# Degradability of Modified Landau-Streater Type Low-Noise Quantum Channels in
High Dimensions

Yun-Feng Lo∗ Department of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, GA, USA
Email<EMAIL_ADDRESS>Yen-Chi Lee∗ and Min-Hsiu Hsieh  Quantum
Computing Research Center
Hon-Hai Research Institute
Taipei, Taiwan
Email: {yen-chi.lee<EMAIL_ADDRESS>
###### Abstract

This paper delves into the degradability of quantum channels, with a specific
focus on high-dimensional extensions of qubit depolarizing channels in low-
noise regimes. We build upon the foundation of $\eta$-approximate degradable
channels, as established by Sutter et al. and Leditzky et al., to introduce
and examine the Modified Landau-Streater (MLS) channels. These channels expand
upon the qubit depolarizing and the recently proposed modified Werner-Holevo
channels by Roofeh and Karimipour, extending them to higher-dimensional
Hilbert spaces (with dimension $d=2j+1$, where $j$ are positive half-
integers). Our investigation centers on their conformity to the
$O(\varepsilon^{2})$ degradability pattern, aligning with and extending
Leditzky et al.’s findings in the $d=2$ case. By replacing the SU($2$)
generators with SU($d$) in our treatment, we may explore the potential
inclusion of generalized Gell-Mann matrices in future research. Our results
enhance the understanding of super-additivity in quantum channels within the
low-noise regime and lay the groundwork for future explorations into
conditions and structures that could lead to $O(\varepsilon^{2})$
degradability across a broader spectrum of quantum channels.

###### Index Terms:

quantum channel, degradability, diamond norm, SU(2) group, depolarizing
channel, Landau-Streater channel

##  I Introduction

††∗Co-first authors.

_Degradable quantum channels_ [1, 2], a crucial subset within the broad
spectrum of completely positive (CP) trace-preserving (TP) linear maps (a.k.a.
quantum channels [3, 4]), play a pivotal role in quantum information theory
[5, 6, 7, 8, 9, 10]. The defining feature of a degradable quantum channel is
the existence of a degrading channel that, after composing with the channel
itself, can simulate (one of) its complementary channel(s) [11] exactly. These
channels enjoy concise, single-letter formula (see [2, 12]) for determining
their quantum and private capacities [13, 14]. In this study, we focus on the
quantum capacity of quantum channels.

In quantum information theory, the channel coherent information $I_{c}$ (see
[15, 16]) is an important measure. It is known that if the channel coherent
information is additive, then the theory of quantum capacity closely aligns
with classical capacity theory [17]. The quantum capacity of a quantum channel
$\mathcal{N}$ is given by the following regularization formula [13]:

$Q(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}Q^{(1)}(\mathcal{N}^{\otimes
n}),$ (1)

where $\mathcal{N}^{\otimes n}$ represents the $n$-fold tensor product of
$\mathcal{N}$, and $Q^{(1)}$ also denotes the channel’s coherent information.
When

$\mathcal{Q}^{(1)}\left(\mathcal{N}^{\otimes
n}\right)=n\mathcal{Q}^{(1)}(\mathcal{N})$

for all $n$, the channel exhibits _weakly additive_ [17] coherent information,
simplifying the calculation of $\mathcal{Q}(\mathcal{N})$ to
$\mathcal{Q}^{(1)}(\mathcal{N})$. However, there exist some channels
satisfying $\mathcal{Q}^{(1)}\left(\mathcal{N}^{\otimes
n}\right)>n\mathcal{Q}^{(1)}(\mathcal{N})$ for a certain $n$, necessitating
the $n\rightarrow\infty$ limit in (1) for determining quantum capacity [18,
19, 16, 20, 21, 22].

Apart from entanglement-breaking channels [23], only a limited number of
channel types, such as (anti-)degradable [2] and PPT channels [24, 25], are
known to exhibit weak additivity [17]. The intuition that low-noise channels
should have limited non-additivity effects motivates an exploration into a
broader definition of degradability. In this context, Sutter et al.’s study
[26] introduces _$\eta$ -approximate degradable channels_, which conform to
the degradability standard to some extent, indicated by a small value
$\eta\geq 0$.111Typically, for a quantum channel $\mathcal{N}$ dependent on a
noise parameter $p$, the degradability parameter $\eta$ is given by
$\eta=cp^{r}$, with $c$ being a constant and $r>1$. For more details, see
[27]. Note, $\eta=0$ indicates an exactly degradable channel. These channels
are identified by the feasibility of finding a degrading channel $\mathcal{D}$
that, when composed with the original channel $\mathcal{N}$, results in a
channel within an $\varepsilon$ margin of (one of the) complementary
channel(s) $\mathcal{N}^{c}$ [11], as measured by the diamond norm [28,
Definition 2.20]. Notably, these approximate degradable channels (see [26,
29]) largely retain the features of fully degradable channels, which is
advantageous in mitigating the super-additivity effect [16, 17, 30] in
calculating quantum (or private) capacities.

To elaborate, the quantum capacity and the coherent information of a quantum
channel, characterized by its degradability222The degradability of a quantum
channel $\mathcal{N}$ is defined to be the smallest possible $\varepsilon$
value in
$\norm{\mathcal{N}^{c}-\mathcal{D}\circ\mathcal{N}}_{\diamond}\leq\varepsilon$
when the optimal degrading map $\mathcal{D}$ is applied.
$\text{dg}(\mathcal{N})=\eta$, are related as follows [27]:

$I_{c}(\mathcal{N})\leq Q(\mathcal{N})\leq I_{c}(\mathcal{N})+f(\eta)$ (2)

where the function $f(\eta)$ is defined as:

$\displaystyle\begin{split}f(\eta):=\frac{\eta}{2}\log(|E|-1)&+\eta\log|E|\\\
&+h\left(\frac{\eta}{2}\right)+\left(1+\frac{\eta}{2}\right)h\left(\frac{\eta}{2+\eta}\right).\end{split}$
(3)

Here, $h(x)=-x\log x-(1-x)\log(1-x)$ represents the binary entropy function
[31], and $|E|$ is the output dimension of any complementary channel
$\mathcal{N}^{c}$. In simpler terms, the degradability parameter $\eta$
influences the extent of the super-additivity effect in determining the
quantum capacity, with an associated error function $f(\eta)$ quantifying this
influence.

Building on the concept of approximate degradability, Leditzky et al.’s
research [27] investigates low-noise quantum channels. They discover that for
an $\varepsilon$-perfect (see the definition in Section II-B) channel
$\mathcal{N}$, where $\norm{\mathcal{N}-\text{Id}}_{\diamond}\leq\varepsilon$
within the $\varepsilon\in[0,2]$ range (where Id denotes the identity
channel), the degradability is upper bounded by $\text{dg}(\mathcal{N})\leq
2\varepsilon^{1.5}$. This estimation is set by choosing the complementary
channel as the degrading map for general low-noise channels. However, in well-
known quantum channels like the qubit depolarizing channel [32, 33] and the
BB84 channel (see [34, 35, 36, 37]), this upper bound is not tight. Leditzky
et al. [27] demonstrate that the upper bound on degradability can be refined
to $\text{dg}(\mathcal{N})\leq O(\varepsilon^{2})$ in the low-noise regime
(i.e., as $\varepsilon\to 0$, where $O(\cdot)$ is the big-oh notation333In
this paper, we denote $f(x)=O(g(x))$ or $f(x)\leq O(g(x))$ whenever there
exists a $C\geq 0$ that does not depend on $x$ (we call $C$ a _leading
constant_) such that $|f(x)|\leq Cg(x)$ for sufficiently small $x\geq 0$,
i.e., as $x\to 0^{+}$.), achieved through a more nuanced choice of the
degrading map [35].

This investigation leads to the ongoing question [27]: Is the
$O(\varepsilon^{2})$ degradability a universal characteristic for all
$\varepsilon$-perfect channels? Or, what channel structures lead to a
degradability parameter scaling as $O(\varepsilon^{2})$ in the low-noise
regime? These questions, as of this writing, remain open and form the
foundation for the upcoming explorations.

###  I-A Preview of our Main Result

In their recent work, Roofeh and Karimipour [38] introduced a modified version
of the Werner-Holevo channel, formulated as:

$\Lambda_{p}[\rho]=(1-p)\rho+p\Lambda_{\text{WH}}[\rho],$ (4)

where $\rho$ represents an arbitrary _quantum state_ (i.e., a unit-trace,
positive semi-definite operator/matrix) throughout this paper, $p\in[0,1]$ is
a parameter that we call the _low-noise_ parameter, and $\Lambda_{\text{WH}}$
symbolizes the Werner-Holevo map [39], defined as

$\rho\longmapsto\Lambda_{\text{WH}}[\rho]=\frac{1}{d-1}\left(\operatorname{tr}(\rho)\mathbbm{1}-\rho^{T}\right),$
(5)

and is applicable in any positive integer dimension $d\geq 2$. For dimension
$d=3$, the channel $\Lambda_{p}$, parameterized by a low-noise factor $p$,
serves as a model for a minor rotation of qutrit states in random directions
and angles.

Our research aims to further expand upon the channel formulated in Eq. (4),
extending its application to any dimension $d:=2j+1$. This extension involves
the use of high-spin matrices $J_{x},J_{y},J_{z}$, commonly employed in high-
energy physics [40]. Notice that matrices $J_{x},J_{y},J_{z}$ are the SU($2$)
generators acting on a $(2j+1)$-dimensional Hilbert space $\mathcal{H}_{2j+1}$
[41]. The unique properties and implications of these matrices within our
model will be detailed in Section II.

To facilitate a clear understanding, we begin by introducing the Landau-
Streater channel of spin $j$ [42, 41], denoted as $\mathcal{L}_{j}$, and
mathematically expressed as:

$\mathcal{L}_{j}[\rho]=\frac{1}{j(j+1)}(J_{x}\rho J_{x}+J_{y}\rho
J_{y}+J_{z}\rho J_{z}).$ (6)

In our following discussion, we will use the indices $x,y,z$ and $1,2,3$
interchangeably.

Building on this, we introduce a low-noise parameter $p$ and propose a family
of channels:

$\mathcal{M}_{j,p}[\rho]=(1-p)\text{Id}_{2j+1}[\rho]+p\mathcal{L}_{j}[\rho],$
(7)

which, for $p\ll 1$, can be regarded as a Landau-Streater type low-noise
channel, generalizing the model from Eq. (4). From now on, we will call this
new family of channels the Modified Landau-Streater (MLS) channels, which
possess a low-noise parameter $p$, and a dimension factor $j$. Notice that
when $j=1$, MLS channel reduces to the modified Werner-Holevo channel (see Eq.
(4)) [38, 41].

Our study’s central goal is to demonstrate that this novel family of channels,
applicable in any dimension, retains the $O(\varepsilon^{2})$ degradability
property. This research thereby broadens the scope of the findings by Leditzky
et al. [27].

###  I-B Organization of this Paper

The structure of this paper is organized as follows: Section II introduces
essential notations and definitions. Section III discusses key properties of
high-spin matrices. In Section IV, we present the six lemmas crucial for
proving our main theorem. Section V is dedicated to stating and proving the
main result, which establishes the $O(\varepsilon^{2})$ degradability of the
MLS channel family. The paper concludes with Section VI, where we summarize
our findings and implications.

##  II Notations and Preliminary

###  II-A The Diamond Norm

For an operator $M\in\mathcal{B}(\mathcal{H})$, where
$\mathcal{B}(\mathcal{H})$ denotes the space of bounded linear operators on
the Hilbert space $\mathcal{H}$, we recall the trace norm $\|M\|_{1}$ and the
max norm $\|M\|_{\max}$ as follows:

$\displaystyle\|M\|_{1}$
$\displaystyle:=\operatorname{tr}\sqrt{M^{\dagger}M},$ (8)
$\displaystyle\|M\|_{\max}$ $\displaystyle:=\max_{i,j}\left|M_{i,j}\right|.$

Note that the max norm is basis-dependent, unlike the trace and the operator
norms.

