# Quantifying dynamical coherence with coherence measures

Sunho Kim<EMAIL_ADDRESS>School of Mathematical Sciences, Harbin
Engineering University, Harbin 150001, China  Chunhe Xiong
<EMAIL_ADDRESS>School of Computer and Computing Science, Zhejiang
University City College, Hangzhou 310015, China  Asutosh Kumar
<EMAIL_ADDRESS>P.G. Department of Physics, Gaya College, Magadh
University, Rampur, Gaya 823001, India  Harish-Chandra Research Institute,
HBNI, Chhatnag Road, Jhunsi, Allahabad 211019, India  Vaidic and Modern
Physics Research Centre, Bhagal Bhim, Bhinmal, Jalore 343029, India  Guijun
Zhang  School of Mathematical Sciences, Zhejiang University, Hangzhou 310027,
China  Junde Wu<EMAIL_ADDRESS>School of Mathematical Sciences, Zhejiang
University, Hangzhou 310027, China

###### Abstract

Quantum coherence, like entanglement, is a fundamental resource in quantum
information. In recent years, remarkable progress has been made in formulating
resource theory of coherence from a broader perspective. The notions of block-
coherence and POVM-based coherence have been established. Certain challenges,
however, remain to be addressed. It is difficult to define incoherent
operations directly, without requiring incoherent states, which proves a major
obstacle in establishing the resource theory of dynamical coherence. In this
paper, we overcome this limitation by introducing an alternate definition of
incoherent operations, induced via coherence measures, and quantify dynamical
coherence based on this definition. Finally, we apply our proposed definition
to quantify POVM-based dynamical coherence.

##  I Introduction

We are quite familiar with the macro-world. In the micro-world, however, a
given physical system may exhibit remarkable properties that are not seen in
the macro-world. These properties of quantum systems offer certain advantages
over classical systems in information processing tasks. Consequently, these
properties may be viewed as “resources” in those tasks. We consider a certain
physical situation and identify the fundamental or practical constraints
associated with it: what are the tools available, what things are free of cost
(these can be spent amply), and what involves a cost (these need to be used
wisely). In quantum physics, things we consider are states and operations. In
a given situation, a particular set of states and operations are free (in the
sense that states can be readily prepared and operations are closed or convex)
while another set of them is viewed as resources because they cannot be
obtained with free states and free operations. Put differently, these objects
cannot be generated in the given setting without incurring a cost. A quantum
resource theory identifies physical processes and quantum states as being
either free, restricted or resources. This is a common feature of any resource
theory.

Several properties of quantum sysetms have been recognized as resources in
quantum information. This recognition is profound as they can be unified and
developed under the same roof of quantum resource theories (QRTs) Chitambar ;
Brandao . Such a unification leads to better understanding of physical
phenomena and new discoveries. This helps in identifying structures and
applications that are common to resource theories in general. Furthermore,
QRTs study what information processing tasks are possible using the restricted
operations. We should note that the structure of resource theories goes far
beyond quantum physics. A basic goal of any resource theory is to quantify the
resource and, with several resources at hand, find necessary and sufficient
conditions that determine whether or not one resource can be converted to
another by the set of free operations. Quantum information theory is regarded
a theory of interconversions among different resources that are classified
from diverse perspectives as classical or quantum, noisy or noiseless, and
static (i.e., quantum states) or dynamic (i.e., quantum channels). The
resource theory framework is versatile and has been adopted in several areas
of quantum information and physics. QRTs provide a structured framework that
quantitatively describes quantum properties such as quantum correlation
Horodecki ; Ollivier , quantum coherence Streltsov1 ; Streltsov2 , quantum
reference frames Bartlett ; Gour , and nonlocality Bell ; Brunner , etc.
Although there is a large degree of freedom in defining the free states and
free operations, striking similarities emerge among different QRTs in terms of
resource measures and resource convertibility. Consequently, various
properties that appear distinct on the surface possess great similarity on a
deeper structural level. For example, quantum coherence in multipartite
systems embodies the essence of entanglement and is a key ingredient for a
plurality of physical phenomena in quantum physics and quantum information.

Quantum coherence marks the departure of quantum physics from classical
physics. Unlike entanglement, coherence can be present in both single and
composite quantum systems. Several studies have been devoted in recent years
to characterize and quantify the coherence in a quantum physical system
Baumgratz ; Girolami ; Lostaglio ; Shao ; Pires ; Rana ; Rastegin ; Napoli ;
Yu ; Luo ; Bu ; Xiong . Measures of coherence have interesting operational
interpretations. Coherence as a basic resource is very promising. It has been
found to offer significant quantum advantages in various quantum information
processing tasks such as quantum biology Huelga , quantum thermodynamics
Mitchison ; Kammerlander ; Korzekwa , quantum algorithms Hillery , and
steering Mondal .

Mathematically, quantum coherence is represented as the presence of the off-
diagonal terms of a density matrix in a preferred basis. In this formalism, it
is a static (state-based) resource. While static coherence is the degree of
superposition present in a state, dynamic coherence is the ability to
generate, preserve or distribute coherence. Dynamic coherence essentially
studies coherence from the viewpoint of quantum operations. Recent studies
have shown that dynamical coherence, like static coherence Streltsov4 ; Ma ;
Tan , can be measured by different dynamical quantum resources Theurer , and
provide advantages in various quantum tasks Ducuara ; Uola .

Since the incoherent (free) states are represented by diagonal matrices in a
fixed orthonormal basis, they can be considered associated with a von Neumann
measurement. From this connection of coherence and quantum measurement, the
resource theory of coherence has been expanded to coherence based on general
projective measurements such as block-coherence Aberg  and coherence based on
positive-operator-valued measures (POVMs) Bischof1 ; Bischof2 , which are
connected with each other through Naimark’s theorem Paulsen ; Watrous . The
Naimark extension, i.e., embedding the states and operations into a higher-
dimensional space allows for a simpler derivation of generic results, and the
possibility of direct implemention of any POVM in an experiment. POVMs
describe the most general quantum measurements and can provide operational
advantages compared with any projective measurement Oszmaniec . Because of the
rich structure of POVMs, the resource theoretic formulation of POVM-based
coherence generalizes and reveals features that are distinct from the standard
resource theory of coherence Bischof1 ; Bischof2 . Moreover, an important
operational interpretation of POVM coherence in terms of cryptographic
randomness gain is given in Bischof2 . In Ref. Kim-povm  authors provided a
connection of POVM-coherence with entanglement. In particular, they showed
that it is possible to convert POVM-coherence into entanglement and suggested
some practical strategies.

