# On coherence of quantum operations by using Choi-Jamiołkowski isomorphism

Xiaorong Wang1, Ting Gao1<EMAIL_ADDRESS>Fengli Yan2
<EMAIL_ADDRESS>1School of Mathematics Science, Hebei Normal
University, Shijiazhuang 050024, China
2 College of Physics, Hebei Key Laboratory of Photophysics Research and
Application, Hebei Normal University, Shijiazhuang 050024, China

###### Abstract

In quantum information, most information processing processes involve quantum
channels. One manifestation of a quantum channel is quantum operation acting
on quantum states. The coherence of quantum operations can be considered as a
quantum resource, which can be exploited to perform certain quantum tasks.
From the viewpoint of Choi-Jamiołkowski isomorphism, we study the coherence of
quantum operations in the framework of resource theory. We define the phase-
out superoperation and give the operation which transforms the Choi-
Jamiołkowski state of a quantum operation to the Choi-Jamiołkowski state of
the another quantum operation obtained by using the phase-out superoperation
to act on the quantum operation. The set of maximally incoherent
superoperations, the set of nonactivating coherent superoperations and the set
of de-phase incoherent superoperations are defined and we prove that these
sets are closed to compound operation and convex combination of quantum
superoperations. Further, we introduce the fidelity coherence measure of
quantum operations and obtain the exact form of the fidelity coherence measure
of the unitary operations on the single qubit.

Keywords: coherence of operations; Choi-Jamiołkowski isomorphism; the phase-
out superoperation; fidelity

###### pacs:

03.67.Mn, 03.65.Ud, 03.67.-a

##  I Introduction

Quantum coherence is not only a fundamental feature of quantum physics, but
also an important research field of quantum information theory. As a kind of
special resource, it plays a significant role in quantum thermodynamics
relixue1 ; relixue2 ; relixue3 ; relixue4 , quantum algorithm jisuanji1 ;
jisuanji2 ; jisuanji3 ; jisuanji4 , quantum meteorology shengwu , etc.
Quantifying and applying coherence is a very interesting work. In 2014,
Baumgratz et al established a rigorous framework of the coherence resource
theory for measuring quantum coherence Baumgratz . After that, the study on
the characterization and measure of quantum coherence has been made a great
leap forward fazhan1 ; fazhan2 ; fazhan3 ; fazhan4 ; fazhan5 ; qigaoyan ;
fuyangao ; zhanggaoyan .

The framework of the coherence resource theory consists of three ingredients:
incoherent states, incoherent operations and coherence measures. Incoherent
states are those quantum states which do not possess any coherence resource.
Incoherent operations are special quantum operations which can not generate
coherence from incoherent states. Coherence measures being used to quantify
the coherence of quantum states are functions mapping quantum states to real
numbers.

In quantum information theory, most information processing processes involve
and rely on quantum channels. One of the manifestation of quantum channel is
the quantum operation acting on quantum states. Therefore, some researchers
investigated the coherence of quantum operations. The works have been done
mainly on characterizing a certain property of quantum operations, such as
entanglement of quantum channels jiuchan1 ; jiuchan2 , channel discrimination
qufen1 ; qufen2 ; qufen3 , channel simulation moni1 ; moni2 ; moni3 ; moni5 ,
quantum memory jiyi1 ; jiyi2 ; jiyi3 , the coherence of quantum operation and
others qita1 ; qita2 ; qita3 ; qita4 . As a matter of fact, the coherence of
quantum operations can be regarded as a resource and applied to the
corresponding quantum information processing.

Coherence resource theory of quantum operations is also composed of three
elements: free operations, free superoperations and coherence measures of
quantum operations. Here the superoperation is a map between operations 123 ;
52 ; Choi1 ; Choi2 . Only free superoperations can transform free operations
to free operations. The coherence measures of operations are functions which
map quantum operations to real numbers.

Quantifying the coherence of operations is a very important and meaningful
work which can enable us to have a deeper understanding of basic physics and
also can provide new ideas for quantum information processing. There are two
main aspects in the research of the coherence of quantum operations. The first
one is to study quantum operations from the perspective that quantum
operations change the coherence of quantum states fangxiangyi1 ; fangxiangyi2
; fangxiangyi3 . The other one is to analyze quantum operations directly.
Coherence resource theory of quantum operations promotes the research strongly
zhijie1 ; zhijie2 ; zhijie3 ; 38 ; fenlei1 ; Korzekwa ; Xu ; Bera . Among
them, Orzekwa et al studied the coherence of quantum operations based on
entropy coherence and 2-norm coherence of quantum operations Korzekwa . Bera
proposed a resource theory framework to quantify the superposition that exists
in any quantum evolutions Bera . Xu defined incoherent channels and incoherent
superchannels, and established a resource theory for quantifying the coherence
of quantum channels Xu .

In this paper we investigate the coherence of quantum operations from the
viewpoint of Choi-Jamiołkowski isomorphism. We first define the phase-out
superoperation and study its properties. Then the three sets of
superoperations are defined, and we prove that these three sets are closed to
compound operation and convex combination of quantum superoperations. Then we
introduce the fidelity coherence measure of quantum operations, and obtain the
exact form of the fidelity coherence measure of the unitary operations on the
single qubit.

##  II Preliminaries

Before we quantify the coherence of quantum operations, an introduction of the
concepts and notation that will be used in the subsequent sections of our
article is necessary. Let $\mathcal{H}^{I}$ and $\mathcal{H}^{O}$ be two
$d$-dimensional Hilbert spaces with $\\{|i\rangle\\}_{i=0}^{d-1}$ and
$\\{|\alpha\rangle\\}_{\alpha=0}^{d-1}$ being the orthonormal bases of
$\mathcal{H}^{I}$ and $\mathcal{H}^{O}$, respectively. We assume that
$\\{|i\rangle\\}_{i=0}^{d-1}$ and $\\{|\alpha\rangle\\}_{\alpha=0}^{d-1}$ are
fixed and adopt the tensor basis $\\{|i\alpha\rangle\\}_{i\alpha}$ as the
fixed basis when we consider the multipartite system with the Hilbert space
$\mathcal{H}^{IO}=\mathcal{H}^{I}\otimes\mathcal{H}^{O}$. Let
$\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ be the set of all
quantum operations from $\mathcal{H}^{I}$ to $\mathcal{H}^{O}$.

