# Universal Method to Estimate Quantum Coherence

Liang-Liang Sun  Hefei National Research Center for Physical Sciences at the
Microscale, University of Science and Technology of China, Hefei, Anhui
230026, China  Sixia Yu 111email<EMAIL_ADDRESS>Hefei National Research
Center for Physical Sciences at the Microscale, University of Science and
Technology of China, Hefei, Anhui 230026, China

(August 29, 2024)

###### Abstract

Coherence is a defining property of quantum theory that accounts for quantum
advantage in many quantum information tasks. Although many coherence
quantifiers have been introduced in various contexts, the lack of efficient
methods to estimate them restricts their applications. In this paper, we
tackle this problem by proposing one universal method to provide measurable
bounds for most current coherence quantifiers. Our method is motivated by the
observation that the distance between the state of interest and its diagonal
parts in reference basis, which lies at the heart of the coherence
quantifications, can be readily estimated by disturbance effect and
uncertainty of the reference measurement. Thus, our method of bounding
coherence provides a feasible and broadly applicable avenue for detecting
coherence, facilitating its further practical applications.

###### pacs:

98.80.-k, 98.70.Vc

##  I I. Introduction

Coherence, captured by the superposition principle, is a defining property of
quantum theory. It underscores almost all the quantum features, such as
symmetry Marvian and Spekkens (2014), entanglement Streltsov et al. (2015);
Chitambar and Hsieh (2016); Zhu et al. (2018), quantum correlation Hu et al.
(2018a); Ma et al. (2016). Coherence also accounts for quantum advantages in
various quantum information processing tasks, such as quantum metrology
Giovannetti et al. (2011); Giorda and Allegra (2017); Kwon et al. (2019) and
quantum cryptography Huttner et al. (1995); Ma et al. (2019). Within a
strictly mathematical framework of resource theory, the significance of
coherence as a resource has been fully appreciated in recent years. Many
aspects of it, ranging from characterization Streltsov et al. (2017),
distillation, and, catalytic Chitambar and Gour (2016a, b); Chitambar and Gour
(2019); Åberg (2014), have been investigated, along with intense analysis of
how coherence plays a role in fundamental physics (see Hu et al. (2018b) for a
review).

Quantifying coherence lies in the heart of coherence resource theory Oi and
Åberg (2006); Girolami (2014); Streltsov et al. (2017); Winter and Yang
(2016); Rastegin (2016); Yu (2017); Liu et al. (2017a). Till now, many methods
have been proposed. The most initiative method is based on state distance, for
example, quantifying coherence with the minimal distance between the state of
interest and the closest coherence free state. The typical examples are the
relative entropy of coherence and the $l_{1}-$norm of coherence Winter and
Yang (2016). Coherence may also be quantified with the distance between the
concerned state and its diagonal parts in the reference basis Liu et al.
(2017b). One example is the coherence of trace-norm Cui et al. (2020); Biswas
et al. (2017). Another method is via the convex roof measure. That is,
provided a quantifier for the pure state, a general mixed state’s quantifier
is constructed via a roof construction, which method leads to the formation of
coherence Yuan et al. (2015) and infidelity coherence measureLiu et al.
(2017a). There are also other quantifiers such as the robustness of coherence
Napoli et al. (2016), and the Wigner-Yanase skew information of coherence Yu
(2017).

While many theoretical works have been devoted to a systematical research of
coherence Baumgratz et al. (2014); Yu et al. (2016); Hu et al. (2018b);
Streltsov et al. (2017), it remains a difficult problem to efficiently
estimate coherence in experiments, which limits the applications of the
quantifications as common tools for quantum information processing. Clearly,
one can perform state tomography and then calculates quantifiers with the
derived quantum density matrix, or estimate coherence by employing normal
witness technique Piani et al. (2016); Wang et al. (2017); Zheng et al. (2018)
and with numerical optimizations Zhang et al. (2018). These methods suffer
from the complexities of mathematics and experiment setup thus lacking
efficiency. Another method is based on spectrum estimation Yu and Gühne
(2019), which commonly needs a few test measurements to obtain a non-trivial
estimation. These methods, unfortunately, are commonly restricted for
estimating the convex roof quantifiers, such as the coherence of formation,
convex roof of infidelity, and convex roof of the Wigner-Yanase skew
information.

To improve the evaluation of coherence in experiments, we report one simple
and feasible detection method, which provides both the upper and the lower
bounds in terms of the reference measurement’s uncertainty and disturbance
effect, respectively. We find that coherence quantifiers are closely related
to the distance between the state of interest and its diagonal parts( written
in the reference basis). The distance can be upper-bounded in terms of
uncertainty according to recent uncertainty-disturbance relations (UDR) Sun et
al. (2022a, b) and lower-bounded according to data processing inequality.
These bounds are formulated with statistics from a universal experiment
scheme. By the method, almost all the current coherence quantifiers of great
interests are immediately bounded as long as one universal experiment setup
output statistics. The quantifiers include the relative entropy of coherence
measure, the coherence of formation, the $l_{1}-$norm of coherence, the
$l_{2}-$ norm of coherence, the trace-norm of coherence, the convex roof of
infidelity coherence, the Wigner-Yanase skew information of coherence and the
convex roof construction, and the robustness of coherence. Thus the method
exhibits merits of simplicity and broad applicability.

The rest of the paper is structured as follows. In section I, we briefly
review state-distance based coherence quantifiers, then provide framework for
upper-bounding and lower-bounding them. In section II, we provide measurable
bounds for most current coherence quantifiers, including the distance-based
quantifiers and some other quantifiers of general interests. In section III,
we compare our method with the one based on spectrum estimation, showing that
our method is experimentally less demanding and more efficient.

##  II II. The Framework of Detecting Coherence

###  II.1 A.The distance-based quantifications of coherence

Coherence is a quantity characterized with respect to one prefixed reference
basis denoted by $\\{|j\rangle\\}$ with the revelent measurement being
referred to as reference measurement. The free states are the ones of the form
$\sigma=\sum_{j}p_{j}|j\rangle\langle j|$. Otherwise, a non-diagonal state
contains coherence. Coherence is commonly quantified with state distance, for
example, with the minimal distance between the state of interest and the
closest incoherent one Chitambar and Gour (2019)

$\displaystyle\mathcal{C}(\rho):=\min_{\sigma\in\mathcal{I}}\mathcal{D}(\rho,\sigma),$
(1)

where $\mathcal{I}$ denotes the set of coherence free states, and
$\mathcal{D}(\rho,\sigma)$ specifies state distance. Apply Eq.(1) to the state
distances of relative entropy, $l_{1}-$norm,the Tsallis relative $\alpha-$
entropies, one can define quantifiers meeting all the criteria of the
coherence resource theory Streltsov et al. (2017). They are referred to as
coherence measures. If the chosen distance measures are trace-norm and
fidelity Shao et al. (2015); Rana et al. (2016), Eq.(1) defines coherence
monotones that meet a specific subset of the criteria of the coherence
resource theory.