A resource theory of dynamical quantum coherence has been formulated recently
by identifying classical channels as the free elements and considering the
preservation of coherence a resource Saxena . They introduced four different
types of free “superchannels”, quantified dynamical coherence using channel-
divergence-based monotones, and provided operational interpretation to these
monotones. As a comparing note to this, we propose that our definition can
also be extended to redefine free superchannels between different dynamical
systems. In present work, however, we focus on complementing the difficulty of
defining POVM-based incoherent operations and quantifying dynamical resources.

The traditional framework based on the premise of a convex set of incoherent
states, however, has certain limitations on quantifying the POVM-based
dynamical coherence in situations where the existence of incoherent states is
not guaranteed. In this paper, we propose a new framework that overcomes
limitations on quantifying POVM-based dynamical coherence.

The paper is organized as follows. Section II introduces POVM-based coherence.
In Sec. III we define incoherent operations induced by coherence measures. The
dynamical coherence is quantified in the newly defined framework in Sec. IV,
and we prove that the proposed measure is faithful. In Sec. V, we apply our
new framework for quantifying dynamical coherence. Section VI concludes with a
summary.

##  II Resource theory of quantum coherence

Since the establishment of the resource theory of quantum coherence by
Baumgratz et al., various interpretations and studies have been conducted.
Moreover, the notion of block coherence associated with a projective
measurement in which the rank of at least one measurement operator exceeds
unity was established in a framework similar to the typical coherence
Baumgratz ; Aberg ; Bischof1 ; Bischof2 . Block coherence is defined with
respect to a projective measurement on the set of quantum states. See Refs.
Aberg ; Bischof1  for details on block coherence and associated measures. On
the contrary, coherence of a quantum state defined with respect to an
arbitrary positive-operator-valued measure (POVM) via its canonical Naimark
extension is called POVM-based coherence. In this paper, we mainly focus on
POVM coherence. Therefore, we briefly recall the resource theory of POVM-based
coherence introduced by Bischof _et al._ They considered coherence defined
with respect to arbitrary POVMs, not limited to projective measurements. The
generalisation requires the following theorem:

  * •

(_Naimark Theorem_) A POVM $\mathbf{E}=\\{E_{i}\\}_{i=0}^{n-1}$ on
$\mathcal{H}^{S_{0}}$ with $n$ outcomes can be extended to a projective
measurement $\mathbf{P}=\\{P_{i}\\}_{i=0}^{n-1}$ on the Naimark space
$\mathcal{H}^{S}$ (with dimension $d_{S}\geq d_{S_{0}}$) such that

$\displaystyle\operatorname{tr}({E_{i}\rho})=\textmd{tr}\big{\\{}P_{i}(\rho\otimes|0\rangle\langle
0|)\big{\\}}$

holds for all states $\rho$ in $\mathcal{S}_{0}$.

Using this theorem, the following two definitions are proposed:

  1. D1

(_POVM-based coherence measure_) A POVM-based coherence measure
$C_{\mathbf{E}}(\rho^{S_{0}})$ for a state $\rho^{S_{0}}$ in $\mathcal{S}_{0}$
is defined in terms of the block coherence of the embedded state
$\Phi[\rho^{S_{0}}]=\rho^{S_{0}}\otimes|0\rangle\langle 0|^{S_{1}}$ with
respect to a canonical Naimark extension $\mathbf{P}$ of the POVM $\mathbf{E}$
as,

$C_{\mathbf{E}}(\rho^{S_{0}}):=C_{\mathbf{P}}(\Phi[\rho^{S_{0}}]),$ (1)

where $C_{\mathbf{P}}(\rho^{S})$ is a unitarily invariant block-coherence
measure on $\mathcal{S}$–the set of quantum states on the Naimark space
$\mathcal{H}^{S}=\mathcal{H}^{S_{0}}\otimes\mathcal{H}^{S_{1}}$.

  2. D2

(_POVM-based incoherent operations_) A (maximally) POVM-based incoherent
operation is defined as
$\Lambda_{\textmd{MPI}}:=\Phi^{-1}\circ\Lambda\circ\Phi$, where $\Lambda$ is a
block incoherent operation with respect to $\mathbf{P}$ that satisfies
$\Lambda[\mathcal{S}_{\Phi}]\subseteq\mathcal{S}_{\Phi}$ for the subset
$\mathcal{S}_{\Phi}\subseteq\mathcal{S}$ of embedded system states
$\Phi[\rho^{S_{0}}]$. Let $\mathcal{O}_{\textmd{PI}}^{\mathbf{E}}$ denotes the
set of POVM($\mathbf{E}$)-based incoherent operations.

The POVM-based coherence measure is well defined by the underlying block-
coherence measure in Eq.(1), and has the following properties:

  1. P1

(_Faithfulness_): $C_{\mathbf{E}}(\sigma^{S_{0}})\geq 0$, and
$C_{\mathbf{E}}(\sigma^{S_{0}})=0$ if and only if

$\sum_{i}\overline{E}_{i}\sigma^{S_{0}}\overline{E}_{i}=\sigma^{S_{0}},$ (2)

where $\overline{E}_{i}$ denotes the projective part of $E_{i}$, _i.e._ , the
projector onto the range of $E_{i}$.

  2. P2

(_Monotonicity_): $C_{\mathbf{E}}(\Lambda_{\textmd{MPI}}[\rho])\leq
C_{\mathbf{E}}(\rho)$ for all POVM-based incoherent operations
$\Lambda_{\textmd{MPI}}$ of the POVM $\mathbf{E}$.

  3. P3

(_Convexity_):
$C_{\mathbf{E}}(\sum_{i}p_{i}\rho_{i}^{S_{0}})\leq\sum_{i}p_{i}C_{\mathbf{E}}(\rho_{i}^{S_{0}})$
for any set of quantum states $\\{\rho_{i}^{S_{0}}\\}$ and probability
distribution $(p_{i})$.