A quantum operation is governed in terms of Choi-Jamiołkowski isomorphism
Choi3 ; Choi4 . An arbitrary operation
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ is fully
characterized by a matrix called the Choi-Jamiołkowski matrix of the operation
$\Phi$

$C_{\Phi}=(\mathbb{I}\otimes\Phi)|\varphi\rangle\langle\varphi|,$ (1)

in $\mathcal{H}^{I}\otimes\mathcal{H}^{O}$. Here
$|\varphi\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|ii\rangle$ is a maximally
entangled state in $d^{2}$-dimensional Hilbert space
$\mathcal{H}^{I}\otimes\mathcal{H}^{I}$, and $\mathbb{I}$ is the identity
operation. The one-to-one correspondence between $C_{\Phi}$ and $\Phi$ is
given by

$\Phi\left(\rho\right)=\operatorname{tr}_{I}\left[\left(\rho^{\mathrm{T}}\otimes\mathbb{I}\right)C_{\Phi}\right],$
(2)

where $\rho^{\mathrm{T}}$ is the transpose of state $\rho$. The quantum
operation $\Phi$ is completely positive and trace preserving if and only if
$C_{\Phi}\geq 0$ and $\mathrm{tr}_{O}(C_{\Phi})=\frac{\mathbb{I}}{d}$. When
$C_{\Phi}\geq 0$ and $\mathrm{tr}_{IO}(C_{\Phi})=1$, the Choi-Jamiołkowski
matrix $C_{\Phi}$ can be viewed as a density operator, a Choi-Jamiołkowski
quantum state in the compound space $\mathcal{H}^{I}\otimes\mathcal{H}^{O}$.
Since Choi-Jamiołkowski isomorphism guarantees an equivalence between $\Phi$
and $C_{\Phi}$, so one can treat quantum operations with the same tools which
were normally used to treat quantum states. Further, the study of the
coherence of the operation in the set
$\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ of all quantum
operations corresponding to Choi-Jamiołkowski states is transformed into the
study of coherence of Choi-Jamiołkowski state.

Assume that $\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$.
It has been shown that $\Phi$ is an incoherent operation (IO) if Xu

$C_{\Phi}=\sum_{i,\alpha=0}^{d-1}\frac{\Phi_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|,$ (3)

where $\Phi_{ii\alpha\alpha}=d\langle
i\alpha|[\mathbb{I}\otimes\Phi(|\varphi\rangle\langle\varphi|)]|i\alpha\rangle=\langle\alpha|\Phi(|i\rangle\langle
i|)|\alpha\rangle.$

Let $\mathcal{F}$ be the set of all incoherent operations in
$\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$, $\mathcal{F}_{C}$ be
the set of all Choi-Jamiołkowski states of incoherent operations. A
superoperation $\tilde{\Omega}$ is a map which transforms the operation $\Phi$
to the operation $\tilde{\Omega}(\Phi)=\Phi_{1}\circ\Phi\circ\Phi_{2}$, where
$\Phi_{1},\Phi_{2}$ are also quantum operations in
$\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$. Let $\mathcal{SO}$
be the set of all superoperators i.e.
$\mathcal{SO}=\\{\tilde{\Omega}|\tilde{\Omega}:\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})\rightarrow\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})\\}$.
A superoperation can be understood as a quantum operation that relates input
Choi-Jamiołkowski states and output Choi-Jamiołkowski states, with an
associated operator-sum representation. That is, for a superoperation
$\tilde{\Omega}$ relating input $\Phi$ and output $\Lambda$ operations, one
may write

$C_{\Lambda}=\Omega\left(C_{\Phi}\right)=\sum_{n}K_{n}C_{\Phi}K_{n}^{\dagger}=\sum_{n}p_{n}C_{\Lambda_{n}},$
(4)

where

$p_{n}=\mathrm{tr}\left(K_{n}C_{\Phi}K_{n}^{\dagger}\right),~{}C_{\Lambda_{n}}=\frac{K_{n}C_{\Phi}K_{n}^{\dagger}}{p_{n}},$
(5)

and $\\{K_{n}\\}$ are the Kraus operators of ${\Omega}$ determined by
$\tilde{\Omega}$, on the Choi-Jamiołkowski state. Obviously, the operation
$\Omega$ has a one-to-one correspondence with the superoperation
$\tilde{\Omega}$.

A superoperation $\tilde{\Omega}$ is a maximally incoherent superoperation if
$\forall~{}\Phi\in\mathcal{F}$ one has $\tilde{\Omega}(\Phi)\in\mathcal{F}$.
We call $\tilde{\Omega}$ an incoherent superoperation if Kraus operators
$\\{K_{m}\\}$ of $\Omega$ determined by $\tilde{\Omega}$ are incoherent for
each $m$ Xu . We use $\mathcal{MISO}$ and $\mathcal{ISO}$ to denote the set of
all maximally incoherent superoperations and the set of all incoherent
superoperations, respectively.

A coherence measure $M$ of quantum operations should satisfy the following
conditions Xu .

(1) Nonnegativity: $M(\Phi)\geq 0$, for any
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$; and
$M(\Phi)=0$ if and only if $\Phi\in\mathcal{F}$.

(2) Monotonicity: $M\left(\tilde{\Omega}(\Phi)\right)\leq M(\Phi)$, for any
$\tilde{\Omega}\in\mathcal{ISO}$,
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$.

(3) Strong monotonicity: $\sum_{n}p_{n}M\left(\Lambda_{n}\right)\leq M(\Phi)$
for any $\tilde{\Omega}\in\mathcal{ISO}$, with $\\{K_{n}\\}$ being an
incoherent expression of $\Omega$ corresponding to $\tilde{\Omega}$,
$p_{n}=\operatorname{tr}\left(K_{n}C_{\Phi}K_{n}^{\dagger}\right)$, and
$C_{\Lambda_{n}}=\frac{K_{n}C_{\Phi}K_{n}^{\dagger}}{p_{n}}$.

