# Decomposition of the regular representation for dihedral quandles

Mohamed Elhamdadi  Department of Mathematics, University of South Florida,
Tampa, FL 33620<EMAIL_ADDRESS>,  Prasad Senesi  Department of
Mathematics, The Catholic University of America, Washington, DC 20064
<EMAIL_ADDRESS>and  Emanuele Zappala  Yale School of Medicine, Yale
University, New Haven, CT 06511<EMAIL_ADDRESS>
###### Abstract.

We decompose the regular quandle representation of a dihedral quandle
$\mathcal{R}_{n}$ into irreducible quandle subrepresentations.

##  1\. Introduction

Let $\mathcal{R}_{n}$ be the dihedral quandle of order $n$, and let
$\mathbb{C}\mathcal{R}_{n}$ be the $\mathbb{C}$-vector space of all
$\mathbb{C}$-valued functions on $\mathcal{R}_{n}$. Then
$\mathbb{C}\mathcal{R}_{n}$ has the structure of a quandle representation of
$\mathcal{R}_{n}$, which we call the regular representation of
$\mathcal{R}_{n}$. In this note we provide, for all positive integers $n$, the
decomposition of this regular representation into irreducible quandle
subrepresentations. This problem was first addressed using the notion of a
quandle ring in [EFT] in 2019, although the results given there are incomplete
and do not address dihedral quandles of odd order. Our main result here was
recently reported by one of the authors at the Knots in Washington Meeting on
April 23, 2022, and will be included in a more comprehensive forthcoming
publication on the representation theory of quandles. The authors would also
like to note the recent arxiv submission [KS], where these results are
obtained with a different approach for the dihedral quandles of even order.

##  2\. Review of racks and quandles

We give a quick review of racks and quandles in this section.

######  Definition 2.1.

[EN, Joyce, Matveev] A rack is a set $X$ provided with a binary operation

$\begin{array}[]{lccc}\triangleright:&X\times X&\longrightarrow&X\\\
&(x,y)&\longmapsto&x\triangleright y\end{array}$

such that

  * (i)

for all $x,y\in X$, there is a unique $z\in X$ such that $y=z\triangleright
x$;

  * (ii)

(right distributivity) for all $x,y,z\in X$, we have $(x\triangleright
y)\triangleright z=(x\triangleright z)\triangleright(y\triangleright z)$.

For $x\in X$, we denote by $R_{x}:X\rightarrow X$ the map given by

$R_{x}(y)=y\triangleright x.$

Observe that property (i) also reads that for any fixed element $x\in X$, the
map $R_{x}$ is a bijection. Also, notice that the right distributivity
condition is equivalent to the relation $R_{x}(y\triangleright
z)=R_{x}(y)\triangleright R_{x}(z)$ for all $y,z\in X$; that is, $R_{x}$ is an
automorphism of the quandle $X$.

Unless otherwise stated, we will always assume our racks (or quandles) to be
finite racks (or quandles).

######  Definition 2.2.

A quandle is a rack such that $x\triangleright x=x,\forall x\in X$.

As each $R_{x}$ permutes the elements of $X$, we can consider the subgroup
$\left\langle R_{x}\right\rangle_{x\in X}$ in $S_{X}$ (the symmetric group on
$X$) generated by the $R_{x}$, which we denote by $\operatorname{Inn}(X)$ (the
group of inner automorphisms of $X$). This group acts on the set $X$, and so
the group action partitions $X$ into distinct orbits. We say the quandle $X$
is connected if $\text{Inn}(X)$ acts transitively on $X$ (and hence there is
only one orbit). If $(X,\triangleright)$ and $(Y,\triangleright)$ are two
quandles, a map $f:X\rightarrow Y$ is a quandle homomorphism if
$f(x\triangleright y)=f(x)\triangleright f(y)$ for all $x,y\in X$.

######  Example 2.3.

Any group $G$ with the operation $x\triangleright y=yxy^{-1}$ is a quandle,
called conjugation quandle and denoted by Conj(G).

######  Example 2.4.

Any group $G$ with the operation $x\triangleright y=yx^{-1}y$ is a quandle,
called the Core quandle of $G$ and denoted Core(G).

######  Example 2.5.

Let $G$ be a group and $f\in{\rm Aut}(G)$. Then one can define a quandle
structure on $G$ by $x\triangleright y=f(xy^{-1})y$. It is called a
generalized Alexander quandle. If $G$ is abelian, the operation becomes
$x\triangleright y=f(x)+(Id-f)(y)$, where $Id$ stands for the identity map.
This quandle is called an Alexander quandle.

######  Example 2.6.

The set $\mathcal{R}_{n}=\left\\{1,2,\ldots,n\right\\}$, with quandle
operation $x\triangleright y=2y-x$ (mod  $n)$ is a quandle, called a
_dihedral_ quandle. This is an Alexander quandle on the the group
$\mathbb{Z}_{n}$ where $f$ is multiplication by $-1$.

For the remainder of this manuscript, $\mathcal{R}_{n}$ will denote the
dihedral quandle of order $n$.

##  3\. Representation theory

As mentioned in the introduction, the results here will be included in a later
manuscript providing more results for the (complex) representation theory of
(finite) quandles. The terminology used here to describe this theory is
adapted from group representation theory, a topic for which there is a wealth
of literature. There are many accessible introductions to this topic; see, for
example, Chapters 4 and 5 of [MR2808160].

All vector spaces henceforth are defined over $\mathbb{C}$. For any positive
integer $k$, we set $\omega_{k}=e^{2\pi i/k}$. A _representation_ of a finite
quandle $(X,*)$ on a finite dimensional complex vector space $V$ is a quandle
homomorphism $\rho:X\rightarrow\text{Conj}(GL(V))$. In other words, for all
$x,y\in X$, we have $\rho(x*y)=\rho(y)\rho(x)\rho(y)^{-1}.$ To simplify the
notation, we will denote $\rho(x)$ by just $\rho_{x}$. Let $V$ and $W$ be two
representations of a quandle $X$. If $\rho^{V}:X\rightarrow\text{Conj}(GL(V))$
and $\rho^{W}:X\rightarrow\text{Conj}(GL(W))$ are two representations of $X$,
then a linear map $\phi:V\rightarrow W$ for which the following square
commutes

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\rho^{V}_{x}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{W}_{x}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{W.}$

for all $x$ is called an intertwiner. If, furthermore, $\phi$ is an
isomorphism then we say that the two representations are isomorphic as
representations of $X$, or _equivalent_. If $\phi:V\hookrightarrow W$ is the
inclusion of the linear subspace $V$ into $W$, then we say that $V$ is a
_subrepresentation_ of $W$. Equivalently, a subspace $U$ of $W$ is a
subrepresentation of $W$ if $U$ is invariant under $\rho^{W}(X)$.

A representation $V$ of a quandle $X$ is called _irreducible_ if its only
quandle subrepresentations are $\\{0\\}$ and $V$. A representation ia called
_indecompasable_ if it cannot be written as direct sum of nontrivial
subrepresentations.

If $\phi:\text{Inn}(X)\rightarrow\text{Aut}(V)$ is a group homomorphism, then
the induced map

$\tilde{\phi}:X\rightarrow\text{Conj}(\text{Aut}(V)),$
$\tilde{\phi}(x):=\phi(R_{x}),$

is a quandle homomorphism. Therefore any group representation of
$\text{Inn}(X)$ induces a quandle representation of $X$.

##  4\. The regular representation of a quandle

For any finite quandle $X$, we denote by $\mathbb{C}X$ the $\mathbb{C}$–vector
space of $\mathbb{C}$-valued functions on $X$; equivalently it is the
$\mathbb{C}$-vector space generated by basis vectors
$\left\\{e_{x}\right\\}_{x\in X}$, whose elements are formal sums
$f=\sum_{x\in X}a_{x}e_{x}$, $a_{x}\in\mathbb{C}$. The regular representation
of $X$ is

$\lambda:X\rightarrow\text{Conj}(GL(\mathbb{C}X)),$

where $\lambda_{t}(f)(x):=f(R_{t}^{-1}(x))$. This action of $X$ is equivalent
to the right $X$–action on $\mathbb{C}X$ given by the linear extension of
$e_{x}\cdot t=e_{R_{t}(x)}=e_{x*t}$. We note that a subspace $W$ of
$\mathbb{C}X$ is a quandle subrepresentation of $\mathbb{C}X$ if and only if
$W$ is a subrepresentation of $\mathbb{C}X$ as a group representation of
$\operatorname{Inn}(X)$. If a distinction is necessary, we will refer to a
vector space as either a quandle representation or a group representation. If
two vector spaces $U,V$ are isomorphic as quandle representations of $X$, we
will write $U\cong_{X}V$; if they are isomorphic as group representations of
some group $G$, we will write $U\cong_{G}V$.

When $|X|\geq 2$, The regular representation always has two nontrivial
subrepresentations, one of which is always irreducible: the one–dimensional
subspace $\mathbb{C}\mathbf{1}$, where $\mathbf{1}$ is the constant function
$\mathbf{1}(x)=1$, for all $x\in X$. Its vector space complement,
$(\mathbb{C}\mathbf{1})^{\perp}$, is also a subrepresentation, since
$(\mathbb{C}\mathbf{1})^{\perp}=\left\\{\sum_{x\in X}a_{x}e_{x}|\;\sum
a_{x}=0\right\\}$, and action by any $t\in X$ only permutes the coefficients
$a_{x}$. So $\mathbb{C}X$ decomposes into a direct sum of subrepresentations

$\mathbb{C}X=\mathbb{C}\mathbf{1}\oplus(\mathbb{C}\mathbf{1})^{\perp}.$

For $x,y\in X$ let $v_{xy}=e_{x}-e_{y}$. If we enumerate
$X=\left\\{x_{1},\ldots,x_{n}\right\\}$, then a basis for
$(\mathbb{C}\mathbf{1})^{\perp}$ is
$\left\\{v_{x_{1}x_{2}},v_{x_{2}x_{3}},\ldots,v_{x_{n-1}x_{n}}\right\\}$.

##  5\. The regular representation of a dihedral quandle

The rest of this paper is devoted to the regular quandle representation of a
dihedral quandle $\mathcal{R}_{n}$. We first explicitly describe some quandle
representations of $\mathcal{R}_{n}$.

Since quandle subrepresentations of $\mathbb{C}\mathcal{R}_{n}$ are in
correspondence with group subrepresentations of $\mathbb{C}\mathcal{R}_{n}$
(under the action of $\text{Inn}(\mathcal{R}_{n})$), we obtain the full
decomposition of $\mathbb{C}\mathcal{R}_{n}$ as a quandle representation of
$\mathcal{R}_{n}$ by finding the decomposition as a group representation of
$\text{Inn}(\mathcal{R}_{n})$. We note that this correspondence between group
and quandle representations does not hold in general, but it is
_characteristic_ of the regular representation. In general, there is no reason
to expect any correspondence or similarity between quandle representations of
a quandle $X$ and group representations of $\text{Inn}(X)$.