We should note here that POVM-based coherence measures are determined via the
block coherence measures selected in the Naimark space and the definition of
POVM-based coherence. These POVM-based coherence measures are, however,
independent of the choice of the Naimark space Bischof1 . Then, the relative
entropy of POVM-based coherence measure $C_{\mathbf{E},r}(\rho^{S_{0}})$ is
derived from the relative entropy of block-coherence measure by Eq.(1). It can
be expressed in the form:
$\displaystyle C_{\mathbf{E},r}(\rho^{S_{0}})$ $\displaystyle=$ $\displaystyle
C_{\mathbf{P},r}(\Phi[\rho^{S_{0}}])=\min_{\sigma\in\mathcal{I}_{\textmd{BI}}}S(\Phi[\rho^{S_{0}}]\parallel\sigma)$
$\displaystyle=$ $\displaystyle
H[\\{p_{i}\\}]+\sum_{i}p_{i}S(\rho_{i})-S(\rho^{S_{0}}),$
where $p_{i}=\operatorname{tr}({E_{i}\rho^{S_{0}}}),\ \rho_{i}$ is the $i$-th
postmeasurement state for a given measurement operator $A_{i}$, _i.e._ ,
$\rho_{i}=(1/p_{i})A_{i}\rho^{S_{0}}A_{i}^{\dagger}$, and the Shannon entropy
$H[\\{p_{i}\\}]=-\sum_{i}p_{i}\log p_{i}$.

Bischof _et al._ have shown in Ref. Bischof1  that these POVM-based coherence
measure and POVM-based incoherent operations are independent of the choice of
Naimark extension. In particular, the set of POVM-based incoherent operations
can be characterized by a semidefinite feasibility problem (SDP). Moreover, it
is still uncertain if POVM-based incoherent operations can be characterized
without Naimark extension; POVM-based incoherent operations are realized only
by the block incoherent operations defined in D2. Unlike general coherence
theory, free states do not materialize or exist in the POVM-based coherence.
Also, it is either limited or impossible to define free operations, using
general resource theory, within a single system. Below we introduce an
alternate definition of incoherent operations to overcome the limitations
mentioned above. These can be applied more conveniently and extensively.

##  III Incoherence operations induced by convex measure of coherence

Our aim here is to establish a new framework for the dynamical coherence
analogous to the static framework. We denote by $\mathcal{D}_{S}$ the set of
quantum states on system $S$. Next we define the incoherent operations induced
by a valid convex measure of coherence. In the theory of coherence established
by Baumgratz et al., incoherent states are required to define incoherent
operations. Here we propose an alternate definition of incoherent operations
that does not require incoherent states necessarily.

######  Definition 1.

We define a completely-positive and trace-preserving (CPTP) map $\Lambda$ an
incoherent operation induced by a convex coherence measure $C$ (CMIO by $C$)
on $S$ if the map $\Lambda$ satisfies
$C\big{(}\Lambda[\rho]\big{)}\leq\mathrm{C}(\rho)$ for all
$\rho\in\mathcal{D}_{S}$. We denote by $\mathcal{O}_{\textmd{CMI}}^{C}$ the
set of incoherent operations induced by $C$.

According to this definition, a given operation is considered incoherent if it
does not increase resources for all quantum states. Moreover, we can see that
$\mathcal{O}_{\textmd{CMI}}^{C}$ is a nonempty convex set which necessarily
includes the identity operation $\mathbf{id}_{S}$, induced by convex measures,
such that $C\big{(}\Lambda[\rho]\big{)}\leq\mathrm{C}(\rho)$. Also, the
definition of incoherent operation induced by measures in the resource theory
of general coherence (in which the existence of incoherent states is assumed)
is consistent with that of the existing incoherent operation. It can be
confirmed by the following proposition.

######  Proposition 2.

In the resource theory of general coherence, a CPTP map $\Lambda$ is a CMIO by
$C$ in Def. 1 if and only if $\Lambda$ is a general incoherent operation. That
is, $\mathcal{O}_{\textmd{CMI}}^{C}=\mathcal{O}_{\textmd{I}}$, where
$\mathcal{O}_{\textmd{I}}$ is the set of incoherent operations based on the
existing definition.

###### Proof.

If $\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C}$ then
$C\big{(}\Lambda[\rho]\big{)}=C(\rho)=0$ for any $\rho\in\mathcal{I}$, i.e.,
$\Lambda[\mathcal{I}]\subset\mathcal{I}$. This implies
$\Lambda\in\mathcal{O}_{\textmd{I}}$. On the other hand, if
$\Lambda\in\mathcal{O}_{\textmd{I}}$ then from the monotonicity of the convex
coherence measure $C$, we have $C\big{(}\Lambda[\rho]\big{)}\leq C(\rho)$ for
all $\rho$. Thus $\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C}$ by Def. 1. ∎

Prop. 2 shows that Def. 1 is equally applicable in the resource theory of
general coherence. Furthermore, we can also define POVM($\mathbf{E}$)-based
incoherent operations on the original single system without considering the
block incoherent operations on the Naimark system and the POVM-based
incoherent states. A CPTP map $\Lambda^{S_{0}}$ on the system $S_{0}$ is a
CMIO by $C_{\mathbf{E}}$, if it satisfies

$\begin{array}[]{ccc}C_{\mathbf{E}}(\Lambda^{S_{0}}[\rho^{S_{0}}])&\leq&C_{\mathbf{E}}(\rho^{S_{0}})\\\
&&\\\ \parallel&&\parallel\\\ &&\\\
C_{\mathbf{P}}(\Phi[\Lambda^{S_{0}}[\rho^{S_{0}}]])&\leq&\quad
C_{\mathbf{P}}(\Phi[\rho^{S_{0}}])\end{array}$ (4)

for any quantum state $\rho^{S_{0}}$, where $\mathbf{P}$ is a Naimark
extension of the POVM on $S_{0}$. This is clearly a simpler and more
convenient form than the definition of POVM-based incoherent operations in D2.
It should be, however, noted here that our new definition applies only to a
subset of $\mathcal{S}$. In other words, the POVM-based CMIOs by
$C_{\mathbf{E}}$ are obtained by applying Def. 1 only to the subset
$\mathcal{S}_{\Phi}(=\\{\Phi[\rho^{S_{0}}]|\rho^{S_{0}}\in\mathcal{S}_{0}\\})$
of $\mathcal{S}$ which implies that the POVM-based CMIO by $C_{\mathbf{E}}$ is
a weaker notion than the POVM-based incoherent operation, _i.e._ ,
$\mathcal{O}_{\textmd{PI}}^{\mathbf{E}}\subseteq\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$
(from Prop. 2, the two definitions are consistent when applied for the whole
set). However, if we can find block incoherent operations satisfying
$\Lambda_{\textmd{MPI}}=\Phi^{-1}\circ\Lambda\circ\Phi$ for all POVM-based
CMIOs by $C_{\mathbf{E}}$, we can prove
$\mathcal{O}_{\textmd{PI}}^{\mathbf{E}}=\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$,
which would mean that the POVM-based CMIOs are independent of the choice of
$C_{\mathbf{E}}$. Here we leave this as an open problem whether or not
$\mathcal{O}_{\textmd{PI}}^{\mathbf{E}}=\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$.