We now discuss the main distance measure we use for channels. For a linear map
$\Phi:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{K})$ between
Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, we define its diamond norm
[28, Definition 2.20] [27] by

$\|\Phi\|_{\diamond}:=\max_{M\in\mathcal{B}\left(\mathbb{C}^{d}\otimes\mathcal{H}\right)}\frac{\left\|\left(\mathrm{id}_{d}\otimes\Phi\right)(M)\right\|_{1}}{\|M\|_{1}}$

where $\operatorname{Id}_{d}$ denotes the identity map on $\mathbb{C}^{d}$
with $d=\operatorname{dim}(\mathcal{H})$.

The following technical lemma upper bounds the diamond norm of an arbitrary
linear map using the max norm of its Choi matrix. We just state it without
proof.

######  Lemma.

For a linear map $\Phi:A\rightarrow B$, we have

$\|\Phi\|_{\diamond}\leq|A||B|^{2}\|\mathcal{J}(\Phi)\|_{\max},$ (9)

where $|A|,|B|$ denote the dimension of the underlying Hilbert spaces with
respect to Alice and Bob respectively.

The proof of this Lemma can be found in [35].

###  II-B The $\varepsilon$-perfectness of MLS Channels

######  Definition ($\varepsilon$-perfectness).

If there exist $\varepsilon\geq 0$ such that a quantum channel $\mathcal{N}$
satisfies $\norm{\mathcal{N}-\text{Id}}_{\diamond}\leq\varepsilon$, we say
that this $\mathcal{N}$ is $\varepsilon$-perfect.444 According to this
definition: $\varepsilon^{\prime}$-perfect implies $\varepsilon$-perfect
provided that $\varepsilon^{\prime}$ is smaller than $\varepsilon$.

In order to show the $\varepsilon$-perfectness of MLS channels, we compute the
following:

$\|\mathcal{M}_{j,p}-\text{Id}_{2j+1}\|_{\diamond}=p\|\mathcal{L}_{j}-\text{Id}_{2j+1}\|_{\diamond}.$
(10)

Figure 1: Depicted here is the relationship between spin $j$ on the horizontal
axis and the diamond distance (ranging from $0$ to $2$) on the vertical axis.
This plot is generated by the SDP described in [28, Proposition 6.20].

We note that the diamond norm of any quantum channel (i.e., CPTP linear map)
is unity [28, Exercise 4.3, 3(b)]. Therefore, we have the following simple
bound due to the triangle equality of norms:

$\|\mathcal{L}_{j}-\text{Id}_{2j+1}\|_{\diamond}\leq\|\mathcal{L}_{j}\|_{\diamond}+\|\text{Id}_{2j+1}\|_{\diamond}=2.$
(11)

From this, we conclude that MLS channels are indeed $\varepsilon$-perfect with
$\varepsilon$ less than or equal to $2p$.

However, we hypothesize that the term
$\|\mathcal{L}_{j}-\text{Id}_{2j+1}\|_{\diamond}$ might exactly equal to $2$.
While we have yet to prove this conjecture analytically, our numerical
experiments provide supportive evidence. This is illustrated in Fig. 1.

##  III Properties of the High-Spin Matrices

General spin operators are traceless Hermitian operators that adhere to the
fundamental commutation relations of angular momentum (see [41] and [40, Chap
3.1.2, Eq. (3.20)]). The relation can be expressed as follows:

$\displaystyle[J_{\alpha},J_{\beta}]=i\hbar\varepsilon_{\alpha\beta\gamma}J_{\gamma}\quad\text{for}\quad\alpha,\beta,\gamma\in\\{1,2,3\\},$
(12)

where $[A,B]$ denotes the commutator of operators $A$ and $B$, defined by
$[A,B]:=AB-BA$. Here, $i$ represents the imaginary unit $\sqrt{-1}$, $\hbar$
is the reduced Planck constant, and $\varepsilon_{\alpha\beta\gamma}$ is the
Levi-Civita symbol of order $3$. For brevity, we can set $\hbar=1$ from now
on, which is just an irrelevant constant in the context of this study.

As noted in [40, Chapter 3.5], many properties of the angular momentum
operators can deduced from the fundamental commutation relations (12). By
convention, $J_{z}$ is chosen to be simultaneously diagonalizable with
$\mathbf{J}^{2}$. Furthermore, for a given _half-integer_ $j$, the eigenvalues
of $J_{z}$ are constrained to the set ${-j,-j+1,\ldots,j-1,j}$, which has a
cardinality of $2j+1$. For convenience, this set will be denoted as
$\llbracket-j;j\rrbracket$.

The above mentioned concept can be thought of as the quantization of angular
momentum. The simultaneous eigenkets (eigenvectors) of $\mathbf{J}^{2}$ and
$J_{z}$ are usually denoted as $\ket{j,m}$ for $m\in\llbracket-j;j\rrbracket$,
assumed to be normalized, where [40, Chapter 3.5.2, Eq. (3.186)] we have

$\displaystyle\begin{cases}\mathbf{J}^{2}\ket{j,m}=j(j+1)\ket{j,m},\\\
J_{z}\ket{j,m}=m\ket{j,m}.\end{cases}$ (13)

Throughout this study, we only consider $j$ to be a _positive_ half-integer,
which is conventionally called the _spin_. That is,
$j\in\mathbb{N}/2:=\\{\frac{1}{2},1,\frac{3}{2},2,\ldots\\}$.

To obtain general spin (or high-spin) matrices, namely, matrix representations
of angular momentum operators, it is conventional to express
$J_{x},J_{y},J_{z}$ on the set of eigenkets
$\\{\ket{j,m}\\}_{m\in\llbracket-j;j\rrbracket}$, which forms an orthonormal
basis of a $(2j+1)$-dimensional Hilbert space we denote by
$\mathcal{H}_{2j+1}$. To this end, it is more convenient to first find matrix
representations of the _ladder operators_ $J_{\pm}:=J_{x}\pm iJ_{y}$ first
[40, Chapter 3.5.1, Eq. (3.157)]. By convention, up to an arbitrary phase
factor, it can be derived that [40, Chapter 3.5.1, Eq. (3.193)]:

$\displaystyle J_{\pm}\ket{j,m}=\sqrt{(j\mp m)(j\pm m+1)}\ket{j,m\pm 1}$ (14)

for $m\in\llbracket-j;j\rrbracket$. Notice that when $m=j$, there is no need
to worry about defining $\ket{j,m+1}$ since the coefficient
$\sqrt{(j-m)(j+m+1)}$ is zero, and similarly when $m=-j$, there is no need to
worry about defining $\ket{j,m-1}$.

With the definition of ladder operators and (14), the matrix representations
of $J_{x},J_{y},J_{z}$ on the eigenbasis
$\\{\ket{j,m}\\}_{m\in\llbracket-j;j\rrbracket}$ can be derived. First, we
obtain

$\displaystyle\begin{split}J_{x}\ket{j,m}=\leavevmode\nobreak\
&\frac{1}{2}\Big{(}\sqrt{j(j+1)-m(m+1)}\ket{j,m+1}\\\
&+\sqrt{j(j+1)-m(m-1)}\ket{j,m-1}\Big{)}\end{split}$ (15)

and

$\displaystyle\begin{split}J_{y}\ket{j,m}=\leavevmode\nobreak\
&\frac{1}{2i}\Big{(}\sqrt{j(j+1)-m(m+1)}\ket{j,m+1}\\\
&-\sqrt{j(j+1)-m(m-1)}\ket{j,m-1}\Big{)}\end{split}$ (16)

for all $m\in\llbracket-j;j\rrbracket$. Therefore, the matrix elements for
$J_{x},J_{y},J_{z}$ on the basis
$\\{\ket{j,m}\\}_{m\in\llbracket-j;j\rrbracket}$ are

$\displaystyle\begin{cases}\bra{j,\ell}J_{x}\ket{j,m}=\frac{1}{2}(\delta_{\ell,m+1}+\delta_{\ell,m-1})\sqrt{j(j+1)-\ell
m},\\\
\bra{j,\ell}J_{y}\ket{j,m}=\frac{i}{2}(\delta_{\ell,m-1}-\delta_{\ell,m+1})\sqrt{j(j+1)-\ell
m},\\\ \bra{j,\ell}J_{z}\ket{j,m}=m\delta_{\ell,m},\end{cases}$ (17)

and also,

$\displaystyle\bra{j,\ell}\mathbf{J}^{2}\ket{j,m}=j(j+1)\delta_{\ell,m},$ (18)

all for $\ell,m\in\llbracket-j;j\rrbracket$, where $\delta_{a,b}$ denotes the
Kronecker delta. Hence, general spin matrices with spin $j$ can be obtained
via the explicit formulas in (17). Note that we also slightly abuse the
notations by regarding $J_{x},J_{y},J_{z}$ as both the angular momentum
_operators_ and their _matrix_ representations via the eigenbasis
$\\{\ket{j,m}\\}_{m\in\llbracket-j;j\rrbracket}$.

##  IV Six Lemmas for Proving
the Main Theorem

In this section, we state six Lemmas essential for deriving our main result
without proof. (Proof details are left to Appendix B.) To streamline the
statements, we employ a strategy that leverages the cyclic permutation
inherent in the fundamental commutation relations (12). This approach involves
a mapping where $x\leftrightarrow 1$, $y\leftrightarrow 2$, and
$z\leftrightarrow 0$, with the indices of $J$ extended to the set of integers,
$\mathbb{Z}$. Specifically, we define

$J_{k}:=J_{k\leavevmode\nobreak\ (\text{mod}\leavevmode\nobreak\ 3)}$ (19)

for any $k\in\mathbb{Z}$. For instance, under this scheme, we have
equivalences such as $J_{x}=J_{-2}=J_{1}=J_{4}$, $J_{y}=J_{-1}=J_{2}=J_{5}$,
and $J_{z}=J_{0}=J_{3}=J_{6}$.

It is important to note that Lemma 1 and Lemma 2 are essentially
reformulations of the fundamental commutation relations and the matrix
representation (18) of $\mathbf{J}^{2}$, respectively, adapted to this
extended index set. This modification is implemented to facilitate their
application in subsequent analyses.

######  Lemma 1.

For any $k\in\mathbb{Z}$, the commutator $[J_{k},J_{k+1}]$ equals $iJ_{k+2}$.

######  Lemma 2.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=k-1}^{k+1}J_{\ell}^{2}=j(j+1)I_{2j+1}$, where
$I_{d}:\mathcal{H}_{d}\to\mathcal{H}_{d}$ denotes the identity operator (i.e.,
the $d\times d$ identity matrix).

The following 4 lemmas can be proved starting from only Lemma 1 and 2. (See
Appendix B.) Note that equivalent results of these 4 lemmas can be found in
[43, Eq. (28)-(29)], where they are stated without proofs.

######  Lemma 3.

$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{\ell}=(j(j+1)-1)J_{k},$ for any
$k\in\mathbb{Z}$.

######  Lemma 4.

$\sum_{\ell=1}^{3}J_{\ell}J^{2}_{k}J_{\ell}=(j(j+1)-3)J^{2}_{k}+j(j+1)I_{2j+1},$
for any $k\in\mathbb{Z}$.

######  Lemma 5.

$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k-1}J_{\ell}=(j(j+1)-3)J_{k}J_{k-1}-iJ_{k-2},$
for any $k\in\mathbb{Z}$.

######  Lemma 6.

$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k+1}J_{\ell}=(j(j+1)-3)J_{k}J_{k+1}+iJ_{k+2},$
for any $k\in\mathbb{Z}$.

