Trade-off relations of geometric coherence

Bingyu Hu and Ming-Jing Zhao∗

School of Science, Beijing Information Science and Technology University,
Beijing, 100192, P. R. China

Abstract Quantum coherence is an important quantum resource and it is
intimately related to various research fields. The geometric coherence is a
coherence measure both operationally and geometrically. We study the trade-off
relation of geometric coherence in qubit systems. We first derive an upper
bound for the geometric coherence by the purity of quantum states. Based on
this, a complementarity relation between the quantum coherence and the
mixedness is established. We then derive the quantum uncertainty relations of
the geometric coherence on two and three general measurement bases in terms of
the incompatibility respectively, which turn out to be state-independent for
pure states. These trade-off relations provide the limit to the amount of
quantum coherence. As a byproduct, the complementarity relation between the
minimum error probability for discriminating a pure-states ensemble and the
mixedness of quantum states is established.

Keywords Geometric coherence, Quantum uncertainty relation, Complementarity
relation

∗Correspondence<EMAIL_ADDRESS>
##  I Introduction

Coherence stemmed from superposition principle is the foundation of
multiparticle interference and plays an important role in physics A.
Streltsov-rev ; M. Hu ; K.D. Wu . After the first resource framework for
quantum coherence was introduced in Ref. T. Baumgratz , the characterization
and quantification of quantum coherence have attracted much attention. Some
coherence measures are proposed to quantify the coherence such as the $l_{1}$
norm of coherence T. Baumgratz , the relative entropy of coherence T.
Baumgratz , the intrinsic randomness of coherence X. Yuan , coherence
concurrence X. Qi , the robustness of coherence C. Napoli , and the fidelity-
based coherence T. Baumgratz ; Liu2017  and so on.

The geometric coherence is also a coherence measure based on the fidelity
Streltsov . It amounts to the maximum bipartite geometric entanglement that
can be generated via incoherent operations applied to the system and an
incoherent ancilla Streltsov . Moreover, the geometric coherence is equal to
the minimum error probability to discriminate a set of pure states with von
Neumann measurement Chunhe Xiong . These geometric characterizations and
operational interpretations make the geometric coherence prominent as the
coherence measure. However, the geometric coherence is rather difficult to
calculate. So some evaluations have been provided Hai-Jun Zhang , which are
tight for some special quantum states. The complementarity relation between
the geometric coherence and the mixedness has been also revealed which
characterizes further the geometric coherence quantitativelyUttam Singh ; Hai-
Jun Zhang ; X.Che .

More than that, the quantum coherence is basis-dependent and changes with
different bases, so it is interesting and meaningful to investigate the
constraints of the quantum coherence on different measurement bases. That is
called the quantum uncertainty relation in the sense that coherence and
quantum uncertainty are dual on the same quantum substrate Luo2017 ; Xiao Yuan
. Specifically, in qubit systems, the quantum uncertainty relations in terms
of the relative entropy of coherence, the coherence of formation and the
$l_{1}$ norm of coherence under any two orthonormal bases are derived Xiao
Yuan . In high dimensional systems, the quantum uncertainty relations of
Tsallis relative entropy of coherence and Rényi relative entropy of coherence
are derived Rastegin2023 ; FuGang Zhang , which are then generalized to the
sandwiched Rényi relative entropy of coherence and unified
($\alpha$,$\beta$)-relative entropy of coherence Haijin Mu . The quantum
uncertainty relations for coherence measure based on the Wigner-Yanase skew
information are also established Shun-Long Luo . More generally, the quantum
uncertainty relation with respect to positive operator-valued measurements is
considered Rastegin2021  and the quantum uncertainty relation in bipartite
systems is also discussed Singh2016 .

In this work, we investigate the trade-off relations of the geometric
coherence in qubit systems. Some basic concepts of the geometric coherence are
presented in Sec. II. In Sec. III, we derive an upper bound for the geometric
coherence in terms of the purity of quantum states. Based on this, a
complementarity relation between the quantum coherence and the mixedness is
established. In Sec. IV, we derive the quantum uncertainty relations of the
geometric coherence on two and three general measurement bases in terms of the
purity of quantum states and the incompatibility of the bases. In Sec. V, we
establish a complementarity relation between the minimum error probability for
discriminating a pure-state ensemble and the mixedness of quantum states. We
conclude in Sec. VI.