Since we have already established a new definition of incoherent operations,
we can consider erecting a theory to quantify dynamic resources from Def. 1.
In the following section, we propose a protocol for quantifying dynamical
resources of incoherent operations induced by measures in the POVM-based
coherence theory.

##  IV Quantifying dynamical coherence induced by POVM-based convex coherence
measures

Let $\mathbf{E}$ be a POVM on $\mathcal{H}^{S}$ and $C_{\mathbf{E}}$ a convex
measure of coherence associated with POVM $\mathbf{E}$. To quantify dynamical
resources based on the measure $C_{\mathbf{E}}$, we define the following
relative function $\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)$ on CPTP maps
$\Theta$ and $\Lambda$ on $\mathcal{H}^{S}$:

$\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda):=\sup_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho]\big{)}\big{\\}}.$
(5)

Since $\mathcal{D}_{S}$ is a closed set, the supremum over the states in this
definition is in effect the maximum, _i.e._ ,

$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)=\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho]\big{)}\big{\\}}.$

Thus, $\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)$ is the maximal relative
increase in the resource that the map $\Theta$ can extract compared with the
map $\Lambda$ for all quantum states. We call the function
$\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)$ the power of $\Theta$ over
$\Lambda$. It should be mentioned here that we focus only on relative
increases and not the relative differences in resources through the two maps,
and therefore the function $\mathfrak{P}_{C_{\mathbf{E}}}$ can assume negative
values. Note that $\mathfrak{P}_{C_{\mathbf{E}}}$ is not symmetric in its
arguments $\Theta$ and $\Lambda$, _i.e.,_
$\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)\neq\mathfrak{P}_{C_{\mathbf{E}}}(\Lambda;\Theta)$.
Moreover, $\mathfrak{P}_{C_{\mathbf{E}}}$ satisfies the convexity:

$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2};\Lambda)$
$\displaystyle\quad=\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}p\Theta_{1}[\rho]+(1-p)\Theta_{2}[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho]\big{)}\big{\\}}$
$\displaystyle\quad\leq\max_{\rho\in\mathcal{D}_{S}}\big{\\{}pC_{\mathbf{E}}\big{(}\Theta_{1}[\rho]\big{)}+(1-p)C_{\mathbf{E}}\big{(}\Theta_{2}[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho]\big{)}\big{\\}}$
$\displaystyle\quad\leq
p\max_{\rho_{1}\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta_{1}[\rho_{1}]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho_{1}]\big{)}\big{\\}}$
$\displaystyle\qquad+\
(1-p)\max_{\rho_{2}\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta_{2}[\rho_{2}]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho_{2}]\big{)}\big{\\}}$
$\displaystyle\quad=p\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{1})+(1-p)\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\Lambda),$
where the first and the last equalities are by definition of
$\mathfrak{P}_{C_{\mathbf{E}}}$, the second inequality is due to the convexity
of $C_{\mathbf{E}}$, and the third inequality is obtained because the sum of
the maximum values is not less than the maximum value for the sum.

Next, we propose a protocol that quantifies dynamical coherence induced by the
measure $C_{\mathbf{E}}$ using this power $\mathfrak{P}_{C_{\mathbf{E}}}$.

######  Definition 3.

For a POVM $\mathbf{E}$ on $\mathcal{H}^{S}$ and $C_{\mathbf{E}}$ the
associated convex measure of coherence, we define

$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta):=\inf_{\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}}|\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda)|.$
(7)

This definition gives us interesting results. First, let us note that
$\mathbf{id}_{S}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ and
$\mathfrak{P}_{C_{\mathbf{E}}}(\Lambda;\mathbf{id}_{S})\leq 0$ when
$\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$. So, for
$\Theta\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ and all
$\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$, we have

$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})$
$\displaystyle\leq$
$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})-\mathfrak{P}_{C_{\mathbf{E}}}(\Lambda;\mathbf{id}_{S})$
$\displaystyle\leq$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Lambda[\rho]\big{)}\big{\\}}$
$\displaystyle=$ $\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\Lambda).$

And if there is at least one quantum state $\rho$ such that
$C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}>C_{\mathbf{E}}(\rho)$, then

$\displaystyle\quad\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})$
$\displaystyle=$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\\{C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}(\rho)\\}$
$\displaystyle\geq$ $\displaystyle
C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}(\rho)>0.$

This means that if the CPTP map
$\Theta\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$, the measure
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)$ represents the maximum
increase in coherence for all quantum states. In contract, if
$\Theta\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$, the relative increase
with itself is zero. That is,

$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)=\left\\{\begin{array}[]{lr}\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})>0&\text{when}~{}~{}\Theta\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}},\\\
\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})=0&\text{when}~{}~{}\Theta\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}.\end{array}\right.$
(8)

Thus, from above results, this function allows us to determine whether a CPTP
map is incoherent or not for the given coherence measure $C_{\mathbf{E}}$.
Next we prove that the measure $\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}$
justly quantifies the dynamical resource of coherence.

######  Theorem 4.

Let $\mathbf{E}$ be a POVM on $\mathcal{H}^{S}$. Then, for any convex measure
$C_{\mathbf{E}}$ of coherence based on $\mathbf{E}$,
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)$ is a convex measure of
dynamical coherence.

###### Proof.

  1. (i)

(Faithfulness) $\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)\geq 0$
from Eqs. (8), and $\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Lambda)=0$
if and only if $\Lambda\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$.