##  V Main Result

To address the degradability of an MLS channel $\mathcal{M}_{j,p}$, our
objective is to identify an effective degrading map $\mathcal{D}_{j,p}$,
contingent on both parameters $j$ and $p$. We aim to demonstrate that the
application of $\mathcal{D}_{j,p}$ subsequent to $\mathcal{M}_{j,p}$ results
in an approximate version of a certain complementary channel to
$\mathcal{M}_{j,p}$ which we denote by $\mathcal{M}_{j,p}^{c}$.

Our hypothesis is that for a given MLS channel $\mathcal{M}_{j,p}$, we can
always select $\mathcal{D}_{j,p}=\mathcal{M}^{\text{c}}_{j,p+ap^{2}}$ ensuring
the target diamond norm is within the order of $O(p^{2})$, given a careful
choice of the parameter $a$. To articulate this more precisely, we present our
main theorem as follows:

######  Theorem 1.

For any spin value $j$, there exists a specific value of $a=2/(j(j+1))$ such
that

$\|\mathcal{M}^{\text{c}}_{j,p}-\mathcal{M}^{\text{c}}_{j,p+ap^{2}}\circ\mathcal{M}_{j,p}\|_{\diamond}=\mathcal{O}(p^{2}).$
(20)

The leading constant depends only on $j$.

In essence, this theorem allows us to reframe our understanding: for a given
dimension factor $j$, the constructed channel family
$\\{\mathcal{M}_{j,p}\\}$, considered as a one-parameter family indexed by
$p$, is degradable to at least the order of $O(p^{2})$.

###### Proof.

(This is just a proof sketch; see Appendix A for a complete proof.) As the
assertion (20) suggests, it should be helpful to write down an explicit form
of a complementary channel of $\mathcal{M}_{j,p}$. As it turns out (see
Appendix A-A), we can choose such a complementary channel of
$\mathcal{M}_{j,p}$ such that its corresponding output to an input quantum
state $\rho$ is given by (21) at the top of the next page,

$\begin{split}\mathcal{M}^{\text{c}}_{j,p}(\rho)&=\left(\begin{array}[]{rrrr}(1-p)\Tr(\rho)&c_{j}(p)\Tr(J_{1}\rho)&c_{j}(p)\Tr(J_{2}\rho)&c_{j}(p)\Tr(J_{3}\rho)\\\
c_{j}(p)\Tr(J_{1}\rho)&p_{j}\Tr(J_{1}J_{1}\rho)&p_{j}\Tr(J_{2}J_{1}\rho)&p_{j}\Tr(J_{3}J_{1}\rho)\\\
c_{j}(p)\Tr(J_{2}\rho)&p_{j}\Tr(J_{1}J_{2}\rho)&p_{j}\Tr(J_{2}J_{2}\rho)&p_{j}\Tr(J_{3}J_{2}\rho)\\\
c_{j}(p)\Tr(J_{3}\rho)&p_{j}\Tr(J_{1}J_{3}\rho)&p_{j}\Tr(J_{2}J_{3}\rho)&p_{j}\Tr(J_{3}J_{3}\rho)\\\
\end{array}\right)\end{split}$ (21)

where we have denoted $c_{j}(p):=\sqrt{p(1-p)/(j(j+1))}$ and
$p_{j}:=p/(j(j+1))$ for brevity. Setting the slightly larger noise parameter
$s:=p+ap^{2}$ for some $a>0$ and applying (21) with this parameter, we can
compose $\mathcal{M}^{\text{c}}_{j,s}$ with $\mathcal{M}_{j,p}$ to obtain a
matrix form for
$\mathcal{M}^{\text{c}}_{j,p+ap^{2}}\circ\mathcal{M}_{j,p}(\rho)$ as

$\displaystyle\mathcal{M}^{\text{c}}_{j,p+ap^{2}}\circ\mathcal{M}_{j,p}(\rho)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\left(\begin{array}[]{l|l|l|l}(1-s)\Tr(\rho)&c_{j}(s)(1-p_{j})\Tr(J_{1}\rho)&c_{j}(s)(1-p_{j})\Tr(J_{2}\rho)&c_{j}(s)(1-p_{j})\Tr(J_{3}\rho)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{1}\rho)&s_{j}(1-3p_{j})\Tr(J_{1}^{2}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{2}J_{1}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{3}J_{1}\rho\right)\\\
&+s_{j}p\Tr(\rho)&-is_{j}p_{j}\Tr\left(J_{3}\rho\right)&+is_{j}p_{j}\Tr\left(J_{2}\rho\right)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{2}\rho)&s_{j}\left(1-3p_{j}\right)\Tr\left(J_{1}J_{2}\rho\right)&s_{j}(1-3p_{j})\Tr(J_{2}^{2}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{3}J_{2}\rho\right)\\\
&+is_{j}p_{j}\Tr\left(J_{3}\rho\right)&+s_{j}p\Tr(\rho)&-is_{j}p_{j}\Tr\left(J_{1}\rho\right)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{3}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{1}J_{3}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{2}J_{3}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{3}^{2}\rho\right)\\\
&-is_{j}p_{j}\Tr\left(J_{2}\rho\right)&+is_{j}p_{j}\Tr\left(J_{1}\rho\right)&+s_{j}p\Tr(\rho)\end{array}\right)$
(29)

(V) at the top of the next page, where we have denoted $s_{j}:=s/(j(j+1))$ for
brevity. (For details, refer to Appendix A-B-1) Then we denote the linear map
$\Phi:=\mathcal{M}^{\text{c}}_{j,p}-\mathcal{M}^{\text{c}}_{j,s}\circ\mathcal{M}_{j,p}$,
whose corresponding output of any input quantum state $\rho$ can be given by
subtracting (V) from (21).

We apply [35, Lemma 1] to upper bound the diamond norm of $\Phi$ and obtain
that $\|\Phi\|_{\diamond}\leq 16(2j+1)\|\mathcal{J}(\Phi)\|_{\text{max}}$,
where
$\mathcal{J}(\Phi):=\sum_{\alpha,\beta\in\llbracket-j;j\rrbracket}\ket{\alpha}\bra{\beta}\otimes\Phi(\ket{\alpha}\bra{\beta})$
is the Choi matrix of $\Phi$ with respect to
$\\{\ket{\alpha}\\}_{\alpha\in\llbracket-j;j\rrbracket}$, an orthonormal basis
for $\mathcal{H}_{2j+1}$. For convenience, we choose
$\ket{\alpha}:=\ket{j,\alpha},\leavevmode\nobreak\
\forall\alpha\in\llbracket-j;j\rrbracket$, i.e., the eigen-kets of $J_{z}$ as
discussed in Sec. III. Moreover, we have
$\|\mathcal{J}(\Phi)\|_{\text{max}}=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\|\Phi(\ket{\alpha}\bra{\beta})\|_{\text{max}}$
thanks to the block structure of the Choi matrix. Then,

$\displaystyle\begin{split}\|\mathcal{J}(\Phi)\|_{\text{max}}=\leavevmode\nobreak\
&\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\max_{\ell,m\in\\{0,1,2,3\\}}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|\\\
=\leavevmode\nobreak\
&\max_{\ell,m\in\\{0,1,2,3\\}}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|,\end{split}$
(30)

where $[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}$ denotes the $(\ell,m)$-th
entry of $\Phi(\ket{\alpha}\bra{\beta})$, where we adopt the convention that
matrix indices start from $0$, not $1$. Thus, to bound
$\|\mathcal{J}(\Phi)\|_{\text{max}}$ and hence $\|\Phi\|_{\diamond}$, we
divide the indices $(\ell,m)$ into 5 cases and bound
$\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|$
respectively; for details, refer to Appendix A-B-2. In summary, we find that
in the low-noise regime, _unless we set $a=2/(j(j+1))$_, the indices
$(\ell,m)$ with either $\ell=0$ or $m=0$ will dominate the other cases (each
having an $\mathcal{O}(p^{2})$ bound) and yield an $\mathcal{O}(p^{3/2})$
bound after application of a Taylor series expansion. However, if we _do_ set
$a=2/(j(j+1))$, these indices instead yield a smaller $\mathcal{O}(p^{5/2})$
bound while all other indices still yield a $\mathcal{O}(p^{2})$ bound. Hence,
with this choice of $a$, we have
$\|\mathcal{J}(\Phi)\|_{\text{max}}=\mathcal{O}(p^{2})$, where the leading
constant only depends on the spin $j$. Altogether, we conclude that
$\|\Phi\|_{\diamond}\leq
16(2j+1)\|\mathcal{J}(\Phi)\|_{\text{max}}=\mathcal{O}(p^{2})$ in the low-
noise regime, where the leading constant again only depends on the spin $j$. ∎

##  VI Conclusion

In this study, we have delved into the intricate world of (approximate)
degradable quantum channels, particularly focusing on their high-dimensional
extensions via SU$(2)$ generators from qubit depolarizing channels. Our work,
drawing on the foundational studies by Sutter et al. [26] and Leditzky et al.
[27], has centered on exploring the concept of $\eta$-approximate degradable
channels, where $\eta=cp^{r}$ for a low-noise parameter $p$ and a certain
positive degree $r$. Understanding these channels is critical in shedding
light on the super-additivity effect in quantum channels operating in low-
noise regimes.

A key aspect of this research has been the development of our primary
experimental family of channels, the MLS family (7). This family is an
extension of both the qubit depolarizing channel and the modified Werner-
Holevo channel, as introduced by Roofeh et al. [38], into higher-dimensional
Hilbert spaces (with dimension $d=2j+1$, where $j$ is the spin, taking
positive half-integer values.) Our investigation aimed to determine whether
the MLS channels, characterized by a low-noise parameter $p$ and adaptable to
arbitrary dimensions $d$, follow the $O(p^{2})$ degradability pattern. This
finding, aligning with the work of Leditzky et al., hints at the potential for
broader applicability of the $O(p^{2})$ degradability characteristic in
general low-noise channels.

Looking ahead, our research paves the way for examining other potential
extensions of qubit depolarizing channels in high-dimensional scenarios. These
include exploring generalized Pauli channels defined in terms of a generalized
Pauli basis, multi-qubit Pauli matrices, and generalized Gell-Mann matrices
(i.e., the SU$(d)$ generators). While Leditzky et al. have confirmed that the
generalized Pauli channels exhibit $O(p^{2})$ degradability, the latter two
extensions calls for further investigation.

Our findings in this paper lay a foundation for future explorations aimed at
identifying general conditions or specific structures for quantum channels to
exhibit $O(p^{2})$ degradability. The long-term goal is to extend this
desirable property to a wider spectrum of quantum channels beyond the
aforementioned generalizations of the qubit depolarizing channel, thereby
enriching our understanding of the super-additivity effect in high-dimensional
low-noise scenarios.

## Acknowledgment

We thank Matthieu Bloch, Felix Leditzky, and Graeme Smith for inspiring
discussions and helpful feedback.