##  II Preliminaries

The resource theory framework of quantum coherence is composed of two basic
elements: incoherent states and incoherent operations. Considering a
$d$-dimensional Hilbert space $\mathcal{H}_{d}$ with an orthonormal basis
$\mathbb{X}=\\{|x_{i}\rangle\\}_{i=1}^{d}$, we call the diagonal quantum state
$\rho=\sum_{i=1}^{d}\lambda_{i}|x_{i}\rangle\langle x_{i}|$ as the incoherent
state. This set of incoherent states is labeled by $\mathcal{I}_{\mathbb{X}}$.
Quantum operation $\Phi(\rho)=\sum_{l}K_{l}\rho K_{l}^{{\dagger}}$ with
$\sum_{l}K_{l}^{{\dagger}}K_{l}=I$ is an incoherent operation if it fulfills
$K_{l}\sigma K_{l}^{{\dagger}}/tr(K_{l}\sigma
K_{l}^{{\dagger}})\in\mathcal{I}_{\mathbb{X}}$ for all
$\sigma\in\mathcal{I}_{\mathbb{X}}$ and for all $l$.

The quantum coherence is degreed by a nonnegative function named the coherence
measure, which should satisfy the following requirements T. Baumgratz :

  1. (C1)

Nonnegativity: $C(\rho)\geq 0$ and $C(\rho)=0$ for all
$\rho\in\mathcal{I}_{\mathbb{X}}$;

  2. (C2)

Monotonicity: $C$ does not increase under the action of incoherent operations:
$C(\Phi(\rho))\leq C(\rho)$ for any incoherent operation $\Phi$;

  3. (C3)

Strong monotonicity: $C$ does not increase on average under selective
incoherent operations: $\sum_{l}p_{l}C(\rho_{l})\leq C(\rho)$ where
$p_{l}=tr(K_{l}\rho K_{l}^{\dagger})$, $\rho_{l}=K_{l}\rho
K_{l}^{\dagger}/p_{l}$, and
$\Phi(\cdot)=\sum_{l}K_{l}(\cdot)K_{l}^{{\dagger}}$ is an incoherent
operation;

  4. (C4)

Convexity: $C(\rho)\leq\sum_{i}p_{i}C(\rho_{i})$ for any
$\rho=\sum_{i}p_{i}\rho_{i}$ and $p_{i}\geq 0$, $\sum_{i}p_{i}=1$.

The geometric coherence is a coherence measure defined as Streltsov

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)=1-\max_{\sigma\in\mathcal{I}_{\mathbb{X}}}F(\rho,\sigma),$
(1)

where the maximum is taken over all incoherent states
$\sigma\in\mathcal{I}_{\mathbb{X}}$, and

$\displaystyle F(\rho,\sigma)=(tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^{2}$ (2)

is the fidelity of states $\rho$ and $\sigma$ Jozsa . Here one should note
that the fidelity in Eq. (2) is the square of the Uhlmann fidelity in Ref.
Uhlmann ; M.A. . For pure states, the geometric coherence is
$C_{g}^{\mathbb{X}}(|\psi\rangle)=1-\max_{|x_{i}\rangle\in\mathbb{X}}|\langle
x_{i}|\psi\rangle|^{2}$, where the maximum is taken over all
$|x_{i}\rangle\in\mathbb{X}=\\{|x_{i}\rangle\\}_{i=1}^{d}$. For high
dimensional mixed states, the geometric coherence is formidably difficult to
calculate. Then Ref. Hai-Jun Zhang  proved the upper and lower bounds of the
geometric coherence by the sub-fidelity and super-fidelity J.A. Miszczak ,
which induce an analytical formula for qubit states as

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)=\frac{1}{2}-\frac{1}{2}\sqrt{1-2(tr(\rho^{2})-\sum\limits_{i=1}\limits^{2}\langle
x_{i}|\rho|x_{i}\rangle^{2})}.$ (3)

Next we shall employ Eq. (3) and discuss the complementarity relation and
quantum uncertainty relation of the geometric coherence in qubit systems.