(4) Convexity:
$M\left(\sum_{n}p_{n}\Phi_{n}\right)\leq\sum_{n}p_{n}M\left(\Phi_{n}\right)$,
where $\\{\Phi_{n}\\}$ are quantum operations belonging to
$\mathcal{O}(\mathcal{H}_{I}\rightarrow\mathcal{H}_{O})$, and $\\{p_{n}\\}$ is
a probability distribution satisfying $p_{n}\geq 0$ and $\sum_{n}p_{n}=1$.

##  III Main results

###  III.1 The phase-out superoperation

It is well-know that in Hilbert space there is the phase-out operation, which
can eliminate the coherence of the quantum state. Similarly, one can define
the phase-out superoperation Xu . Since the incoherent states depend on the
choice of the basis of the Hilbert space, so does the phase-out operation.

Definition 1. A superoperation $\tilde{\Theta}\in\mathcal{SO}$ is called the
phase-out superoperation if $\tilde{\Theta}(\Phi)$ is a incoherent operation
for any quantum operation $\Phi$. The phase-out superoperation is implemented
by

$\displaystyle\tilde{\Theta}(\Phi)=\Delta^{O}\circ\Phi\circ\Delta^{I},$ (6)

where $\Delta^{O}$ and $\Delta^{I}$ are the phase-out operations on Hilbert
space $\mathcal{H}^{O}$ and $\mathcal{H}^{I}$ respectively.

According to Definition $1$, we have

Theorem 1.1. The specific form of the quantum operation $\Theta$ corresponding
to the phase-out superoperation $\tilde{\Theta}$ is

$\displaystyle
C_{\tilde{\Theta}(\Phi)}=\Theta(C_{\Phi})=\sum_{i,\alpha=0}^{d-1}\langle
i\alpha|C_{\Phi}|i\alpha\rangle|i\alpha\rangle\langle i\alpha|.$ (7)

Proof. For any quantum operation
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$, the Choi-
Jamiołkowski state corresponding to $\Phi$ is

$C_{\Phi}=\sum_{i,j,\alpha,\beta=0}^{d-1}\frac{\Phi_{ij\alpha\beta}}{d}|i\alpha\rangle\langle
j\beta|,$ (8)

where $\Phi_{ij\alpha\beta}=d\langle
i\alpha|[\mathbb{I}\otimes\Phi(|\varphi\rangle\langle\varphi|)]|j\beta\rangle=\langle\alpha|\Phi(|i\rangle\langle
j|)|\beta\rangle$. According to the definition of the phase-out
superoperation, we know that the new operation $\tilde{\Theta}(\Phi)$ is an
incoherent operation, then the Choi-Jamiołkowski state corresponding to
$\tilde{\Theta}(\Phi)$ must be the diagonal in the fixed basis
$\\{|i\alpha\rangle\\}_{i,\alpha=0}^{d-1}$, i.e.,

$\displaystyle C_{\tilde{\Theta}(\Phi)}$
$\displaystyle=[\mathbb{I}\otimes(\Delta^{O}\circ\Phi\circ\Delta^{I})]|\varphi\rangle\langle\varphi|$
(9)
$\displaystyle=\sum_{i,\alpha=0}^{d-1}\frac{\tilde{\Theta}(\Phi)_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|,$

where

$\displaystyle\tilde{\Theta}(\Phi)_{ii\alpha\alpha}$ $\displaystyle=$
$\displaystyle d\langle
i\alpha|\mathbb{I}\otimes\tilde{\Theta}(\Phi)(|\varphi\rangle\langle\varphi|)|i\alpha\rangle$
$\displaystyle=$
$\displaystyle\langle\alpha|\tilde{\Theta}(\Phi)(|i\rangle\langle
i|)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha|\Delta^{O}\circ\Phi\circ\Delta^{I}(|i\rangle\langle
i|)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha|\Delta^{O}\circ\Phi(|i\rangle\langle
i|)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha|\Delta(\sum_{\beta,\gamma}\langle\beta|\Phi(|i\rangle\langle
i|)|\gamma\rangle|\beta\rangle\langle\gamma|)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha|(\sum_{\beta}\langle\beta|\Phi(|i\rangle\langle
i|)|\beta\rangle|\beta\rangle\langle\beta|)|\alpha\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha|\Phi(|i\rangle\langle i|)|\alpha\rangle$
$\displaystyle=$ $\displaystyle\Phi_{ii\alpha\alpha}.$

Thus,

$\displaystyle C_{\tilde{\Theta}(\Phi)}$
$\displaystyle=[\mathbb{I}\otimes(\Delta^{O}\circ\Phi\circ\Delta^{I})]|\varphi\rangle\langle\varphi|$
(11)
$\displaystyle=\sum_{i,\alpha=0}^{d-1}\frac{\Phi_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|.$

Obviously the Kraus operators of $\Theta$ are
$\\{M_{i\alpha}=|i\alpha\rangle\langle i\alpha|\\}^{d-1}_{i,\alpha=0}$,
because

$\displaystyle\Theta\left(C_{\Phi}\right)$
$\displaystyle=\sum_{i\alpha}M_{i\alpha}C_{\Phi}M_{i\alpha}^{\dagger}$ (12)
$\displaystyle=\sum_{i\alpha}|i\alpha\rangle\langle
i\alpha|\left(\sum_{i^{\prime},j^{\prime},\alpha^{\prime},\beta^{\prime}=0}^{d-1}\frac{\Phi_{i^{\prime}j^{\prime}\alpha^{\prime}\beta^{\prime}}}{d}|i^{\prime}\alpha^{\prime}\rangle\langle
j^{\prime}\beta^{\prime}|\right)|i\alpha\rangle\langle i\alpha|$
$\displaystyle=\sum_{i,\alpha=0}^{d-1}\frac{\Phi_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|$ $\displaystyle=\sum_{i,\alpha=0}^{d-1}\langle
i\alpha|C_{\Phi}|i\alpha\rangle|i\alpha\rangle\langle i\alpha|$
$\displaystyle=C_{\tilde{\Theta}(\Phi)}.$

It is easy to check that $\Theta$ is a resource destroying map yingshe . The
essence of the quantum operation $\Theta$ acting on any Choi-Jamiołkowski
state is that all the non-diagonal elements in the selected base are
completely eliminated and a new Choi-Jamiołkowski state with only diagonal
elements is created.