By virtue of the aforementioned correspondence, and the fact that the group
$\text{Inn}(\mathcal{R}_{n})$ is a dihedral group, we will consider in detail
the representation theory of dihedral groups. We will let $D_{m}$ be the
dihedral group of order $2m$. Then it is shown in [EMR] that

$\text{Inn}(\mathcal{R}_{n})\cong\begin{cases}D_{n/2},&n\text{ even},\\\
D_{n},&n\text{ odd}.\end{cases}$

We describe the (isomorphism classes of) finite–dimensional irreducible group
representations of the dihedral group $D_{m}$ here, to facilitate statement of
results below. The dihedral group $D_{m}$ has a presentation given by

$D_{m}=\left\langle\alpha,\beta\;\;|\;\;|\alpha|=|\beta|=2,\;|\alpha\beta|=m\right\rangle.$

We fix generators $\alpha,\beta$ for this presentation. The classification of
the finite–dimensional irreducible group representations of $D_{m}$ falls into
two cases: $m$ even and $m$ odd. If $\lambda,\mu\in\mathbb{C}$, we will denote
by $\mathbb{C}(\lambda,\mu)$ the one–dimensional group representation of
$D_{m}$ on which $\alpha$ and $\beta$ act by scalar multiplication by
$\lambda$ and $\mu$, respectively. And for nonzero $\lambda\in\mathbb{C}$, we
will denote by $W(\lambda)$ the two–dimensional group representation of
$D_{m}$ for which the matrix representations of $\alpha$ and $\beta$ are

$\alpha\mapsto\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\;\;\;\;\beta\mapsto\begin{bmatrix}0&\lambda\\\
\lambda^{-1}&0\end{bmatrix}.$

With this notation fixed, we provide the classification of the
finite–dimensional irreducible group representations of $D_{m}$. All such
group representations are either 1– or 2– dimensional, and are listed here:

$D_{m}$ |  $m$ even |  $m$ odd
---|---|---
1-dimensional |  $\mathbb{C}(1,1)$, $\mathbb{C}(1,-1)$, |
group representations |  $\mathbb{C}(-1,1)$, $\mathbb{C}(-1,-1)$ |  $\mathbb{C}(1,1)$, $\mathbb{C}(-1,-1)$
2-dimensional |  $W(\omega_{m}^{s})$ , |  $W(\omega_{m}^{s})$ ,
group representations | $1\leq s\leq m/2-1$ | $1\leq s\leq(m-1)/2$

We will denote by $\Gamma(D_{m})$ the set of (isomorphism classes of)
finite–dimensional irreducible group representations of $D_{m}$. For example,
$D_{8}$ has 4 representations of dimension 1 and 4 irreducible representations
of dimension 2, and they are

$\Gamma(D_{8})=\left\\{\mathbb{C}(\pm 1,\pm 1),\;\;\mathbb{C}(\pm 1,\mp
1),\;\;W(\omega_{8}),\;\;W(\omega_{8}^{2}),\;\;W(\omega_{8}^{3})\right\\},$

while $D_{9}$ has 2 representations of dimension 1 and 3 irreducible
representations of dimension 2, and they are

$\Gamma(D_{9})=\left\\{\mathbb{C}(\pm 1,\pm
1),\;\;W(\omega_{9}),\;\;W(\omega_{9}^{2}),\;\;W(\omega_{9}^{3}),\;\;W(\omega_{9}^{4})\right\\}.$

Since $\text{Inn}(\mathcal{R}_{n})$ is a dihedral group generated by $R_{1}$
and $R_{2}$, these group representations also provide us with examples of
1-dimensional and 2-dimensional irreducible quandle representations of
$\mathcal{R}_{n}$. We will make this precise as follows: let
$\mathbb{C}(\lambda,\mu)$ be the 1-dimensional representation of
$\text{Inn}(\mathcal{R}_{n})$ for which $R_{1}$ and $R_{2}$ act by scalar
multiplication by $\lambda$ and $\mu$, respectively, and let $W(\lambda)$ be
the two-dimensional representation of $\text{Inn}(\mathcal{R}_{n})$ for which
$R_{1}$ and $R_{2}$ have matrix representations

$R_{1}\mapsto\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\;\;\;\;R_{2}\mapsto\begin{bmatrix}0&\lambda\\\
\lambda^{-1}&0\end{bmatrix}.$

Now we consider the regular quandle representation $\mathbb{C}\mathcal{R}_{n}$
of $\mathcal{R}_{n}$ and the subrepresentation
$(\mathbb{C}\mathbf{1})^{\perp}$, in cases $n$ even or $n$ odd. In either
case, $(\mathbb{C}\mathbf{1})^{\perp}$ is spanned by
$\left\\{v_{ij}\right\\}_{1\leq i,j\leq n}$.

If $n=2r$ is even, $\mathcal{R}_{n}$ has two orbits: the even orbit
$\left\\{2,4,\ldots,2r\right\\}$, and the odd orbit
$\left\\{1,3,\ldots,2r-1\right\\}$. Let

$\Phi_{n,0}=\text{sp}\left\\{v_{ij}\right\\}_{i\neq j\text{
even}},\;\;\;\Delta_{n,0}=\left\\{v_{24},v_{46},\ldots,v_{2r-2\;2r}\right\\},$

and

$\Phi_{n,1}=\text{sp}\left\\{v_{ij}\right\\}_{i\neq j\text{
odd}},\;\;\;\Delta_{n,1}=\left\\{v_{13},v_{35},\ldots,v_{2r-3\;2r-1}\right\\}.$

We call $\Phi_{n,0}$ and $\Phi_{n,1}$ the even and odd subspaces,
respectively, of $(\mathbb{C}\mathbf{1})^{\perp}$. For $i=0,1$, $\Delta_{n,i}$
is a basis for $\Phi_{n,i}$.

If $n$ is odd, $\mathcal{R}_{n}$ has a single orbit. In this case, we set

$\Phi_{n}=\text{sp}\left\\{v_{ij}\right\\}_{i\neq
j},\;\;\;\Delta_{n}=\left\\{v_{12},v_{23},\ldots,v_{n-1\;n}\right\\},$

and $\Delta_{n}$ is a basis for $\Phi_{n}$. Because the orbits of
$\mathcal{R}_{n}$ are as described above, all of the subspaces $\Phi_{n,i}$,
$\Phi_{n}$ are quandle subrepresentations of $\mathbb{C}\mathcal{R}_{n}$.

######  Theorem 5.1.

If $n$ is even, then the decomposition of the regular representation
$\mathbb{C}\mathcal{R}_{n}$ of $\mathcal{R}_{n}$ into irreducible quandle
subrepresentations is, for $n\equiv 0\bmod 4$,

$\mathbb{C}\mathcal{R}_{n}\cong_{\mathcal{R}_{n}}\mathbb{C}(1,1)^{\oplus
2}\oplus\mathbb{C}(\pm 1,\mp
1)\bigoplus_{s=1}^{n/4-1}W(\omega_{n/2}^{s})^{\oplus 2},$

and for $n\equiv 2\bmod 4$,

$\mathbb{C}\mathcal{R}_{n}\cong_{\mathcal{R}_{n}}\mathbb{C}(1,1)^{\oplus
2}\bigoplus_{s=1}^{(n-2)/4}W(\omega_{n/2}^{s})^{\oplus 2}.$

If $n$ is odd, then the decomposition of $\mathbb{C}\mathcal{R}_{n}$ into
irreducible quandle subrepresentations of $\mathcal{R}_{n}$ is

$\mathbb{C}\mathcal{R}_{n}\cong_{\mathcal{R}_{n}}\mathbb{C}(1,1)\bigoplus_{s=1}^{(n-1)/2}W(\omega_{n}^{2s}).$

###### Proof.

For arbitrary $n$, let
$\hat{\mathbf{1}}=\sum_{i=1}^{n}(-1)^{i}e_{i}\in(\mathbb{C}\mathbf{1})^{\perp}$.
Then, as already pointed out before, $\mathbb{C}\mathcal{R}_{n}$ decomposes as
$\mathbb{C}\mathcal{R}_{n}=\mathbb{C}\mathbf{1}\oplus(\mathbb{C}\mathbf{1})^{\perp}$,
and
$\mathbb{C}\mathbf{1}\cong_{\mathcal{R}_{n}}\mathbb{C}\hat{\mathbf{1}}\cong_{\mathcal{R}_{n}}\mathbb{C}(1,1)$.
We proceed distinguishing the cases $n$ even or $n$ odd.

Case 1: $n$ is even. Let $n=2r$. In this case, each $\Phi_{n,i}$ (for $i=0,1)$
is a subrepresentation of $\mathbb{C}\mathcal{R}_{n}$ of dimension $r-1$, and
$(\mathbb{C}\mathbf{1})^{\perp}$ decomposes into
$\mathcal{R}_{n}$-subrepresentations as

$(\mathbb{C}\mathbf{1})^{\perp}=\mathbb{C}\hat{\mathbf{1}}\oplus\Phi_{n,0}\oplus\Phi_{n,1}.$

Now set
$[a_{1},\ldots,a_{r-1}]_{0}=\displaystyle{\sum_{i=1}^{r-1}{a_{i}}v_{2i,2i+2}}$
(the coordinate vector in the basis $\Delta_{n,0}$). The right multiplication
operators $R_{1}$ and $R_{2}$ together generate all of
$\text{Inn}(\mathcal{R}_{n})\cong D_{n/2}$, and with respect to the basis
$\Delta_{n,0}$, the matrix representations of these operators on $\Phi_{n,0}$
are

$\left[R_{1}\right]_{\Delta_{n,0}}=\begin{bmatrix}0&0&\cdots&0&-1\\\
0&0&\cdots&-1&0\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
0&-1&\cdots&0&0\\\
-1&0&\cdots&0&0\end{bmatrix},\;\;\;\;\left[R_{2}\right]_{\Delta_{n,0}}=\begin{bmatrix}1&0&\cdots&0&0\\\
1&0&\cdots&0&-1\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\\\ 1&-1&\cdots&0&0\end{bmatrix}.$ (1)

For example, for $n=12$, we have

$R_{1}\left(\left[a_{1},a_{2},a_{3},a_{4},a_{5}\right]_{0}\right)=\left[-a_{5},-a_{4},-a_{3},-a_{2},-a_{1}\right]_{0},$
$R_{2}\left(\left[a_{1},a_{2},a_{3},a_{4},a_{5}\right]_{0}\right)=\left[a_{1},a_{1}-a_{5},a_{1}-a_{4},a_{1}-a_{3},a_{1}-a_{2}\right]_{0}.$

For $1\leq s\leq r-1$, let

$\mathbf{u}_{s}:=\left[1-\omega_{r}^{s},1-(\omega_{r}^{s})^{2},\cdots,1-(\omega_{r}^{s})^{r-1}\right]_{0}=\sum_{i=1}^{r-1}\left(1-(\omega_{r}^{s})^{i}\right)v_{2i,2i+2},\;\text{
and }\;\mathbf{v}_{s}:=R_{1}(\mathbf{u}_{s}).$