##  III The complementarity relation of the geometric coherence

In this section we shall discuss the complementarity relation of the geometric
coherence in qubit systems. Throughout the paper, we denote
$\mathcal{P}=tr(\rho^{2})$ as the purity of quantum state $\rho$. The purity
satisfies $\frac{1}{2}\leq\mathcal{P}\leq 1$. $\mathcal{P}=1$ if and only if
$\rho$ is pure. $\mathcal{P}=\frac{1}{2}$ if and only if $\rho$ is maximally
mixed, $\rho=I/2$. We also refer to
$\mathbb{B}_{\rho}=\\{|\phi_{1}\rangle,|\phi_{2}\rangle\\}$ as the basis
composed by the orthonormal eigenvectors of the nonmaximally mixed state
$\rho$. Then the geometric coherence $C_{g}^{\mathbb{X}}$ and the purity
$\mathcal{P}$ are factually linked together closely.

######  Theorem 1.

For any qubit state $\rho$ and any orthonormal basis
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$, the geometric coherence
$C_{g}^{\mathbb{X}}(\rho)$ is bounded from above by the purity $\mathcal{P}$
as

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)\leq\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}}.$
(4)

The inequality becomes an equality if and only if $\langle
x_{1}|\rho|x_{1}\rangle=\langle x_{2}|\rho|x_{2}\rangle=\frac{1}{2}$.

###### Proof.

Since $2(\langle x_{1}|\rho|x_{1}\rangle^{2}+\langle
x_{2}|\rho|x_{2}\rangle^{2})\geq(\langle x_{1}|\rho|x_{1}\rangle+\langle
x_{2}|\rho|x_{2}\rangle)^{2}=1$, so by Eq. (3),
$C_{g}^{\mathbb{X}}(\rho)=\frac{1}{2}-\frac{1}{2}\sqrt{1-2(\mathcal{P}-\langle
x_{1}|\rho|x_{1}\rangle^{2}-\langle
x_{2}|\rho|x_{2}\rangle^{2})}\leq\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}}$.
The inequality becomes an equality if and only if $\langle
x_{1}|\rho|x_{1}\rangle=\langle x_{2}|\rho|x_{2}\rangle=\frac{1}{2}$. This
completes the proof.

Theorem 1 tells us that
$C_{g}^{\mathbb{X}}(\rho)=\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}}$
if and only if $\langle x_{1}|\rho|x_{1}\rangle=\langle
x_{2}|\rho|x_{2}\rangle=\frac{1}{2}$, which means the density matrix of $\rho$
in the orthonormal basis $\mathbb{X}$ has the equal diagonal entries $1/2$.
This class of quantum states are called the assisted maximally coherent states
in Ref. MZhao  and can be decomposed as the convex combination of maximally
coherent states $\frac{1}{\sqrt{2}}(|x_{1}\rangle+e^{{\rm
i}\theta}|x_{2}\rangle)$. This also means the geometric coherence
$C_{g}^{\mathbb{X}}(\rho)$ reaches the upper bound
$\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}}$ if and only if $\rho$ is
the maximally mixed state or for nonmaximally mixed states, the orthonormal
basis $\mathbb{B}_{\rho}=\\{|\phi_{1}\rangle,|\phi_{2}\rangle\\}$ and the
orthonormal basis $\mathbb{X}=\\{|x_{1}\rangle,|x_{2}\rangle\\}$ are mutually
unbiased, $|\langle\phi_{i}|x_{j}\rangle|^{2}=1/2$, for
$|\phi_{i}\rangle\in\mathbb{B}_{\rho}$ and $|x_{j}\rangle\in\mathbb{X}$,
$\forall i,j$.