The computability of quantifier $\mathcal{C}(\rho)$ is generally hard except
the state distance, for which, $\mathcal{C}(\rho)=\mathcal{D}(\rho,\rho_{d})$,
where $\rho_{d}$ specifies the diagonal parts of $\rho$ in the reference
basis. One may simply define an easily computable quantifier as the distance
between $\rho$ and $\rho_{d}$ Liu et al. (2017b):

$\displaystyle\tilde{\mathcal{C}}(\rho):=\mathcal{D}(\rho,\rho_{d}),$ (2)

which can be understood as the disturbance caused by reference measurement in
$\rho$ as $\rho_{d}$ is just the post-measurement state. For some distance
measures, Eq.(2) defines a better monotone than the one defined by Eq.(1). One
example is trace-norm Cui et al. (2020), for which, Eq.(2) defines a
quantifier that can satisfy more criteria (than the defined by Eq.(1)) and
also allows a physically well-motivated interpretation as the capability to
exhibit interference visibility Biswas et al. (2017).
$\tilde{\mathcal{C}}(\rho)$ does not involve a minimization process.
Therefore,

$\displaystyle\tilde{\mathcal{C}}(\rho)\geq\mathcal{C}(\rho).$

One may also define coherence quantifier via a convex roof technique Yuan et
al. (2015). That is, provided a quantifier for pure state, one can define a
mixed state’s quantifier via a convex roof construction. For example, one may
define the pure state coherence via Eq.(1), then the convex roof construction
is

$\displaystyle\mathcal{C}^{\prime}(\rho):=\textstyle\min_{\\{f_{i},|\phi_{i}\rangle\\}}\sum_{i}f_{i}\cdot\mathcal{C}(\phi_{i}),$
(3)

where the minimization is taken over all the possible pure state
decompositions of $\rho=\sum f_{i}|\phi_{i}\rangle\langle\phi_{i}|$. This
definition has its advantage, e.g., when applied to infidelity, Eq.(3) can
define a measure Liu et al. (2017a) for coherence while Eq.(1) defines only a
monotone.

The above definitions have led to many quantifiers and also induce the bounds
for the quantifiers defined in other ways. In the following, we first
introduce a framework for bounding them.

Figure 1: Two-slit experiment and coherence detection protocol. Particles with
and without coherence leads to different statistics when subjected to a test
measurement.

###  II.2 B. The framework of detecting coherence

Detecting the nonclassical properties, such as entanglement, coherence, and
randomness, is to lower-bound them using the statistics coming from
experiments. For coherence, it is instructive to recall its early illustration
based on the two-slit experiment which is shown in Fig.1A, where the reference
measurement is the path detectors that erases the coherence between paths, and
the screen is an incompatible measurement that verifies the coherence in terms
of the change of interference fringes due to the destroy of coherence. Here,
we would like to exploit how this idea applies to many other coherence
quantifiers. We consider a similar measurement setting as shown in Fig1.B. It
consists of one reference measurement $\mathcal{R}$ and one following test
measurement $\mathcal{B}$. The reference measurement updates an input state
$\rho$ into an incoherent state, $i.e.$, its diagonal parts $\rho_{d}$. It
then is subject to a following measurement
$\mathcal{B}=\\{|b_{j}\rangle\langle b_{j}|\\}$, giving rise to a distribution
$\mathbf{q^{\prime}}=\\{q^{\prime}_{j}=\operatorname{tr}(\rho_{d}\cdot|b_{j}\rangle\langle
b_{j}|)\\}$. This is a typical sequential measurement scheme that can be
readily realised with off-the-shelf instruments Pryde et al. (2005); Kaneda et
al. (2014); Weston et al. (2013); Marques et al. (2014). If without the
measurement $\mathcal{R}$, directly performing $\mathcal{B}$ on $\rho$ yields
a distribution
$\mathbf{q}=\\{q_{j}=\operatorname{tr}(\rho\cdot|b_{j}\rangle\langle
b_{j}|)\\}$. Distance between $\mathbf{q}$ and $\mathbf{q}^{\prime}$ can be
understood as the disturbance introduced by the reference measurement in
$\mathcal{B}$. One may choose $\mathcal{B}$ as the one maximally incompatible
with the reference measurement, $i.e.$, $\forall i,j$, $|\langle
i|b_{j}\rangle|^{2}=\frac{1}{d}$ with $d$ specifying dimension of the revelent
Hilbert’s space. This setting commonly can ensure a significant distance
between $\mathbf{q}$ and $\mathbf{q}^{\prime}$, which is in favour of
coherence estimation. We also note that $\mathbf{q}^{\prime}$ actually does
not require performing a real test measurement after the reference
measurement. As $\rho_{d}$ is determined by the reference measurement’s
distribution as $\rho_{d}=\sum_{i}p_{i}|i\rangle\langle i|$ and
$\mathbf{p}=\\{p_{i}=\operatorname{tr}(\rho\cdot|i\rangle\langle i|)\\}$, one
can directly calculate $\mathbf{q}^{\prime}$ via Born’s rule. For example, the
probability when $\mathcal{B}:=\\{|b_{j}\rangle\langle b_{j}|\\}$ is
$q^{\prime}_{i}=\operatorname{tr}(\rho_{d}\cdot|b_{i}\rangle\langle
b_{j}|)=\sum_{i}c_{ij}p_{i}$ is given by $\mathbf{p}$, where $c_{ij}=|\langle
i|b_{j}\rangle|^{2}$. In this way, only two independent measurements, namely,
$\mathcal{R}$ and $\mathcal{B}$, are sufficient for giving the statistics
$\mathbf{p}$, $\mathbf{q}$, and $\mathbf{q}^{\prime}$. In the following, the
estimation of coherence quantifiers only involves these distributions.

_Lower-bounding coherence_ First, the coherence quantifiers of the form Eq.(2)
can be estimated according to data processing inequality, which states that
distance between states, say $\rho$ and $\rho_{d}$, are no less than the
corresponding classical distance between statistics coming from another
measurement, say, $\mathcal{B}$, performed on them:

$\displaystyle\mathcal{D}(\rho,\rho_{d})\geq\operatorname{D}[\mathbf{q},\mathbf{q}^{\prime}].$

Immediately, we obtain a lower bound for the coherence measure

$\displaystyle\tilde{\mathcal{C}}(\rho)\geq\operatorname{D}[\mathbf{q},\mathbf{q}^{\prime}].$
(4)

We highlight a useful property of the classical distance, namely, the
convexity of classical distance $\operatorname{D}$,

$\textstyle\sum_{i}f_{i}\cdot\operatorname{D}[\mathbf{q}_{i},\mathbf{q}^{\prime}_{i}]\geq\operatorname{D}[\sum_{i}f_{i}\cdot\mathbf{q}_{i},\sum_{i}f_{i}\cdot\mathbf{q}_{i}^{\prime}],$

which will be used for bounding $\mathcal{C}^{\prime}(\rho)$.