  2. (ii)

(Convexity) We need to prove that
$\quad\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2})\leq
p\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{1})+(1-p)\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{2})$
for $0\leq p\leq 1$. If
$p\Theta_{1}+(1-p)\Theta_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$,
the above inequality is true from the faithfulness of
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}$. If
$p\Theta_{1}+(1-p)\Theta_{2}\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$,
at least one of the operations $\Theta_{1}$ and $\Theta_{2}$ is not included
in $\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ because
$\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ is a convex set. Let us first
consider the case when
$\Theta_{1}\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ and
$\Theta_{2}\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$. Then

$\displaystyle\quad\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2})$
$\displaystyle\qquad\qquad=\mathfrak{P}_{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2};\mathbf{id}_{S})$
$\displaystyle\qquad\qquad\leq
p\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{1};\mathbf{id}_{S})+(1-p)\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\mathbf{id}_{S})$
$\displaystyle\qquad\qquad=p\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{1})+(1-p)\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{2}),$

where the first and the last equalities are due to Eq.(8), and for the
inequality we use the convexity (LABEL:eq:the_convexity_of_power) of
$\mathfrak{P}_{C_{\mathbf{E}}}$. Next, without loss of generality, we assume
that $\Theta_{1}\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ and
$\Theta_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$. Then we have the
following chain of equalities and inequalities:

$\displaystyle\quad\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2})$
$\displaystyle\qquad\qquad=\mathfrak{P}_{C_{\mathbf{E}}}(p\Theta_{1}+(1-p)\Theta_{2};\mathbf{id}_{S})$
$\displaystyle\qquad\qquad\leq
p\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{1};\mathbf{id}_{S})+(1-p)\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\mathbf{id}_{S})$
$\displaystyle\qquad\qquad\leq
p\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{1};\mathbf{id}_{S})+(1-p)\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\Theta_{2})$
$\displaystyle\qquad\qquad=p\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{1})+(1-p)\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta_{2}).$

Here again, the first and the last equalities follow from Eq.(8), the first
inequality is due to the convexity (LABEL:eq:the_convexity_of_power) of
$\mathfrak{P}_{C_{\mathbf{E}}}$, and the second inequality is obtained because
$\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\mathbf{id}_{S})\leq\mathfrak{P}_{C_{\mathbf{E}}}(\Theta_{2};\Theta_{2})=0$
for $\Theta_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$.

  3. (iii)

(Monotonicity) For
$\Phi_{1},\Phi_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$, we show
that
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Phi_{1}\Theta\Phi_{2})\leq\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)$
for any CPTP map $\Theta$. If
$\Theta\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$, then
$\Phi_{1}\Theta\Phi_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ because
$C_{\mathbf{E}}\big{(}\Phi_{1}\Theta\Phi_{2}[\rho]\big{)}\leq
C_{\mathbf{E}}\big{(}\Theta\Phi_{2}[\rho]\big{)}\leq
C_{\mathbf{E}}\big{(}\Phi_{2}[\rho]\big{)}\leq C_{\mathbf{E}}(\rho)$ for all
$\rho\in\mathcal{D}_{S}$. Therefore, we have
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Phi_{1}\Theta\Phi_{2})=\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)=0$.
And, if $\Phi_{1}\Theta\Phi_{2}\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$
for any CPTP map $\Theta$, we can easily see that
$0=\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Phi_{1}\Theta\Phi_{2})\leq\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta)$.
Finally, we consider the case
$\Phi_{1}\Theta\Phi_{2}\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ (it
is possible only when
$\Theta\notin\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$). In this case, we
have
$\displaystyle\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Phi_{1}\Theta\Phi_{2})$
$\displaystyle=$
$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Phi_{1}\Theta\Phi_{2};\mathbf{id}_{S})$
$\displaystyle=$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Phi_{1}\Theta\Phi_{2}[\rho]\big{)}-C_{\mathbf{E}}(\rho)\big{\\}}$
$\displaystyle\leq$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Phi_{1}\Theta\Phi_{2}[\rho]\big{)}-C_{\mathbf{E}}\big{(}\Phi_{2}[\rho]\big{)}\big{\\}}$
$\displaystyle\leq$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Phi_{1}\Theta[\rho]\big{)}-C_{\mathbf{E}}(\rho)\big{\\}}$
$\displaystyle\leq$
$\displaystyle\max_{\rho\in\mathcal{D}_{S}}\big{\\{}C_{\mathbf{E}}\big{(}\Theta[\rho]\big{)}-C_{\mathbf{E}}(\rho)\big{\\}}$
$\displaystyle=$
$\displaystyle\mathfrak{P}_{C_{\mathbf{E}}}(\Theta;\mathbf{id}_{S})=\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}(\Theta),$
where we use $C_{\mathbf{E}}\big{(}\Phi[\rho]\big{)}\leq C_{\mathbf{E}}(\rho)$
for $\Phi\in\mathcal{O}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ in the third and
fifth lines, and inequality in the fourth line follows from an increase of the
set of states over which we maximize because
$\Phi_{2}[\mathcal{D}_{S}]\subset\mathcal{D}_{S}$.

∎

##  V Application of Dynamical Coherence Measure for a POVM with four elements

The advantages of $\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E}}}$ are that
dynamical resources can be quantified through a relatively simple calculation,
as in Eqs. (8), and that an extensive application is possible even in the
special cases where free states are not given. In this section we illustrate
this for certain situations. Let
$\mathbf{E}=\\{\frac{1}{2}|\phi_{k}\rangle\langle\phi_{k}|\\}_{k=0}^{3}$ be a
POVM on $\mathcal{H}^{S_{0}}=\mathcal{C}^{2}$ with
$\ket{\phi_{k}}=1/\sqrt{2}(\ket{0}+\omega^{k}\ket{1})$, where $\omega=\exp(\pi
i/2)$, and $\mathbf{P}=\\{|\varphi_{k}\rangle\langle\varphi_{k}|\\}_{k=0}^{3}$
be a Naimark extension of it on
$\mathcal{H}^{S}(=\mathcal{H}^{S_{0}}\otimes\mathcal{H}^{S_{1}})=\mathcal{C}^{4}$
(here $\mathcal{C}^{4}$ is equivalent to
$\mathcal{C}^{2}\otimes\mathcal{C}^{2}$ with the identification
$\ket{0}^{\mathcal{C}^{4}}=\ket{00}^{\mathcal{C}^{2}\otimes\mathcal{C}^{2}},\ket{1}^{\mathcal{C}^{4}}=\ket{01}^{\mathcal{C}^{2}\otimes\mathcal{C}^{2}},\ket{2}^{\mathcal{C}^{4}}=\ket{10}^{\mathcal{C}^{2}\otimes\mathcal{C}^{2}}$
and
$\ket{3}^{\mathcal{C}^{4}}=\ket{11}^{\mathcal{C}^{2}\otimes\mathcal{C}^{2}}$),
where