So we get the specific form of the quantum operation $\Theta$ corresponding to
the phase-out superoperation $\tilde{\Theta}$ is

$\Theta(C_{\Phi})=\sum_{i,\alpha=0}^{d-1}\langle
i\alpha|C_{\Phi}|i\alpha\rangle|i\alpha\rangle\langle
i\alpha|=C_{\tilde{\Theta}(\Phi)}.$ (13)

This completes the proof of Theorem 1.1.

Theorem 1.2. If a quantum operation
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ is a
completely positive trace preserving operation (CPTP), then
$\tilde{\Theta}(\Phi)$ is a CPTP.

Proof. Obviously, if a quantum operation
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ is a CPTP,
then $\Phi$ satisfies

$C_{\Phi}\geq
0,~{}~{}~{}~{}~{}\mathrm{tr}_{O}(C_{\Phi})=\frac{\mathbb{I}}{d},$

where $d$ is the dimension of the Hilbert space $\mathcal{H}^{I}$.

It is easy to obtain

$\displaystyle\mathrm{tr}_{O}(C_{\Phi})$
$\displaystyle=\mathrm{tr}_{O}\left(\frac{1}{d}\sum_{i,j=0}^{d-1}|i\rangle\langle
j|\otimes\sum_{\alpha,\beta=0}^{d-1}\Phi_{ij\alpha\beta}|\alpha\rangle\langle\beta|\right)$
(14) $\displaystyle=\left(\frac{1}{d}\sum_{i,j=0}^{d-1}|i\rangle\langle
j|\right)\mathrm{tr}\left(\sum_{\alpha,\beta=0}^{d-1}\Phi_{ij\alpha\beta}|\alpha\rangle\langle\beta|\right)$
$\displaystyle=\sum_{i,j=0}^{d-1}\frac{\sum_{\alpha=0}^{d-1}\Phi_{ij\alpha\alpha}}{d}|i\rangle\langle
j|.$

As $\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$ is a CPTP,
so one must have

$\sum_{i,j=0}^{d-1}\frac{\sum_{\alpha=0}^{d-1}\Phi_{ij\alpha\alpha}}{d}|i\rangle\langle
j|=\frac{\mathbb{I}}{d}.$ (15)

The above equation means that

$\sum_{\alpha=0}^{{}^{d-1}}\Phi_{ij\alpha\alpha}=\delta_{ij}.$ (16)

The Choi-Jamiołkowski state corresponding to $\tilde{\Theta}(\Phi)$ is

$C_{\tilde{\Theta}(\Phi)}=\sum_{i,\alpha=0}^{d-1}\frac{\tilde{\Theta}(\Phi)_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|=\sum_{i,\alpha=0}^{d-1}\frac{\Phi_{ii\alpha\alpha}}{d}|i\alpha\rangle\langle
i\alpha|.$ (17)

So it is easy to obtain

$\displaystyle\mathrm{tr}_{O}(C_{\tilde{\Theta}(\Phi)})$
$\displaystyle=\mathrm{tr}_{O}\left(\frac{1}{d}\sum_{i=0}^{d-1}|i\rangle\langle
i|\otimes\sum_{\alpha=0}^{d-1}\tilde{\Theta}(\Phi)_{ii\alpha\alpha}|\alpha\rangle\langle\alpha|\right)$
(18) $\displaystyle=\left(\frac{1}{d}\sum_{i=0}^{d-1}|i\rangle\langle
i|\right)\mathrm{tr}\left(\sum_{\alpha=0}^{d-1}\tilde{\Theta}(\Phi)_{ii\alpha\alpha}|\alpha\rangle\langle\alpha|\right)$
$\displaystyle=\sum_{i=0}^{d-1}\frac{\sum_{\alpha=0}^{d-1}\tilde{\Theta}(\Phi)_{ii\alpha\alpha}}{d}|i\rangle\langle
i|$
$\displaystyle=\sum_{i=0}^{d-1}\frac{\sum_{\alpha=0}^{d-1}\Phi_{ii\alpha\alpha}}{d}|i\rangle\langle
i|$ $\displaystyle=\frac{\mathbb{I}}{d}.$

Thus we have

$C_{\tilde{\Theta}(\Phi)}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{tr}_{O}(C_{\tilde{\Theta}(\Phi)})=\frac{\mathbb{I}}{d},$
(19)

which means that $\tilde{\Theta}(\Phi)$ is a CPTP. Theorem 1.2 has been
proved.

###  III.2 The three sets of quantum superoperations

We now give the three sets of quantum superoperations based on the phase-out
superoperation $\tilde{\Theta}$.

Let $\tilde{\Omega}$ be a quantum superoperation. Suppose that
$\tilde{\Omega}$ satisfies

$\displaystyle\tilde{\Omega}\circ\tilde{\Theta}=\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta},$
(20)

where $\circ$ is the composition of superoperations. The compound
superoperation $\tilde{\Omega}\circ\tilde{\Theta}$ and
$\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}$ acting on the any
quantum operation $\Phi$, generate the following quantum operations

$\displaystyle\tilde{\Omega}\circ\tilde{\Theta}(\Phi)=\tilde{\Omega}(\tilde{\Theta}(\Phi)),$
(21)
$\displaystyle\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}(\Phi)=\tilde{\Theta}(\tilde{\Omega}(\tilde{\Theta}(\Phi))).$

Since Eq.(20) holds, so any output operation of the superoperation
$\tilde{\Omega}\circ\tilde{\Theta}$ can be regarded as an output operation of
the phase-out superoperation $\tilde{\Theta}$. Thus, all output operations of
superoperation $\tilde{\Omega}\circ\tilde{\Theta}$ are incoherent operations.
In other words, the set of incoherent operations is closed under
$\tilde{\Omega}$ if $\tilde{\Omega}$ satisfies Eq.(20). Therefore we call
condition Eq.(20) the nongenerating coherent conditions. The superoperations
satisfying this condition are called nongenerating coherent superoperations.
Usually, nongenerating coherent superoperations is also called maximally
incoherent superoperations fenlei1 ; jiyi2 . The set of maximally incoherent
superoperations is denoted as $\mathcal{MISO}$.