Second, we consider the convex roof-based coherence quantifier. We note that
for pure state a coherence quantifier $\mathcal{C}^{\prime}(\phi)$ is always a
function of distribution $\mathbf{p}_{\phi}$. This is because the diagonal
elements of a pure state are sufficient to determine coherence quantifier as
they determine a pure state up to some relative phases. These phases are
inessential in quantifying coherence as they can be modified freely with
reversible incoherent operation of phase shifting. Noting that the maximum
coherent state $|\phi\rangle=\frac{1}{d}\sum_{i}|i\rangle$ and the zero-
coherence pure state $|i\rangle\langle i|$ exhibit the maximum and the
minimal, respectively, uncertainty of reference measurement in the give basis.
It is therefore reasonable to assume that the coherence quantifier
$\mathcal{C}^{\prime}(\phi)$ for pure state is positively related with
uncertainty $\delta_{\mathcal{D}}(\mathbf{p})$ (the subscript means that
uncertainty can be related to state distance $\mathcal{D}$) and may be lower-
bounded with uncertainty or a function of it (whose definition is left in the
next section). We find that, if a pure state coherence
$\mathcal{C}^{\prime}(\phi)$ allows a lower bound in terms of a convex and
monotonically increasing function of $\delta_{\mathcal{D}}(\mathbf{p}_{\phi})$
specified by $g(\delta_{\mathcal{D}})$, the constructed convex roof-based
coherence quantifier $\mathcal{C}^{\prime}(\rho)$ can be lower-bounded as

$\displaystyle\mathcal{C}^{\prime}(\rho)\geq
g(\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})).$ (5)

It needs to be stressed that $\mathcal{C}^{\prime}(\rho)\geq
g(\delta_{\mathcal{D}}(\mathbf{p}))$ generally does not hold for a mixed state
$\rho$ due to the concavity of $\delta_{\mathcal{D}}$, namely,
$\mathcal{C}^{\prime}(\rho)\geq\sum_{i}f_{i}\cdot
g(\delta_{\mathcal{D}}(\mathbf{p}_{i}))\geq
g(\sum_{i}f_{i}\cdot\delta_{\mathcal{D}}(\mathbf{p}_{i}))$ while
$g(\sum_{i}f_{i}\cdot\delta_{\mathcal{D}}(\mathbf{p}_{i}))\leq
g(\delta_{\mathcal{D}}(\mathbf{p}))$, where
$\sum_{i}\mathbf{p}_{i}=\mathbf{p}$ and
$\sum_{i}f_{i}\cdot\mathbf{p}_{i}=\mathbf{p}$. The key idea behind Eq.(5) is
to relax a concave uncertainty measure $\delta_{\mathcal{D}}$ into a convex
disturbance measure $\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})$ using
UDRs Sun et al. (2022b) stating that one measurement’s uncertainty in terms
of, say $\delta_{\mathcal{D}}(\mathbf{p})$, is no less than its disturbance
effect in the measured state $\rho$ and in the subsequent test measurement
$\mathcal{B}$:

$\delta_{\mathcal{D}}(\mathbf{p})\geq\mathcal{D}(\rho,\rho_{d})\geq\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime}).$

Then, we have
$g(\sum_{i}f_{i}\cdot\operatorname{D}(\mathbf{q}_{i},\mathbf{q}^{\prime}_{i}))\geq
g(\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime}))$ with
$\sum_{i}f_{i}\cdot\mathbf{q}_{i}=\mathbf{q}$ and
$\sum_{i}f_{i}\cdot\mathbf{q}^{\prime}_{i}=\mathbf{q}^{\prime}$, leading to a
lower-bound for coherence quantifiers. We left the proof of Eq.(5) in SM. It
can be seen that the function of $g(\cdot)$ provides a way of finding the
lower-bound of coherence measure in terms of disturbance
$\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})$. In the next section, we
shall show that $g(\cdot)$ can always be found for the existing convex roof-
based coherence quantifiers.

_Estimation of upper-bounds_ In general, upper bounds for quantum properties
are not as useful as the lower bounds since they may be way larger than the
actual value and thus are commonly ignored in theory and experiment. Here, we
can obtain the estimation of the upper bound with the outcome distribution of
the reference measurement for free, i.e., without introducing extra experiment
settings, and most importantly the resulting upper bound may assist the
estimation of coherence in our framework.

It follows from the UDRs that the upper bounds of the quantifiers of the form
Eq.(1) or Eq.(2) are given as

$\delta(\mathbf{p})\geq\tilde{\mathcal{C}}(\rho)\geq\mathcal{C}(\rho).$ (6)

The upper bound of $\mathcal{C}^{\prime}(\rho)$ is given as

$\delta(\mathbf{p})\geq\sum_{i}f_{i}\cdot\delta(\mathbf{p}_{i})\geq\sum_{i}f_{i}\cdot\mathcal{C}(\phi_{i})=\mathcal{C}^{\prime}(\rho),$
(7)

Thus, one reference measurement is sufficient for upper-bounding the three
kinds of coherence quantifiers.

Till now, with the distributions of $\mathbf{p}$, $\mathbf{q}$, and
$\mathbf{q}^{\prime}$, both the upper and the lower bounds are obtained. A
possible large gap between them roughly indicates that: (i) the state of
interest contains little coherence, and (ii) the setting of $\mathcal{B}$ is
not well-chosen. Then the lower bound may be optimized by choosing other
settings of $\mathcal{B}$ or one may almost confirm that the state of interest
contains little coherence. In this way the upper bound assists the estimation
of lower bound.

##  III III. Detecting Coherence in Various contexts

In the following, we use the above framework to estimate coherence quantifiers
having general interests.

###  III.1 A. The relative entropy of coherence measure and the coherence of
formation

First, we consider the relative entropy of coherence Baumgratz et al. (2014)
and the coherence of formation Åberg (2006); Oi and Åberg (2006); Yuan et al.
(2015), which are defined by applying Eq.(1) and Eq.(3) to the relative
entropy

$S(\rho\|\sigma):=\operatorname{Tr}(\rho\log\rho-\rho\log\sigma).$

Relative entropy of coherence is a legitimate measure. It has operational
meaning as asymptotic coherence distillation rate Winter and Yang (2016) and
also quantifies the quantum randomness under the quantum adversaries (with
independent measurements) Yuan et al. (2015); Liu et al. (2018); Yuan et al.
(2019). The quantifier is defined as

$\mathcal{C}_{re}(\rho):=\min_{\sigma\in\mathcal{I}}S(\rho\|\sigma)=S(\rho\|\rho_{d}).$

The coherence of formation has an interpretation of the asymptotic coherence
dilution rate Winter and Yang (2016). It also quantifies the quantum
randomness under the classical adversaries (with independent measurements)
Yuan et al. (2015); Liu et al. (2018); Yuan et al. (2019). The quantifier
reads

$\mathcal{C}^{\prime}_{re}(\rho):=\min_{\\{f_{i},\phi_{i}\\}}\sum_{i}f_{i}\cdot
S(\phi_{i}\|\phi_{i,d}).$

The UDR corresponding to the relative entropy is given as

$\operatorname{H}(\mathbf{p})\geq
S(\rho\|\rho_{d})\geq\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime}),$

where the Shannon entropy $\operatorname{H}(\mathbf{p}):=-\sum_{i}p_{i}\log
p_{i}$ defines measurement uncertainty $\delta_{re}(\mathbf{p})$ and the
relative entropy $S(\rho\|\rho_{d})$ defines disturbance in quantum state and
the classical relative entropy
$\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime}):=\sum_{i}q_{i}\log
q_{i}-\sum_{i}q_{i}\log q^{\prime}_{i}$ defines the disturbance in measurement
$\mathcal{B}$ denoted with $\rm D_{re}(\mathbf{q},\mathbf{q}^{\prime})$. Based
on the general arguments just provided, we immediately have Sun et al. (2022b)

$\operatorname{H}(\mathbf{p})\geq\mathcal{C}_{re}(\rho),\quad\mathcal{C}^{\prime}_{re}(\rho)\geq\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime}).$
(8)

For $\mathcal{C}_{re}(\rho)$, the bounds are obvious. For
$\mathcal{C}^{\prime}_{re}(\rho)$, we have
$\mathcal{C}^{\prime}_{re}(|\phi\rangle)=\operatorname{H}(\mathbf{p}_{\phi})$,
which leads to a definition of convex and monotonically increasing $g$
function as $g(x)=x$. Then a lower bound for
$\mathcal{C}^{\prime}_{re}(\rho)$, namely,
$\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime})$, follows from Eq.(5) and
uncertainty disturbance relation Sun et al. (2022b).

