, M is a homogeneous function of $(T_{B},S_{B})$ with an
order of $\left(1+\delta/2\right)$, and of $(\tilde{\Phi},\,\tilde{Q})$,
$(\mu,\,C)$, and $(\Omega,\,J)$ with an order of 1. This result contradicts
the nature of the Smarr relation in the RPS formalism in normal cases [17, 18,
19, 20, 21], where $M$ is a homogeneous function of order 1 of both
quantities. The homogeneity property of the Smarr relation in RPS
thermodynamics is violated by the effects of quantum gravity, represented by
the fractal parameter. Mathematically, these effects are determined by the
addition of the element $(T_{B},S_{B})$ with the factor, $\delta/2$,
signifying a departure from the homogeneity of the Smarr relation in the RPS
formalism. For $\delta=0$, we recover the Smarr relation of the ordinary RPS
formalism [17]. We can construct the first law for charged AdS black holes,
where $J=0$, by differentiating the mass function, where
$M=M(S_{B},\tilde{Q},C)$.

$dM=T_{B}dS_{B}+\tilde{\Phi}d\tilde{Q}+\mu dC.$ (14)

##  3 Charged AdS Black Holes

In this section, we study the effects of the fractal structure on charged
black holes in the Anti-de Sitter spacetime in the context of restricted phase
space thermodynamics. The action describing Einstein-Maxwell theory in
4-dimensional AdS spacetime, is given by

$I=-\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R-2\Lambda-
F_{\mu\nu}F^{\mu\nu}\right),$ (15)

where R is the Ricci scalar, $\Lambda=-3/l^{2}$ is the cosmological constant,
and $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the Maxwell
field, with the electromagnetic potential given by

$A^{\mu}=(\frac{Q}{r},0,0,0).$ (16)

The metric for a charged AdS black holes is given by

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (17)

where the metric function is defined as

$f(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}.$ (18)

These black holes under consideration possess two horizons, denoted as
$r_{\pm}$. The event horizon, $r_{+}$, represents the exterior radius, while
$r_{-}$ represents the interior horizon. The expression for the black hole’s
mass, obtained by solving the equation $f(r_{+})=0$, writes

$M=\frac{r_{+}}{2G}\left(1+\frac{GQ^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{l^{2}}\right).$
(19)

The Hawking temperature of the charged black hole and the Bekenstein-Hawking
entropy are given by

$T=\frac{f^{\prime}\left(r_{+}\right)}{4\pi}=\frac{r_{+}^{2}+3r_{+}^{4}l^{-2}-GQ^{2}}{4\pi
r_{+}^{3}},$ (20) $S=\frac{\pi r_{+}^{2}}{G},$ (21)

respectively. Due to the quantum gravity effects on the event horizon area,
the entropy of the black hole becomes the Barrow entropy

$S_{B}=\left(\frac{\pi r_{+}^{2}}{G}\right)^{1+\frac{\delta}{2}},$ (22)

and the Barrow temperature, from Eq. (11), is expressed as

$T_{B}=\frac{r_{+}^{2}+3r_{+}^{4}l^{-2}-GQ^{2}}{2\pi
r_{+}^{3}\left(2+\delta\right)\left(\frac{\pi
r_{+}^{2}}{G}\right)^{\frac{\delta}{2}}}.$ (23)

For $\delta=0$, we recover the Hawking temperature and the Bekenstein-Hawking
entropy. However, in the case of the most deformed fractal structure, i.e.
$\delta=1$, we obtain $S_{B}\propto r_{+}^{3}$. This is similar to the entropy
of black hole in $(4+1)$-dimensions [41]. This result, due to a maximum
fractal structure, breaks the holographic principle which states that the
entropy of a black hole is proportional to its surface. From Eq. (13), we
obtain the Smarr relation of a charged AdS black hole with Barrow correction
in the RPS thermodynamics as

$M=\left(1+\frac{\delta}{2}\right)T_{B}\,S_{B}+\tilde{\Phi}\tilde{Q}+\mu C,$
(24)

and the first law is

$dM=T_{B}dS_{B}+\tilde{\Phi}d\tilde{Q}+\mu dC.$ (25)

Hereafter, we will denote the Barrow quantities $(S_{B},\,T_{B},)$ by
$(S,\,T)$, respectively.