Then
$\mathbf{v}_{s}=\displaystyle{\sum_{i=1}^{r-1}\left((\omega_{r}^{s})^{r-i}-1\right)v_{2i,2i+2}}$,
and

$\displaystyle R_{2}(\mathbf{u}_{s})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{r-1}\left((\omega_{r}^{s})^{r+1-i}-\omega_{r}^{s}\right)v_{2i,2i+2}$
$\displaystyle=$
$\displaystyle\omega_{r}^{s}\sum_{i=1}^{r-1}\left((\omega_{r}^{s})^{r-i}-1\right)v_{2i,2i+2}$
$\displaystyle=$ $\displaystyle\omega_{r}^{s}\mathbf{v}_{s}.$

Since all $R_{i}$ are involutions, we also have
$R_{2}(\mathbf{v}_{s})=(\omega_{r}^{-1})^{s}\mathbf{u}_{s}$. Let $U_{s,0}$ be
the subspace of $\Phi_{n,0}$ spanned by $\mathbf{u}_{s}$ and $\mathbf{v}_{s}$.
The above identities show that each $U_{s,0}$ is a quandle subrepresentation
of $\Phi_{n,0}$. If $\mathbf{u}_{s}$ and $\mathbf{v}_{s}$ are linearly
dependent, then $U_{s,0}$ is a 1-dimensional quandle subrepresentation. And if
$\mathbf{u}_{s}$ and $\mathbf{v}_{s}$ are linearly independent, the matrix
representations of $R_{1}$ and $R_{2}$ with respect to the basis
$\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}$ are

$\left[R_{1}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\;\;\;\;\left[R_{2}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&\omega_{r}^{s}\\\
\omega_{r}^{-s}&0\end{bmatrix},$

which gives us $U_{s,0}\cong W(\omega_{r}^{s})$ (both an
$\mathcal{R}_{n}$-quandle and an $\text{Inn}(\mathcal{R}_{n})$-group
isomorphism). From the identity
$(\omega_{r}^{r-s})^{\ell}=(\omega_{r}^{s})^{-\ell}$ (for any integers
$\ell,s$), we obtain $\mathbf{u}_{s}=-\mathbf{v}_{r-s}$, for $1\leq s\leq
r-1$. Therefore $U_{s,0}=U_{r-s,0}$, $1\leq s\leq r-1$. And since each
$W(\omega_{r}^{s})$ is an irreducible group representation for
$\text{Inn}(\mathcal{R}_{n})$ for $1\leq s\leq r-1$, it follows that
$U_{s,0}\cong_{\mathcal{R}_{n}}U_{v,0}$ if and only if $s=v$, for $1\leq
s,v\leq\lfloor r/2\rfloor$, and $U_{s,0}\cap U_{v,0}=\left\\{0\right\\}$ for
$1\leq s\neq v\leq\lfloor r/2\rfloor$. We summarize these results here:

For $r$ even, $U_{\ell,0}\cap U_{s,0}=\left\\{0\right\\}$ for $1\leq\ell\neq
s\leq r/2$, and $\displaystyle{\dim(U_{s,0})=\begin{cases}2,&1\leq s<r/2\\\
1,&s=r/2\end{cases}}$ .

For $r$ odd, $U_{\ell,0}\cap U_{s,0}=\left\\{0\right\\}$ for $1\leq\ell\neq
s\leq(r-1)/2$, and $\dim(U_{s,0})=2$ for all $s$.

Hence we obtain a decomposition of $\Phi_{n,0}$ into irreducible quandle
subrepresentations:

$\Phi_{n,0}\cong_{\mathcal{R}_{n}}\bigoplus_{s=1}^{\lfloor
r/2\rfloor}U_{s,0}\cong_{\mathcal{R}_{n}}\begin{cases}W(\omega_{r})\oplus\cdots\oplus
W\left(\omega_{r}^{r/2-1}\right)\oplus\mathbb{C}(-1,1),&r\;\text{even},\\\
W(\omega_{r})\oplus\cdots\oplus
W\left(\omega_{r}^{(r-1)/2}\right),&r\;\text{odd}.\end{cases}$

The decomposition of $\Phi_{n,1}$ is achieved in an similar fashion - with a
few small changes. We first replace $R_{1}$, $R_{2}$ with $R_{n/2}$ and
$R_{1}$, respectively. The matrix representations of these transformations,
with respect to the basis $\Delta_{n,1}$, are given by

$\left[R_{n/2}\right]_{\Delta_{n,1}}=\begin{bmatrix}0&0&\cdots&0&-1\\\
0&0&\cdots&-1&0\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
0&-1&\cdots&0&0\\\
-1&0&\cdots&0&0\end{bmatrix},\;\;\;\;\left[R_{1}\right]_{\Delta_{n,1}}=\begin{bmatrix}1&0&\cdots&0&0\\\
1&0&\cdots&0&-1\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\\\ 1&-1&\cdots&0&0\end{bmatrix}.$ (2)

Then we define

$\displaystyle\mathbf{u}_{s}$ $\displaystyle:=$
$\displaystyle\left[1-\omega_{r}^{s},1-(\omega_{r}^{s})^{2},\cdots,1-(\omega_{r}^{s})^{r-1}\right]_{1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{r-1}\left(1-(\omega_{r}^{s})^{i}\right)v_{2i-1,2i+1},\;\;\;\;\;\text{
and }\;\mathbf{v}_{s}:=\omega_{r}^{-s}R_{n/2}(\mathbf{u}_{s}).$

Then we can show that $R_{1}(\mathbf{u}_{s})=\mathbf{v}_{s}$, and since

$R_{2}=R_{n/2+2}=R_{n/2\,\triangleright
1}=R_{1}R_{n/2}R_{1}^{-1}=R_{1}R_{n/2}R_{1},$

we have
$R_{2}(\mathbf{u}_{s})=R_{1}R_{n/2}R_{1}(\mathbf{u}_{s})=\omega_{r}^{s}\mathbf{v}_{s}$.
For $1\leq s\leq r-1$, we define $U_{s,1}$ to be the subspace of $\Phi_{n,1}$
spanned by $\mathbf{u}_{s}$ and $\mathbf{v}_{s}$. Then, as above, we have

$\dim(U_{s,1})=\begin{cases}1,&s=r/2,\,\,r\text{ even}\\\ 2,&s\neq
r/2,\end{cases}$

and for $s\neq r/2$, the matrix representations of $R_{1}$ and $R_{2}$ are

$\left[R_{1}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\;\;\;\;\left[R_{2}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&\omega_{r}^{s}\\\
\omega_{r}^{-s}&0\end{bmatrix}.$

This shows

$\Phi_{n,1}\cong\bigoplus_{s=1}^{\lfloor
r/2\rfloor}U_{s,1}\cong_{\mathcal{R}_{n}}\begin{cases}W(\omega_{r})\oplus\cdots\oplus
W\left(\omega_{r}^{r/2-1}\right)\oplus\mathbb{C}(1,-1),&r\;\text{even},\\\
W(\omega_{r})\oplus\cdots\oplus
W\left(\omega_{r}^{(r-1)/2}\right),&r\;\text{odd}.\end{cases}$

For $r$ even ($n\equiv 0\bmod 4$), the one-dimensional quandle
subrepresentations of $\mathbb{C}\mathcal{R}_{n}$ are:

$\mathbb{C}\mathbf{1}\cong\mathbb{C}(1,1),\;\;\;\;\mathbb{C}\hat{\mathbf{1}}\cong\mathbb{C}(1,1),\;\;\;U_{r/2,0}\cong\mathbb{C}(-1,1),\;\;\;U_{r/2,1}\cong\mathbb{C}(1,-1),$

and the two-dimensional irreducible quandle subrepresentations are
$U_{s,0}\cong U_{s,1}\cong W(\omega_{r}^{s})$, $1\leq s\leq r/2-1$.

For $r$ odd ($n\equiv 2\bmod 4)$, the one-dimensional quandle
subrepresentations of $\mathbb{C}\mathcal{R}_{n}$ are:

$\mathbb{C}\mathbf{1}\cong\mathbb{C}(1,1)\;\;\text{ and
}\;\;\mathbb{C}\hat{\mathbf{1}}\cong\mathbb{C}(1,1),$

and the two-dimensional irreducible quandle subrepresentations are
$U_{s,0}\cong U_{s,1}\cong W(\omega_{r}^{s})$, $1\leq
s\leq(r~{}-~{}1)~{}/~{}2$.

Case 2: $n$ odd. Let $n=2r+1$. With respect to the basis
$\Delta_{n}=\left\\{v_{12},v_{23},\ldots,v_{n-1,n}\right\\}$, the elements
$R_{1}$, $R_{2}\in\operatorname{Inn}(\mathcal{R}_{n})$ have matrix
representations

$\left[R_{1}\right]_{\Delta_{n}}=\begin{bmatrix}0&\cdots&0&-1&1\\\
0&\cdots&-1&0&1\\\ \vdots&\reflectbox{$\ddots$}&\vdots&\vdots&\vdots\\\
-1&0&\cdots&0&1\\\
0&0&\cdots&0&1\end{bmatrix},\;\;\;\;\left[R_{2}\right]_{\Delta_{n}}=\begin{bmatrix}1&0&\cdots&0&0\\\
1&0&\cdots&0&-1\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\\\ 1&-1&\cdots&0&0\end{bmatrix}.$

For $1\leq s\leq n-1$, let

$\mathbf{u}_{s}=\left[1-\omega_{n}^{s},1-(\omega_{n}^{s})^{2},\cdots,1-(\omega_{n}^{s})^{n-1}\right],\;\;\;\text{
and }\;\;\mathbf{v}_{s}=R_{1}(\mathbf{u}_{s}).$

We can then show that
$\mathbf{v}_{s}=\omega_{n}^{-s}\left(\omega_{n}^{-s}-1,(\omega_{n}^{-s})^{2}-1,\ldots,(\omega_{n}^{-s})^{n-1}-1\right)$.