###  III.2 B. The $l_{1}-$norm, the $l_{2}-$ norm, and the trace-norm of
coherence

_The $l_{1}-$norm of coherence._ The $l_{1}-$norm of coherence quantifies the
maximum entanglement that can be created from coherence under incoherent
operations acting on the system and an incoherent ancilla Zhu et al. (2018).
It has been used to investigate the speed-up of quantum computation Hillery
(2016); Shi et al. (2017), wave-particle duality Hu et al. (2018b); Bera et
al. (2015); Cheng and Hall (2015), and uncertainty principle Yuan et al.
(2017). The quantifier is defined via Eq.(1) as Baumgratz et al. (2014)

$\mathcal{C}_{l_{1}}(\rho)=\min_{\sigma\in\mathcal{I}}\mathcal{D}_{l_{1}}(\rho,\sigma)=\mathcal{D}_{l_{1}}(\rho,\rho_{d}).$

where the $l_{1}-$norm

$\mathcal{D}_{l_{1}}(\rho,\sigma^{\prime})=\textstyle\sum_{i,j}|\rho_{ij}-\sigma^{\prime}_{ij}|$

with $\rho_{ij}$ and $\sigma^{\prime}_{ij}$ specifying matrixes’ elements.

The UDR corresponding to the $l_{1}-$norm distance is

$\|\mathbf{p}\|_{\frac{1}{2}}-1\geq\mathcal{D}_{l_{1}}(\rho,\rho_{d}),$

where $\|\mathbf{p}\|_{x}=(\sum_{i}p^{x}_{i})^{\frac{1}{x}}$. The inequality
is because that $\sum_{i\neq j}|\rho_{ij}|\leq\sum_{i\neq
j}\sqrt{p_{i}p_{j}}=\|\mathbf{p}\|_{\frac{1}{2}}-1$. Based on Eq.(6) and the
UDR, the $\mathcal{C}_{l_{1}}(\rho)$ is estimated via

$\|\mathbf{p}\|_{\frac{1}{2}}-1\geq\mathcal{C}_{l_{1}}(\rho)\geq
2|\mathbf{q}-\mathbf{q}^{\prime}|,$ (9)

where
$|\mathbf{q}-\mathbf{q}^{\prime}|:=\frac{1}{2}\sum_{i}|q_{i}-q^{\prime}_{i}|$
is the Kolmogorov distance. The lower bound side is due to $\sum_{i\neq
j}|\rho_{ij}|=\sum_{i>j}$$\operatorname{tr}|\varrho_{ij}|\geq\operatorname{tr}|\rho-\rho_{d}|\geq
2|\mathbf{q}-\mathbf{q}^{\prime}|$, where
$\operatorname{tr}|A|=\operatorname{tr}\sqrt{AA^{\dagger}}$ and
$\varrho_{ij}:=|j\rangle\langle j|\rho|i\rangle\langle i|+|i\rangle\langle
i|\rho|j\rangle\langle j|$ and $\sum_{i\neq
j}\operatorname{tr}|\varrho_{ij}|\geq\operatorname{tr}|\rho-\rho_{d}|$ is due
to the convexity.

_The $l_{2}-$norm of coherence._ The $l_{2}-$norm of coherence Chitambar and
Gour (2016b); Streltsov et al. (2017) has an operational interpretation as
state uncertainty Sun and Luo (2021) and is also employed to study non-
classical correlations Sun et al. (2017). The quantifier reads

$\mathcal{C}_{l_{2}}(\rho):=\min_{\sigma\in\mathcal{I}}\mathcal{D}_{l_{2}}(\rho,\sigma)=\mathcal{D}_{l_{2}}(\rho,\rho_{d}),$

where the $l_{2}-$norm or the (squared) Hilbert-Schmidt distance
$\mathcal{D}_{l_{2}}(\rho,\sigma):=\operatorname{tr}(\rho-\sigma)^{2}.$

The UDR corresponding to $l_{2}-$norm is

$1-\|\mathbf{p}\|^{2}_{2}\geq\mathcal{D}_{l_{2}}(\rho,\rho_{d}),$

which is due to
$\operatorname{tr}(\rho-\rho_{d})^{2}$$=\operatorname{tr}(\rho^{2}-\rho\cdot\rho_{d})\leq
1-\|\mathbf{p}\|^{2}_{2}$. Based on Eq.(6) and Eq.(4),
$\mathcal{C}_{l_{2}}(\rho)$ is bounded as

$\displaystyle
1-\|\mathbf{p}\|^{2}_{2}\geq\mathcal{C}_{l_{2}}(\rho)\geq\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2},$
(10)

where the lower bound is due to data processing inequality with measurement
$\mathcal{B}$ being required to be projective Ozawa (2000).

_Trace-norm of coherence._ The trace-norm of coherence has an interpretation
of interference visibility and reads Cui et al. (2020)

$\displaystyle\tilde{\mathcal{C}}_{tr}(\rho)=\rm\mathcal{D}_{tr}(\rho,\rho_{d})$

where the distance of trace-norm
$\mathcal{D}_{tr}(\rho,\sigma)=\textstyle\rm\frac{1}{2}tr|\rho-\sigma|.$
According to UDR corresponding to this distance Sun et al. (2022b)

$\displaystyle\sqrt{1-\|\mathbf{p}\|^{2}_{2}}\geq\rm\mathcal{D}_{tr}(\rho,\rho_{d}),$

we have

$\displaystyle\sqrt{1-\|\mathbf{p}\|^{2}_{2}}\geq\tilde{\mathcal{C}}_{tr}(\rho)\geq|\mathbf{q}-\mathbf{q}^{\prime}|.$
(11)

Again, the lower bound is due to data processing inequality.

###  III.3 C. Convex roof coherence of infidelity

Now, we consider the convex roof coherence of infidelity Liu et al. (2017a)

$\mathcal{C}^{\prime}_{if}(\rho):=\textstyle\min_{\\{f_{i},|\phi_{i}\rangle\\}}\sum_{i}f_{i}\cdot\mathcal{C}_{if}(\phi_{i}),$

where the pure state coherence is quantified as

$\mathcal{C}_{if}(\phi)=\min_{\sigma\in\mathcal{I}}\mathcal{D}_{if}(\phi,\sigma)$

with the infidelity $\mathcal{D}_{if}(\rho,\sigma):=\sqrt{1-\rm
F(\rho,\sigma)}$ and $\operatorname{F}(\rho,\sigma)=[\rm
Tr(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})]^{2}$. The UDR corresponding to
infidelity is Sun et al. (2022b).