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##  Appendix A Calculations to Establish the $O(p^{2})$-degradability for MLS
Channels under Arbitrary $j$

###  A-A The Complementary Channel(s) of the MLS Channel

Recall that we defined the Modified Landau-Streater (MLS) quantum channel as
follows:

$\displaystyle\begin{split}\mathcal{M}_{j,p}(\rho):&=(1-p)\rho+p\frac{1}{j(j+1)}(J_{x}\rho
J_{x}+J_{y}\rho J_{y}+J_{z}\rho J_{z})\\\
&=\sum_{\ell=0}^{3}K^{(\ell)}_{j,p}\rho K^{(\ell)\dagger}_{j,p},\end{split}$
(31)

where the Kraus operators $K^{(\ell)}_{j,p}$ are

$\displaystyle K^{(0)}_{j,p}$
$\displaystyle=\sqrt{1-p}I,\quad\text{where}\leavevmode\nobreak\ I:=I_{2j+1};$
(32) $\displaystyle K^{(\ell)}_{j,p}$
$\displaystyle=\sqrt{\frac{p}{j(j+1)}}J_{\ell},\quad\text{for}\leavevmode\nobreak\
\ell\in\\{1,2,3\\}.$ (33)

The validity of the MLS _channel_ being a CPTP (linear) map follows from that
of the LS channel, which is proven in [41]. The validity of the Kraus
representation in (31) of the MLS channel can be seen by noting that each of
the $\text{SU}(2)$ generator
$\\{J_{\ell}\\}_{\ell=1}^{3}=\\{J_{x},J_{y},J_{z}\\}$ is Hermitian. From [6,
Chapter 5.2, Eq. (5.53)], we can explicitly compute one of a complementary
channel of $\mathcal{M}_{j,p}$, which we denote by
$\mathcal{M}^{\text{c}}_{j,p}$, as the following map:
$\mathcal{M}^{\text{c}}_{j,p}:\mathcal{B}(\mathcal{H}_{2j+1})\to\mathcal{B}(\mathbb{C}^{4})$
where

$\displaystyle\begin{split}\mathcal{M}^{\text{c}}_{j,p}(\rho)&=\sum_{\ell=0}^{3}\sum_{m=0}^{3}\Tr\left(K^{(\ell)}_{j,p}\rho
K^{(m)\dagger}_{j,p}\right)\ket{\ell}\bra{m}\\\
&=\left(\begin{array}[]{rrrr}(1-p)\Tr(\rho)&c_{j}(p)\Tr(J_{1}\rho)&c_{j}(p)\Tr(J_{2}\rho)&c_{j}(p)\Tr(J_{3}\rho)\\\
c_{j}(p)\Tr(J_{1}\rho)&p_{j}\Tr(J_{1}J_{1}\rho)&p_{j}\Tr(J_{2}J_{1}\rho)&p_{j}\Tr(J_{3}J_{1}\rho)\\\
c_{j}(p)\Tr(J_{2}\rho)&p_{j}\Tr(J_{1}J_{2}\rho)&p_{j}\Tr(J_{2}J_{2}\rho)&p_{j}\Tr(J_{3}J_{2}\rho)\\\
c_{j}(p)\Tr(J_{3}\rho)&p_{j}\Tr(J_{1}J_{3}\rho)&p_{j}\Tr(J_{2}J_{3}\rho)&p_{j}\Tr(J_{3}J_{3}\rho)\\\
\end{array}\right),\end{split}$ (34)

where $\\{\ket{\ell}\\}_{\ell=0}^{3}$ is chosen to be the computational basis
of $\mathbb{C}^{4}$. Note that in (34) we applied the cyclic property of
trace, the generators $J_{x},J_{y},J_{z}$ being Hermitian, and for brevity we
introduced the following auxiliary function

$c_{j}(p):=\sqrt{\frac{p(1-p)}{j(j+1)}}\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ 0\leq p\leq 1,$ (35)

and the auxiliary noise parameter $p_{j}:=\frac{p}{j(j+1)}.$ It is known that
for a fixed quantum channel, the uniqueness of its complementary channel is
only up to an isometry on the environment’s system [6].

###  A-B Approximate Degradability of MLS Channels

In this part, we aim to show that the MLS channel $\mathcal{M}_{j,p}$ is
approximately degradable, defined in [27], with degradability parameter
$\text{dg}(\mathcal{M}_{j,p})\leq\mathcal{O}(p^{2})$, where the leading
coefficient depends on the spin $j$. Recall that the degradability parameter
of a general quantum channel $\mathcal{N}:A\to B$ with environment $E$ is
defined as [27]:

$\displaystyle\text{dg}(\mathcal{N}):=\inf\\{\eta\in[0,2]:\exists\leavevmode\nobreak\
\text{quantum channel}\leavevmode\nobreak\ \mathcal{D}:B\to
E\leavevmode\nobreak\ \text{such that}\leavevmode\nobreak\
\|\mathcal{N}^{\text{c}}-\mathcal{D}\circ\mathcal{N}\|_{\diamond}\leq\eta\\},$
(36)

where $\|\cdot\|_{\diamond}$ denotes the diamond norm [6].

We followed the method proposed by Leditzky et al. [27] and employ the
“slightly noisier" complementary MLS channel $\mathcal{M}^{\text{c}}_{j,s}$ as
an approximately degrading map $\mathcal{D}$ for the MLS channel
$\mathcal{M}_{j,p}$, where $s:=p+ap^{2}$ for some $a\geq 0$ to be determined.
To this end, we need to bound from above the diamond norm

$\displaystyle\|\mathcal{M}^{\text{c}}_{j,p}-\mathcal{D}\circ\mathcal{M}_{j,p}\|_{\diamond}=\|\mathcal{M}^{\text{c}}_{j,p}-\mathcal{M}^{\text{c}}_{j,s}\circ\mathcal{M}_{j,p}\|_{\diamond}=\|\mathcal{M}^{\text{c}}_{j,p}-\Psi\|_{\diamond}=\|\Phi\|_{\diamond},$
(37)

where for brevity we denoted, for a fixed set of system parameters $j,p$ and
tunable parameter $a$,

$\displaystyle\Psi$
$\displaystyle:=\mathcal{M}^{\text{c}}_{j,s}\circ\mathcal{M}_{j,p};$ (38)
$\displaystyle\Phi$ $\displaystyle:=\mathcal{M}^{\text{c}}_{j,p}-\Psi.$ (39)

We notice that $\Psi$, being a composition of two quantum channels, is still a
quantum channel, while $\Phi$ is no longer guaranteed to be completely
positive (CP) since it is the difference of two CP maps.

####  A-B1 Computation of Matrix Representation of $\Psi(\rho)$ and
$\Phi(\rho)$

First, we compute the quantum channel $\Psi$ explicitly by fixing an input
operator $\rho$ and write

$\displaystyle\Psi(\rho)=:\psi=\sum_{\ell=0}^{3}\sum_{m=0}^{3}\psi_{\ell,m}\ket{\ell}\bra{m}.$
(40)

The $(\ell,m)$th entry of $\psi$ is, by (34),

$\displaystyle\begin{split}\psi_{\ell,m}&=\Tr\left(K^{(\ell)}_{j,s}\mathcal{M}_{j,p}(\rho)K^{(m)\dagger}_{j,s}\right)\\\
&\overset{(a)}{=}\Tr\left(K^{(\ell)}_{j,s}\sum_{k=0}^{3}K^{(k)}_{j,p}\rho
K^{(k)\dagger}_{j,p}K^{(m)\dagger}_{j,s}\right)\\\
&=\sum_{k=0}^{3}\Tr\left(K^{(\ell)}_{j,s}K^{(k)}_{j,p}\rho
K^{(k)\dagger}_{j,p}K^{(m)\dagger}_{j,s}\right)\\\
&\overset{(b)}{=}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}K^{(m)}_{j,s}K^{(\ell)}_{j,s}K^{(k)}_{j,p}\rho\right),\end{split}$
(41)

where in (a) we applied the Kraus representation in (31) for
$\mathcal{M}_{j,p}$, and in (b) we used the cyclic property of trace and the
fact that $I,J_{x},J_{y},J_{z}$, and hence the Kraus operators $K^{(m)}_{j,s}$
and $K^{(k)}_{j,p}$, are Hermitian.

To find a matrix representation for

$\displaystyle\psi=\sum_{\ell=0}^{3}\sum_{m=0}^{3}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}K^{(m)}_{j,s}K^{(\ell)}_{j,s}K^{(k)}_{j,p}\rho\right)\ket{\ell}\bra{m},$
(42)

recall that

$\displaystyle K^{(\ell)}_{j,p}=\begin{cases}\sqrt{\bar{p}}I,&\ell=0;\\\
\sqrt{p_{j}}J_{\ell},&\ell\in\\{1,2,3\\},\end{cases}$ (43)

where $\bar{p}:=1-p$. Similarly, we have

$\displaystyle K^{(\ell)}_{j,s}=\begin{cases}\sqrt{\bar{s}}I,&\ell=0;\\\
\sqrt{s_{j}}J_{\ell},&\ell\in\\{1,2,3\\},\end{cases}$ (44)

where $\bar{s}:=1-s$ and, for brevity, we also denote another auxiliary noise
parameter

$\displaystyle s_{j}:=\frac{s}{j(j+1)}.$ (45)

Note that for $p\geq 0$ small enough, we have $\bar{s}\geq 0$, so that
$\sqrt{\bar{s}}$ is real.

For the calculation of $\psi_{\ell,m}$ through (41), we divide $(\ell,m)$ into
5 cases:

  * •

$\ell=m=0$:

$\displaystyle\psi_{0,0}=\bar{s}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}K^{(k)}_{j,p}\rho\right)=\bar{s}\Tr\left(\sum_{k=0}^{3}K^{(k)}_{j,p}\rho
K^{(k)\dagger}_{j,p}\right)=\bar{s}\Tr\left(\mathcal{M}_{j,p}(\rho)\right)=\bar{s}\Tr(\rho),$
(46)

where the last equality follows from the quantum channel $\mathcal{M}_{j,p}$
being trace-preserving.

  * •

$\ell=0$ and $m\in\\{1,2,3\\}$: Since $K^{(0)}_{j,s}$ is a multiple of $I$, it
commutes with $K^{(m)}_{j,s}$ for any $m\in\\{1,2,3\\}$, and thus

$\displaystyle\begin{split}\psi_{\ell,m}&=\psi_{m,\ell}=\sqrt{s_{j}\bar{s}}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}K^{(k)}_{j,p}\rho\right)=\sqrt{s_{j}\bar{s}}\left(\Tr\left(K^{(0)}_{j,p}J_{m}K^{(0)}_{j,p}\rho\right)+\sum_{k=1}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}K^{(k)}_{j,p}\rho\right)\right)\\\
&=c_{j}(s)\left(\bar{p}\Tr(J_{m}\rho)+p_{j}\Tr\left(\sum_{k=1}^{3}J_{k}J_{m}J_{k}\rho\right)\right)=c_{j}(s)\left(\bar{p}\Tr(J_{m}\rho)+p_{j}\Tr\left((j(j+1)-1)J_{m})\rho\right)\right)\\\
&=c_{j}(s)(1-p_{j})\Tr\left(J_{m}\rho\right),\end{split}$ (47)

where the penultimate equality follows from Lemma 3.

  * •

$\ell=m\in\\{1,2,3\\}$:

$\displaystyle\begin{split}\psi_{\ell,m}&=s_{j}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}^{2}K^{(k)}_{j,p}\rho\right)=s_{j}\left(\Tr\left(K^{(0)}_{j,p}J_{m}^{2}K^{(0)}_{j,p}\rho\right)+\sum_{k=1}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}^{2}K^{(k)}_{j,p}\rho\right)\right)\\\
&=s_{j}\left(\bar{p}\Tr\left(J_{m}^{2}\rho\right)+p_{j}\Tr\left(\sum_{k=1}^{3}J_{k}J_{m}^{2}J_{k}\rho\right)\right)=s_{j}\left(\bar{p}\Tr\left(J_{m}^{2}\rho\right)+p_{j}\Tr\left(\left((j(j+1)-3)J_{m}^{2}+j(j+1)I\right)\rho\right)\right)\\\
&=s_{j}p\Tr(\rho)+s_{j}(1-3p_{j})\Tr(J_{m}^{2}\rho),\end{split}$ (48)

where penultimate equality follows from Lemma 4.