Let $U_{s}$ be the subspace of $\Phi_{n}$ spanned by $\mathbf{u}_{s}$ and
$\mathbf{v}_{s}$. We can show that $\mathbf{u}_{s}=\mathbf{v}_{n-s}$, hence
$U_{s}=U_{n-s}$. Furthermore, we have

$\displaystyle R_{2}(\mathbf{u}_{s})$ $\displaystyle=$ $\displaystyle
R_{2}(1-\omega_{n}^{s},1-(\omega_{n}^{s})^{2},\ldots,1-(\omega_{n}^{s})^{n-1})$
$\displaystyle=$
$\displaystyle(1-\omega_{n}^{s},(\omega_{n}^{s})^{n-1}-\omega_{n}^{s},(\omega_{n}^{s})^{n-2}-\omega_{n}^{s},\ldots,(\omega_{n}^{s})^{2}-\omega_{n}^{s})$
$\displaystyle=$
$\displaystyle\omega_{n}^{2s}\omega_{n}^{-s}(\omega_{n}^{-s}-1,(\omega_{n}^{-s})^{2}-1,\ldots,(\omega_{n}^{-s})^{n-1}-1)$
$\displaystyle=$ $\displaystyle\omega_{n}^{2s}\mathbf{v}_{s}.$

Therefore, if $\mathbf{u}_{s}$ and $\mathbf{v}_{s}$ are linearly independent,
the matrix representations of $R_{1}$ and $R_{2}$ with respect to the basis
$\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}$ of $U_{s}$ are again

$\left[R_{1}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\;\;\;\;\left[R_{2}\right]_{\left\\{\mathbf{u}_{s},\mathbf{v}_{s}\right\\}}=\begin{bmatrix}0&\omega_{n}^{2s}\\\
\omega_{n}^{-2s}&0\end{bmatrix},$

We can show linear independence of $\mathbf{u}_{s}$ and $\mathbf{v}_{s}$ for
$1\leq s\leq r$, which gives us the decomposition

$(\mathbb{C}\mathbf{1})^{\perp}=\bigoplus_{s=1}^{r}U_{s},$

and we obtain the desired decomposition into irreducible representations

$\mathbb{C}\mathcal{R}_{n}\cong\mathbb{C}(1,1)\bigoplus_{s=1}^{r}W(\omega_{n}^{2s}).$

∎

### Examples

The decomposition of $\mathbb{C}\mathcal{R}_{n}$ given in Theorem 5.1 falls
into 3 cases: $n\equiv 0\bmod 4$, $n\equiv 2\bmod 4$, and $n$ odd. We provide
three examples below, one for each of these cases, wherein we provide the
decomposition and all irreducible subrepresentations of
$\mathbb{C}\mathcal{R}_{10}$, $\mathbb{C}\mathcal{R}_{11}$, and
$\mathbb{C}\mathcal{R}_{12}$:

Regular quandle representation of $\mathcal{R}_{10}:$

$\mathbb{C}\mathcal{R}_{10}=\mathbb{C}\mathbf{1}\oplus\overbrace{\mathbb{C}\hat{\mathbf{1}}\oplus\underbrace{U_{1,0}\oplus U_{2,0}}_{\Phi_{10,0}}\oplus\underbrace{U_{1,1}\oplus U_{2,1}}_{\Phi_{10,1}}}^{(\mathbb{C}\mathbf{1})^{\perp}}$ subrep of $\mathbb{C}\mathcal{R}_{10}$ |  | irrep of $D_{5}$ | generated by | dimension
---|---|---|---|---
$\mathbb{C}\mathbf{1}$ | $\cong$ | $\mathbb{C}(1,1)$ | $\hskip 36.135pt\mathbf{1}$ | 1
$\mathbb{C}\hat{\mathbf{1}}$ | $\cong$ | $\mathbb{C}(1,1)$ |  $\hat{\mathbf{1}}$ | 1
$U_{1,0}$ | $\cong$ | $W(\omega_{5})$ | $\sum_{i=1}^{4}\left(1-\omega_{5}^{i}\right)v_{2i,2i+2}$ | 2
$U_{2,0}$ | $\cong$ | $W(\omega_{5}^{2})$ | $\sum_{i=1}^{4}\left(1-\omega_{5}^{2i}\right)v_{2i,2i+2}$ | 2
$U_{1,1}$ | $\cong$ | $W(\omega_{5})$ | $\sum_{i=1}^{4}\left(1-\omega_{5}^{i}\right)v_{2i-1,2i+1}$ | 2
$U_{2,1}$ | $\cong$ | $W(\omega_{5}^{2})$ | $\sum_{i=1}^{4}\left(1-\omega_{5}^{2i}\right)v_{2i-1,2i+1}$ | 2

Regular quandle representation of $\mathcal{R}_{11}:$

$\mathbb{C}\mathcal{R}_{11}=\mathbb{C}\mathbf{1}\oplus\overbrace{U_{1}\oplus U_{2}\oplus U_{3}\oplus U_{4}\oplus U_{5}}^{(\mathbb{C}\mathbf{1})^{\perp}}$ subrep of $\mathbb{C}\mathcal{R}_{11}$ |  | irrep of $D_{11}$ | generated by | dimension
---|---|---|---|---
$\mathbb{C}\mathbf{1}$ | $\cong$ | $\mathbb{C}(1,1)$ |  $\mathbf{1}$ | 1
$U_{1}$ | $\cong$ | $W(\omega_{11}^{2})$ | $\sum_{i=1}^{8}\left(1-\omega_{11}^{2i}\right)e_{i}$ | 2
$U_{2}$ | $\cong$ | $W(\omega_{11}^{4})$ | $\sum_{i=1}^{8}\left(1-\omega_{11}^{4i}\right)e_{i}$ | 2
$U_{3}$ | $\cong$ | $W(\omega_{11}^{6})$ | $\sum_{i=1}^{8}\left(1-\omega_{11}^{6i}\right)e_{i}$ | 2
$U_{4}$ | $\cong$ | $W(\omega_{11}^{8})$ | $\sum_{i=1}^{8}\left(1-\omega_{11}^{8i}\right)e_{i}$ | 2
$U_{5}$ | $\cong$ | $W(\omega_{11}^{10})$ | $\sum_{i=1}^{8}\left(1-\omega_{11}^{10i}\right)e_{i}$ | 2

Regular quandle representation of $\mathcal{R}_{12}:$

$\mathbb{C}\mathcal{R}_{12}=\mathbb{C}\mathbf{1}\oplus\overbrace{\mathbb{C}\hat{\mathbf{1}}\oplus\underbrace{U_{1,0}\oplus U_{2,0}\oplus U_{3,0}}_{\Phi_{12,0}}\oplus\underbrace{U_{1,1}\oplus U_{2,1}\oplus U_{3,1}}_{\Phi_{12,1}}}^{(\mathbb{C}\mathbf{1})^{\perp}}$ subrep of $\mathbb{C}\mathcal{R}_{12}$ |  | irrep of $D_{6}$ | generated by | dimension
---|---|---|---|---
$\mathbb{C}\mathbf{1}$ | $\cong$ | $\mathbb{C}(1,1)$ | $\hskip 36.135pt\mathbf{1}$ | 1
$\mathbb{C}\hat{\mathbf{1}}$ | $\cong$ | $\mathbb{C}(1,1)$ |  $\hat{\mathbf{1}}$ | 1
$U_{1,0}$ | $\cong$ | $W(\omega_{6})$ | $\sum_{i=1}^{5}\left(1-\omega_{6}^{i}\right)v_{2i,2i+2}$ | 2
$U_{2,0}$ | $\cong$ | $W(\omega_{6}^{2})$ | $\sum_{i=1}^{5}\left(1-\omega_{6}^{2i}\right)v_{2i,2i+2}$ | 2
$U_{3,0}$ | $\cong$ | $\mathbb{C}(-1,1)$ | $\sum_{i=1}^{5}\left(1-(-1)^{i}\right)v_{2i,2i+2}$ | 1
$U_{1,1}$ | $\cong$ | $W(\omega_{6})$ | $\sum_{i=1}^{5}\left(1-\omega_{6}^{i}\right)v_{2i-1,2i+1}$ | 2
$U_{2,1}$ | $\cong$ | $W(\omega_{6}^{2})$ | $\sum_{i=1}^{5}\left(1-\omega_{6}^{2i}\right)v_{2i-1,2i+1}$ | 2
$U_{3,1}$ | $\cong$ | $\mathbb{C}(1,-1)$ | $\sum_{i=1}^{5}\left(1-(-1)^{i}\right)v_{2i-1,2i+1}$ | 1

## References

  * [1]
ElhamdadiMohamedMacquarrieJenniferRestrepoRicardoAutomorphism groups of
quandlesJ. Algebra Appl.11201211250008, 9ISSN 0219-4988Review
MathReviewsDocument@article{EMR, author = {Elhamdadi, Mohamed}, author =
{Macquarrie, Jennifer}, author = {Restrepo, Ricardo}, title = {Automorphism
groups of quandles}, journal = {J. Algebra Appl.}, volume = {11}, date =
{2012}, number = {1}, pages = {1250008, 9}, issn = {0219-4988}, review =
{\MR{2900878}}, doi = {10.1142/S0219498812500089}}

  * [3]
ElhamdadiMohamedFernandoNerangaTsvelikhovskiyBorisRing theoretic aspects of
quandlesJ. Algebra526201915166–187Review MathReviews@article{EFT, author =
{Elhamdadi, Mohamed}, author = {Fernando, Neranga}, author = {Tsvelikhovskiy,
Boris}, title = {Ring theoretic aspects of quandles}, journal = {J. Algebra},
volume = {526}, date = {2019}, number = {15}, pages = {166-187}, review =
{\MR{3915329}}}

  * [5]
ElhamdadiMohamedNelsonSamQuandles—an introduction to the algebra of
knotsStudent Mathematical Library74American Mathematical Society, Providence,
RI2015x+245ISBN 978-1-4704-2213-4Review MathReviews@book{EN, author =
{Elhamdadi, Mohamed}, author = {Nelson, Sam}, title = {Quandles—an
introduction to the algebra of knots}, series = {Student Mathematical
Library}, volume = {74}, publisher = {American Mathematical Society,
Providence, RI}, date = {2015}, pages = {x+245}, isbn = {978-1-4704-2213-4},
review = {\MR{3379534}}}

  * [7]
EtingofPavelGolbergOlegHenselSebastianLiuTiankaiSchwendnerAlexVaintrobDmitryYudovinaElenaIntroduction
to representation theoryStudent Mathematical Library59With historical
interludes by Slava GerovitchAmerican Mathematical Society, Providence,
RI2011viii+228ISBN 978-0-8218-5351-1Review MathReviewsDocument@book{MR2808160,
author = {Etingof, Pavel}, author = {Golberg, Oleg}, author = {Hensel,
Sebastian}, author = {Liu, Tiankai}, author = {Schwendner, Alex}, author =
{Vaintrob, Dmitry}, author = {Yudovina, Elena}, title = {Introduction to
representation theory}, series = {Student Mathematical Library}, volume =
{59}, note = {With historical interludes by Slava Gerovitch}, publisher =
{American Mathematical Society, Providence, RI}, date = {2011}, pages =
{viii+228}, isbn = {978-0-8218-5351-1}, review = {\MR{2808160}}, doi =
{10.1090/stml/059}}

  * [9]
JoyceD.A classifying invariant of knots, the knot quandleJ. Pure Appl.
Algebra231982137–65ISSN 0022-4049Review MathReviewsDocument@article{Joyce,
author = {Joyce, D.}, title = {A classifying invariant of knots, the knot
quandle}, journal = {J. Pure Appl. Algebra}, volume = {23}, date = {1982},
number = {1}, pages = {37–65}, issn = {0022-4049}, review = {\MR{638121}}, doi
= {10.1016/0022-4049(82)90077-9}}

  * [11]
MatveevS. V.Distributive groupoids in knot theoryRussianMat. Sb.
(N.S.)119(161)1982178–88, 160ISSN 0368-8666Review MathReviews@article{Matveev,
author = {Matveev, S. V.}, title = {Distributive groupoids in knot theory},
language = {Russian}, journal = {Mat. Sb. (N.S.)}, volume = {119(161)}, date =
{1982}, number = {1}, pages = {78–88, 160}, issn = {0368-8666}, review =
{\MR{672410}}}