##  4 Equations of State

In this section, we will delve further into the study of charged AdS black
holes, incorporating influences of quantum gravity through the framework of
RPS thermodynamics. We will establish thermodynamic parameters and formulate
the equations of state.
Using Eq. (22), the black hole mass, Eq (19), in terms of extensive variables
and the fractal parameter, $\delta$, becomes

$M\left(S,\,C,\,\tilde{Q},\,\delta\right)=\frac{\pi
CS^{\frac{2}{\delta+2}}+\pi^{2}\tilde{Q}^{2}+S^{\frac{4}{\delta+2}}}{2\pi^{3/2}l\sqrt{CS^{\frac{2}{\delta+2}}}}.$
(26)

Through the first law, (25), all thermodynamic quantities can be obtained as
follows

$T=\left(\frac{\partial M}{\partial
S}\right)_{C,\tilde{Q},\delta},\qquad\tilde{\Phi}=\left(\frac{\partial
M}{\partial\tilde{Q}}\right)_{S,C,\delta},\qquad\mu=\left(\frac{\partial
M}{\partial C}\right)_{S,\tilde{Q},\delta}.$ (27)

Finally, we determine the equations of state describing charged AdS black
holes in the presence of quantum gravity effects

$T=\frac{\pi
CS^{\frac{2}{\delta+2}}-\pi^{2}\tilde{Q}^{2}+3S^{\frac{4}{\delta+2}}}{2\pi^{3/2}(\delta+2)lS\sqrt{CS^{\frac{2}{\delta+2}}}},$
(28)
$\tilde{\Phi}=\frac{\sqrt{\pi}\tilde{Q}}{l\sqrt{CS^{\frac{2}{\delta+2}}}},$
(29) $\mu=-\frac{-\pi
CS^{\frac{2}{\delta+2}}+\pi^{2}\tilde{Q}^{2}+S^{\frac{4}{\delta+2}}}{4\pi^{3/2}Cl\sqrt{CS^{\frac{2}{\delta+2}}}},$
(30)

From Eq. (26) and Eqs. (28)-(30), it is evident that $M$, $T$, $\tilde{\Phi}$,
and $\mu$ depend on the extensive variables of $S$, $\tilde{Q}$, and $C$. When
the variables $S$, $\tilde{Q}$, and $C$ undergo rescaling, namely
$S\rightarrow\lambda S$, $\tilde{Q}\rightarrow\lambda\tilde{Q}$,
$C\rightarrow\lambda C$, $M$ looses its first-order homogeneity. Furthermore,
$T$, $\tilde{\Phi}$, and $\mu$ no longer exhibit zero-order homogeneity,
contradicting the findings in [17]. In the absence of a fractal structure,
$\delta=0$, we recover the equation of state for the standard RPS formalism
[17]. Conversely, when the structure of black hole horizons exhibit its
maximum fractal structure, $\delta=1$, we obtain

$T=\frac{\pi
CS^{2/3}-\pi^{2}\tilde{Q}^{2}+3S^{4/3}}{6\pi^{3/2}lS\sqrt{CS^{2/3}}},$ (31)
$\tilde{\Phi}=\frac{\sqrt{\pi}\tilde{Q}}{l\sqrt{CS^{2/3}}},$ (32)
$\mu=-\frac{-\pi
CS^{2/3}+\pi^{2}\tilde{Q}^{2}+S^{4/3}}{4\pi^{3/2}Cl\sqrt{CS^{2/3}}}.$ (33)

Through the equations of state, Eqs. (28)-(30), the effect of the fractal
structure parameter appears clear, as it changes the degrees of extensive
variables, $(S,\,\tilde{Q},\,C)$.

##  5 Thermodynamic Processes and Phase Transitions

In this section, we will investigate the impact of the fractal structure on
thermodynamic processes, critical phenomena, and the stability of black holes.
Our approach involves determining the critical parameters by solving the
following system,

$\left(\frac{\partial T}{\partial
S}\right)_{C,\tilde{Q},\delta}=0,\qquad\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{C,\tilde{Q},\delta}=0.$ (34)

From Eqs.(28) and (34), we find, for $\delta\neq 1$, the critical entropy
$S_{c}$

$S_{c}=\left(\frac{\pi\,C\,(\delta+1)}{6(1-\delta)}\right)^{\frac{\delta}{2}+1},$
(35)

and the critical re-scaled electric charge $\tilde{Q}_{c}$

$\tilde{Q}_{c}=\frac{C(\delta+1)}{2\sqrt{\left(3-3\delta\right)\left(\delta+3\right)}}.$
(36)

Hence, at critical points, the critical temperature $T_{c}$ writes

$T_{c}=\frac{2^{\frac{\delta+3}{2}}\sqrt{3(\delta+1)}}{l\,(\delta+1)(\delta+2)(\delta+3)\pi^{\frac{\delta}{2}+1}\sqrt{1-\delta}\left(\frac{C(\delta+1)}{3-3\delta}\right)^{\delta/2}}.$
(37)