  * •

$(\ell,m)\in\\{(1,2),(2,3),(3,1)\\}$:

$\displaystyle\begin{split}\psi_{\ell,m}&=s_{j}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}J_{\ell}K^{(k)}_{j,p}\rho\right)=s_{j}\left(\Tr\left(K^{(0)}_{j,p}J_{m}J_{\ell}K^{(0)}_{j,p}\rho\right)+\sum_{k=1}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}J_{\ell}K^{(k)}_{j,p}\rho\right)\right)\\\
&=s_{j}\left(\bar{p}\Tr\left(J_{m}J_{\ell}\rho\right)+p_{j}\Tr\left(\sum_{k=1}^{3}J_{k}J_{m}J_{\ell}J_{k}\rho\right)\right)=s_{j}\left(\bar{p}\Tr\left(J_{m}J_{\ell}\rho\right)+p_{j}\Tr\left(\left((j(j+1)-3)J_{m}J_{\ell}-iJ_{\ell-1}\right)\rho\right)\right)\\\
&=-is_{j}p_{j}\Tr(J_{\ell-1}\rho)+s_{j}(1-3p_{j})\Tr(J_{m}J_{\ell}\rho),\end{split}$
(49)

where penultimate equality follows from Lemma 5.

  * •

$(\ell,m)\in\\{(1,3),(2,1),(3,2)\\}$:

$\displaystyle\begin{split}\psi_{\ell,m}&=s_{j}\sum_{k=0}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}J_{\ell}K^{(k)}_{j,p}\rho\right)=s_{j}\left(\Tr\left(K^{(0)}_{j,p}J_{m}J_{\ell}K^{(0)}_{j,p}\rho\right)+\sum_{k=1}^{3}\Tr\left(K^{(k)}_{j,p}J_{m}J_{\ell}K^{(k)}_{j,p}\rho\right)\right)\\\
&=s_{j}\left(\bar{p}\Tr\left(J_{m}J_{\ell}\rho\right)+p_{j}\Tr\left(\sum_{k=1}^{3}J_{k}J_{m}J_{\ell}J_{k}\rho\right)\right)=s_{j}\left(\bar{p}\Tr\left(J_{m}J_{\ell}\rho\right)+p_{j}\Tr\left(\left((j(j+1)-3)J_{m}J_{\ell}+iJ_{\ell+1}\right)\rho\right)\right)\\\
&=is_{j}p_{j}\Tr(J_{\ell+1}\rho)+s_{j}(1-3p_{j})\Tr(J_{m}J_{\ell}\rho),\end{split}$
(50)

where penultimate equality follows from Lemma 6.

Combining the 5 cases considered, we have a matrix representation for
$\Psi(\rho)=\psi$, namely,

$\displaystyle\psi=\left(\begin{array}[]{l|l|l|l}(1-s)\Tr(\rho)&c_{j}(s)(1-p_{j})\Tr(J_{1}\rho)&c_{j}(s)(1-p_{j})\Tr(J_{2}\rho)&c_{j}(s)(1-p_{j})\Tr(J_{3}\rho)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{1}\rho)&s_{j}(1-3p_{j})\Tr(J_{1}^{2}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{2}J_{1}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{3}J_{1}\rho\right)\\\
&+s_{j}p\Tr(\rho)&-is_{j}p_{j}\Tr\left(J_{3}\rho\right)&+is_{j}p_{j}\Tr\left(J_{2}\rho\right)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{2}\rho)&s_{j}\left(1-3p_{j}\right)\Tr\left(J_{1}J_{2}\rho\right)&s_{j}(1-3p_{j})\Tr(J_{2}^{2}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{3}J_{2}\rho\right)\\\
&+is_{j}p_{j}\Tr\left(J_{3}\rho\right)&+s_{j}p\Tr(\rho)&-is_{j}p_{j}\Tr\left(J_{1}\rho\right)\\\
\hline\cr
c_{j}(s)(1-p_{j})\Tr(J_{3}\rho)&s_{j}(1-3p_{j})\Tr\left(J_{1}J_{3}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{2}J_{3}\rho\right)&s_{j}(1-3p_{j})\Tr\left(J_{3}^{2}\rho\right)\\\
&-is_{j}p_{j}\Tr\left(J_{2}\rho\right)&+is_{j}p_{j}\Tr\left(J_{1}\rho\right)&+s_{j}p\Tr(\rho)\end{array}\right).$
(58)

Then, using (34) and (58), for the same fixed input operator $\rho$, we can
calculate that

$\displaystyle\phi:=\Phi(\rho)=\mathcal{M}^{\text{c}}_{j,p}(\rho)-\psi$ (59)

has the following matrix form

$\displaystyle\left(\begin{array}[]{l|l|l|l}(s-p)\Tr(\rho)&(c_{j}(p)-c_{j}(s)(1-p_{j}))&(c_{j}(p)-c_{j}(s)(1-p_{j}))&(c_{j}(p)-c_{j}(s)(1-p_{j}))\\\
&\cdot\Tr(J_{1}\rho)&\cdot\Tr(J_{2}\rho)&\cdot\Tr(J_{3}\rho)\\\
\hline\cr(c_{j}(p)-c_{j}(s)(1-p_{j}))&(p_{j}-s_{j}(1-3p_{j}))\Tr(J_{1}^{2}\rho)&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{2}J_{1}\rho\right)&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{3}J_{1}\rho\right)\\\
\cdot\Tr(J_{1}\rho)&-s_{j}p\Tr(\rho)&+is_{j}p_{j}\Tr\left(J_{3}\rho\right)&-is_{j}p_{j}\Tr\left(J_{2}\rho\right)\\\
\hline\cr(c_{j}(p)-c_{j}(s)(1-p_{j}))&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{1}J_{2}\rho\right)&(p_{j}-s_{j}(1-3p_{j}))\Tr(J_{2}^{2}\rho)&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{3}J_{2}\rho\right)\\\
\cdot\Tr(J_{2}\rho)&-is_{j}p_{j}\Tr\left(J_{3}\rho\right)&-s_{j}p\Tr(\rho)&+is_{j}p_{j}\Tr\left(J_{1}\rho\right)\\\
\hline\cr(c_{j}(p)-c_{j}(s)(1-p_{j}))&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{1}J_{3}\rho\right)&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{2}J_{3}\rho\right)&(p_{j}-s_{j}(1-3p_{j}))\Tr\left(J_{3}^{2}\rho\right)\\\
\cdot\Tr(J_{3}\rho)&+is_{j}p_{j}\Tr\left(J_{2}\rho\right)&-is_{j}p_{j}\Tr\left(J_{1}\rho\right)&-s_{j}p\Tr(\rho)\end{array}\right),$
(68)

which gives a concrete description of the map
$\Phi:\mathcal{B}(\mathcal{H}_{2j+1})\to\mathcal{B}(\mathbb{C}^{4})$.

####  A-B2 Bounding from above the diamond norm of $\Phi$

Using the closed-form expression (68), we can bound from above the diamond
norm of the linear map $\Phi$. By applying [35, Lemma 1], we have

$\displaystyle\|\Phi\|_{\diamond}\leq(2j+1)\cdot
4^{2}\cdot\|\mathcal{J}(\Phi)\|_{\text{max}}=16(2j+1)\max_{\alpha,\beta\in\llbracket-j;j\rrbracket}\|\Phi(\ket{\alpha}\bra{\beta})\|_{\text{max}},$
(69)

where $\mathcal{J}(\Phi)$ is the Choi matrix of $\Phi$, i.e.,

$\displaystyle\mathcal{J}(\Phi)=\sum_{\alpha,\beta\in\llbracket-j;j\rrbracket}\ket{\alpha}\bra{\beta}\otimes\Phi(\ket{\alpha}\bra{\beta}),$
(70)

$\\{\ket{\alpha}\\}_{\alpha\in\llbracket-j;j\rrbracket}$ is an orthonormal
basis for $\mathcal{H}_{2j+1}$, and
$\|M\|_{\text{max}}:=\max_{k,\ell}|M_{k,\ell}|$ for a matrix
$M=[M_{k,\ell}]_{k,\ell}$. In particular, we choose
$\\{\ket{\alpha}\\}_{\alpha\in\llbracket-j;j\rrbracket}$ to be the eigenbasis
of $J_{z}$, i.e.,

$\displaystyle\ket{\alpha}:=\ket{j,\alpha}\quad\text{for}\leavevmode\nobreak\
\alpha\in\llbracket-j;j\rrbracket.$ (71)

We notice that

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\|\Phi(\ket{\alpha}\bra{\beta})\|_{\text{max}}&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\max_{\ell,m\in\\{0,1,2,3\\}}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|\\\
&=\max_{\ell,m\in\\{0,1,2,3\\}}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|,\end{split}$
(72)

where $[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}$ denotes the $(\ell,m)$th
entry of $\Phi(\ket{\alpha}\bra{\beta})$ as in (68) when we set
$\rho=\ket{\alpha}\bra{\beta}$. Thus, to bound $\|\Phi\|_{\diamond}$ via (69),
we may first calculate or bound
$\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|$
for fixed $(\ell,m)$, and then maximize over all $(\ell,m)$’s. To this end, we
divide $(\ell,m)$ into 5 cases:

  * •

$\ell=m=0$: We have

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{0,0}|&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|(s-p)\Tr(\ket{\alpha}\bra{\beta})|=ap^{2}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}\ket{\alpha}|\\\
&=ap^{2},\end{split}$ (73)

where the maximum is achieved when $\alpha=\beta$.

  * •

$\ell=0\neq m$ or $m=0\neq\ell$: Since
$[\Phi(\ket{\alpha}\bra{\beta})]_{0,m}=[\Phi(\ket{\alpha}\bra{\beta})]_{m,0}$
for $m\in\\{1,2,3\\}$, it suffices to consider $\ell=0\neq m$. We have

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{0,m}|&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|(c_{j}(p)-c_{j}(s)(1-p_{j}))\Tr(J_{m}\ket{\alpha}\bra{\beta})|\\\
&=|c_{j}(p)-c_{j}(s)(1-p_{j})|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}J_{m}\ket{\alpha}|\\\
&=\left|\sqrt{\frac{p(1-p)}{j(j+1)}}-\sqrt{\frac{(p+ap^{2})(1-p-ap^{2})}{j(j+1)}}\left(1-\frac{p}{j(j+1)}\right)\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}J_{m}\ket{\alpha}|\\\
&=\left|\frac{2-j(j+1)a}{2(j(j+1))^{3/2}}p^{3/2}+\mathcal{O}(p^{5/2})\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}J_{m}\ket{\alpha}|\quad\leavevmode\nobreak\
\text{as}\leavevmode\nobreak\ p\to 0,\end{split}$ (74)

where in the penultimate equality we applied the definition of $c_{j}(\cdot)$,
$p_{j}$ and $s:=p+ap^{2}$, and in the last equality we applied Taylor
expansion for $c_{j}(p)-c_{j}(s)(1-p_{j})$ round $p=0$. As for the term
$\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}J_{m}\ket{\alpha}|$,
applying (17), we have

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\bra{\beta}J_{m}\ket{\alpha}|&=\begin{cases}\frac{1}{2}\max\limits_{\beta\in\llbracket-j;j\rrbracket}\max\left\\{\sqrt{j(j+1)-\beta(\beta+1)},\sqrt{j(j+1)-\beta(\beta-1)}\right\\},&m\in\\{1,2\\},\\\
\max\limits_{\beta\in\llbracket-j;j\rrbracket}|\beta|,&m=3,\end{cases}\\\
&=\begin{cases}\begin{cases}\frac{1}{2}\sqrt{j(j+1)},&j\in\mathbb{N},\\\
\frac{1}{2}\left(j+\frac{1}{2}\right),&j\in((\mathbb{N}/2)\setminus\mathbb{N}),\end{cases}&m\in\\{1,2\\},\\\
j,&m=3,\end{cases}\\\ &\leq j,\leavevmode\nobreak\ \text{with inequality if
and only if}\leavevmode\nobreak\ m=3\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \alpha=\beta\in\\{-j,j\\}.\end{split}$ (75)

Notice that if we do _not_ set $a=2/(j(j+1))$, then we will have

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{0,m}|=\mathcal{O}(p^{3/2})\quad\leavevmode\nobreak\
\text{as}\leavevmode\nobreak\ p\to 0,\quad\leavevmode\nobreak\ \text{for
all}\leavevmode\nobreak\ m\in\\{1,2,3\\}.\end{split}$ (76)

  * •

$\ell=m\in\\{1,2,3\\}$: We have

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{m,m}|&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|(p_{j}-s_{j}(1-3p_{j}))\Tr(J_{m}^{2}\ket{\alpha}\bra{\beta})-ps_{j}\Tr(\ket{\alpha}\bra{\beta})\right|\\\
&\leq\left|p_{j}-s_{j}(1-3p_{j})\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|\Tr(J_{m}^{2}\ket{\alpha}\bra{\beta})\right|+ps_{j}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|\Tr(\ket{\alpha}\bra{\beta})\right|\\\
&=\left|\frac{3-j(j+1)a}{(j(j+1))^{2}}p^{2}+\frac{3a}{(j(j+1))^{2}}p^{3}\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|\Tr(J_{m}^{2}\ket{\alpha}\bra{\beta})\right|+\frac{p^{2}+ap^{3}}{j(j+1)}\\\
&=\mathcal{O}(p^{2})\quad\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\
p\to 0,\quad\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
m\in\\{1,2,3\\},\end{split}$ (77)

regardless of the value of $a$.