  * [13]
KaurDilpreettitle=Decomposition of quandle rings of dihedral quandlesSingh,
PushpendraarXiv:2206.03162, 2022 @article{KS, author = {Kaur, Dilpreet},
author = {{Singh, Pushpendra} title={Decomposition of quandle rings of
dihedral quandles}}, status = {arXiv:2206.03162, 2022 }}

  * [15]


# Bounds on positive operator-valued measure based coherence of superposition

Meng-Li Guo  School of Mathematical Sciences, Capital Normal University,
Beijing 100048, China  Jin-Min Liang  School of Mathematical Sciences, Capital
Normal University, Beijing 100048, China  Bo Li  School of Computer and
Computing Science, Hangzhou City University, Hangzhou 310015, China  Shao-Ming
Fei<EMAIL_ADDRESS>School of Mathematical Sciences, Capital Normal
University, Beijing 100048, China  Max–Planck-Institute for Mathematics in the
Sciences, Leipzig 04103, Germany  Zhi-Xi Wang<EMAIL_ADDRESS>School of
Mathematical Sciences, Capital Normal University, Beijing 100048, China

###### Abstract

Quantum coherence is a fundamental feature of quantum physics and plays a
significant role in quantum information processing. By generalizing the
resource theory of coherence from von Neumann measurements to positive
operator-valued measures (POVMs), POVM-based coherence measures have been
proposed with respect to the relative entropy of coherence, the $l_{1}$ norm
of coherence, the robustness of coherence and the Tsallis relative entropy of
coherence. We derive analytically the lower and upper bounds on these POVM-
based coherence of an arbitrary given superposed pure state in terms of the
POVM-based coherence of the states in superposition. Our results can be used
to estimate range of quantum coherence of superposed states. Detailed examples
are presented to verify our analytical bounds.

##  I Introduction

Originated from the superposition principle of quantum mechanics, quantum
coherence and entanglement are important quantum resources in quantum
information processing and quantum computation [1, 2, 3]. However, in general
the entanglement of a superposed pure state cannot be simply expressed as a
linear summation of the entanglement of the individual states in the
superposition. Linden _et al_. first investigated the relations between the
entanglement of a superposed pure state and the entanglement of the states in
the superposition [4] and Gour _et al_. provided the upper and lower bounds on
superposed entanglement based on the Von Neumann entropy of the reduced states
[5] and the entanglement measure concurrence [6, 7].

Baumgratz, Cramer and Plenio [8] first proposed the resource theory of
coherence, established a rigorous coherence quantification framework (BCP
framework), and identified computable measures of coherence. Based on the BCP
framework, coherence has witnessed theoretically and experimentally fruitful
progress [9, 10, 11, 12, 13, 14, 15]. For a quantum system associated with a
$d$-dimensional Hilbert space $H$, the BCP framework takes into account the
coherence defined by an orthogonal basis $\\{|j\rangle\\}_{j=1}^{d}$, which we
call standard coherence. A standard orthogonal basis
$\\{|j\rangle\\}_{j=1}^{d}$ corresponds to a rank-$1$ projective measurement
$\\{|j\rangle\langle j|\\}_{j=1}^{d}$. Recently, Bischof, Kampermann and Bruß
[16, 17] generalized the conventional framework of coherence to the case of
general positive operator-valued measures (POVMs), by replacing the projective
measurements with POVMs. Recently, several POVM-based coherence measures have
been proposed [16, 17, 18], such as relative entropy of POVM-based coherence
$C_{r}(\rho,E)$, $l_{1}$-norm of POVM-based coherence $C_{l_{1}}(\rho,E)$,
robustness of POVM-based coherence $C_{rob}(\rho,E)$, and POVM-based coherent
based on Tsallis relative entropy $C_{T,\lambda}(\rho,E)$. Moreover, maximum
relative entropy of coherence for quantum channels has been introduced in
[20].

Similar to the case of quantum entanglement, the superposition of two
incoherent states may give rise to a maximally coherent state, for instance,
$|\Omega_{1}\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ in the
computational basis $|0\rangle,|1\rangle$. While the superposition of two
maximally coherent states may produce an incoherent state, for example,
$|\Omega_{2}\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$, where
$|\pm\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm|1\rangle)$. In [21, 22, 23, 24]
the coherence for superposition states of two mutual orthogonal states has
been discussed.

In this work, we investigate the POVM-based coherence of superposition states.
More specifically, given a superposition state,

$\displaystyle|\Omega\rangle=\sum_{k=1}^{n}\alpha_{k}|\phi_{k}\rangle,$ (1)

we explore the relationship between the POVM-based coherence of the
superposition state $|\Omega\rangle$ and the POVM-based coherence of the
states $|\phi_{k}\rangle$ in the superposition. We focus on four POVM-based
coherence measures $C_{r}(\rho,E)$, $C_{l_{1}}(\rho,E)$, $C_{rob}(\rho,E)$,
and $C_{T,\lambda}(\rho,E)$ and present the upper and lower bounds of the
inequalities satisfied by these coherence measures. We illustrate our results
by detailed single and two-qubit examples.

##  II POVM-based coherence of superposition states

In order to quantitatively describe the coherence resources contained in a
given quantum state, a variety of coherence measures has been introduced from
different perspectives. Before giving our main results, we first recall the
definitions of $C_{r}(\rho,E)$, $C_{l_{1}}(\rho,E)$, $C_{rob}(\rho,E)$ and
$C_{T,\lambda}(\rho,E)$ with respect to a positive operator-valued measure
given by $d$ measurement operators $E=\\{E_{j}\geq 0\\}_{j=1}^{d}$,
$\sum_{j}E_{j}=I$ with $I$ the identity operator.

###  II.1 The POVM-based coherence measures

The relative entropy coherence measure is a commonly used and well-defined
coherence measure [8], which is tightly related to the optimal distillation
rate in standard coherent distillation [11] and the minimum amount of noise
for complete decoherence [9, 25]. Given a POVM $E$, the relative entropy POVM-
based coherence is defined by [16, 17]

$\displaystyle
C_{r}(\rho,E)=\sum^{d}_{j=1}S(\sqrt{E_{j}}\rho\sqrt{E_{j}})-S(\rho),$

where $S(X)=-\textrm{Tr}(X\log X)$ is the von Neumann entropy for positive
semidefinite matrix $X$. For a pure state $|\phi\rangle$, $C_{r}(\rho,E)$ can
be expressed as

$\displaystyle
C_{r}(\phi,E)=\sum^{d}_{j=1}S(\sqrt{E_{j}}|\phi\rangle\langle\phi|\sqrt{E_{j}}).$
(2)

The $l_{1}$-norm POVM-based coherence $C_{l_{1}}(\rho,E)$ of a density matrix
$\rho$ is defined by [17, 18],

$\displaystyle C_{l_{1}}(\rho,E)=\sum^{d}_{i\neq
j=1}||\sqrt{E_{i}}\rho\sqrt{E_{j}}||_{\textrm{tr}},$

with $||X||_{\textrm{tr}}=\textrm{Tr}\sqrt{X^{{\dagger}}X}$ the trace norm of
matrix $X$. For a pure state $|\phi\rangle$, the $l_{1}$-norm POVM-based
coherence is given by

$\displaystyle C_{l_{1}}(\phi,E)=\sum^{d}_{i\neq
j=1}\textrm{Tr}|\sqrt{E_{i}}|\phi\rangle\langle\phi|\sqrt{E_{j}}|.$ (3)

The robust coherence is closely related to coherence sightings and can be used
to quantify the advantages of quantum states in phase discrimination tasks.
The robust POVM-based coherence $C_{rob}(\rho,E)$ can be expressed as [17],

$\displaystyle C_{rob}(\rho,E)=\min_{\tau\in S^{{}^{\prime}}}\left\\{s\geq
0:s\tau_{i,j}=-\sqrt{E_{i}}\rho\sqrt{E_{j}},\forall i\neq j\right\\}.$

where $\tau=\sum_{i,j}\tau_{i,j}\otimes|i\rangle\langle j|$ as
$\tau=\sum_{i,j}|\phi_{i}\rangle\langle\phi_{j}|\otimes|i\rangle\langle
j|+\sum_{i}(I-|\phi_{i}\rangle\langle\phi_{i}|)\otimes|i\rangle\langle i|$.
Note that for pure states $|\phi\rangle$, the robust POVM-based coherence
$C_{rob}(\rho,E)$ and the $l_{1}$-norm POVM-based coherence satisfy the
following relation,

$\displaystyle C_{rob}(\phi,E)=C_{l_{1}}(\phi,E).$

The Tsallis relative entropy of POVM-based coherence $C_{T,\lambda}(\rho,E)$
introduced in [18] is given by

$C_{T,\lambda}(\rho,E)=\frac{1}{\lambda-1}\Big{\\{}\sum^{d}_{j=1}\textrm{Tr}[(\sqrt{E_{j}}\rho^{\lambda}\sqrt{E_{j}})^{1/\lambda}]-1\Big{\\}},$
(4)

for $\lambda\in(0,1)\cup(1,2]$.

For further use, we extend the above definition of the POVM-based coherence of
a pure state to the coherence of two pure states, by analog to the case of
quantum entanglement [7]. Let $|\phi\rangle$ and $|\psi\rangle$ be two pure
states. We define $C_{l_{1}}(\phi,\psi)$ and $C_{T,\lambda}(\phi,\psi)$ as the
POVM coherence of $|\phi\rangle$ and $|\psi\rangle$ by

$\displaystyle C_{l_{1}}(\phi,\psi,E)=\sum^{d}_{i\neq
j=1}\textrm{Tr}|\sqrt{E_{i}}|\phi\rangle\langle\psi|\sqrt{E_{j}}|$ (5)

and

$\displaystyle
C_{T,\lambda}(\phi,\psi,E)=\frac{1}{\lambda-1}\Big{\\{}\sum_{j=1}^{d}\textrm{Tr}[(\sqrt{E_{j}}(|\phi\rangle\langle\psi|)^{\lambda}\sqrt{E_{j}})^{1/\lambda}]-1\Big{\\}}.$

Notice that according to the above definitions, we have
$C(\phi,\phi,E)=C(\phi,E)$.

###  II.2 The upper and lower bounds of POVM-based coherence measures

In the following, we use the relative entropy POVM-based coherence to
investigate the relationship between the coherence of two arbitrary pure
states and the POVM-based coherence of their superposed state.