When considering the maximum fractal level ($\delta=1$), the critical
parameters diverge, indicating the absence of any critical phenomena.
Furthermore, we can determine the entropy at which first-order phase
transitions occur by solving the following equation

$\left(\frac{\partial T}{\partial S}\right)_{C,\tilde{Q},\delta}=0,$ (38)

after excluding the imaginary solutions, we find two entropy values

$S_{1}=\left(\frac{\pi}{6}\right)^{1+\frac{\delta}{2}}\left(\frac{C+C\delta+\sqrt{C^{2}(1+\delta)^{2}+12\tilde{Q}^{2}(-1+\delta)(3+\delta)}}{1-\delta}\right)^{{1+\frac{\delta}{2}}}$
(39)

and

$S_{2}=\left(\frac{\pi}{6}\right)^{1+\frac{\delta}{2}}\left(\frac{C+C\delta-\sqrt{C^{2}(1+\delta)^{2}+12\tilde{Q}^{2}(-1+\delta)(3+\delta)}}{1-\delta}\right)^{{1+\frac{\delta}{2}}}$
(40)

Figure 1: $T-S$ curves for different values of $\delta$ and $\tilde{Q}$.

In Fig. 1, where we plot the normalized Barrow temperature $T/T_{c}$ as a
function of the normalized Barrow entropy $S/S_{c}$, three distinct types of
$T$-$S$ curves can be identified. The first type, characterized by
$\tilde{Q}<\tilde{Q}_{c}$, corresponds to a first-order phase transition
between small and large black holes while passing through an intermediate
black hole. The second type, occurs when the electric charge of the black hole
reaches its critical value, indicating a second-order phase transition from
small black holes to large black holes. For the last type, where
$\tilde{Q}>\tilde{Q}_{c}$, there are no phase transitions. We concluded that,
within the RPS formalism with the effect of quantum gravity on the event
horizon area, the thermodynamics of charged AdS black holes exhibit
similarities to the thermodynamics of a Van der Waals fluid. The value of the
central charge does not affect the behavior of thermodynamics and the
temperature evolution. This property is similar to “the law of corresponding
states” in thermodynamics for ordinary matter, where the thermodynamic
behavior is completely independent of the number of particles.

Figure 2: Critical entropy and phase transition entropy as a functions of the
fractal parameter, with $C=1$

In Fig. 2, we plot the evolution of the critical entropy $S_{c}$
characterizing second-order phase transitions and the entropies of first-order
phase transitions $S_{1}$ and $S_{2}$ in terms of the fractal parameter.We
observe that as the value of $\delta$ increases, both the critical entropy
$S_{c}$ and the entropies of phase transitions also increase. When the
deformation reaches its maximum value $\delta=1$, the critical quantities
$(S_{c},\tilde{Q}_{c},T_{c})$ and the entropies of phase transitions
$(S_{1},S_{2})$ diverge, indicating that the phase transitions do not occur
under these circumstances. The deformation caused by quantum gravity delays
phase transitions, critical phenomena, and even prevents them from occurring
when the deformation reaches its maximum.Thus, this deformation due to quantum
gravity effects on the event horizon behaves as a “phase transitions
resistance”.

Furthermore, the nature of phase transitions is described by studying the free
energy. We express the Barrow free energy, denoted as $F$, as follows

$F\left(S,\,C,\,\tilde{Q},\,\delta\right)=M\left(S,\,C,\,\tilde{Q},\,\delta\right)-TS.$
(41)

Next, we intend to plot the normalized Barrow free energy $F/F_{c}$ against
the normalized Barrow temperature $T/T_{c}$, where $F_{c}$ is the value of the
free energy that corresponds to the critical quantities.

Figure 3: $F-T$ curves for different values of $\delta$ and $\tilde{Q}$.

Fig. 3 illustrates the evolution of Barrow free energy, $F$, as a function of
the Barrow temperature for different values of the electric charge and the
fractal parameter. When $\tilde{Q}<\tilde{Q}_{c}$, a swallowtail pattern is
observed, indicating a first-order phase transition. However, as the fractal
parameter increases, the area of the swallowtail decreases. However, this
pattern does not appear in other cases, $\tilde{Q}\geq\tilde{Q}_{c}$,
regardless of the temperature value.