  * •

$(\ell,m)\in\\{(1,2),(2,3),(3,1)\\}$:

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|(p_{j}-s_{j}(1-3p_{j}))\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})+is_{j}p_{j}\Tr(J_{\ell-1}\ket{\alpha}\bra{\beta})\right|\\\
&\leq|p_{j}-s_{j}(1-3p_{j})|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})|+|is_{j}p_{j}|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{\ell-1}\ket{\alpha}\bra{\beta})|\\\
&=\left|\frac{3-j(j+1)a}{(j(j+1))^{2}}p^{2}+\frac{3a}{(j(j+1))^{2}}p^{3}\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})|\\\
&\quad+\frac{p^{2}+ap^{3}}{(j(j+1))^{2}}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{\ell-1}\ket{\alpha}\bra{\beta})|\\\
&=\mathcal{O}(p^{2})\quad\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\
p\to 0,\quad\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
m\in\\{1,2,3\\},\end{split}$ (78)

regardless of the value of $a$.

  * •

$(\ell,m)\in\\{(1,3),(2,1),(3,2)\\}$:

$\displaystyle\begin{split}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|&=\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\left|(p_{j}-s_{j}(1-3p_{j}))\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})-is_{j}p_{j}\Tr(J_{\ell+1}\ket{\alpha}\bra{\beta})\right|\\\
&\leq|p_{j}-s_{j}(1-3p_{j})|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})|+|-is_{j}p_{j}|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{\ell+1}\ket{\alpha}\bra{\beta})|\\\
&=\left|\frac{3-j(j+1)a}{(j(j+1))^{2}}p^{2}+\frac{3a}{(j(j+1))^{2}}p^{3}\right|\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{m}J_{\ell}\ket{\alpha}\bra{\beta})|\\\
&\quad+\frac{p^{2}+ap^{3}}{(j(j+1))^{2}}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|\Tr(J_{\ell+1}\ket{\alpha}\bra{\beta})|\\\
&=\mathcal{O}(p^{2})\quad\leavevmode\nobreak\ \text{as}\leavevmode\nobreak\
p\to 0,\quad\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\
m\in\\{1,2,3\\},\end{split}$ (79)

regardless of the value of $a$.

Recall that we still need to maximize over $(\ell,m)$ to get the final upper
bound on $\|\Phi\|_{\diamond}$ via (69). In terms of the order in $p$ in the
low-noise regime ($p\to 0$), since in all but case 2) above the term
$\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|$
is of order $p^{2}$, whereas in the second case the term is of order $p^{3/2}$
if we do _not_ choose $a=2/(j(j+1))$, the situation calls for choosing

$\displaystyle a=\frac{2}{j(j+1)},$ (80)

so that we can achieve

$\displaystyle\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}\|\Phi(\ket{\alpha}\bra{\beta})\|_{\text{max}}$
$\displaystyle=\max_{\ell,m\in\\{0,1,2,3\\}}\max\limits_{\alpha,\beta\in\llbracket-j;j\rrbracket}|[\Phi(\ket{\alpha}\bra{\beta})]_{\ell,m}|=\mathcal{O}(p^{2})\quad\leavevmode\nobreak\
\text{as}\leavevmode\nobreak\ p\to 0.$ (81)

We notice that when $j=1/2$, this choice of $a$ becomes $a=8/3$, which
recovers the result of [27] for qubit depolarizing channels. Then, via (69),
we have

$\displaystyle\|\mathcal{M}^{\text{c}}_{j,p}-\mathcal{M}^{\text{c}}_{j,s}\circ\mathcal{M}_{j,p}\|_{\diamond}=\|\Phi\|_{\diamond}\leq\mathcal{O}(p^{2})\quad\leavevmode\nobreak\
\text{as}\leavevmode\nobreak\ p\to 0,$ (82)

so that the degradability parameter of $\mathcal{M}_{j,p}$ satisfies

$\displaystyle\text{dg}(\mathcal{M}_{j,p})\leq\mathcal{O}(p^{2})\quad\leavevmode\nobreak\
\text{as}\leavevmode\nobreak\ p\to 0,$ (83)

which is a generalization of the result of [27] for general spin $j$.

##  Appendix B Proofs of Lemmas 1 to 6

For this work to be self-contained, we provide detailed proofs of Lemmas 1 to
6 in this Appendix.

######  Lemma 1.

For any $k\in\mathbb{Z}$, the commutator $[J_{k},J_{k+1}]$ equals $iJ_{k+2}$.

###### Proof.

The Lemma is a restatement of the fundamental commutation relations of angular
momentum (12). To see why, take $k=1,2,3$ one by one. Statement of Lemma 1 is
equivalent to

$\displaystyle\begin{split}[J_{1},J_{2}]&=iJ_{3},\\\
[J_{2},J_{3}]&=iJ_{4}=iJ_{1},\\\
[J_{3},J_{1}]&=[J_{3},J_{4}]=iJ_{5}=iJ_{2},\end{split}$ (84)

respectively, which corresponds to (12) when
$\varepsilon_{\alpha\beta\gamma}=+1$. Negating the three commutators (84) and
noticing that $-[A,B]=[B,A]$ by definition, we have

$\displaystyle[J_{2},J_{1}]=-iJ_{3},\leavevmode\nobreak\ \leavevmode\nobreak\
[J_{3},J_{2}]=-iJ_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
[J_{1},J_{3}]=-iJ_{2},$ (85)

respectively, which corresponds to (12) when
$\varepsilon_{\alpha\beta\gamma}=-1$. The extension to any three consecutive
integers $k=\ell,\ell+1,\ell+2$ in $\mathbb{Z}$ is immediate. Thus the Lemma
is proved. ∎

######  Lemma 2.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=k-1}^{k+1}J_{\ell}^{2}=j(j+1)I_{2j+1}$, where
$I_{d}:\mathcal{H}_{d}\to\mathcal{H}_{d}$ denotes the identity operator (i.e.,
the $d\times d$ identity matrix).

###### Proof.

The relation

$\displaystyle\sum_{\ell=1}^{3}J_{\ell}^{2}=j(j+1)I_{2j+1}$ (86)

is a restatement of the matrix representation (18) of $\mathbf{J}^{2}$, noting
that by definition $\mathbf{J}^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$. Extending
the summation range in (86) to any three consecutive integers $\ell=k-1,k,k+1$
in $\mathbb{Z}$ is immediate. Thus the Lemma is proved. ∎

######  Lemma 3.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{\ell}=(j(j+1)-1)J_{k}$.

###### Proof.

We have

$\displaystyle\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{\ell}$
$\displaystyle=\sum_{\ell=k-1}^{k+1}J_{\ell}J_{k}J_{\ell}$ (87)
$\displaystyle=(J_{k-1}J_{k})J_{k-1}+J_{k}^{3}+(J_{k+1}J_{k})J_{k+1}$ (88)
$\displaystyle\overset{(a)}{=}(J_{k}J_{k-1}+iJ_{k+1})J_{k-1}+J_{k}^{3}+(J_{k}J_{k+1}-iJ_{k+2})J_{k+1}$
(89) $\displaystyle=J_{k}\sum_{\ell=k-1}^{k+1}J_{\ell}^{2}+i[J_{k+1},J_{k+2}]$
(90) $\displaystyle\overset{(b)}{=}j(j+1)J_{k}+i^{2}J_{k+3}$ (91)
$\displaystyle=(j(j+1)-1)J_{k},$ (92)

where (a) follows from Lemma 1 and (b) follow from both Lemma 2 and 1. Thus
the Lemma is proved. ∎

######  Lemma 4.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=1}^{3}J_{\ell}J^{2}_{k}J_{\ell}=(j(j+1)-3)J^{2}_{k}+j(j+1)I_{2j+1}$.

###### Proof.

We have

$\displaystyle\sum_{\ell=1}^{3}J_{\ell}J_{k}^{2}J_{\ell}$
$\displaystyle=\sum_{\ell=k-1}^{k+1}J_{\ell}J_{k}^{2}J_{\ell}$ (93)
$\displaystyle=(J_{k-1}J_{k})(J_{k}J_{k-1})+J_{k}^{4}+(J_{k+1}J_{k})(J_{k}J_{k+1})$
(94)
$\displaystyle\overset{(a)}{=}(J_{k}J_{k-1}+iJ_{k+1})(J_{k-1}J_{k}-iJ_{k+1})+J_{k}^{4}+(J_{k}J_{k+1}-iJ_{k+2})(J_{k+1}J_{k}+iJ_{k+2})$
(95)
$\displaystyle=J_{k}\left(\sum_{\ell=k-1}^{k+1}J_{\ell}^{2}\right)J_{k}-i^{2}J_{k+1}^{2}-i^{2}J_{k+2}^{2}+i\left(J_{k+1}J_{k-1}J_{k}-J_{k}J_{k-1}J_{k+1}+J_{k}J_{k+1}J_{k+2}-J_{k+2}J_{k+1}J_{k}\right)$
(96)
$\displaystyle\overset{(b)}{=}j(j+1)J_{k}^{2}+J_{k+1}^{2}+J_{k+2}^{2}+i\left(J_{k+1}J_{k+2}J_{k}-J_{k}J_{k+2}J_{k+1}+J_{k}J_{k+1}J_{k+2}-J_{k+2}J_{k+1}J_{k}\right)$
(97)
$\displaystyle=(j(j+1)-1)J_{k}^{2}+(J_{k}^{2}+J_{k+1}^{2}+J_{k+2}^{2})+i\left([J_{k+1},J_{k+2}]J_{k}+J_{k}[J_{k+1},J_{k+2}]\right)$
(98)
$\displaystyle\overset{(c)}{=}(j(j+1)-1)J_{k}^{2}+j(j+1)I_{2j+1}+i^{2}\left(J_{k+3}J_{k}+J_{k}J_{k+3}\right)$
(99) $\displaystyle=(j(j+1)-3)J_{k}^{2}+j(j+1)I_{2j+1},$ (100)

where (a) follows from Lemma 1, (b) follows from Lemma 2 and (c) follows from
both Lemma 2 and 1. Thus the Lemma is proved. ∎

######  Lemma 5.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k-1}J_{\ell}=(j(j+1)-3)J_{k}J_{k-1}-iJ_{k-2}$.

###### Proof.

We have

$\displaystyle\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k-1}J_{\ell}$
$\displaystyle\overset{(a)}{=}\sum_{\ell=1}^{3}J_{\ell}(J_{k-1}J_{k}-iJ_{k+1})J_{\ell}$
(101)
$\displaystyle=\sum_{\ell=1}^{3}J_{\ell}J_{k-1}J_{k}J_{\ell}-i\sum_{\ell=1}^{3}J_{\ell}J_{k+1}J_{\ell}$
(102)
$\displaystyle\overset{(b)}{=}(j(j+1)-3)J_{k-1}J_{k}+iJ_{k+1}-i(j(j+1)-1)J_{k+1}$
(103) $\displaystyle=(j(j+1)-3)J_{k-1}J_{k}-i(j(j+1)-2)J_{k+1}$ (104)
$\displaystyle\overset{(c)}{=}(j(j+1)-3)(J_{k}J_{k-1}+iJ_{k+1})-i(j(j+1)-2)J_{k+1}$
(105) $\displaystyle=(j(j+1)-3)J_{k}J_{k-1}-iJ_{k+1}$ (106)
$\displaystyle=(j(j+1)-3)J_{k}J_{k-1}-iJ_{k-2},$ (107)

where (a) follows from Lemma 1, (b) follows from both Lemma 5 amd 3, and (c)
follows from Lemma 1. Thus the Lemma is proved. ∎

######  Lemma 6.

For any $k\in\mathbb{Z}$, we have
$\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k+1}J_{\ell}=(j(j+1)-3)J_{k}J_{k+1}+iJ_{k+2}$.