Before we present our two main results (Theorem 1), we briefly review the
properties of the von-Neumann entropy $S(\rho)$ in [19, 4] and provide a
slight improvement of it. The authors in [4] have used two properties of
$S(\rho)$:

$|\alpha_{1}|^{2}S(\rho)+|\alpha_{2}|^{2}S(\sigma)\leq
S(|\alpha_{1}|^{2}\rho+|\alpha_{2}|^{2}\sigma)$ (6)

and

$\displaystyle
S(|\alpha_{1}|^{2}\rho+|\alpha_{2}|^{2}\sigma)\leq|\alpha_{1}|^{2}S(\rho)+|\alpha_{2}|^{2}S(\sigma)+h_{2}(|\alpha_{1}|^{2}).$

where $h_{2}(x)=-x\log x-(1-x)\log(1-x)$ and
$\left|\alpha_{1}\right|^{2}+\left|\alpha_{2}\right|^{2}=1$. Now, consider
$|\Omega\rangle=\alpha_{1}|\phi\rangle+\alpha_{2}|\psi\rangle$, we obtain

$\displaystyle
S(\sqrt{E_{j}}|\Omega\rangle\langle\Omega|\sqrt{E_{j}})\leq|\alpha_{1}|^{2}S(\sqrt{E_{j}}|\phi\rangle\langle\phi|\sqrt{E_{j}})+|\alpha_{2}|^{2}S(\sqrt{E_{j}}|\psi\rangle\langle\psi|\sqrt{E_{j}})+h_{2}(|\alpha_{1}|^{2}).$

Using Eq. (2), we get,

$\displaystyle
C_{r}(\Omega,E)\leq|\alpha_{1}|^{2}C_{r}(\phi,E)+|\alpha_{2}|^{2}C_{r}(\psi,E)+h_{2}({|\alpha_{1}|^{2}}),$
(7)

Based on the above results, we have the following conclusion.

###### Theorem 1

Given two different pure states $|\phi\rangle$ and $|\psi\rangle$, denote
$|\Omega\rangle=\alpha|\phi\rangle+\beta|\psi\rangle$, where $\alpha$ and
$\beta$ are complex numbers. The relative entropy POVM-based coherence
$C_{r}(\Omega^{{}^{\prime}},E)$ of the state
$|\Omega^{{}^{\prime}}\rangle=\frac{|\Omega\rangle}{||\Omega||}$, with
$||\Omega||=|\langle\Omega|\Omega\rangle|$ the normalization constant, has an
upper bound

$\displaystyle p_{1}[\mu C_{r}(\phi,E)+(1-\mu)C_{r}(\psi,E)+h_{2}(\mu)],$ (8)

and a lower bound $L=\max\\{L_{1},L_{2},0\\}$ with

$\displaystyle
L_{1}=p_{2}C_{r}(\phi,E)-\frac{1-\nu}{\nu}C_{r}(\psi,E)-\frac{1}{\nu}h_{2}(\nu),$
$\displaystyle
L_{2}=p_{3}C_{r}(\psi,E)-\frac{1-\xi}{\xi}C_{r}(\phi,E)-\frac{1}{\xi}h_{2}(\xi),$

where $h_{2}(x)=-x\log x-(1-x)\log(1-x)$,

$\displaystyle
p_{1}=\frac{(1-\mu)|\alpha|^{2}+\mu|\beta|^{2}}{\mu(1-\mu)||\Omega||^{2}},~{}~{}\mu=\frac{|\alpha|^{2}}{\cos^{2}\theta},$
$\displaystyle
p_{2}=\frac{(1-\nu)|\alpha|^{2}}{(1-\nu)||\Omega||^{2}+\nu|\beta|^{2}},~{}~{}\nu=\frac{\sin^{2}\theta||\Omega||^{2}}{\sin^{2}\theta||\Omega||^{2}+|\beta|^{2}\cos^{2}\theta},$
$\displaystyle
p_{3}=\frac{(1-\xi)|\beta|^{2}}{(1-\xi)||\Omega||^{2}+\xi|\alpha|^{2}},~{}~{}\xi=\frac{\sin^{2}\theta||\Omega||^{2}}{\sin^{2}\theta||\Omega||^{2}+|\alpha|^{2}\cos^{2}\theta}$

for $0<\mu,\nu,\xi<1$ and
$\frac{|\alpha|^{2}}{\cos^{2}\theta}+\frac{|\beta|^{2}}{\sin^{2}\theta}=1$.

Proof We prove the theorem by introducing an auxiliary system. First we prove
the upper bound (8). Consider the following bipartite state in systems $A$ and
$B$,

$\displaystyle|\chi\rangle^{AB}=\sqrt{\mu}|\phi\rangle^{A}|0\rangle^{B}+\sqrt{1-\mu}|\psi\rangle^{A}|1\rangle^{B}.$

According to (7), the relative entropy POVM-based coherence of $|\chi\rangle$
can be expressed as

$\displaystyle C_{r}(\chi,E)\leq\mu
C_{r}(\phi,E)+(1-\mu)C_{r}(\psi,E)+h_{2}(\mu).$

Measuring the ancillary system $B$ with Kraus operators

$\displaystyle K_{1}=|0\rangle(\cos\theta e^{i\omega_{1}}\langle 0|+\sin\theta
e^{i\omega_{2}}\langle 1|),$ $\displaystyle K_{2}=|1\rangle(-\sin\theta
e^{-i\omega_{2}}\langle 0|+\cos\theta e^{-i\omega_{1}}\langle 1|),$

one gets the collapsed state

$\displaystyle|\chi_{1}\rangle|0\rangle=\left(\sqrt{\frac{\mu}{p}}\cos\theta
e^{i\omega_{1}}|\phi\rangle+\sqrt{\frac{1-\mu}{p}}\sin\theta
e^{i\omega_{2}}|\psi\rangle\right)|0\rangle,$

with probability

$\displaystyle p=\|\sqrt{\mu}\cos\theta
e^{i\omega_{1}}|\phi\rangle+\sqrt{1-\mu}\sin\theta
e^{i\omega_{2}}|\psi\rangle\|^{2},$ (9)

and the collapsed state

$\displaystyle|\chi_{2}\rangle|1\rangle=\left(\sqrt{\frac{1-\mu}{1-p}}\cos\theta
e^{-i\omega_{1}}|\psi\rangle-\sqrt{\frac{\mu}{1-p}}\sin\theta
e^{i\omega_{2}}|\phi\rangle\right)|1\rangle,$

with probability $1-p$. As the measurement can be seen as an incoherent
operation, we obtain the following inequality,

$\displaystyle pC_{r}(\chi_{1},E)+(1-p)C_{r}(\chi_{2},E)\leq\mu
C_{r}(\phi,E)+(1-\mu)C_{r}(\psi,E)+h_{2}(\mu).$

Since $C_{r}(\chi_{2},E)\geq 0$, we have

$\displaystyle C_{r}(\chi_{1},E)\leq\frac{1}{p}[\mu
C_{r}(\phi,E)+(1-\mu)C_{r}(\psi,E)+h_{2}(\mu)].$ (10)

Now we set $|\chi_{1}\rangle=|\Omega^{{}^{\prime}}\rangle$. We get

$\displaystyle\frac{\alpha}{||\Omega||}=\sqrt{\frac{\mu}{p}}\cos\theta
e^{i\omega_{1}},~{}~{}\frac{\beta}{||\Omega||}=\sqrt{\frac{1-\mu}{p}}\sin\theta
e^{i\omega_{2}}.$

It is straightforward to derive that

$\displaystyle
p=\frac{\|\Omega\|^{2}\mu(1-\mu)}{(1-\mu)|\alpha|^{2}+\mu|\beta|^{2}}.$ (11)

Substituting (11) into (10), we have

$\displaystyle C_{r}(\Omega^{{}^{\prime}},E)\leq p_{1}\left[\mu
C_{r}(\phi,E)+(1-\mu)C_{r}(\psi,E)+h_{2}(\mu)\right],$

with $0<\mu<1$, $\mu=\frac{|\alpha|^{2}}{\cos^{2}\theta}$ and
$p_{1}=\frac{(1-\mu)|\alpha|^{2}+\mu|\beta|^{2}}{\mu(1-\mu)\|\Omega\|^{2}}$.

To prove the lower bound, we can consider the following bipartite states,

$\displaystyle|\chi^{{}^{\prime}}\rangle^{AB}$ $\displaystyle=$
$\displaystyle\sqrt{\nu}|\Omega^{{}^{\prime}}\rangle^{A}|0\rangle^{B}+\sqrt{1-\nu}|\psi\rangle^{A}|1\rangle^{B},$
$\displaystyle|\chi^{{}^{\prime\prime}}\rangle^{AB}$ $\displaystyle=$
$\displaystyle\sqrt{\xi}|\Omega^{{}^{\prime}}\rangle^{A}|0\rangle^{B}+\sqrt{1-\xi}|\phi\rangle^{A}|1\rangle^{B},$

Similar to the proof of the upper bound (8), one obtains easily the lower
bound $L$. $\Box$

Concerning the $l_{1}$-norm of the POVM-based coherence measure, we have the
following conclusion.

###### Theorem 2

Given $|\Omega\rangle=\sum^{n}_{k=1}\alpha_{k}|\phi_{k}\rangle$ with
$\alpha_{k}$ complex numbers, the POVM-based coherence of the superposed sate
$|\Omega^{{}^{\prime}}\rangle=\frac{|\Omega\rangle}{||\Omega||}$ has an upper
bound

$\displaystyle\|\Omega\|^{-2}\Bigg{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{l_{1}}(\phi_{k},E)+M\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|\Bigg{)},$ (12)

and a lower bound

$\displaystyle\|\Omega\|^{-2}\Bigg{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{l_{1}}(\phi_{k},E)-M\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|\Bigg{)},$ (13)

where
$M=(d-1)\sum^{d}_{i=1}\Big{\|}\sqrt{E_{i}}|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|\Big{\|}_{\textrm{tr}}$.

Proof  From the definition of the $l_{1}$-norm POVM-based coherence (3), we
have

$\displaystyle\|\Omega\|^{2}C_{l_{1}}(\Omega^{{}^{\prime}},E)$
$\displaystyle=\sum^{d}_{i\neq
j=1}\text{Tr}\Big{|}\sqrt{E_{i}}\sum^{n}_{kk^{{}^{\prime}}=1}\alpha_{k}\alpha_{k^{{}^{\prime}}}|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|\sqrt{E_{j}}\Big{|}$
$\displaystyle\leq\sum^{n}_{k=1}|\alpha_{k}|^{2}\sum^{d}_{i\neq
j=1}\text{Tr}\Big{|}\sqrt{E_{i}}|\phi_{k}\rangle\langle\phi_{k}|\sqrt{E_{j}}\Bigg{|}+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|\sum^{d}_{i\neq
j=1}\text{Tr}\Big{|}\sqrt{E_{i}}|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|\sqrt{E_{j}}\Big{|}$
$\displaystyle=\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{l_{1}}(\phi_{k},E)+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|C_{l_{1}}(\phi_{k},\phi_{k^{{}^{\prime}}},E)$
$\displaystyle\leq\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{l_{1}}(\phi_{k},E)+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|M,$

where the first inequality is due to $|a+b|\leq|a|+|b|$, the second inequality
is based on the Theorem 2 in Ref. [26] with
$M=(d-1)\sum^{d}_{i=1}\Big{\|}\sqrt{E_{i}}|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|\Big{\|}_{\textrm{tr}}$.