$\displaystyle\ket{\varphi_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}\ket{0}+\ket{1}+\sqrt{2}\ket{2}\big{)}$
$\displaystyle\ket{\varphi_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}\ket{0}+i\ket{1}-\exp(\frac{\pi
i}{4})\ket{2}+\exp(\frac{\pi i}{4})\ket{3}\big{)}$
$\displaystyle\ket{\varphi_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}\ket{0}-\ket{1}-\sqrt{2}i\ket{3}\big{)}$
$\displaystyle\ket{\varphi_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}\ket{0}-i\ket{1}-\exp(-\frac{\pi
i}{4})\ket{2}+\exp(\frac{3\pi i}{4})\ket{3}\big{)}.$

Using the formula (LABEL:eq:3), the relative entropy of POVM-based coherence
for quantum state $\rho\in\mathcal{D}_{S_{0}}$ is

$C_{\mathbf{E},r}(\rho)=C_{\mathbf{P},r}(\Phi[\rho])=H[\\{p_{k}\\}]-S(\rho),$
(10)

where $p_{k}=\frac{\bra{\phi_{k}}\rho\ket{\phi_{k}}}{2}.$

Figure 1:  (a) Graph for
$C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)$.
Coherence is quantified via $C_{\mathbf{E},r}$ for pure states
$|\psi_{\theta,\phi}\rangle$. We have a minimum of $1.5$ at
$\theta=\frac{\pi}{2}$ and $\phi=0,\frac{\pi}{2},\pi$ or $\frac{3\pi}{2}$, and
a maximum of $2$ at $\theta=0$ or $\pi$. (b) Graph for
$C_{\mathbf{E},r}(\Theta_{\textmd{max}}[|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|])$.
The unitary operation $\Theta_{\textmd{max}}$ causes a maximum increment of
$0.5$ in coherence where $\theta=\frac{\pi}{2}$ and $\phi=0$ or $\pi$.

We first investigate how the degree of POVM-based coherence is distributed for
the pure quantum states. Consider a state
$\ket{\psi_{\theta,\phi}}=\cos{(\frac{\theta}{2})}\ket{0}+\exp(i\phi)\sin{(\frac{\theta}{2})}\ket{1}\
(0\leq\theta\leq\pi,\ 0\leq\phi<2\pi)$ on the Bloch sphere surface. We have

$C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)=H[\\{p_{k}\\}],$
(11)

where $p_{k}=|\cos{(\frac{\theta}{2})}+\exp\\{k(\frac{\pi
i}{2})-i\phi\\}\sin{(\frac{\theta}{2})}|^{2}$ for $k=0,1,2,3$ (see Fig. 1(a)).
Regardless of $\phi$, when $\theta=0$ or $\pi$, we have the maximal POVM-based
coherence
$C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)=2$
(the maximal POVM-based coherence is obtained in pure states due to the
convexity of $C_{\mathbf{E},r}$). On the contrary, when $\theta=\frac{\pi}{2}$
and $\phi=0,\frac{\pi}{2},\pi$ or $\frac{3\pi}{2}$, we find
$C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)=1.5$
as the minimum value for the pure states.

Now, we consider the degree of POVM-based coherence in the mixed states. For
given $\ket{\psi_{\theta,\phi}}$, let there exists another pure state
$\ket{\psi^{\prime}_{\theta,\phi}}$ such that
$\langle\psi_{\theta,\phi}|\psi^{\prime}_{\theta,\phi}\rangle=0$. Then, every
quantum state can be represented in the following form:

$\rho_{p,\theta,\phi}=p|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|+(1-p)|\psi^{\prime}_{\theta,\phi}\rangle\langle\psi^{\prime}_{\theta,\phi}|,$
(12)

with $0\leq p\leq 1,0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$. When $p$ is
fixed, we have the following maximum and minimum values of
$C_{\mathbf{E},r}(\rho_{p,\theta,\phi})$:

$\displaystyle\max_{\theta,\phi}\\{C_{\mathbf{E},r}(\rho_{p,\theta,\phi})\\}=2-H(p)\quad\
\ \ $ $\displaystyle\textmd{when}\ \theta=0\ \textmd{or}\ \pi,$
$\displaystyle\min_{\theta,\phi}\\{C_{\mathbf{E},r}(\rho_{p,\theta,\phi})\\}=1.5-\frac{H(p)}{2}\quad$
$\displaystyle\textmd{when}\ \theta=\frac{\pi}{2},\phi=0\ \textmd{or}\
\frac{\pi}{2},$

where $H(p)=S(\rho_{p,\theta,\phi})=-p\log p-(1-p)\log(1-p)$. Thus, we obtain
an inequality

$1.5-\frac{H(p)}{2}\leq C_{\mathbf{E},r}(\rho_{p,\theta,\phi})\leq 2-H(p)$
(13)

for any quantum state $\rho_{p,\theta,\phi}$ with fixed $p$. Moreover, we
obtain the minimal POVM-based coherence
$C_{\mathbf{E},r}(\rho_{p,\theta,\phi})=1$ when $p=\frac{1}{2}$. We hereby
confirm that the POVM-based incoherent state for $\mathbf{E}$ does not exist
in $\mathcal{D}_{S}$.