$\sqrt{1-\|\mathbf{p}\|_{2}^{2}}\geq\mathcal{D}_{if}(\rho,\rho_{d}).$

By Eq.(7), the coherence measure acquires an upper bound as
$\sqrt{1-\|\mathbf{p}\|_{2}^{2}}$.

To lower-bound the measure using Eq.(5), we note that
$\mathcal{C}_{if}(\phi)\geq\frac{\sqrt{2}}{2}\sqrt{1-\|\mathbf{p}\|^{2}_{2}}:=\frac{\sqrt{2}}{2}\delta_{if}(\mathbf{p})$
(See SM), which leads to a definition of $g$ function as
$g(x)=\frac{\sqrt{2}}{2}x$. By Eq(5), we have
$\textstyle\mathcal{C}^{\prime}_{if}(\rho)\geq\frac{\sqrt{2}}{2}\operatorname{D}_{if}(\mathbf{q},\mathbf{q}^{\prime})$,
where
$\operatorname{D}_{if}(\mathbf{q},\mathbf{q}^{\prime}):=\textstyle\sqrt{1-(\sum_{i}\sqrt{q_{i}q^{\prime}_{i}})^{2}}.$
Thus, we finally have

$\displaystyle\sqrt{1-\|\mathbf{p}\|_{2}^{2}}\geq\mathcal{C}^{\prime}_{if}(\rho)\geq\textstyle\frac{\sqrt{2}}{2}\operatorname{D}_{if}(\mathbf{q},\mathbf{q}^{\prime}).$
(12)

Table 1: Measurable lower bounds for coherence quantifiers. Quantifier | $\mathcal{C}_{re}$ | $\mathcal{C}_{l_{1}}$ | $\mathcal{C}_{l_{2}}$ | $\tilde{\mathcal{C}}_{tr}$ | $\mathcal{C}^{\prime}_{if}$ | $C_{s}$ | $C^{\prime}_{s}$ | $C_{rob}$
---|---|---|---|---|---|---|---|---
Upper Bound | $\operatorname{H}(\mathbf{p})\quad$ | $\|\mathbf{p}\|_{\frac{1}{2}}-1\quad$ | $1-\|\mathbf{p}\|^{2}_{2}\quad$ | $\sqrt{1-\|\mathbf{p}\|^{2}_{2}}\quad$ | $\sqrt{1-\|\mathbf{p}\|^{2}_{2}}\quad$ | $1-\|\mathbf{p}\|^{2}_{2}\quad$ | $1-\|\mathbf{p}\|^{2}_{2}$ | $\|\mathbf{p}\|_{\frac{1}{2}}-1$
Lower Bound | $\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime})\quad$ | $2|\mathbf{q}-\mathbf{q}^{\prime}|\quad$ | $\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}\quad$ | $|\mathbf{q}-\mathbf{q}^{\prime}|\quad$ | $\frac{\sqrt{2}}{2}\operatorname{D}_{if}(\mathbf{q},\mathbf{q}^{\prime})\quad$ | $\frac{1}{2}\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}\quad$ | $\operatorname{D}^{2}_{if}(\mathbf{q},\mathbf{q}^{\prime})\quad$ | $\frac{\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}}{\|\mathbf{p}\|_{\infty}}$

Till now, we have estimated many distance-based coherence quantifiers. In the
following, we shall use the method to estimate the quantifiers going beyond
the above definitions. The following quantifiers shall be specified by $C$
instead of $\mathcal{C}$ for the sake of specification.

###  III.4 D. The coherence of Wigner-Yanase skew information and the convex
roof construction

Coherence can also be quantified based on quantum Fisher information, which is
a basic concept in the field of quantum metrology that places the fundamental
limit on the information accessible by performing measurement on quantum
state. Two remarkable quantifiers are the Wigner-Yanase skew information of
coherence and the convex roof construction based on it.

_The Wigner-Yanase skew information of coherence._ The Skew information
coherence is a legitimate coherence measure and defined as Yu (2017)

$C_{s}(\rho)=\textstyle\sum^{d}_{1=j}\operatorname{I}(\rho,|j\rangle\langle
j|),$

where $\operatorname{I}(\rho,|j\rangle\langle
j|)\equiv-\frac{1}{2}\operatorname{Tr}([\rho,|j\rangle\langle j|])^{2}$
represents the Wigner-Yanase skew information subject to the projector
$|j\rangle\langle j|$. This measure has ever been estimated with a spectrum
estimation method employing the standard overlap measurement technique Yu
(2017), where one needs to perform $2d-2$ measurements to estimate the
coherence of a $d-$dimensional system. In the following, our method can reduce
the number to three.

First, we can reexpress the measure as $\textstyle
C_{s}(\rho)=1-\sum_{j}\langle j|\sqrt{\rho}|j\rangle^{2}$ Yu (2017), then an
upper bound readily follows as

$\displaystyle\textstyle C_{s}(\rho)$ $\displaystyle=$ $\displaystyle
1-\sum_{j}\langle j|\sqrt{\rho}|j\rangle^{2}\leq 1-\textstyle\sum_{j}\langle
j|\rho|j\rangle^{2}$ $\displaystyle=$ $\displaystyle
1-\|\mathbf{p}\|_{2}^{2}.$

In order to derive a lower-bound, we use an inequality
$C_{s}(\rho)\geq\frac{1}{2}\mathcal{C}_{l_{2}}(\rho)$ Yu (2017) whose bound
has already been bounded in Eq.(10). Then, we have

$\displaystyle 1-\|\mathbf{p}\|_{2}^{2}\geq
C_{s}(\rho)\geq\frac{1}{2}\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}.$ (13)

_The Convex Roof Construction._ With $C_{s}(\phi)$ quantifying coherence of a
pure state $|\phi\rangle$, the Wigner-Yanase skew information can lead to a
convex roof construction of coherence as Li et al. (2021)

$\displaystyle
C^{\prime}_{s}(\rho)=\textstyle\min_{\\{f_{i},\phi_{i}\\}}\sum_{i}f_{i}\cdot
C_{s}(\phi_{i}).$

This quantifier can be equivalently defined via the quantum Fisher information
(up to an inessential factor) with respective to reference measurement Yu
(2013); Li et al. (2021):

$\displaystyle
C^{\prime}_{s}(\rho)=\frac{1}{4}\sum_{j}\operatorname{F}(\rho,|j\rangle\langle
j|),$

where $\operatorname{F}(\rho,|j\rangle\langle
j|):=\sum_{k,l}2\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}|c_{kl}|^{2}$
specifies quantum information of $\rho$ subject to the measurement projector
$|j\rangle\langle j|$ and $c_{kl}=\langle\phi_{k}|j\rangle\langle
j|\phi_{l}\rangle$ with $|\phi_{n}\rangle$ being the $n$th eigenvector of
$\rho$ and $\lambda_{n}$ being the weight.