Next, by using the inverse triangle inequality,
$|a+b|\geq||a|-|b||\geq|a|-|b|$, similar to the proof of the upper bound we
can easily get (13). $\Box$

The robustness of coherence [27, 17] plays an important role in the
characterization of quantum states in phase discrimination. For pure states,
the robustness of POVM-based coherence is equivalent to the $l_{1}$-norm of
POVM-based coherence. Therefore, Theorem 2 also gives rise to the bounds for
the robustness of POVM-based coherence.

Concerning the POVM-based coherence of Tsallis relative entropy
$C_{T,\lambda}(\rho,E)$, we present the following lemma.

###### Lemma 1

For $\lambda\in(0,1)\cup(1,2]$, we have

$\displaystyle
C_{T,\lambda}(\rho,E)\leq-\ln_{\lambda}\Bigg{(}\frac{1}{[d\sum^{d}_{j=1}\textrm{Tr}(\sqrt{E_{j}}\rho^{2}\sqrt{E_{j}})]^{1/\lambda}}\Bigg{)},$

where $\ln_{\lambda}x=\frac{x^{1-\lambda}-1}{1-\lambda}$.

Proof. Due to (4), for $\lambda\in(0,1)\cup(1,2]$ according to [28, 29], we
obtain

$\displaystyle
C_{T,\lambda}(\rho,E)\leq\frac{[d^{\lambda-1}\sum^{d}_{j=1}\textrm{Tr}(\sqrt{E_{j}}\rho^{\lambda}\sqrt{E_{j}})]^{1/\lambda}-1}{\lambda-1}.$
(14)

Applying the Jensen’s inequality, we have

$\displaystyle\frac{[\sum^{d}_{j=1}\textrm{Tr}(\sqrt{E_{j}}\rho^{\lambda}\sqrt{E_{j}})]^{1/\lambda}}{\lambda-1}\leq\frac{[(\sum^{d}_{j=1}\textrm{Tr}(\sqrt{E_{j}}\rho^{2}\sqrt{E_{j}}))^{1/\lambda}]^{\lambda-1}}{\lambda-1}.$
(15)

Combining (15) and (14), we complete the proof. $\Box$

Based on the above results, we have the following conclusion.

###### Theorem 3

Given a state $|\Omega\rangle=\sum^{n}_{k=1}\alpha_{k}|\phi_{k}\rangle$ with
$\alpha_{k}$ the complex numbers. The POVM-based coherence of the normalized
superposed state
$|\Omega^{{}^{\prime}}\rangle=\frac{|\Omega\rangle}{\|\Omega\|}$ has an upper
bound

$\displaystyle\|\Omega\|^{-2}\Big{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{T,\lambda}(\phi_{k},E)-\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|\ln_{\lambda}X\Big{)}+\frac{1}{\|\Omega\|^{2}(\lambda-1)}\Big{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|-\|\Omega\|^{2}\Big{)}$

for $\lambda\in(0,1)\cup(1,2]$, where

$\displaystyle
X=1/\Big{(}d\sum^{d}_{j=1}\textrm{Tr}[\sqrt{E_{j}}(|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|)^{2}\sqrt{E_{j}}]\Big{)}^{\frac{1}{\lambda}},$
(16)

and a lower bound $L=\max\\{L_{1},0\\}$ for $\lambda\in(1,2]$, where

$\displaystyle\|\Omega\|^{2}L_{1}=\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{T,\lambda}(\phi_{k},E)+N\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|+\frac{1}{(\lambda-1)}\Big{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|-\|\Omega\|^{2}\Big{)}$

and

$\displaystyle
N=\frac{1}{\lambda-1}\Big{\\{}\sum^{d}_{j=1}\textrm{Tr}[(\sqrt{E_{j}})^{\frac{1}{\lambda}}|\phi_{k}\rangle\langle\phi_{k^{{}^{\prime}}}|(\sqrt{E_{j}})^{\frac{1}{\lambda}}]-1\Big{\\}}.$

Proof Let $\\{|i\rangle_{i=1}^{m}\\}$ be a set basic vectors such that
$|\phi_{k}\rangle=\sum_{i=1}^{m}b^{k}_{i}|i\rangle$. We have
$|\Omega\rangle=\sum^{n}_{k=1}\sum^{m}_{i}\alpha_{k}b^{k}_{i}|i\rangle$ and

$\displaystyle\|\Omega\|^{2}C_{T,\lambda}(\Omega^{{}^{\prime}},E)=$
$\displaystyle\frac{1}{\lambda-1}\Bigg{\\{}\sum^{d}_{j=1}\textrm{Tr}\Bigg{[}\sqrt{E_{j}}\Big{[}\sum^{n}_{kk^{{}^{\prime}}=1}\sum^{m}_{ii^{{}^{\prime}}=1}\alpha_{k}b^{k}_{i}\alpha_{k^{{}^{\prime}}}b^{k^{{}^{\prime}}}_{i^{{}^{\prime}}}|i\rangle\langle
i^{{}^{\prime}}|\Big{]}^{\lambda}\sqrt{E_{j}}\Bigg{]}^{\frac{1}{\lambda}}-\|\Omega\|^{2}\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda-1}\Bigg{\\{}\sum^{n}_{k=1}|\alpha_{k}|^{2}\sum^{d}_{j=1}\textrm{Tr}\Bigg{[}\sqrt{E_{j}}\Big{[}\sum^{m}_{ii^{{}^{\prime}}=1}|b^{k}_{i}b^{k}_{i^{{}^{\prime}}}|\cdot|i\rangle\langle
i^{{}^{\prime}}|\Big{]}^{\lambda}\sqrt{E_{j}}\Bigg{]}^{\frac{1}{\lambda}}-\|\Omega\|^{2}$
$\displaystyle+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k}^{{}^{\prime}}|\sum^{d}_{j=1}\textrm{Tr}\Bigg{[}\sqrt{E_{j}}\Big{[}\sum^{m}_{ii^{{}^{\prime}}=1}|b^{k}_{i}b^{k^{{}^{\prime}}}_{i^{{}^{\prime}}}|\cdot|i\rangle\langle
i^{{}^{\prime}}|\Big{]}^{\lambda}\sqrt{E_{j}}\Bigg{]}^{\frac{1}{\lambda}}\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\sum^{n}_{k=1}|\alpha_{k}|^{2}C_{T,\lambda}(\phi_{k},E)+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|C_{T,\lambda}(\phi_{k},\phi_{k^{{}^{\prime}}},E)$
$\displaystyle+\frac{1}{\lambda-1}\Big{(}\sum^{n}_{k=1}|\alpha_{k}|^{2}+\sum^{n}_{k\neq
k^{{}^{\prime}}=1}|\alpha_{k}\alpha_{k^{{}^{\prime}}}|-\|\Omega\|^{2}\Big{)}.$

Based on the Lemma 1, we obtain the upper bound in theorem.

Next, by using the Araki-Lieb-Thirring Inequality [30],
$\textrm{Tr}(A^{r}B^{r}A^{r})^{q}\geq\textrm{Tr}(ABA)^{rq}$ for $r\geq 1$ and
$q\geq 0$, we have

$\displaystyle C_{T,\lambda}(\phi_{k},\phi_{k^{{}^{\prime}}},E)$
$\displaystyle=\frac{1}{\lambda-1}\Bigg{\\{}\sum^{d}_{j=1}\textrm{Tr}\Big{[}(E_{j}^{\frac{1}{2\lambda}})^{\lambda}(|\phi_{k}\rangle\langle\phi_{k}^{{}^{\prime}}|)^{\lambda}(E_{j}^{\frac{1}{2\lambda}})^{\lambda}\Big{]}^{1/\lambda}-1\Bigg{\\}}$
$\displaystyle\geq\frac{1}{\lambda-1}\Bigg{\\{}\sum^{d}_{j=1}\textrm{Tr}\Big{[}(\sqrt{E_{j}})^{1/\lambda}|\phi_{k}\rangle\langle\phi_{k}^{{}^{\prime}}|(\sqrt{E_{j}})^{1/\lambda}\Big{]}-1\Bigg{\\}},$

where $\lambda\in(1,2]$. Then, we obtain the lower bound in theorem. $\Box$

##  III Numerical results

In this section, we demonstrate numerically our results by investigating
different superposition states. We consider a single-qubit case for the
$l_{1}$ coherence measure and two-qubit case for the relative entropy and the
Tsallis relative entropy coherence measures. In both cases, we plot the
relations between exact value, upper bound, and lower bound. Crucially, a POVM
of $n$-qubit system is expressed as $4^{n}$ linear independent positive
operators $\\{E_{i}=A_{i}A_{i}^{{\dagger}}\\}_{i=0}^{4^{n}-1}$, which can be
obtained according to the Naimark theorem [31, 32]. Specially, each
measurement operator $A_{i}$ embedded in a larger unitary can be found via a
projective measurement in the standard computational basis of a larger Hilbert
space.

Consider a POVM with respect to a single-qubit system
$\\{E_{i}=A_{i}A_{i}^{{\dagger}}\\}_{i=0}^{3}$, which can be realized by
performing a two-qubit unitary
$U=\sum_{ijkl=0}^{3}u_{ij}^{kl}|ij\rangle\langle kl|$ on the qubit system and
an ancillary system. Implementing $U$ on an initial state
$\rho_{a}\otimes|0\rangle\langle 0|_{b}$, one measures the two qubits system
in the standard computational basis $\\{|m\rangle\\}_{m=0}^{3}$. Denote
$(q_{a},q_{b})$ the measurement outcome with $q_{a},q_{b}\in\\{0,1\\}$. The
corresponding probability of each outcome is given by

$\displaystyle P_{(q_{a},q_{b})}$ $\displaystyle=\langle
q_{a}q_{b}|U(\rho_{a}\otimes|0\rangle\langle
0|_{b})U^{{\dagger}}|q_{a}q_{b}\rangle$
$\displaystyle=\sum_{kl=0}^{3}u_{q_{a}q_{b}}^{kl}\langle
kl|(\rho_{a}\otimes|0\rangle\langle
0|_{b})\sum_{kl=0}^{3}(u_{q_{a}q_{b}}^{kl})^{*}|kl\rangle$
$\displaystyle=\sum_{k=0}^{3}u_{q_{a}q_{b}}^{k0}\langle
k|\rho_{a}\sum_{k=0}^{3}(u_{q_{a}q_{b}}^{k0})^{*}|k\rangle$
$\displaystyle=\textrm{Tr}[|E_{q_{a}q_{b}}\rangle\langle
E_{q_{a}q_{b}}|\rho_{a}],$ (17)

where $|E_{(q_{a}q_{b})}\rangle=\sum_{k=0}^{3}u_{q_{a}q_{b}}^{k0}|k\rangle$.
Hence, the measurement operators of each POVM is given by
$E_{i}=E_{(q_{a}q_{b})}=|E_{(q_{a}q_{b})}\rangle\langle E_{(q_{a}q_{b})}|$ for
$i=0,\cdots,3$, where $i$ has a binary representation $i=2q_{a}+q_{b}$. In
particular, let us we consider the following two-qubit unitary $U$,