Furthermore, it is worthy to emphasize that the upper bound of geometric
coherence in Theorem 1 is basis-independent. It just depends on the purity of
quantum states. This is a remarkable difference between other basis-dependent
upper bounds. Now we make a comparison between our upper bound in Theorem 1
and the upper bounds derived in Ref. Chunhe Xiong  in evaluating the geometric
coherence.

######  Example 1.

For maximally coherent mixed states in a qubit system Uttam Singh

$\displaystyle\rho_{m}=\frac{1-q}{2}I+q|\psi^{+}\rangle\langle\psi^{+}|,$ (5)

where $q\in[0,1]$, $|\psi^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
is the maximally coherent state in the computational basis
$\\{|0\rangle,|1\rangle\\}$. If we fix $\mathbb{X}=\\{|0\rangle,|1\rangle\\}$,
we find our upper bound in Eq. (4) is exactly the geometric coherence

$\displaystyle C_{g}^{\mathbb{X}}(\rho_{m})=\frac{1-\sqrt{1-q^{2}}}{2}.$ (6)

The upper bounds derived in Ref. Chunhe Xiong  are $l_{3}=\frac{q^{2}}{2}$ and
$l_{4}=q$. Obviously, both $l_{3}<C_{g}^{\mathbb{X}}(\rho_{m})$ and
$l_{4}<C_{g}^{\mathbb{X}}(\rho_{m})$, so our upper bound has an advantage over
the upper bounds in Ref. Chunhe Xiong  for this example.

Additionally, Theorem 1 also indicates a complementarity relation between the
geometric coherence and the mixedness by utilizing the normalized linear
entropy $S_{L}(\rho)=\frac{d}{d-1}(1-\mathcal{P})$ as the mixedness quantifier
Nicholas A .

######  Corollary 1.

For any qubit state $\rho$ and any orthonormal basis
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$, the geometric coherence
$C_{g}^{\mathbb{X}}(\rho)$ and the mixedness $S_{L}(\rho)$ should satisfy

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+\frac{1}{2}\sqrt{S_{L}(\rho)}\leq\frac{1}{2}.$ (7)

The inequality becomes an equality if and only if $\langle
x_{1}|\rho|x_{1}\rangle=\langle x_{2}|\rho|x_{2}\rangle=\frac{1}{2}$.

This complementarity relation imposes a fundamental limit on the amount of
coherence that can be extracted from quantum states with equal mixedness. For
any given mixedness $S_{L}(\rho)$ of quantum states ($S_{L}(\rho)\neq 1$, i.e.
$\rho\neq I/2$) and given reference basis $\mathbb{X}$, the quantum states
such that the orthonormal basis $\mathbb{B}_{\rho}$ and the reference basis
$\mathbb{X}$ are mutually unbiased can reach the maximal possible geometric
coherence.

##  IV The quantum uncertainty relation of geometric coherence

In this section, we present the quantum uncertainty relations of geometric
coherence on two and three general measurement bases respectively. These
quantum uncertainty relations depend on the purity of quantum state and
incompatibility of bases. For any two orthonormal bases
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$ and
$\mathbb{Y}=\\{|y_{i}\rangle,i=1,2\\}$ in qubit systems, the incompatibility
Maassen  between $\mathbb{X}$ and $\mathbb{Y}$ is

$\displaystyle\mathfrak{c}=\max_{|x_{i}\rangle\in\mathbb{X},|y_{j}\rangle\in\mathbb{Y}}|\langle
x_{i}|y_{j}\rangle|^{2},$ (8)

which is the maximum overlap of the two measurements. It describes the
commutativity between two orthonormal bases. The incompatibility satisfies
$\frac{1}{2}\leq\mathfrak{c}\leq 1$. Two orthonormal bases
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$ and
$\mathbb{Y}=\\{|y_{i}\rangle,i=1,2\\}$ are called compatible if
$\mathfrak{c}=1$, that is, $\\{|y_{1}\rangle,|y_{2}\rangle\\}=\\{e^{{\rm
i}\theta_{1}}|x_{1}\rangle,e^{{\rm i}\theta_{2}}|x_{2}\rangle\\}$. Moreover,
we also need the following lemma proved in Ref. Xiao Yuan  for the
characterization of the feasible domain of the optimization problem next.