Given that
$C_{s}(\phi)=1-\|\mathbf{p}(\phi)\|^{2}_{2}=\delta^{2}_{if}(\mathbf{p})$, we
can define $g(\delta_{if})=\delta_{if}^{2}$. By Eq.(5), the lower bound of
$C^{\prime}_{s}(\rho)$ follows as

$C_{s}(\rho)\geq\operatorname{D}^{2}_{if}(\mathbf{q},\mathbf{q^{\prime}}).$

Due to the convexity of $1-\|\mathbf{a}\|^{2}_{2}$ with respect to
distribution $\mathbf{a}$, an upper bound is immediately obtained as

$\displaystyle 1-\|\mathbf{p}\|^{2}_{2}\geq C^{\prime}_{s}(\rho).$

Thus, the quantifier is bounded as

$\displaystyle 1-\|\mathbf{p}\|^{2}_{2}\geq
C^{\prime}_{s}(\rho)\geq\operatorname{D}^{2}_{if}(\mathbf{q},\mathbf{q^{\prime}})$
(14)

###  III.5 F. The robustness of coherence

One important coherence monotone is the robustness of coherence Napoli et al.
(2016), which quantifies the advantage enabled by a quantum state in a phase
discrimination task. For a given state $\rho$, it is defined as the minimal
mixing required to make the state incoherent

$\displaystyle C_{Ro}(\rho)=\min_{\tau}\left\\{s\geq
0\bigg{|}\frac{\rho+s\tau}{1+s}:=\sigma\in\mathcal{I}\right\\}$

where $\tau$ is a general quantum state. With the inequality Napoli et al.
(2016)

$C_{Ro}(\rho)\geq\frac{\mathcal{D}_{l_{2}}(\rho,\rho_{d})}{\|\rho_{d}\|_{\infty}}.$

where note that $\|\rho_{d}\|_{\infty}=\|\mathbf{p}\|_{\infty}$ is just the
maximum element in $\mathbf{p}$. It follows Eq.(10) that

$\displaystyle
C_{Ro}(\rho)\geq\frac{\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}}{\|\mathbf{p}\|_{\infty}}.$

With the inequality $C_{Ro}(\rho)\leq\mathcal{C}_{l_{1}}(\rho)$ and Eq.(9), we
have

$\displaystyle\|\mathbf{p}\|_{\frac{1}{2}}-1\geq
C_{Ro}(\rho)\geq\frac{\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}}{\|\mathbf{p}\|_{\infty}}.$
(15)

The robustness of coherence has ever been detected based on the witness method
Piani et al. (2016); Wang et al. (2017); Zheng et al. (2018), where the
mathematics of constructing witness and experiment setup are generally
complex.

Finally we summarize the obtained measurable lower bounds in Table.I. These
bounds are all formulated in terms of the statistics $\mathbf{p}$,
$\mathbf{q}$, and $\mathbf{q}^{\prime}$. As the essential quantity, the
disturbance $\operatorname{D}$ can readily be measured by performing a
sequential measurements scheme, which has been well-developed in the study of
error-disturbance relations Pryde et al. (2005); Kaneda et al. (2014); Weston
et al. (2013); Marques et al. (2014). In our approach, as the lower-bounds are
smooth functions of experimental statistics, the statistics errors due to
imperfection of implementation of measurement or state preparation are one
order smaller comparing with the lower-bounds. These aspects make our protocol
quite feasible.

It is also of practical interests to consider the tightness of these bounds.
We find that the bounds of coherence measures $\mathcal{C}_{l_{2}}$ and
$\tilde{\mathcal{C}}_{tr}$ can be saturated for any input state when the test
measurement $\mathcal{B}$ is taken as the eigenvectors of $\rho-\rho_{d}$, and
the bounds for the convex-roof based measure $C^{\prime}_{s}$ can be saturated
if the concerned state is pure, and the bound of $\mathcal{C}_{l_{1}}$ is
tight in qubit case (when $\mathcal{B}$ is taken as the eigenvectors of
$\rho-\rho_{d}$). The bounds for other quantifiers either cannot not be non-
trivially saturated or only be saturated for some specific states.

##  IV IV. Efficiency Argument

The previous coherence estimation protocols commonly apply to only a few
coherence quantifiers. The collective measurement protocol Yuan et al. (2020)
applies to the relative entropy of coherence and the $l_{2}-$norm of
coherence. The witness method Piani et al. (2016); Wang et al. (2017); Zheng
et al. (2018); Ma et al. (2021) applies to the robustness of coherence, the
$l_{1}-$norm, and the $l_{2}-$norm of coherence. One quite simple and
efficient method is based on spectrum estimation via majorization theory Yu
and Gühne (2019), which can be used to estimate the $l_{1}-$norm, $l_{2}-$norm
of coherence, the robustness of coherence, and the relative entropy of
coherence. This method employs a similar measurement scheme to ours. In the
following, we compare our approach with it.

###  IV.1 A. Comparison with the method based on spectrum estimation

The spectrum estimation method is based on the theory of majorization Yu and
Gühne (2019). A probability $\mathbf{a}$ is said to majorize a probability
distribution $\mathbf{b}$, specified as $\mathbf{a}\succeq\mathbf{b}$, if
their elements satisfy
$\sum^{k}_{i=1}a^{\downarrow}_{i}\geq\sum^{k}_{i=1}b^{\downarrow}_{i}$
$\forall k$, where the superscript means that the elements are arranged in a
descending ordered, namely,
$\mathbf{a}^{\downarrow}=(a^{\downarrow}_{1},a^{\downarrow}_{2},\cdots
a^{\downarrow}_{d})$,
$\mathbf{b}^{\downarrow}=(b^{\downarrow}_{1},b^{\downarrow}_{2},\cdots
b^{\downarrow}_{d})$ with $a_{i}\geq a_{i+1}$ and $b_{i}\geq b_{i+1}$.
Clearly, the spectrum of $\rho$, specified as
$\boldsymbol{\lambda}=(\lambda_{1},\lambda_{2}\cdots\lambda_{d})$, majorizes
distribution from any projection measurement, say $\mathcal{B}$, performed on
$\rho$, $i.e.$, $\boldsymbol{\lambda}\succeq\mathbf{q}$. By the Shur convexity
theorem, $\operatorname{H}(\mathbf{a})\geq\operatorname{H}(\mathbf{b})$ if
$\mathbf{b}\succeq\mathbf{a}$. Thus,
$\mathcal{C}_{r}(\rho)=\operatorname{H}(\mathbf{p})-\operatorname{H}(\boldsymbol{\lambda})\geq\operatorname{H}(\mathbf{p})-\operatorname{H}(\mathbf{q})$
with $\mathbf{p}$ and $\mathbf{q}$ being distributions from the reference
measurement and $\mathcal{B}$, which provides a nontrivial lower bound if
$\mathbf{q}\succ\mathbf{p}$. Generally, as state of interest is unknown one
needs to try a few settings of $\mathcal{B}$ to ensure
$\mathbf{q}\succ\mathbf{p}$. In this paper, we deal with the disturbance
effect, namely, $\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})$, which is
zero iff $\rho-\rho_{d}$ is perpendicular with all the elements of
$\mathcal{B}$ simultaneously. The settings resulting in such a failure lie in
a space of measure zero. Therefore, our method almost always works even the
$\mathcal{B}$ is chosen arbitrarily. As a simple illustration, assume that
$\rho$ is given as
$|\phi\rangle=\sin\frac{\pi}{8}|0\rangle+\cos\frac{\pi}{8}|1\rangle$ and the
reference basis is $\\{|0\rangle,|1\rangle\\}$, immediately,
$\mathbf{p}=\\{\sin^{2}\frac{\pi}{8},\cos^{2}\frac{\pi}{8}\\}$. By the
spectrum estimation method, a nontrivial estimation, namely,
$\operatorname{H}(\mathbf{q})-\operatorname{H}(\mathbf{p})>0$ requires that
$|\langle\phi|\mathcal{B}|\phi\rangle|>\cos^{2}\frac{\pi}{8}-\sin^{2}\frac{\pi}{8}=\frac{\sqrt{2}}{2}$.
With our method, $\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})\neq 0$
requires that $\mathcal{B}\neq\sigma_{z},\sigma_{y}$, which is much weaker
than the above requirement.