$\displaystyle
U(\bm{\theta})=\textrm{CNOT}\cdot\Big{[}R_{y}(\theta_{1})\otimes
R_{y}(\theta_{2})\Big{]},$ (18)

parameterized by $\bm{\theta}=(\theta_{1},\theta_{2})$, where
$R_{y}(\theta_{j})=e^{-\imath\theta_{j}\sigma_{y}/2}$, $\imath^{2}=-1$, and
CNOT is the CNOT-gate. Setting $\bm{\theta}=(0.301723,0.011681)$ we have a
POVM given by

$\displaystyle|E_{0}\rangle=(u_{00}^{00},u_{00}^{10})^{{\dagger}},~{}~{}|E_{1}\rangle=(u_{01}^{00},u_{01}^{10})^{{\dagger}},$
$\displaystyle|E_{2}\rangle=(u_{10}^{00},u_{10}^{10})^{{\dagger}},~{}~{}|E_{3}\rangle=(u_{11}^{00},u_{11}^{10})^{{\dagger}}.$

Given two $1$-qubit states
$|\phi\rangle=e^{-\imath\theta_{j}0.432/2}|0\rangle$ and
$|\psi\rangle=e^{-\imath\theta_{j}0.618/2}|0\rangle$, we calculate the exact
value, upper bound, and lower bound of $l_{1}$ coherence of the superposition
state

$\displaystyle|\Omega\rangle=\alpha|\phi\rangle+\beta|\psi\rangle,~{}~{}|\Omega^{{}^{\prime}}\rangle=\frac{|\Omega\rangle}{\|\Omega\|},$
(19)

in which the coefficients $\alpha,\beta$ are random scalar drawn from the
uniform distribution in the interval $(0,1)$. We run our procedure $10$ times
by randomly choosing the coefficients $\alpha,\beta$. Fig. (1.b) shows the
upper and lower bounds of
$C_{l_{1}}(|\Omega^{{}^{\prime}}\rangle\langle\Omega^{{}^{\prime}}|,E)$.

Figure 1: (a) The relative entropy of POVM-based coherence. (b) The $l_{1}$ of
POVM-based coherence.

For two-qubit case, consider POVMs with $16$ measurement operators,
$\\{E_{i}=A_{i}A_{i}^{{\dagger}}\\}_{i=0}^{15}$, which can be realized by
performing a unitary
$V=\sum_{\bm{i}\bm{j}=0}^{3}v_{i_{1}i_{2}i_{3}i_{4}}^{j_{1}j_{2}j_{3}j_{4}}|\bm{i}\rangle\langle\bm{j}|$
on a four-qubit system, where the index $\bm{i}$ has binary representation
$\bm{i}=i_{1}i_{2}i_{3}i_{4}$. Similar to the single qubit case, it is easily
verified that each POVM has a form
$E_{i}=|E_{(q_{a}q_{b}q_{c}q_{d})}\rangle\langle E_{(q_{a}q_{b}q_{c}q_{d})}|$
where $(q_{a}q_{b}q_{c}q_{d})$ denote the measurement outcomes,
$i=2^{3}q_{a}+2^{2}q_{b}+2q_{c}+q_{d}$, and

$\displaystyle|E_{(q_{a}q_{b}q_{c}q_{d})}\rangle=\sum_{j_{1}j_{2}=0}^{1}v_{q_{a}q_{b}q_{c}q_{d}}^{j_{1}j_{2}00}|j_{1}j_{2}\rangle.$
(20)

Let us consider

$\displaystyle
V(\bm{\gamma})=\prod_{i,j=0}^{2}\textrm{CNOT}_{i,i+1}\cdot\Big{[}R_{y}(\gamma_{1})\otimes
R_{y}(\gamma_{2})\otimes R_{y}(\gamma_{3})\otimes R_{y}(\gamma_{4})\Big{]},$
(21)

given by the parameter vector
$\bm{\gamma}=(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})=(0.30173,0.01168,0.53991,0.09537,0.14651)$.

Given two $2$-qubit states

$\displaystyle|\phi_{1}\rangle=U(\bm{\theta}_{1})|00\rangle,~{}|\psi_{1}\rangle=U(\bm{\theta}_{2})|00\rangle,~{}|\psi_{2}\rangle=U(\bm{\theta}_{3})|00\rangle,$

where parameters $\bm{\theta}_{1}=(0.4827,0.3760)$,
$\bm{\theta}_{2}=(0.9394,0.2212)$ and $\bm{\theta}_{3}=(0.1557,0.8190)$. We
calculate the exact values, upper bounds, and lower bounds of relative entropy
coherence of the superposition state

$\displaystyle|\Omega_{1}\rangle=\alpha|\phi_{1}\rangle+\beta|\psi_{1}\rangle,~{}~{}|\Omega_{1}^{{}^{\prime}}\rangle=\frac{|\Omega_{1}\rangle}{\|\Omega_{1}\|},$
(22)

and Tsallis relative entropy coherence of the superposition state

$\displaystyle|\Omega_{2}\rangle=\alpha|\phi_{1}\rangle+\beta|\psi_{2}\rangle,~{}~{}|\Omega_{2}^{{}^{\prime}}\rangle=\frac{|\Omega_{2}\rangle}{\|\Omega_{2}\|},$
(23)

in which the coefficients $\alpha,\beta$ are random scalar drawn from the
uniform distribution in the interval $(0,1)$. We run our procedure $10$ times
by randomly choosing the coefficients $\alpha,\beta$. Fig. (1a) shows the
bounds of the relative entropy of POVM-based coherence. Only upper bound of
the Tsallis relative entropy is shown in Fig. (2a) when $\lambda=0.3$. Fig.
(2b) plots the upper and lower bounds of the Tsallis relative entropy of POVM
coherence when the parameter $\lambda=1.5$. Clearly, the numerical results are
good agree with our theoretical analysis.

Figure 2: The Tsallis relative entropy of POVM coherence.

##  IV Conclusion

The superposition principle of quantum states is fundamental in quantum
mechanics. Generally, quantum objects exhibit both the wave and particle
nature. In particular, when the POVM $E$ is taken to be the von Neumann
projective measurement, the coherence $C_{l_{1}}(\rho,E)$ stands for the wave
property of quantum objects. In this case the coherence $C_{l_{1}}(\rho,E)$
and the distinguishability of which-path information (a measure of particle-
property) in multi-path interference satisfy a trade-off relation [33], which
has been verified in a generalized multi-path delayed-choice experiment on a
large-scale quantum nanophotonic chip [34].

Based on the POVM-based coherence measures $C_{r}(\rho,E)$,
$C_{l_{1}}(\rho,E)$, $C_{rob}(\rho,E)$ and $C_{T,\lambda}(\rho,E)$, we haver
studied coherence of superposition for arbitrary dimensional pure states. We
have explored the relationship between the POVM-based coherence of a
superposed state and the POVM-based coherence of the states in the
superposition, and derived analytically the lower and upper bounds. To
illustrate our results, we have presented detailed examples related to the
POVM-based $l_{1}$ norm coherence for the one-qubit case and the POVM-based
relative entropy coherence for the two-qubit case. Our results can be used to
estimate the range of quantum coherence of superposed states, and may
highlight investigations on superposition of coherence for quantum channels
[20].

Acknowledgments This work is supported by the National Natural Science
Foundation of China (NSFC) under Grants 12075159, 12171044 and 12175147;
Beijing Natural Science Foundation (Grant No. Z190005); the Academician
Innovation Platform of Hainan Province; Shenzhen Institute for Quantum Science
and Engineering, Southern University of Science and Technology (No.
SIQSE202001).

## References

  * [1] Jozsa R and Linden N 2003 Proc. R. Soc. A 459, 2011
  * [2] Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81, 865
  * [3] Designolle S, Uola R, Luoma K and Brunner N 2021 Phys. Rev. Lett. 126, 220404
  * [4] Linden N, Popescu S, Smolin J A 2006 Phys. Rev. Lett. 97, 100502
  * [5] Gilad G 2007 Phys. Rev. A 76, 052320
  * [6] Niset J and Cerf N J 2007 Phys. Rev. A 76, 042328
  * [7] Akhtarshenas S J 2011 Phys. Rev. A 83, 042306
  * [8] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113, 140401
  * [9] Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89, 041003
  * [10] Hu M L, Hu X, Wang J, Peng Y, Zhang Y R, Fan H 2018 Phys. Rep. 762, 1-100
  * [11] Winter A and Yang D 2016 Phys. Rev. Lett. 116, 120404
  * [12] Bu K F, U Singh, Fei S M, Pati A K, Wu J D 2017 Phys. Rev. Lett. 119, 150405
  * [13] Zhao H Q and Yu C S 2018 Sci. Rep. 8, 299
  * [14] Xiong C H, Kumar A, Wu J D 2018 Phys. Rev. A 98, 032324
  * [15] Guo M L, Jin Z X, Li B, Hu B, Fei S M 2020 Quant. Inf. Process 19, 382
  * [16] Bischof F, Kampermann H, Bruß D 2019 Phys. Rev. Lett 123, 110402
  * [17] Bischof F, Kampermann H, Bruß D 2021 Phys. Rev. A 103, 032429
  * [18] Xu J W, Shao L H, Fei S M 2020 Phys. Rev. A 102, 012411
  * [19] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information(Cambridge: Cambridge University Press)pp.155-162
  * [20] Jin Z X, Yang L M, Fei S M, Li-Jost X, Wang Z X, Long G L, Qiao C F 2021 Sci. China Phys. Mech. & Astron. 64, 280311
  * [21] Yue Q L, Liu F, Yu C H, Wang X L, Gao F 2016 arXiv preprint arXiv: 1605.04067
  * [22] Liu F, Li F 2016 Quantum. Inf. Process 15, 4203–4209
  * [23] Yue Q L, Gao F, Wen Q Y, Zhang W W 2017 Sci. Rep 7 4006
  * [24] Yuwen S S, Shao L H, Xi Z J 2019 Commun. Theor. Phys 71 9
  * [25] Singh U, Bera M N, Misra A, Pati A K 2015 arXiv preprint arXiv:1506.08186
  * [26] Xu J W, Zhang L, Fei S M 2022 Quant. Inf. Process 21 39
  * [27] Piani M, Cianciaruso M, Bromley T R, Napoli C, Johnston N, Adesso G 2016 Phys. Rev. A 93 042107
  * [28] Rastegin A E 2016 Phys. Rev. A 93 032136
  * [29] Rastegin A E 2013 Eur. Phys. J. D 67 269
  * [30] Audenaert K M 1990 Lett. Math Phys 19 167
  * [31] Peres A, _2006 Quantum Theory: Concepts and Methods_(Springer Science $\&$ Business Media, New York)pp.55-60
  * [32] Decker T, Janzing D and Rötteler M 2005 J. Math. Phys. (N.Y.) 46 012104
  * [33] Roy P, Qureshi T. Path, predictability and quantum coherence in multi-slit interference. Phys. Scr. 94, 095004 (2019).
  * [34] Chen X, Deng Y, Liu S, Pramanik T, Mao J, Bao J, Zhai C, Yuan H, Guo J, Fei S M, Tang B, Yang Y, Li Z, He Q, Gong Q, Wang J 2021 Nature Commun 12 2712