Next, we quantify dynamical resources using the measure
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}$ defined in the preceding
section. We consider increasing the degree of POVM-based coherence via the
unitary operations $\Theta$, _i.e._ , $\Theta(\rho)=U\rho U^{\dagger}$ with
$U$ a unitary operator. We first note that any unitary operation $\Theta$ is
only involved in changing $\theta$ and $\phi$ without changing $p$, _i.e._ ,
$\Theta(\rho_{p,\theta,\phi})=\rho_{p,\theta^{\prime},\phi^{\prime}}\
(0\leq\theta^{\prime}\leq\pi,0\leq\phi^{\prime}<2\pi)$. Thus, for any state
$\rho_{p,\theta,\phi}$, we have

$\displaystyle
C_{\mathbf{E},r}\big{(}\Theta[\rho_{p,\theta,\phi}]\big{)}-C_{\mathbf{E},r}(\rho_{p,\theta,\phi})$
$\displaystyle\leq$ $\displaystyle 2-H(p)-\big{(}1.5-\frac{H(p)}{2}\big{)}$
$\displaystyle=$ $\displaystyle 0.5-\frac{H(p)}{2}\leq 0.5,$

where the first inequality is due to Ineq. (13). Therefore,
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}(\Theta)\leq 0.5$ for any
unitary operation $\Theta$, and the equality is saturated with the following
unitary operator:

$\displaystyle U_{\textmd{max}}=|0\rangle\frac{\langle 0|+\langle
1|}{\sqrt{2}}+|1\rangle\frac{\langle 0|-\langle 1|}{\sqrt{2}}.$

The unitary operator serves to move symmetrically the states on Bloch sphere
around the axis from $\cos(\frac{\pi}{8})\ket{0}+\sin(\frac{\pi}{8})\ket{1}$
to $\cos(\frac{3\pi}{8})\ket{0}-\sin(\frac{3\pi}{8})\ket{1}$, and when
$\Theta_{\textmd{max}}$ is a unitary operation defined by $U_{\textmd{max}}$,
_i.e._ , $\Theta_{\textmd{max}}[\rho]=U_{\textmd{max}}\rho
U_{\textmd{max}}^{\dagger}$, the change in POVM-based coherence for the states
on Bloch sphere by $\Theta_{\textmd{max}}$ is given by the difference in Fig.
1(a) and Fig. 1(b). In particular, when $p=0$ or $1,\theta=\frac{\pi}{2}$ and
$\phi=0$ or $\pi$, the increase in the POVM-based coherence due to
$\Theta_{\textmd{max}}$ amounts to $0.5$ which means
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}(\Theta_{\textmd{max}})=0.5$.
Since the action of the unitary operator $U_{\textmd{max}}$ is to move
symmetrically the states on Bloch sphere, the overall mean density of POVM-
based coherence remains unchanged, and an increase in the POVM-based coherence
can be expected only for the subset of the quantum states.

We also find the incoherent operations induced by measure $C_{\mathbf{E},r}$
through the cycle of degree of POVM-based coherence for $\theta$ and $\phi$.
Fig. 1(a) shows that the unitary operations, which symmetrically move the
states on Bloch sphere around some axis, maintain the POVM-based coherence.
Let us consider, for example, a unitary operation
$\Lambda_{\textmd{min}}[\rho]=U_{\textmd{min}}\rho U_{\textmd{min}}^{\dagger}$
for $\rho\in\mathcal{D}_{S}$ and a unitary operator
$U_{\textmd{min}}=|0\rangle\langle 1|+|1\rangle\langle 0|$. Then

$\displaystyle U_{\textmd{min}}\ket{\psi_{\theta,\phi}}$ $\displaystyle=$
$\displaystyle
e^{i\phi}\sin(\frac{\theta}{2})\ket{0}+\cos(\frac{\theta}{2})\ket{1}$
$\displaystyle=$
$\displaystyle\cos(\frac{\pi-\theta}{2})\ket{0}+e^{i(2\pi-\phi)}\sin(\frac{\pi-\theta}{2})\ket{1},$

and
$C_{\mathbf{E},r}(|\psi_{\pi-\theta,2\pi-\phi}\rangle\langle\psi_{\pi-\theta,2\pi-\phi}|)=C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)$
from Fig. 1(a). Hence,

$C_{\mathbf{E},r}\big{(}\Lambda_{\textmd{min}}[|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|]\big{)}=C_{\mathbf{E},r}(|\psi_{\theta,\phi}\rangle\langle\psi_{\theta,\phi}|)$
(14)

for $0\leq\theta\leq\pi,\ 0\leq\phi<2\pi$. Above result is also valid for any
mixed state obtained by combining such pure states, i.e.,
$C_{\mathbf{E},r}\big{(}\Lambda_{\textmd{min}}[\rho]\big{)}=C_{\mathbf{E},r}(\rho)$
for $\rho\in\mathcal{D}_{S}$, because $\Lambda_{\textmd{min}}$ preserves the
entropies after the implementation of $\mathbf{E}$ without changing phase.
Therefore,
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}(\Lambda_{\textmd{min}})=0$.
This means that $\Lambda_{\textmd{min}}$ is an incoherent operation induced by
the coherence measure $C_{\mathbf{E},r}$ (CMIO by $C_{\mathbf{E},r}$). Again,
the same result is seen for a unitary operator
$U_{\textmd{min}}^{\prime}=|0\rangle\langle 0|-|1\rangle\langle 1|$.

Furthermore, consider the mixed unitary operation
$\Theta_{\textmd{mixed}}(\rho)=\sum_{i}p_{i}\Theta_{i}(\rho)$ for some unitary
operations $(\Theta_{i})$ and a probability distribution $(p_{i})$. Then
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}(\Theta_{\textmd{mixed}})\leq\sum_{i}p_{i}\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}(\Theta_{i})\leq
0.5$ follows from the convexity of
$\mathfrak{C}_{\textmd{CMI}}^{C_{\mathbf{E},r}}$. Likewise, we find the
following CMIO by $C_{\mathbf{E},r}$:

$\displaystyle\Lambda_{\textmd{mixed}}=p\Lambda_{\textmd{min}}+(1-p)\Lambda^{\prime}_{\textmd{min}},$

where
$\Lambda^{\prime}_{\textmd{min}}[\rho]=U^{\prime}_{\textmd{min}}\rho(U^{\prime}_{\textmd{min}})^{\dagger}$
for $\rho\in\mathcal{D}_{S}$, and $0<p<1$.