###  IV.2 B Numerical results for the qubit case

The lower bounds provided in Table.I work very well. As the second
illustration of efficiency, we consider the qubit case. For the sake of
computability, we let $\rho$ be a pure state. How well a quantifier is
estimated can be naturally quantified with the ratio of the estimate to the
exact value. For each quantifier, we calculate the average of the ratio over
the randomly chosen measurement $\mathcal{B}$ and randomly chosen pure state
(See SM for details) with two different methods, $i.e.$, our method and the
one based on spectrum estimation. The averages are listed in the following
table, where the one based on our method is specified by $Q^{C}_{D}$ and by
$Q^{C}_{S}$ the other method.

Table 2: Efficiencies in estimating coherence of qubit states. Measure | $\mathcal{C}_{re}$ | $\mathcal{C}_{l_{1}}$ | $\mathcal{C}_{l_{2}}$ | $\tilde{\mathcal{C}}_{tr}$ | $\mathcal{C}^{\prime}_{if}$ | $C_{s}$ | $C^{\prime}_{s}$ | $C_{rob}$
---|---|---|---|---|---|---|---|---
$Q^{C}_{D}$ | $0.36$ | $\frac{2}{\pi}$ | $\frac{1}{2}$ | $\frac{2}{\pi}$ | $0.22$ | $0.25$ | $0.31$ | $0.28$
$Q^{C}_{S}$ | $0.17$ | $0.29$ | $0.20$ | $/$ | $/$ | $/$ | $/$ | $0.20$

It can be seen that our method enjoys wider applicability and higher
efficiency.

##  V IV. Conclusion

In conclusion, we have provided a universal and straightforward method to
estimate coherence. It enables us to give measurable bounds for many
quantifiers of general interest, where all the bounds are expressed as
functions of experiment accessible data $\mathbf{p}$, $\mathbf{q}$, and
$\mathbf{q}^{\prime}$ without involving cumbersome mathematics. This is
advantageous over the previous methods, which only applies to one or a few
measures and cannot apply to the quantifiers based on convex roof
construction. Our approach exhibits many desired features: experiment
friendly, broad applicability, and mathematical simplicity therefore serves as
an efficient coherence detecting method.

For the possible further researches in the quantum foundation, we note that
disturbance effect is one basic concept in the quantum foundation that closely
relates to nonlocality, uncertainty principle, and the security of quantum
cryptography. Thus, the framework may inspire novel connection among these
concepts, for example, nonlocality and coherence.

_Acknowledgement —_ Supports from Guangdong Provincial Key Laboratory Grant
No.2019B121203002 and fundings SIQSE202104 are acknowledged.

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##  Appendix A Supplemental Material.

###  A.1 A. Proof of Eq.(5)

For any pure state ensemble, for example, the one achieving the minimal of
convex roof-based measure, specified by
$\rho=\sum_{i}f_{i}\cdot|\phi_{i}\rangle\langle\phi_{i}|$, we have
$\sum_{i}f_{i}\cdot\mathbf{q_{i}}=\mathbf{q}$ and
$\sum_{i}f_{i}\cdot\mathbf{q_{i}}^{\prime}=\mathbf{q}^{\prime}$. For
$\mathcal{C}^{\prime}(\rho)$, we have

$\displaystyle\mathcal{C}^{\prime}(\rho)$ $\displaystyle=$
$\displaystyle\sum_{i}f_{i}\cdot\mathcal{C}(\phi_{i})\geq\sum_{i}f_{i}\cdot
g(\delta_{\mathcal{D}}(\mathbf{p}_{i}))$ (16) $\displaystyle\geq$
$\displaystyle\sum_{i}f_{i}\cdot
g(\operatorname{D}(\mathbf{q}_{i},\mathbf{q}^{\prime}_{i}))\geq
g(\sum_{i}f_{i}\cdot\operatorname{D}(\mathbf{q}_{i},\mathbf{q}^{\prime}_{i}))$
$\displaystyle\geq$ $\displaystyle
g(\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})).$

The first inequality is due to the definition of $g$ function. The second
inequality is due to UDRs, which relax a concave quantity
$\delta_{\mathcal{D}}(\mathbf{p})$ into a convex one
$\operatorname{D}(\mathbf{q},\mathbf{q}^{\prime})$ . The third is due to the
convexity of $g(\cdot)$. The fourth is due to the convexity of the classical
distance $\operatorname{D}$ and the monotonicity assumption of $g(\cdot)$.

##  Appendix B B. Proof of Eq.(12)

To lower-bound the coherence measure, let us first consider a pure state case,
for example, $|\phi\rangle=\sum_{i}\sqrt{p_{i}(\phi)}e^{i\psi_{i}}|i\rangle$,
we have

$\displaystyle\mathcal{C}_{if}(\phi)=\min_{\sigma\in\mathcal{I}}\mathcal{D}_{if}(\rho,\sigma)=\sqrt{1-p}$

where $p=\max_{i}\\{|p_{0}(\phi)|,\cdots,|p_{d-1}(\phi)|\\}$. Note that,

$\sqrt{2(1-p)}\geq\textstyle\sqrt{(1+p)(1-p)}\geq\sqrt{1-\|\mathbf{p}(\phi)\|_{2}^{2}}.$

The UDR corresponding to infidelity is

$\sqrt{1-\|\mathbf{p}\|_{2}^{2}}\geq\operatorname{D}_{if}(\mathbf{q}(\phi),\mathbf{q}^{\prime}(\phi_{d})).$

where
$\operatorname{D}_{if}(\mathbf{q}(\phi),\mathbf{q}^{\prime}(\phi_{d})):=\sum_{j}\sqrt{q_{j}q^{\prime}_{j}}$
is classical infidelity. Then we have
$\mathcal{C}_{if}(\phi)\geq\frac{\sqrt{2}}{2}\sqrt{1-\|\mathbf{p}\|^{2}_{2}}:=\frac{\sqrt{2}}{2}\delta_{if}(\mathbf{p})$,
which leads to a definition of $g$ function as $g(x)=\frac{\sqrt{2}}{2}x$. The
lower bound of $\mathcal{C}^{\prime}_{if}(\rho)$ follows from Eq(16)

$\displaystyle\mathcal{C}^{\prime}_{if}(\rho)\geq\textstyle\frac{\sqrt{2}}{2}\rm
D_{if}(\mathbf{q},\mathbf{q}^{\prime}).$

To obtain an upper bound, we use Eq.(7) and the UDR $\rm
D_{if}(\rho,\rho_{d})\leq\sqrt{1-\|\mathbf{p}\|_{2}^{2}}$ then have

$\displaystyle\mathcal{C}^{\prime}_{if}(\rho)\leq\textstyle\sum_{i}f_{i}\cdot\sqrt{1-\|\mathbf{p}_{i}\|^{2}}\leq\sqrt{1-\|\mathbf{p}\|_{2}^{2}}$

where we have used the concavity of $\sqrt{1-\|\mathbf{p}\|_{2}^{2}}$.