Next, we consider the following dual form of the nongenerating coherent
condition:

$\displaystyle\tilde{\Theta}\circ\tilde{\Omega}=\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}.$
(22)

Think of the output operation $\tilde{\Theta}(\Phi)$ of an input operation
$\Phi$, the free part of $\Phi$. Apparently, $\tilde{\Omega}$ cannot make use
of the resource stored in any input operation $\Phi$ to affect the free part
$\tilde{\Theta}(\Phi)$ if $\tilde{\Omega}$ satisfies the condition Eq.(22). We
call this condition the nonactivating coherent condition. The superoperations
satisfying this condition are called resource nonactivating coherent
superoperations. The set of nonactivating coherent superoperations is denoted
as $\mathcal{MISO}^{\ast}$.

Another interpretation is that the superoperations in the set
$\mathcal{MISO}^{\ast}$ never break up a family: members of the same family
must be mapped to the same target family (not necessarily the original one).

In general, the nongenerating coherent condition and the nonactivating
coherent condition are independent of each other and do not influence each
other.

After that, we consider the case where the quantum superoperation satisfy not
only the nongenerating coherent condition, but also the nonactivating coherent
condition:

$\displaystyle\tilde{\Omega}\circ\tilde{\Theta}=\tilde{\Theta}\circ\tilde{\Omega}=\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}.$
(23)

We call this condition the exchangeable resource condition. The
superoperations satisfying this condition are called exchangeable resource
superoperations. We also call exchangeable resource superoperations de-phase
incoherent superoperations fenlei1 . The set of de-phase incoherent
superoperations is denoted as $\mathcal{DISO}$.

It is easy to obtain the following conclusions for the sets $\mathcal{MISO}$,
$\mathcal{MISO}^{\ast}$ and $\mathcal{DISO}$.

(1) The relationship between $\mathcal{MISO}$, $\mathcal{MISO}^{\ast}$ and
$\mathcal{DISO}$ is $\mathcal{DISO}=\mathcal{MISO}\cap\mathcal{MISO^{\ast}}.$

(2) For any quantum superoperation $\tilde{\Omega}$, the following
relationship exists

$\displaystyle\tilde{\Theta}\circ\tilde{\Omega}\in\mathcal{MISO},~{}~{}~{}~{}~{}~{}\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}\in\mathcal{MISO}.$
(24)

(3) If a quantum superoperation $\tilde{\Omega}\in\mathcal{MISO}^{\ast}$, then
there exists the following relationship

$\displaystyle\tilde{\Theta}\circ\tilde{\Omega}\in\mathcal{MISO^{\ast}},~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\Omega}\circ\tilde{\Theta}\in\mathcal{MISO^{\ast}}.$
(25)

Furthermore, we also have

Theorem 2.1. $\mathcal{MISO}$, $\mathcal{MISO}^{\ast}$ and $\mathcal{DISO}$
are closed to compound operation and convex combination of quantum
superoperations.

Proof. First, we prove that $\mathcal{MISO}$ is closed to the compound
operation and convex combination of quantum superoperations.

For any quantum superoperations
$\tilde{\Omega}_{1},\tilde{\Omega}_{2}\in\mathcal{MISO}$, there are

$\displaystyle\tilde{\Omega}_{1}\circ\tilde{\Theta}=\tilde{\Theta}\circ\tilde{\Omega}_{1}\circ\tilde{\Theta},$
(26)
$\displaystyle\tilde{\Omega}_{2}\circ\tilde{\Theta}=\tilde{\Theta}\circ\tilde{\Omega}_{2}\circ\tilde{\Theta}.$

Thus we have

$\displaystyle(\tilde{\Omega}_{1}\circ\tilde{\Omega}_{2})\circ\tilde{\Theta}$
$\displaystyle=\tilde{\Omega}_{1}\circ(\tilde{\Theta}\circ\tilde{\Omega}_{2}\circ\tilde{\Theta})$
(27)
$\displaystyle=\tilde{\Theta}\circ\tilde{\Omega}_{1}\circ\tilde{\Theta}\circ\tilde{\Omega}_{2}\circ\tilde{\Theta}$
$\displaystyle=\tilde{\Theta}\circ(\tilde{\Omega}_{1}\circ\tilde{\Omega}_{2})\circ\tilde{\Theta}.$

The above equation means that
$\tilde{\Omega}_{1}\circ\tilde{\Omega}_{2}\in\mathcal{MISO}$. Therefore
$\mathcal{MISO}$ is closed to the compound operation.

If there is a quantum superoperation $\tilde{\Omega}$ which satisfies
$\tilde{\Omega}=p\tilde{\Omega}_{1}+(1-p)\tilde{\Omega}_{2}$, where $0\leq
p\leq 1$, then we have

$\displaystyle\tilde{\Omega}\circ\tilde{\Theta}$
$\displaystyle=(p\tilde{\Omega}_{1}+(1-p)\tilde{\Omega}_{2})\circ\tilde{\Theta}$
(28)
$\displaystyle=p\tilde{\Omega}_{1}\circ\tilde{\Theta}+(1-p)\tilde{\Omega}_{2}\circ\tilde{\Theta}$
$\displaystyle=p\tilde{\Theta}\circ\tilde{\Omega}_{1}\circ\tilde{\Theta}+(1-p)\tilde{\Theta}\circ\tilde{\Omega}_{2}\circ\tilde{\Theta}$
$\displaystyle=\tilde{\Theta}\circ(p\tilde{\Omega}_{1}+(1-p)\tilde{\Omega}_{2})\circ\tilde{\Theta}$
$\displaystyle=\tilde{\Theta}\circ\tilde{\Omega}\circ\tilde{\Theta}.$

Hence $\tilde{\Omega}\in\mathcal{MISO}$. That indicates that $\mathcal{MISO}$
is closed to the convex combination of quantum superoperations.