One key objective of black hole thermodynamics is to investigate the stability
of black holes. The stability of black holes is determined by analyzing their
heat capacity. A positive heat capacity indicates stability, whereas a
negative value implies instability. In this paper, we study the effect of
quantum gravity on the thermodynamic behavior of black holes, including their
stability. To achieve this, we introduce the Barrow heat capacity, denoted as
$\xi$, which incorporates the influences of quantum gravity, as expressed by
the following relation.

$\xi=T\,\left(\frac{\partial S}{\partial T}\right)_{C,\tilde{Q},\delta}.$ (42)

Through Eqs. (31) and (42), we can plot the evolution of the Barrow heat
capacity as a function of the entropy in various scenarios involving electric
charge and fractal parameter.

Figure 4: Variation of $\xi/C$ as a function of $S/S_{c}$ for different values
of the fractal parameter and electric charge, with $C=1$

Fig. 4, represents the evolution of the heat capacity divided by the central
charge as a function of the entropy divided by the critical entropy. The
central charge is always positive, so the sign of $\xi/C$ determines the sign
of the heat capacity. In Fig. 4, three sub-figures are presented, each
corresponding to a particular case of electric charge. For the first case,
i.e. $\tilde{Q}<\tilde{Q}_{c}$, indicates a first-order phase transition among
Small/Intermediate/Large black holes. We observe that both small and large
black holes demonstrate stability, while intermediate black holes exhibits
instability. In the second case, $\tilde{Q}=\tilde{Q}_{c}$, i.e. a second-
order phase transition between Small/Large black holes, we find stability in
both small and large black holes is observed. In the case of
$\tilde{Q}>\tilde{Q}_{c}$, black holes are stable. When considering the impact
of a fractal structure on the stability of black holes, we notice that as
$\delta$ increases, the region of intermediate black holes i.e. unstable
region also expands, and the value of the heat capacity increases for
$\tilde{Q}\geq\tilde{Q}_{c}$.

##  6 Discussions and Conclusions

In our analysis, we have reformulated the thermodynamics of RPS, considering
the influence of quantum gravity effects. Barrow [24] concluded that the area
of the event horizon is subject to deformations generated by quantum gravity
effects and modeled using the fractal structure. This deformation leads to a
modification in the expression for the area of the event horizon, consequently
reformulating the entropy of black holes as Barrow entropy.

This Study of charged AdS black holes within the framework of the RPS
formalism has allowed us to uncover the influence of quantum gravity and its
impact on the thermodynamic properties of black holes. Our study was conducted
using the RPS formalism, which is a development version of Visser’s
thermodynamics with certain restrictions. We considered the AdS radius as a
constant while treating Newton’s constant as a variable. This thermodynamics
is constructed within the framework of the AdS/CFT correspondence.

Through the entropy and temperature in fractal structure, we find the Smarr
relation and the first law. In the RPS thermodynamics with the fractal
structure, the black hole mass is interpreted as internal energy. While the
Smarr relation does not remain homogeneous of the first order, due to the
appearance of the fractal structure parameter in the relation. In addition, if
the deformation resulting from the effects of quantum gravity is at its
maximum, the Barrow entropy of black holes will be proportional to the cubic
size of black holes as in five dimensions [41].

We have selected charged AdS black holes as an application and we have derived
thermodynamic quantities and equations of state in terms of the parameter that
characterizes the fractal structure. Additionally, we have investigated
critical quantities within this context. After studying the evolution of the
temperature of black holes and their heat capacity in terms of the entropy for
different values of the electric charge and the fractal parameter, we have
reached the following conclusion: For the case $\tilde{Q}>\tilde{Q}_{c}$, no
phase transition occurs and black holes are stable. Second-order phase
transitions occur between small and large black holes, if the electric charge
equals the critical one. For, $\tilde{Q}<\tilde{Q}_{c}$, first-order phase
transitions occur between small and large black holes, passing through
intermediate black holes, where small and large black holes are stable and the
intermediate black holes are unstable. This behavior of thermodynamics is
similar to that of Van der Waals fluids. One important effect of the fractal
structure on thermodynamic properties is that at maximum complexity and
deformation, the critical quantities diverge. In other words, critical
phenomena do not occur. The closer the parameter is to 1, the more delayed the
phase transition becomes. Deformations caused by the effect of quantum gravity
on the event horizon hinder the phase transition. Therefore, these
deformations represent a “resistance to phase transitions”. In the absence of
deformations, the situation is associated with Reissner-Nordström AdS black
holes as outlined in [17].

In upcoming papers, we will extend our examination of how quantum gravity
impacts black holes to encompass various types of black holes and diverse
gravity theories.

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