##  Appendix C C. Tightness argument

Measure | $\mathcal{C}_{re}$ | $\mathcal{C}_{l_{1}}$ | $\mathcal{C}_{l_{2}}$ | $\mathcal{C}^{\prime}_{if}$ | $\tilde{\mathcal{C}}_{tr}$ | $C^{\prime}_{s}$ | $C_{s}$ | $C_{rob}$
---|---|---|---|---|---|---|---|---
Exact value $C(\phi)$ | $\quad\mathrm{H}(\sin^{2}\frac{\theta}{2})$ | $\quad\sin\theta$ | $\quad\frac{1}{2}\sin^{2}\theta$ | $\quad\min\\{\sin\frac{\theta}{2},\cos\frac{\theta}{2}\\}$ | $\quad\frac{1}{2}\sin^{2}\theta$ | $\quad\frac{1}{2}\sin^{2}\theta$ | $\quad\textstyle\sin\theta$ |  $\sin\theta$
$Q^{C}_{D}$ | $0.36$ | $\frac{2}{\pi}$ | $\frac{1}{2}$ | $\frac{2}{\pi}$ | $0.22$ | $0.25$ | $0.31$ | $0.28$
$Q^{C}_{M}$ | $0.17$ | $0.29$ | $0.20$ | $/$ | $/$ | $/$ | $/$ | $0.20$
$Q^{{}^{\prime}C}_{M}$ | $0.20$ | $0.32$ | $0.22$ | $/$ | $/$ | $/$ | $/$ | $0.22$
Table 3: Comparing the efficiencies of our method and the one based on the
spectrum estimation.

For the sake of computability, we consider an arbitrary pure state
$\rho=|\phi\rangle\langle\phi|$ with
$|\phi\rangle=\sin\frac{\theta}{2}|0\rangle+\cos\frac{\theta}{2}e^{i\psi}|1\rangle$
and $0\leq\theta\leq\pi$ and $0\leq\psi\leq 2\pi$. The exact value of the
coherence is specified by $C(\phi)$. We choose the test measurement
$\mathcal{B}$ as the one maximally incompatible with the reference
measurement. Its setting thus is determined up to a relative phase as
$\\{\frac{1}{\sqrt{2}}|0\rangle+\frac{e^{i\psi^{\prime}}}{\sqrt{2}}|1\rangle;\frac{1}{\sqrt{2}}|0\rangle-\frac{e^{i\psi^{\prime}}}{\sqrt{2}}|1\rangle\\}$,
and none of them is of priority. We average the ratio
$\frac{L^{C}_{D,\mathcal{B}}}{C(\phi)}$ over all the possible $\mathcal{B}$ as

$\frac{\bar{L}^{C}_{D,\mathcal{B}}}{C(\phi)}:=\frac{1}{2\pi}\int^{2\pi}_{0}\frac{L^{C}_{D,\mathcal{B}}}{C(\phi)}d\psi^{\prime},$

where one lower bound $L^{C}_{D,\mathcal{B}}$ is given under the choice of
$\mathcal{B}$. As state of interest is unknown, we access the average
performance of the protocol over all the pure states as
$\frac{\bar{L}^{C}_{D,\mathcal{B}}}{C(\phi)}$.

$Q^{C}_{D}:=\frac{1}{4\pi}\int^{2\pi}_{0}\int^{\pi}_{0}\frac{\bar{L}^{C}_{D,\mathcal{B}}}{C(\phi)}\sin\theta
d\theta d\psi.$

The $Q^{C}_{D}$ for the various quantifiers are calculated and listed in the
following table. Using the above maximally incompatible measurement implies a
greater disturbance for $\mathcal{C}_{l_{1}}$, $\mathcal{C}_{l_{2}}$,
$\tilde{\mathcal{C}}_{tr}$, $C_{s}$, and $C_{rob}$, than using a random choice
of the test measurement $\mathcal{B}$ along direction
$\vec{B}=\\{\sin\alpha\sin\psi^{\prime\prime},\sin\alpha\cos\psi^{\prime\prime},\cos\alpha\\}$
with $0\leq\alpha\leq\pi$ and $0\leq\psi^{\prime\prime}\leq 2\pi$. This is
because typical quantities such as $|\mathbf{q}-\mathbf{q}^{\prime}|$ or
$\|\mathbf{q}-\mathbf{q}^{\prime}\|^{2}_{2}$ involved in lower bounds, which
are $|\sin\alpha\sin\theta\cos(\psi^{\prime}-\psi^{\prime\prime})|$ or
$\frac{1}{2}|\sin\alpha\sin\theta\cos(\psi^{\prime}-\psi^{\prime\prime})|^{2}$,
respectively, attain their maximums at $\alpha=\frac{\pi}{2}$ and
$\psi^{\prime}=\psi^{\prime\prime}$. We also calculate $Q^{C}_{D}$ for the
other three quantifiers, namely, $C_{re}$, $\mathcal{C}^{\prime}_{if}$, and
$C^{\prime}_{s}$ using general choice of $\mathcal{B}$, and obtain the average
lower bounds as $0.266$, $0.365$, and $0.234$.

We also calculate the average performance $Q^{C}_{M}$ using the method based
on the spectrum estimation. One key difference is that the test measurement
$\mathcal{B}$ is not need to be the one maximum incompatible with the
reference measurement. This is because in the protocol based on spectrum
estimation the closer the setting of $\mathcal{B}$ to the eigenvectors of
state of interest, the better the estimate is. However, the state of interest
is unknown and so do its eigenvectors. Therefore there is no reasonable to
choose the test measurement $\mathcal{B}$ as the one maximum incompatible with
the reference measurement. According to this protocol, $\mathcal{B}$ is taken
randomly as $\mathcal{B}=\vec{\sigma}\cdot\vec{B}$ and
$\vec{B}=\\{\sin\alpha\sin\psi^{\prime\prime},\sin\alpha\cos\psi^{\prime\prime},\cos\alpha\\}$
with $0\leq\alpha\leq\pi$ and $0\leq\psi^{\prime\prime}\leq 2\pi$ and
$\vec{\sigma}$ being the Pauli matrix. The average performance is defined as

$Q^{C}_{M}:=\frac{1}{4\pi}\int^{2\pi}_{0}\int^{\pi}_{0}\frac{\bar{L}^{C}_{M,\mathcal{B}}}{C(\phi)}\sin\theta
d\theta d\psi.$

with $\frac{\bar{L}^{C}_{M,\mathcal{B}}}{C(\phi)}$ being defined as

$\frac{\bar{L}^{C}_{M,\mathcal{B}}}{C(\phi)}:=\frac{1}{4\pi}\int^{2\pi}_{0}\int^{\pi}_{0}\frac{L^{C}_{M,\mathcal{B}}}{C(\phi)}\sin\alpha
d\alpha d\psi^{\prime\prime}.$

We have calculated the coherence quantifiers that this method is applicable
to. As a comparison, we also calculate the performance when $\mathcal{B}$
taking the measurement maximally incompatible with the reference measurement,
which is specified with $Q^{{}^{\prime}C}_{M}$. By the table, it is shown that
our method enjoys higher efficiency and wider applicability.