######  Lemma 1.

Let $|x\rangle$, $|z\rangle$, $|r\rangle$ be any three $d$-dimensional
normalized vectors, $a=|\langle x|r\rangle|^{2}$, $b=|\langle
z|r\rangle|^{2}$, $c=|\langle z|x\rangle|^{2}$. Then $a$, $b$, $c$ satisfy

$\displaystyle a+b$ $\displaystyle\leq 1+\sqrt{c}\ \ \text{and}\ \ |a-b|$
$\displaystyle\leq\sqrt{1-c}.$

Especially, when $d=2$, we have one more inequality that

$\displaystyle 1-\sqrt{c}\leq a+b.$

###  IV.1 The quantum uncertainty relation of geometric coherence on two
general measurement bases

Based on the incompatibility of two orthonormal bases, the quantum uncertainty
relation of the geometric coherence on two general measurement bases can be
derived.

######  Theorem 2.

For any qubit state $\rho$ and two orthonormal bases
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$ and
$\mathbb{Y}=\\{|y_{i}\rangle,i=1,2\\}$, the geometric coherence of $\rho$
satisfies the following quantum uncertainty relation:

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)\geq\frac{1}{2}[1-\sqrt{1+4(2\mathcal{P}-1)(\mathfrak{c}-\sqrt{\mathfrak{c}})}],$
(9)

where $\mathcal{P}$ is the purity of the quantum state $\rho$, and
$\mathfrak{c}$ is the incompatibility of $\mathbb{X}$ and $\mathbb{Y}$. The
equality holds true if and only if $\rho$ is the maximally mixed state or for
nonmaximally mixed states, the three bases $\mathbb{X}$, $\mathbb{Y}$ and
${\mathbb{B}}_{\rho}$ are all compatible.

###### Proof.

Without loss of generality, we suppose $\mathfrak{c}=|\langle
x_{1}|y_{1}\rangle|^{2}$. For any given qubit state $\rho$ with spectral
decomposition
$\rho=p|\phi_{1}\rangle\langle\phi_{1}|+(1-p)|\phi_{2}\rangle\langle\phi_{2}|$,
we denote $a=|\langle\phi_{1}|x_{1}\rangle|^{2}$ and
$b=|\langle\phi_{1}|y_{1}\rangle|^{2}$, using Eq. (3), we have

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)=1-\frac{1}{2}[\sqrt{1+4(2\mathcal{P}-1)(a^{2}-a)}+\sqrt{1+4(2\mathcal{P}-1)(b^{2}-b)}].$

Let $f(x)=\sqrt{1+4(2\mathcal{P}-1)(x^{2}-x)}$, $x\in[0,1]$, it is easy to
verify that $f(x)$ is convex and symmetric about $x=\frac{1}{2}$. Furthermore
$f$ decreases when $x\leq\frac{1}{2}$ and increases when $x\geq\frac{1}{2}$.
Now we focus on the maximum of the function $g(a,b)=f(a)+f(b)$, which is
symmetric about $a=\frac{1}{2}$ and $b=\frac{1}{2}$. Here we only need to
consider the case $a,b\in[0,\frac{1}{2}]$. Hence we get the following
optimization problem:

$\displaystyle\max\quad g(a,b)=f(a)+f(b)$ (10)
$\displaystyle\begin{array}[]{r@{\quad}r@{}l@{\quad}l}s.t.&&a+b\in[1-\sqrt{\mathfrak{c}},1+\sqrt{\mathfrak{c}}]\\\
&&b-a\in[-\sqrt{1-\mathfrak{c}},\sqrt{1-\mathfrak{c}}]\\\
&&a,b\in[0,\frac{1}{2}]\\\ \end{array}$

by Lemma 1. The feasible domain of the optimization problem is depicted in
Fig. 1.