Similarly, one can prove that $\mathcal{MISO}^{\ast}$ and $\mathcal{DISO}$ are
closed to the compound operation and convex combination of quantum
superoperations. Therefore Theorem 2.1 holds.

###  III.3 The fidelity coherence measure of operations

How to measure the coherence of operations is very important in the study of
quantum coherence resources. The coherence of operations have been measured by
the relative entropy coherence measure Bera ; Xu , the $l_{1}$ norm coherence
measure Bera ; Xu  and the robustness coherence measure moni5 . Next we will
give another measure, the fidelity coherence measure of operations and provide
some examples.

Definition 2. For any two quantum operations
$\Phi,\Lambda\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$, the
fidelity $F(\Phi,\Lambda)$ of the operations $\Phi$ and $\Lambda$ is defined
as

$F(\Phi,\Lambda)=F(C_{\Phi},C_{\Lambda}),$ (29)

where $C_{\Phi}$ and $C_{\Lambda}$ are Choi-Jamiołkowski states corresponding
to $\Phi$ and $\Lambda$ respectively, $F(C_{\Phi},C_{\Lambda})$ is the
fidelity of $C_{\Phi}$ and $C_{\Lambda}$.

By Ref. Xu , we have

Lemma 1. If $M$ is a coherence measure for quantum states in the Baumgratz-
Cramer-Plenio (BCP) framework Baumgratz , then

$M(\Phi)\equiv
M(C_{\Phi}),~{}~{}~{}~{}~{}\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$
(30)

is a coherence measure for quantum operations.

According to the fidelity of quantum operations and Lemma $1$, we can define
the fidelity coherence measure of quantum operations as follows.

Definition 3. For any quantum operation
$\Phi\in\mathcal{O}(\mathcal{H}^{I}\rightarrow\mathcal{H}^{O})$, if the Choi-
Jamiołkowski state $C_{\Phi}$ is a pure state, the fidelity coherence measure
of the operation $\Phi$ is defined as

$M_{\mathrm{f}}(\Phi)=\min_{C_{\Lambda}\in\mathfrak{{F_{C}}}}\sqrt{1-F(C_{\Phi},C_{\Lambda})},$
(31)

where
$F(\rho,\sigma)=(\operatorname{tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}))^{2}$
is the Uhlmann fidelity baozhendu , $\mathfrak{{F_{C}}}$ is the set of pure
incoherent states in the Hilbert space
$\mathcal{H}^{I}\otimes\mathcal{H}^{O}$. Then we can define convex-roof
extended fidelity coherence measure of the operations for the general case as

$M_{\mathrm{f}}(\Phi)=\min_{\\{p_{n},\Phi_{n}\\}}\sum_{n}p_{n}M_{\mathrm{f}}(\Phi_{n}),$
(32)

where the minimum is taken over all the ensembles $\\{p_{n},\Phi_{n}\\}$
realizing $\Phi$, i.e., $\Phi=\sum_{n}p_{n}\Phi_{n}$, for any $n$,
$C_{\Phi_{n}}$ is the pure Choi-Jamiołkowski state corresponding to the
operation $\Phi_{n}$.

For the general unitary operations on a single qubit we have the following
result.

Theorem 3.1. The fidelity coherence measure of the general form of unitary
operations

$\displaystyle
U=\mathrm{e}^{\mathrm{i}\alpha}\begin{bmatrix}\mathrm{e}^{-\mathrm{i}\frac{\beta}{2}}&0\\\
0&\mathrm{e}^{\mathrm{i}\frac{\beta}{2}}\end{bmatrix}\begin{bmatrix}\mathrm{cos}\frac{\gamma}{2}&-\mathrm{sin}\frac{\gamma}{2}\\\
\mathrm{sin}\frac{\gamma}{2}&\mathrm{cos}\frac{\gamma}{2}\end{bmatrix}\begin{bmatrix}\mathrm{e}^{-\mathrm{i}\frac{\delta}{2}}&0\\\
0&\mathrm{e}^{\mathrm{i}\frac{\delta}{2}}\end{bmatrix}$ (33)

on a single qubit is

$M_{\mathrm{f}}(U)=\min\left\\{\sqrt{1-\frac{|\mathrm{cos}\frac{\gamma}{2}|^{2}}{2}},~{}\sqrt{1-\frac{|\mathrm{sin}\frac{\gamma}{2}|^{2}}{2}}\right\\}.$
(34)

Here $\alpha,\beta,\gamma$ and $\delta$ are any real numbers.

Proof. It is easy to derive that the general form of unitary operation $U$ on
a single qubit can be written as

$\displaystyle U=\begin{bmatrix}a&-b\\\
\mathrm{e}^{2\mathrm{i}\alpha}b^{\ast}&\mathrm{e}^{2\mathrm{i}\alpha}a^{\ast}\end{bmatrix},$
(35)

where

$a=\mathrm{e}^{\mathrm{i}(\alpha-\frac{\beta}{2}-\frac{\delta}{2})}\mathrm{cos}\frac{\gamma}{2},~{}~{}~{}~{}~{}~{}b=\mathrm{e}^{\mathrm{i}(\alpha-\frac{\beta}{2}+\frac{\delta}{2})}\mathrm{sin}\frac{\gamma}{2}.$
(36)