###### Proof.

We have

$\displaystyle\sum_{\ell=1}^{3}J_{\ell}J_{k}J_{k+1}J_{\ell}$
$\displaystyle=\sum_{\ell=k-1}^{k+1}J_{\ell}J_{k}J_{k+1}J_{\ell}$ (108)
$\displaystyle=(J_{k-1}J_{k})(J_{k+1}J_{k+2})+J_{k}^{2}(J_{k+1}J_{k})+(J_{k+1}J_{k})J_{k+1}^{2}$
(109)
$\displaystyle\overset{(a)}{=}(J_{k}J_{k-1}+iJ_{k+1})(J_{k+2}J_{k+1}+iJ_{k+3})+J_{k}^{2}(J_{k}J_{k+1}-iJ_{k+2})+(J_{k}J_{k+1}-iJ_{k+2})J_{k+1}^{2}$
(110)
$\displaystyle=J_{k}\left(\sum_{\ell=k-1}^{k+1}J_{\ell}^{2}\right)J_{k+1}+i^{2}J_{k+1}J_{k}+i\left(J_{k}J_{k-1}J_{k}+J_{k+1}J_{k+2}J_{k+1}-J_{k}^{2}J_{k+2}-J_{k+2}J_{k+1}^{2}\right)$
(111)
$\displaystyle\overset{(b)}{=}j(j+1)J_{k}J_{k+1}-J_{k+1}J_{k}+i\left(J_{k}[J_{k-1},J_{k}]+[J_{k+1},J_{k+2}]J_{k+1}\right)$
(112)
$\displaystyle\overset{(c)}{=}(j(j+1)-1)J_{k}J_{k+1}+[J_{k},J_{k+1}]+i^{2}\left(J_{k}J_{k+1}+J_{k+3}J_{k+1}\right)$
(113) $\displaystyle=(j(j+1)-3)J_{k}J_{k+1}+iJ_{k+2},$ (114)

where (a) follows from Lemma 1, (b) follows from Lemma 2, and (c) follows from
Lemma 1. Thus the Lemma is proved. ∎


# Temperature dependent graphene suspension due to thermal Casimir interaction

Anh D. Phan$,^{1}$ Lilia M. Woods$,^{1}$ D. Drosdoff$,^{1}$ I. V.
Bondarev$,^{2}$ and N. A. Viet3 1Department of Physics, University of South
Florida, Tampa, Florida 33620, USA  2Physics Department, North Carolina
Central, Durham, North Carolina 27707, USA  3Institute of Physics, 10 Daotan,
Badinh, Hanoi, Vietnam

###### Abstract

Thermal effects contributing to the Casimir interaction between objects are
usually small at room temperature and they are difficult to separate from
quantum mechanical contributions at higher temperatures. We propose that the
thermal Casimir force effect can be observed for a graphene flake suspended in
a fluid between substrates at the room temperature regime. The properly chosen
materials for the substrates and fluid induce a Casimir repulsion. The balance
with the other forces, such as gravity and buoyancy, results in a stable
temperature dependent equilibrium separation. The suspended graphene is a
promising system due to its potential for observing thermal Casimir effects at
room temperature.

###### pacs:

Valid PACS appear here

Casimir interactions between objects arise due to electromagnetic
fluctuations. The Casimir force is universal and it is present at all length
scales. Investigations in the past several years directed towards
understanding the fundamental nature of this interaction and its role
especially in miniature devices have been particularly intense 1 . In most
cases, the Casimir force is attractive, and as a result close proximity
between materials can lead to unwanted effects due to stiction in micro and
nanomechanical systems 2 . Thus, finding ways to reduce the magnitude of the
force or even make it repulsive is an important field of research 3 ; 4 ; 5 ;
28 ; 29 ; 30 .

In most vacuum separated materials, the Casimir interaction arise due to
quantum mechanical effects ($T=0$ $K$). As the temperature is elevated, the
force changes due to changes in the photon thermal distribution, but such
effects are usually small at submicron scales and room temperature, and they
are difficult to observe 12 ; 6 . Recently, it has been proposed to utilize
objects suspended in a fluid, in which the corrections due to thermal
fluctuations in the Casimir force can be observed 7 ; 8 . This method is
particularly intriguing since it relies on the balance between attractive and
repulsive contributions to the force arising from the dielectric response of
the materials.

Graphene is an atomically thin planar sheet of carbon atoms arranged in a
honeycomb lattice. Its recent isolation has generated much scientific interest
9 . The Casimir interaction involving graphene has also been considered 27 ;
10 ; 11 ; 16 ; 17 ; 24 . The unique properties of this material, originating
from the linear in wave vector low energy dispersion together with its 2D
geometry have given rise to several unusual features. In particular, it was
found that the graphene Casimir interaction is strongly temperature dependent.
In the quantum mechanical limit, the force has the same distance dependence
($~{}d^{-4}$) as compared to the one between perfect metals, but with a
significantly reduced strength 4 , which is also distinct from 2D metals and
insulators 24 . At room temperature the interaction is dominated by thermal
fluctuations in contrast to regular metals and dielectrics in vacuum, in which
case it is dominated by the quantum fluctuations 27 ; 11 .

In this Letter, we investigate a graphene sheet suspended in a fluid between
substrates. We study how the Casimir interaction is influenced by the
substrates. In addition, by using the balance between the Casimir force,
gravity and bouyancy, we demonstrate the existence of a stable temperature
dependent equilibrium of the suspended graphene. The chosen materials are
suitable for the detection of purely thermal effects in the micron and
submicron scale. This is particluarly encouraging for future experimental
developments utilizing strong temperature dependent Casimir interactions.

Figure 1: (Color online) Dielectric function $\epsilon(i\xi)$ as a function of
frequency $\xi$ ($eV$) for several materials. Insert: Schematics of a planar
sheet (graphene) immersed in fluid (BB) between two slab substrates (top slab
- $Au$; bottom slab - $SiC$, $SiO_{2}$, or teflon).

The studied system is shown schematically in the insert of Fig. 1. It consists
of a graphene sheet immersed in a fluid sandwiched between a lower (of
thickness $D$) and an upper semi-infinite substrate. For such a layered
configuration, the Casimir force per unit area can be calculated in each layer
4 ; 10 ; 26 . The force per unit area in layer $2^{\prime}$ is effectively the
one between graphene and the bottom substrate given by

$\displaystyle
F(d_{1})=-\dfrac{k_{B}T}{\pi}\sum_{n=0}^{\infty}{{}^{\prime}}\int_{0}^{\infty}q(i\xi_{n})k_{\perp}dk_{\perp}$
$\displaystyle\times\left(\dfrac{R_{TE}^{+}(i\xi_{n})R_{TE}^{-}(i\xi_{n})}{e^{2q(i\xi_{n})d_{1}}-R_{TE}^{+}(i\xi_{n})R_{TE}^{-}(i\xi_{n})}\right.$
$\displaystyle\left.+\dfrac{R_{TM}^{+}(i\xi_{n})R_{TM}^{-}(i\xi_{n})}{e^{2q(i\xi_{n})d_{1}}-R_{TM}^{+}(i\xi_{n})R_{TM}^{-}(i\xi_{n})}\right),$
(1)

where $k_{\perp}$ is the 2D wave vector,
$q(i\xi_{n})=\sqrt{k_{\perp}^{2}+\varepsilon_{m}(i\xi_{n})(\xi_{n}/c)^{2}}$,
and $\xi_{n}=2n\pi k_{B}T/\hbar$ are the Matsubara frequencies. The prime in
the sum of Eq. (1) indicates that $n=0$ term is multiplyed by 1/2.
$R_{TM}(i\xi_{n})$ and $R_{TE}(i\xi_{n})$ are the effective reflection
coefficients for the transverse ellectric (TE) and transverse magnetic (TM)
polarizations of the electromagnetic field.

$R_{TE,TM}^{+,-}$ correspond to the effective boundary conditions due to the
objects above layer $2^{\prime}$ ($+$ subscipt) and below layer $2^{\prime}$
($-$ subscipt)4 ; 26 . For the system given in Fig. 1 they are derived as
follows:

$\displaystyle
R_{TM,TE}^{-}=r_{TM,TE}^{-}\frac{1-e^{-2k_{1}D}}{1-(r_{TM,TE}^{-})^{2}e^{-2k_{1}D}},$
$\displaystyle
r^{-}_{TM}=\frac{{\varepsilon_{1}q-\varepsilon_{m}k_{1}}}{{\varepsilon_{1}q+\varepsilon_{m}k_{1}}},r^{-}_{TE}=\frac{q-k_{1}}{q+k_{1}},$
$\displaystyle
R^{+}_{TM}=\frac{r_{TM}^{\sigma}+r_{TM}^{t}(1-2r_{TM}^{\sigma})e^{-2qd_{2}}}{1-r_{TM}^{\sigma}r_{TM}^{t}e^{-2qd_{2}}},$
$\displaystyle
R^{+}_{TE}=\frac{r_{TE}^{\sigma}+r_{TE}^{t}(1+2r_{TE}^{\sigma})e^{-2qd_{2}}}{1-r_{TE}^{\sigma}r_{TE}^{t}e^{-2qd_{2}}},$
$\displaystyle r_{TM}^{\sigma}=\frac{2\pi\sigma
q/\xi_{n}}{\varepsilon_{m}+2\pi\sigma
q/\xi_{n}},r_{TE}^{\sigma}=-\frac{2\pi\xi_{n}\sigma/c^{2}}{q+2\pi\xi_{n}\sigma/c^{2}},$
$\displaystyle
r_{TM}^{t}=\frac{{\varepsilon_{3}q-\varepsilon_{m}k_{3}}}{{\varepsilon_{3}q+\varepsilon_{m}k_{3}}},r_{TE}^{t}=\frac{q-k_{3}}{q+k_{3}},$
(2)

where
$k_{1,3}=\sqrt{k_{\perp}^{2}+\varepsilon_{1,3}(i\xi_{n})(\xi_{n}/c)^{2}}$ and
$\sigma$ is 2D graphene conductivity.

For this study, the material for the bottom substrate is $SiO_{2}$, $SiC$, or
teflon (PTFE), the top one is $Au$, while the medium fluid is bromobenzene
(BB). We assume that the thickness of the $Au$ substrate to be much greater
than the $Au$ skin depth ($\approx 22$ $nm$). Under this condition, the
Casimir interaction is not influenced by the thickness, as shown by others 31
, thus we take it to be semi-infinite. $\varepsilon_{1,3,m}(i\xi)$ are taken
from available experiments with Lorentz and Drude (for $Au$) models fitted
parameters 3 ; 5 ; 13 .

The dielectric function for graphene is found as $\varepsilon_{g}({\bf
k},i\xi)=1+2\pi k\sigma(i\xi)/\xi$ 25 . The graphene conductivity is
calculated from the Kubo formalism using a two-band Dirac model 14 .