Figure 1: The yellow regions represent the feasible domain of the optimization
problem in Eq. (10). When $\mathfrak{c}$ belongs to $[0.5,0.75]$, the feasible
domain is in shape of the subfigure (a) with $A_{1}(0,\frac{1}{2})$,
$A_{2}(0,1-\sqrt{\mathfrak{c}})$, $A_{3}(1-\sqrt{\mathfrak{c}},0)$,
$A_{4}(\frac{1}{2},0)$ and $A_{5}(\frac{1}{2},\frac{1}{2})$ as the extreme
points. When $\mathfrak{c}$ belongs to $(0.75,1]$, the feasible domain is in
shape of the subfigure (b) with $B_{1}(0,\sqrt{1-\mathfrak{c}})$,
$B_{2}(0,1-\sqrt{\mathfrak{c}})$, $B_{3}(1-\sqrt{\mathfrak{c}},0)$,
$B_{4}(\sqrt{1-\mathfrak{c}},0)$,
$B_{5}(\frac{1}{2},\frac{1}{2}-\sqrt{1-\mathfrak{c}})$,
$B_{6}(\frac{1}{2},\frac{1}{2})$ and
$B_{7}(\frac{1}{2}-\sqrt{1-\mathfrak{c}},\frac{1}{2})$ as the extreme points.
In the subfigure (a), we choose $\mathfrak{c}=0.5$ as an example. In the
subfigure (b), we choose $\mathfrak{c}=0.9$ as an example.

Because $g$ is a convex function of $a,b$ and the feasible domain is a convex
polygon, the maximum of function $g$ must be obtained at the extreme points of
the feasible domain.

When $\mathfrak{c}$ belongs to $[0.5,0.75]$, the extreme points of the
feasible domain are $A_{1}(0,\frac{1}{2})$, $A_{2}(0,1-\sqrt{\mathfrak{c}})$,
$A_{3}(1-\sqrt{\mathfrak{c}},0)$, $A_{4}(\frac{1}{2},0)$ and
$A_{5}(\frac{1}{2},\frac{1}{2})$ in the subfigure (a) of Fig. 1. Therefore we
have

$\displaystyle\max
g(a,b)=\max\\{f(0)+f(1-\sqrt{\mathfrak{c}}),f(0)+f(\frac{1}{2}),f(\frac{1}{2})+f(\frac{1}{2})\\}.$

Since $0\leq 1-\sqrt{\mathfrak{c}}<\frac{1}{2}$, so the maximum of $g$ is
definitely $f(0)+f(1-\sqrt{\mathfrak{c}})$.

When $\mathfrak{c}$ belongs to $(0.75,1]$, the extreme points of the feasible
domain are $B_{1}(0,\sqrt{1-\mathfrak{c}})$, $B_{2}(0,1-\sqrt{\mathfrak{c}})$,
$B_{3}(1-\sqrt{\mathfrak{c}},0)$, $B_{4}(\sqrt{1-\mathfrak{c}},0)$,
$B_{5}(\frac{1}{2},\frac{1}{2}-\sqrt{1-\mathfrak{c}})$,
$B_{6}(\frac{1}{2},\frac{1}{2})$ and
$B_{7}(\frac{1}{2}-\sqrt{1-\mathfrak{c}},\frac{1}{2})$ in the subfigure (b) of
Fig. 1. Therefore we have

$\displaystyle\max
g(a,b)=\max\\{f(0)+f(1-\sqrt{\mathfrak{c}}),f(0)+f(\sqrt{1-\mathfrak{c}}),f(\frac{1}{2})+f(\frac{1}{2}-\sqrt{1-\mathfrak{c}}),f(\frac{1}{2})+f(\frac{1}{2})\\}.$

Since
$|1-\sqrt{\mathfrak{c}}-\frac{1}{2}|\geq|\sqrt{1-\mathfrak{c}}-\frac{1}{2}|$,
so the maximum of $g$ is definitely
$f(0)+f(1-\sqrt{\mathfrak{\mathfrak{c}}})$.