The Choi-Jamiołkowski state corresponding to the unitary matrix $U$ is

$\displaystyle C_{U}$ $\displaystyle=(\mathbb{I}\otimes
U)\left(\frac{1}{2}\sum_{i,j=0}^{1}|ii\rangle\langle
jj|\right)(\mathbb{I}\otimes U)^{\dagger}$ (37)
$\displaystyle=\frac{1}{2}\begin{bmatrix}aa^{\ast}&\mathrm{e}^{-2\mathrm{i}\alpha}ab&-ab^{\ast}&\mathrm{e}^{-2\mathrm{i}\alpha}aa\\\
\mathrm{e}^{2\mathrm{i}\alpha}a^{\ast}b^{\ast}&bb^{\ast}&-\mathrm{e}^{2\mathrm{i}\alpha}b^{\ast}b^{\ast}&ab^{\ast}\\\
-a^{\ast}b&-\mathrm{e}^{-2\mathrm{i}\alpha}bb&bb^{\ast}&-\mathrm{e}^{-2\mathrm{i}\alpha}ab\\\
\mathrm{e}^{2\mathrm{i}\alpha}a^{\ast}a^{\ast}&a^{\ast}b&-\mathrm{e}^{2\mathrm{i}\alpha}a^{\ast}b^{\ast}&aa^{\ast}\\\
\end{bmatrix}$
$\displaystyle=\frac{1}{\sqrt{2}}(a|00\rangle+\mathrm{e}^{2\mathrm{i}\alpha}b^{\ast}|01\rangle-b|10\rangle+\mathrm{e}^{2\mathrm{i}\alpha}a^{\ast}|11\rangle)$
$\displaystyle~{}~{}~{}~{}\times\frac{1}{\sqrt{2}}(a^{\ast}\langle
00|+\mathrm{e}^{-2\mathrm{i}\alpha}b\langle 01|-b^{\ast}\langle
10|+\mathrm{e}^{-2\mathrm{i}\alpha}a\langle 11|).$

Because the Choi-Jamiołkowski state $C_{U}$ is a pure state, the fidelity
coherence measure of $U$ is

$M_{\mathrm{f}}(U)=\min\left\\{\sqrt{1-\frac{|a|^{2}}{2}},~{}\sqrt{1-\frac{|b|^{2}}{2}}\right\\}=\min\left\\{\sqrt{1-\frac{|\mathrm{cos}\frac{\gamma}{2}|^{2}}{2}},~{}\sqrt{1-\frac{|\mathrm{sin}\frac{\gamma}{2}|^{2}}{2}}\right\\}.$
(38)

The proof of Theorem 3.1 is completed.

Obviously, we have the following result.

Corollary 3.2. For any unitary operation $U$ on a single qubit, the range of
values of $M_{\mathrm{f}}(U)$ is $[\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2}]$.

Corollary 3.3. The fidelity coherence measure $M_{\mathrm{f}}(\Phi_{\max})$ of
the the maximally coherent operation
$\Phi_{\max}=\sum_{i,\alpha=0}^{1}\frac{1}{\sqrt{2}}\mathrm{e}^{\mathrm{i}\theta_{i\alpha}}|\alpha\rangle\langle
i|$ on a single qubit Xu , is largest in the unitary operations on a single
qubit. Here $\theta_{i\alpha}$ is a real number.

Proof. The Choi-Jamiołkowski state corresponding to $\Phi_{\max}$ is

$\displaystyle
C_{\Phi_{\max}}=\sum_{i,j,\alpha,\beta=0}^{1}\frac{1}{4}\mathrm{e}^{\mathrm{i}(\theta_{i\alpha}-\theta_{j\beta}})|i\alpha\rangle\langle
j\beta|.$ (39)

Because $C_{\Phi_{\max}}$ is a pure state, it is easy to derive that the
fidelity coherence measure of $\Phi_{\max}$ is

$\displaystyle
M_{\mathrm{f}}(\Phi_{\max})=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}.$ (40)

By using Corollary 3.2, we know that the fidelity coherence measure of the the
maximally coherent operation is largest in the unitary operations on a single
qubit. This ends the proof.

##  IV Conclusion

Based on Choi-Jamiołkowski isomorphism we investigated the coherence of
quantum operations in the resource theory. We first defined the phase-out
superoperation and gave the form of operation which transforms the Choi-
Jamiołkowski state of the quantum operation $\Phi$ to the Choi-Jamiołkowski
state of the quantum operation $\tilde{\Theta}(\Phi)$ obtained by using the
phase-out superoperation $\tilde{\Theta}$ to act on $\Phi$. By using phase-out
superoperation we defined the set of maximally incoherent superoperations, the
set of nonactivating coherent superoperations and the set of de-phase
incoherent superoperations and proved that these sets are closed to compound
operation and convex combination of quantum superoperations. Further, the
fidelity coherence measure of quantum operations was introduced. The fidelity
coherence measure of the unitary operations on the single qubit has been
calculated. We hope the research will lead to a better understanding of the
coherence of quantum operations.