##  VI Conclusion and Summary

We presented an alternate definition of incoherent operations, induced by
coherence measures, to overcome the limitations of the traditional framework
of POVM-based resource theory. This is not only consistent with the definition
of existing incoherent operations in the resource theory of general coherence,
but also makes it simpler to determine whether or not operations are
incoherent, even if the free states are not given or cannot be determined.
Moreover, we proposed a protocol that allow us to quantify dynamical coherence
when no free states are given, based on the newly defined incoherent
operation. Finally, as an example, we applied our dynamical coherence theory
in the case of POVM-based coherence.

###### Acknowledgements.

This project is supported by the National Natural Science Foundation of China
(Grants No. 12050410232, No. 12031004, and No. 61877054).

## References

  * (1) E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  * (2) F. G. S. L. Brandão and G. Gour, Reversible framework for quantum resource theories, Phys. Rev. Lett. 115, 070503 (2015).
  * (3) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * (4) H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  * (5) A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource Rev. Mod. Phys. 89, 041003 (2017).
  * (6) A. Streltsov, S. Rana, P. Boes, and J. Eisert, Structure of the resource theory of quantum coherence, Phys. Rev. Lett. 119, 140402 (2017).
  * (7) S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Rev. Mod. Phys. 79, 555 (2007).
  * (8) G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New J. Phys. 10, 033023 (2008).
  * (9) J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964).
  * (10) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  * (11) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * (12) D. Girolami, Observable measure of quantum coherence in finite dimensional systems, Phys. Rev. Lett. 113, 170401 (2014).
  * (13) M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Quantum coherence, time-translation symmetry, and thermodynamics, Phys. Rev. X 5, 021001 (2015).
  * (14) L. H. Shao, Z. Xi, H. Fan, and Y. Li, Fidelity and trace-norm distances for quantifying coherence, Phys. Rev. A 91, 042120 (2015).
  * (15) D. P. Pires, L. C. Céleri, and D. O. Soares-Pinto, Geometric lower bound for a quantum coherence measure, Phys. Rev. A 91, 042330 (2015).
  * (16) S. Rana, P. Parashar, and M. Lewenstein, Trace-distance measure of coherence, Phys. Rev. A 93, 012110 (2016).
  * (17) A. E. Rastegin, Quantum-coherence quantifiers based on the Tsallis relative entropies, Phys. Rev. A 93, 032136 (2016).
  * (18) C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Robustness of coherence: An operational and observable measure of quantum coherence, Phys. Rev. Lett. 116, 150502 (2016).
  * (19) X. D. Yu, D. J. Zhang, G. F. Xu, and D. M. Tong, Alternative framework for quantifying coherence, Phys. Rev. A 94, 060302 (2016).
  * (20) S. Luo and Y. Sun, Partial coherence with application to the monotonicity problem of coherence involving skew information, Phys. Rev. A 96, 022136 (2017).
  * (21) K. Bu, U. Singh, S.-M. Fei, A. K. Pati, and J. Wu, Maximum relative entropy of coherence: An operational coherence measure, Phys. Rev. Lett. 119, 150405 (2017).
  * (22) C. Xiong, A. Kumar, and J. Wu, Family of coherence measures and duality between quantum coherence and path distinguishability, Phys. Rev. A 98, 032324 (2018).
  * (23) S. F. Huelga and M. B. Plenio, Vibrations, quanta and biology, Contemp. Phys. 54, 181 (2013).
  * (24) M. T. Mitchison, M. P. Woods, J. Prior, and M. Huber, Coherence-assisted single-shot cooling by quantum absorption refrigerators, New J. Phys. 17, 115013 (2015).
  * (25) P. Kammerlander and J. Anders, Coherence and measurement in quantum thermodynamics, Sci. Rep. 6, 22174 (2016).
  * (26) K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, The extraction of work from quantum coherence, New J. Phys. 18, 023045 (2016).
  * (27) M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation, Phys. Rev. A 93, 012111 (2016).
  * (28) D. Mondal, T. Pramanik, and A. K. Pati, Nonlocal advantage of quantum coherence, Phys. Rev. A 95, 010301 (2017).
  * (29) A. Streltsov, U. Singh, H. S. Dhar, M. Bera, and G. Adesso, Measuring quantum coherence with entanglement, Phys. Rev. Lett. 115, 020403 (2015).
  * (30) J. Ma, B. Yadin, D. Girolami, V. Vedral, and M. Gu, Converting coherence to quantum correlations, Phys. Rev. Lett. 116, 160407 (2016).
  * (31) K. C. Tan, S. Choi, H. Kwon, and H. Jeong, Coherence, quantum Fisher information, superradiance, and entanglement as interconvertible resources, Phys. Rev. A 97, 052304 (2018).
  * (32) T. Theurer, S Satyajit, and M. B. Plenio, Quantifying dynamical coherence with dynamical entanglement, Phys. Rev. Lett. 125, 130401 (2020).
  * (33) A. F. Ducuara and P. Skrzypczyk, Operational interpretation of weight-based resource quantifiers in convex quantum resource theories, Phys. Rev. Lett. 125, 110401 (2020).
  * (34) R. Uola, T. Bullock, T. Kraft, J. P. Pellonpää, and N. Brunner, All quantum resources provide an advantage in exclusion tasks, Phys. Rev. Lett. 125, 110402 (2020).
  * (35) J. Åberg, Quantifying superposition, arXiv:quant-ph/0612146.
  * (36) F. Bischof, H. Kampermann, and D. Bruß, Resource theory of coherence based on positive-operator-valued measures, Phys. Rev. Lett. 123, 110402 (2019).
  * (37) F. Bischof, H. Kampermann, and D. Bruß, Quantifying coherence with respect to general quantum measurements, Phys. Rev. A 103, 032429 (2021).
  * (38) V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, Cambridge, England, 2003).
  * (39) J. Watrous, Theory of Quantum Information (Institute for Quantum Computing, University of Waterloo, Waterloo, Canada, 2008).
  * (40) M. Oszmaniec, L. Guerini, P.Wittek, and A. Acín, Phys. Rev. Lett. 119, 190501 (2017).
  * (41) S. Kim, C. Xiong, A. Kumar, and J. Wu, Converting coherence based on positive-operator-valued measures into entanglement, Phys. Rev. A 103, 052418 (2021).
  * (42) G. Saxena, E. Chitambar, and G. Gour, Dynamical resource theory of quantum coherence, Phys. Rev. Research 2, 023298 (2020).