Combining the two cases above, we derive that the geometric coherence of
$\rho$ on two general measurement bases $\mathbb{X}$ and $\mathbb{Y}$ should
satisfy

$\displaystyle C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)$
$\displaystyle=$ $\displaystyle 1-\frac{1}{2}[f(a)+f(b)]$ $\displaystyle\geq$
$\displaystyle 1-\frac{1}{2}[f(0)+f(1-\sqrt{\mathfrak{c}})]$ $\displaystyle=$
$\displaystyle\frac{1}{2}[1-\sqrt{1+4(2\mathcal{P}-1)(\mathfrak{c}-\sqrt{\mathfrak{c}})}].$

Now we consider the saturation of the quantum uncertainty relation. First, if
$\mathcal{P}=\frac{1}{2}$, then the lower bound of the geometric coherence
$C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)$ as well as the geometric
coherence $C_{g}^{\mathbb{X}}(\rho)$ and $C_{g}^{\mathbb{Y}}(\rho)$ vanishes.
So Eq. (9) becomes an equality for the maximally mixed state. For nonmaximally
mixed state, by the analysis above we know the geometric coherence
$C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)$ reaches its lower bound
when (1) $a=1$, $b=\sqrt{\mathfrak{c}}$; (2) $a=0$, $b=1-\sqrt{\mathfrak{c}}$;
(3) $a=1-\sqrt{\mathfrak{c}}$, $b=0$; (4) $a=\sqrt{\mathfrak{c}}$, $b=1$.
Thanks to the symmetry of function $f$, we only need to consider the case
$a=1$, $b=\sqrt{\mathfrak{c}}$. The condition $a=1$ implies
$\\{|\phi_{1}\rangle,|\phi_{2}\rangle\\}=\\{e^{i\alpha_{1}}|x_{1}\rangle,e^{i\alpha_{2}}|x_{2}\rangle\\}$.
The condition $b=\sqrt{\mathfrak{c}}$ associated with $\mathfrak{c}=|\langle
x_{1}|y_{1}\rangle|^{2}=|\langle\phi_{1}|y_{1}\rangle|^{2}=\sqrt{\mathfrak{c}}$
involves $\mathfrak{c}=1$, which means
$\\{|y_{1}\rangle,|y_{2}\rangle\\}=\\{e^{i\beta_{1}}|x_{1}\rangle,e^{i\beta_{2}}|x_{2}\rangle\\}$.
Therefore three bases $\mathbb{X}$, $\mathbb{Y}$ and ${\mathbb{B}}_{\rho}$ are
all compatible. This completes the proof.

Theorem 2 provides a quantum uncertainty relation for the geometric coherence
with two measurement bases, in terms of the purity of quantum states and the
incompatibility of the two bases. The equality in (9) holds true only for
maximally mixed states or three compatible bases $\mathbb{X}$, $\mathbb{Y}$
and ${\mathbb{B}}_{\rho}$ for nonmaximally mixed states. This is equivalent to
that the quantum state $\rho$ is incoherent simultaneously under the bases
$\mathbb{X}$ and $\mathbb{Y}$. Conversely, for any nonmaximally mixed states
and two incompatible measurement bases $\mathbb{X}$ and $\mathbb{Y}$, the
lower bound in Eq. (9) is always strictly positive. Particularly, if the
purity is specified to the maximum, $\mathcal{P}=1$, the quantum uncertainty
relation is just relied on the incompatibility of the two bases. This
demonstrates the quantum uncertainty relation is state-independent for any
pure state $|\psi\rangle$, that is,

$\displaystyle
C_{g}^{\mathbb{X}}(|\psi\rangle)+C_{g}^{\mathbb{Y}}(|\psi\rangle)\geq\frac{1}{2}[1-\sqrt{1+4(\mathfrak{c}-\sqrt{\mathfrak{c}})}].$
(11)

Now we consider an example to illustrate the evaluation of the geometric
coherence.