$\sigma(i\xi)=\frac{2e^{2}k_{B}T\ln
2}{\pi\hbar^{2}\xi}+\frac{e^{2}\xi}{\pi}\int_{0}^{\infty}\frac{\tanh[\epsilon/2k_{B}T]d\epsilon}{\epsilon^{2}+(\hbar\xi)^{2}},$
(3)

The first term corresponds to intraband and the second term - to interband
transitions. In the infrared and low optical regime ($\leq$ 3 eV), $\sigma$
takes a universal value $\sigma_{0}=e^{2}/4\hbar$ characterizing the optical
tranparency of graphene. This has been demonstrated experimentally even in
room temperatures 15 .

Figure 2: (Color online) $F/F_{0}$ as a function of $d_{1}$ for (a) silica;
(b) teflon; (c) $SiC$; (d) the BB liquid phase temperature regime for the
three substrates. $F_{0}=\pi^{2}\hbar c/\left(240d_{1}^{4}\right)$ is the
Casimir force between two perfect metal plates. The bottom substrate is semi-
infinite. The BB liquid phase exists for $T=[242,429]K$.

Using Eq.(1-3), we calculate the Casimir force between the submerged in BB
graphene and the bottom substrate. The results are shown in Fig. 2 (a,b,c) for
several separations of the top $Au$ substrate. For $SiO_{2}$ and PTFE, the
force is found to be repulsive, while for $SiC$ \- it is attarctive for all
$d_{1}$ and $d_{2}$ distances. This is understood by examining the dielectric
response of the materials and their relative contribuion in the reflection
coefficients. For silica and teflon and $\xi<10$ $eV$,
$\varepsilon_{1}(i\xi)<\varepsilon_{m}(i\xi)<\varepsilon_{eff}(i\xi)$, where
$\varepsilon_{eff}$ is the effective dielectric function for graphene,
bromobenzene in layer $2"$ and the $Au$ substrate. Since $\varepsilon_{g}$ and
$\varepsilon_{BB}$ are much smaller than $\varepsilon_{Au}$, the effective
materials response above layer $2^{\prime}$ is reduced as compared to the one
for $Au$. This particular ordering in $\varepsilon$ 5 ; 6  results in
$R_{TM}^{-}<0$, thus $F$ changes sign. Stronger repulsion is achieved by
making $R_{TM}^{-}$ more negative either by bringing the $Au$ substrate closer
and/or by choosing PTFE instead of $SiO_{2}$. For $SiC$, however, this
ascending in dielectric response ordering is not maintained as
$\varepsilon_{SiC}>\varepsilon_{BB}$, and one obtains attraction for all
separations. We also find that for $d_{2}>80$ $nm$, the influence of the $Au$
is minimal and the force is changed little upon taking away the top substrate.

Graphene has an important effect on the temperature dependence of the Casimir
interaction. Thermal effects in most materials become important at separations
larger than the characteristic length $\lambda=\hbar c/(k_{B}T)$. For
graphene, however, $\lambda$ is reduced by the fine structure constant
$\alpha=1/137$ 27 , thus at room temperature thermal fluctuations effects are
seen at distances greater than $25-30$ $nm$. In that case, the interaction is
dominated by the $n=0$ term in Eq.(1). This is typical for graphene/graphene
10  or graphene/other materials interactions 4 ; 11 . The results shown in
Fig.2 were performed using the full expression Eq.(1), although for
$d_{2}>0.1$ $\mu m$, the $n=0$ dominates the force. In Fig.2 (d), we also show
how the force changes for the temperature region where BB liquid phase exists.
The force modulations are not very significant, although $F$ at higher $T$ is
more repulsive for PTFE and $SiO_{2}$ and more attractive for $SiC$.

The repulsive interaction together with the strong thermal effects in the
Casimir force can be used to create equilibrium separations of the submerged
in BB graphene sheet above PTFE or $SiO_{2}$ substrates. Besides the Casimir
interaction for the configuration in Fig. 1 (insert), the graphene experiences
other forces - buoyancy force and gravity attraction. To illustrate the
equilibrium configuration, we focus on the calculation of the total energy of
the suspended graphene:

$\displaystyle E(d_{1})=E_{c}+E_{g}+E_{b}$ $\displaystyle
E_{c}=\frac{k_{B}T}{2\pi}\sum_{n=0}^{\infty}{{}^{\prime}}\int_{0}^{\infty}k_{\perp}dk_{\perp}\left(\ln\left[1-R_{TM}^{+}R_{TM}^{-}e^{-2qd_{1}}\right]\right.$
$\displaystyle\left.+\ln\left[1-R_{TE}^{+}R_{TE}^{-}e^{-2qd_{1}}\right]\right).$
(4)

where $E_{c}$ is the Casmir energy found from Eq. (10. The energy due to
gravity is $E_{g}=\rho_{g}gd_{1}$ with $\rho_{g}=7.6\times 10^{-7}$ $kg/m^{2}$
being the surface mass density of graphene18 . The buoyancy energy is
$E_{b}=-\rho_{b}gN_{0}Vd_{1}$, where $\rho_{b}=1.5\times 10^{-3}$ $kg/m^{3}$
is the BB volume mass density, $g$ is the gravitational acceleration, $N_{0}$
is the number of carbon atoms per unit area and $V$ is volume of a carbon
atom.

We consider the case when the top substrate is not present. For distances at
the submicron scale, gravity becomes important attracting the graphene sheet
downward. The buoyancy acts upward together with the Casimir repulsion. The
balance between these forces is captured by calculating the total energy from
Eq.(4). Results for a $50\times 50$ $\mu m^{2}$ graphene sheet are shown in
Fig.3, where the equilibrium graphene/substrate separation $a_{0}$ corresponds
to the minimum of $E/(k_{B}T)$ vs $d_{1}$. $a_{0}$ is in the submicron range
and it becomes larger as the thikness of the bottom substrate is increased.
This behavior is also shown in the insert of Fig.3 for different temperatures.
One finds that for PTFE, $a_{0}=1.1-2.0$ $\mu m$ , while for $SiO_{2}$
$a_{0}=0.8-1.5$ $\mu m$ for 242-420 $K$. We note that the Brownian motion will
cause random fluctuations around $a_{0}$. As a result, it is possible for
stiction to occur. The probability for such a process is $\sim
e^{-E/(k_{B}T)}$ 7 ; 8 . For the suspended graphene, the probablity is small,
thus the Brownian motion is not expected to be important.

Figure 3: (Color online) (Top) Total energy of the suspended graphene in terms
of $k_{B}T$ as a function of $d_{1}$ for various thicknesses of the bottom
substrate. Inserts show the equilibrium distance $a_{0}$ vs the thickness $D$
for silica and teflon for different temperatures.

For graphene separations in the submicron and micron scales and large
temperatures, the contribution to the Casimir energy $E_{C}$ comes almost
entirely from the $n=0$ term in the Matsubara summation due to the reduced
characteristic length $\lambda$, as discussed earlier. We consider a bottom
substrate with width $D=800$ $nm$. Evaluating the $n=0$ term in Eq.(4), we
find $E_{c}=k_{B}T\gamma/(4\pi d_{1}^{2})$, where $\gamma\approx 0.028$ is a
constant. Note that $\gamma=0.038$ for $D=\infty$. Using this dominant term
together with Eq.(4), we can describe the graphene/substrate energy around the
equilibrium distance $a_{0}$ by a Taylor series of $E$.

In Fig.4, we show $E/(k_{B}T)$ as a function of the separation $d_{1}$
calculated via the full expression (Eq.(4)) and the Taylor series expansion by
retaining the first several terms. Fig.4 shows that in a rather wide range
around the equilibrium, the total energy can be described by just the first
term in the Taylor series as
$E(d_{1})-E(a_{0})\approx\frac{3k_{B}T\gamma}{4\pi
a_{0}^{4}}(d_{1}-a_{0})^{2}$, where $E(a_{0})$ is the total energy at the
equilibrium distance, found to be
$a_{0}=\left({k_{B}T\gamma}/{2\pi(\rho_{g}g-\rho_{b}gN_{0}V)}\right)^{1/3}$.

This result is especially useful in relating the Casimir interaction to a
simple harmonic-like oscillatory behavior of the submerged graphene. From
Eq.(5), one finds that the frequency of oscillations is

$\omega\sim\sqrt{\frac{3k_{B}T\gamma}{2\pi\rho_{g}a_{0}^{4}}}.$ (5)

The frequency is in the typical for a mechanical system kHz regime (about 2 to
5 kHz) for the studied temperature regime and substrates.

As the graphene/substrate distance is decreased, the Casimir repulsion becomes
dominant and $E$ rapidly increases. As the separation increases, the gravity
attraction dominates and $E$ increases again. The balance between these three
forces results in an equilibrium distance between the submerged graphene with
respect to the substrate.

Figure 4: (Color online) Total energy of the $50\times 50$ $\mu m^{2}$
suspeded graphene in terms of $k_{B}T$ vs $d_{1}$ calculated by the full
expression from Eq.(4) and by the first four terms in its Taylor series for
$D=800$ $nm$.

This study shows that graphene experiences suspension in bromobenzene above
substrates providing the dielectric functions of the layers (insert Fig. 1)
are in ascending (discending) order. The suspension is characterized by a
temperature-dependent equilibrium separation in the micron range. The balance
between gravity, buoyancy, and the Casimir force yields one equilibrium
$a_{0}$ which changes at a rate $~{}2-3$ $nm/K$ \- a substantial rate at the
room temperature.

We compare this setting to the sphere/plate geometry in Ref.7 ; 8 , proposed
for achieving tempereture-dependent Casimir effect. It was shown that a
temperature dependent stable equilibrium of the suspended sphere due to the
Casimir force alone is obtained by making the integrand in Eq. (1)
oscillatory. This is done by choosing materials with dielectric crossings
occuring at sufficiently small Matsubara frequencies close to room
temperature. Note that here temperature fluctuations appear together with the
quantum mechanical contributions. Inlcuding gravity and buoyancy results in
the appearance of additional stable or unstable equilibriua, which maybe
difficult to measure experimentally due to the possibility of sticktion. At
the same time, for suspended graphene the Casimir force alone does not result
in an equlibrium sepration. The inlcusion of gravity and buoyancy is necessary
to ensure the desired balance. Thus we find only one stable equilibrium, which
is $\sim T^{1/3}$. Another important point is that due to the significantly
reduced thermal characteristic length of graphene, only thermal fluctuations
at the micron and submicron range are relavant for room temperatures. This is
truly unique since in regular dielectrics and metals such effects are seen at
much larger $T$ and $d_{1}$. We note that the suspended in fluid sphere as
well as the suspended in fluid graphene attest to the richness of the
temperature dependent Casimir phenomenon, which is yet to be explored
experimentally.

Furthermore, the fact that the graphene Casimir force is essentially thermal
for such separations and small $T$ yields a relatively simple estimate for the
oscillatory-like frequency around the stable separation. Around the
equilibrium the suspended graphene behaves like a strongly temperature-
dependent Hooke-like spring with characteristics directly related to the
thermal Casimir force. This system is a example, which can be used by
experimentalists to measure submicron separation changes over a few hundred
Kelvin temperature regimes entirely due to thermal Casimir effects. Such
distance scale measurements are feasible with available techniques, such as
traditional setups for Casimir measurements 19 , setups using suspended
Michelson interferrometers 20 , total internal reflection microscopy in
suspensions 21 , or other techniques based on transition electron microscopy,
suited for oscillatory measurements 22 ; 23 .

The authors are indebted to Dr. Alexandro Rodriguez (MIT) for discussions.
Financial support from the Department of Energy under Contract No. DE-
FG02-06ER46297 is also acknowledged. I.V.B. was supported by the NSF-
HRD-0833184 and ARO-W911NF-11-1-0189 grants. N.A.V. was supported by the
Nafosted Grant No. 103.06-2011.51.

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