## References

  * (1) M. Lostaglio, D. Jennings, and T. Rudolph, Thermodynamic resource theories, non-commutativity and maximum entropy principles, New J. Phys. 19, 043008 (2017).
  * (2) F. Brand$\tilde{a}$o, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. USA 112, 3275 (2015).
  * (3) P. $\acute{\mathrm{C}}$wikli$\acute{\mathrm{n}}$ski, M. Studzi$\acute{\mathrm{n}}$ski, M. Horodecki, and J. Oppenheim, Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics, Phys. Rev. Lett. 115, 210403 (2015).
  * (4) A. Misra, U. Singh, S. Bhattacharya, and A. K. Pati, Energy cost of creating quantum coherence, Phys. Rev. A 93, 052335 (2016).
  * (5) H. L. Shi, S. Y. Liu, X. H. Wang, W. L. Yang, Z. Y. Yang, and H. Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A 95, 032307 (2017).
  * (6) M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation, Phys. Rev. A 93, 012111 (2016).
  * (7) A. E. Rastegin, On the role of dealing with quantum coherence in amplitude amplification, arXiv:1703.10118.
  * (8) N. Anand and A. K. Pati, Coherence and entanglement monogamy in the discrete analogue of analog grover search, arXiv:1611.04542.
  * (9) V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222-229 (2011).
  * (10) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * (11) J. $\mathrm{{\AA}}$berg, Quantifying superposition, arXiv:quant-ph/0612146.
  * (12) A. Winter and D. Yang, Operational resource theory of coherence, Phys. Rev. Lett. 116, 120404 (2016).
  * (13) Y. Peng, Y. Jiang, and H. Fan, Maximally coherent states and coherence-preserving operations, Phys. Rev. A 93, 032326 (2016).
  * (14) M. L. Hu, X. Hu, J. Wang, Y. Peng, Y. R. Zhang, and H. Fan, Quantum coherence and geometric quantum discord, Phys. Rep. 1, 762-764 (2018).
  * (15) A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * (16) X. F. Qi, T. Gao, and F. L. Yan, Measuring coherence with entanglement concurrence, J. Phys. A: Math. Theor. 50, 285301 (2017).
  * (17) L. X. Fu, F. L. Yan, and T. Gao, The block-coherence measures and the coherence measures based on positive-operator-valued measures, arXiv: 2108.04405.
  * (18) L. M. Zhang, T. Gao, and F. L. Yan, Transformations of multilevel coherent states under coherence-preserving operations, Sci. China-Phys. Mech. Astron. 64, 260313 (2021).
  * (19) C. H. Bennett, A. W. Harrow, D. W. Leung, and J. A. Smolin, On the capacities of bipartite Hamiltonians and unitary gates, IEEE Trans. Inf. Theory 49, 1895 (2003).
  * (20) E. Kaur and M. M. Wilde, Amortized entanglement of a quantum channel and approximately teleportation-simulable channels, J. Phys. A: Math. Theor. 51, 035303 (2017).
  * (21) S. Pirandola and C. Lupo, Ultimate precision of adaptive noise estimation, Phys. Rev. Lett. 8, 100502 (2017).
  * (22) M. Berta, C. Hirche, E. Kaur, and M. M. Wilde, Amortized channel divergence for asymptotic quantum channel discrimination, arXiv:1808.01498.
  * (23) S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, Fundamental limits to quantum channel discrimination, npj Quantum Inf. 5, 50 (2019).
  * (24) M. Berta, F. G. S. L. Brand$\tilde{a}$o, M. Christandl, and S. Wehner, Entanglement cost of quantum channels, IEEE Trans. Inf. Theory 59, 6779 (2013).
  * (25) S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Fundamental limits of repeaterless quantum communications, Nat. Commun. 8, 15043 (2017).
  * (26) M. M. Wilde, Entanglement cost and quantum channel simulation, Phys. Rev. A 98, 042338 (2018).
  * (27) M. G. D$\acute{\i}$az, K. Fang, X. Wang, M. Rosati, M. Skotiniotis, J. Calsamiglia, and A. Winter, Using and reusing coherence to realize quantum processes, Quantum 2, 100 (2018).
  * (28) D. Rosset, F. Buscemi, and Y. C. Liang, Resource theory of quantum memories and their faithful verification with minimal assumptions, Phys. Rev. X 8, 021033 (2018).
  * (29) T. Simnacher, N. Wyderka, C. Spee, X. D. Yu, and O. Gühne, Certifying quantum memories with coherence, Phys. Rev. A 99, 062319 (2019).
  * (30) X. Yuan, Y. Liu, Q. Zhao, B. Regula, J. Thompson, and M. Gu, Universal and operational benchmarking of quantum memories, arXiv:1907.02521.
  * (31) J. H. Hsieh, S. H. Chen, and C. M. Li, Quantifying quantum-mechanical processes, Sci. Rep. 7, 13588 (2017).
  * (32) S. H. Chen, H. Lu, Q. C. Sun, Q. Zhang, Y. A. Chen, and C. M. Li, Discriminating quantum correlations with networking quantum teleportation, Phys. Rev. Research 2, 013043 (2020).
  * (33) C. C. Kuo, S. H. Chen, W. T. Lee, H. M. Chen, H. Lu, and C. M. Li, Quantum process capability, Sci. Rep. 9, 20316 (2019).
  * (34) E. Wolfe, D. Schmid, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Quantifying bell: The resource theory of nonclassicality of common-cause boxes, arXiv:1903.06311.
  * (35) Z. W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure, arXiv:1904.04201.
  * (36) G. Gour and A. Winter, How to quantify a dynamical quantum resource, Phys. Rev. Lett. 123, 150401 (2019).
  * (37) G. Gour and C. M. Scandolo, The entanglement of a bipartite channel, arXiv:1907.02552.
  * (38) G. Gour, Comparison of quantum channels by superchannels, IEEE. Trans. Inf. Theory 65, 5880-5904 (2019).
  * (39) G. Chiribella, G. M. DAriano, and P. Perinotti, Transforming quantum operations: Quantum supermaps, Europhys. Lett. 83, 30004 (2008).
  * (40) K. F. Bu, A. Kumar, L. Zhang, and J. Wu, Cohering power of quantum operations, Phys. Lett. A 381, 1670-1676 (2017).
  * (41) X. Y. Hu, Channels that do not generate coherence, Phys. Rev. A 94, 012326 (2016).
  * (42) K. B. Dana, M. G. D$\acute{\i}$az, M. Mejatty, and A. Winter, Resource theory of coherence: Beyond states, Phys. Rev. A 95, 062327 (2017).
  * (43) C. Datta, S. Sazim, A. K. Pati, and P. Agrawal, Coherence of quantum channels, Ann. Phys. (NY) 397, 243 (2018).
  * (44) Y. Liu and X. Yuan, Operational resource theory of quantum channels, arXiv:1904.02680.
  * (45) T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Quantifying operations with an application to coherence, Phys. Rev. Lett. 122, 190405 (2019).
  * (46) K. Orzekwa, S. Czach$\acute{\mathrm{o}}$rski, Z. Puchala, and K. $\dot{Z}$yczkowski, Coherifying quantum channels, New J. Phys. 20, 043028 (2018).
  * (47) M. N. Bera, Quantifying superpositions of quantum evolutions, Phys. Rev. A 100, 042307 (2019).
  * (48) J. W. Xu, Coherence of quantum channels, Phys. Rev. A 100, 052311 (2019).
  * (49) G. Saxena, E. Chitambar, and G. Gour, Dynamical resource theory of quantum coherence, Phys. Rev. Research 2, 023298 (2020).
  * (50) M. D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 (1975).
  * (51) A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
  * (52) Z. W. Liu, X. Hu, and S. Lloyd, Resource destroying maps, Phys. Rev. Lett. 118, 060502 (2017).
  * (53) A. Uhlmann, The ”transition probability” in the state space of a $\ast$-algebra, Rep. Math. Phys. 9, 273-279 (1976).