######  Example 2.

For the maximally coherent mixed states in Eq. (5) in qubit systems,
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ can be evaluated
from above by Eq. (4) and from below by Eq. (9) as

$\displaystyle\frac{1}{2}[1-\sqrt{1+4q^{2}(\mathfrak{c}-\sqrt{\mathfrak{c}})}]\leq
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})\leq
1-\sqrt{1-q^{2}}.$ (12)

From this equation we can see the upper bound is independent with the
orthonormal bases while the lower bound increases when the incompatibility
$\mathfrak{c}$ decreases.

First we fix
$\mathbb{X}=\mathbb{B}_{\rho_{m}}=\\{\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)\\}$
composed by the eigenvectors of $\rho_{m}$ and
$\mathbb{Y}=\\{\frac{1}{\sqrt{5}}(|0\rangle+2|1\rangle),\frac{1}{\sqrt{5}}(-2|0\rangle+|1\rangle)\\}$
with the incompatibility $\mathfrak{c}=\frac{9}{10}$. Then
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ can be evaluated
by Eq. (12) as

$\displaystyle\frac{1}{2}[1-\sqrt{1+\frac{6(3-\sqrt{10})q^{2}}{5}}]\leq
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})\leq
1-\sqrt{1-q^{2}}.$ (13)

In fact by Eq. (3) we have

$\displaystyle
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})=C_{g}^{\mathbb{Y}}(\rho_{m})=\frac{1}{2}(1-\sqrt{1-\frac{9}{25}q^{2}}).$
(14)

The geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ and the upper and
lower bounds in Eq. (13) are plotted in the first subfigure in Fig. 2.

Second, we consider
$\mathbb{X}=\\{\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle),\frac{1}{\sqrt{2}}(|0\rangle-i|1\rangle)\\}$
and keep the basis
$\mathbb{Y}=\\{\frac{1}{\sqrt{5}}(|0\rangle+2|1\rangle),\frac{1}{\sqrt{5}}(-2|0\rangle+|1\rangle)\\}$
with the incompatibility $\mathfrak{c}=\frac{1}{2}$. Then
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ can be evaluated
by Eq. (12) as

$\displaystyle\frac{1}{2}[1-\sqrt{1+2(1-\sqrt{2})q^{2}}]\leq
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})\leq
1-\sqrt{1-q^{2}},$ (15)

In this case, by Eq. (3) we have

$\displaystyle
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})=\frac{1}{2}(1-\sqrt{1-q^{2}})+\frac{1}{2}(1-\sqrt{1-\frac{9}{25}q^{2}}).$
(16)

The geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ and the upper and
lower bounds in Eq. (15) are plotted in the second subfigure in Fig. 2.

Since the incompatibility of the first set of orthonormal bases is greater
than that of the second, so the lower bound of the geometric coherence under
the second set of orthonormal bases is greater. This makes the region between
the upper bound and lower bound in subfigure (b) in Fig. 2 smaller. So the
evaluation of the geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ is more precise
with smaller incompatibility.

Figure 2: The evaluation of the geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$. In the subfigure
(a), we choose
$\mathbb{X}=\\{\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)\\}$
and
$\mathbb{Y}=\\{\frac{1}{\sqrt{5}}(|0\rangle+2|1\rangle),\frac{1}{\sqrt{5}}(-2|0\rangle+|1\rangle)\\}$.
The green dot-dashed line is
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ in Eq. (14). The
red solid line and blue dashed line are the lower and upper bounds in Eq.
(13). In the subfigure (b), we choose
$\mathbb{X}=\\{\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle),\frac{1}{\sqrt{2}}(|0\rangle-i|1\rangle)\\}$
and
$\mathbb{Y}=\\{\frac{1}{\sqrt{5}}(|0\rangle+2|1\rangle),\frac{1}{\sqrt{5}}(-2|0\rangle+|1\rangle)\\}$.
The green dot-dashed line is
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})$ in Eq. (16). The
red solid line and blue dashed line are the lower and upper bounds in Eq.
(15).

###  IV.2 The quantum uncertainty relation of geometric coherence with three
measurement bases

For any three orthonormal bases $\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$,
$\mathbb{Y}=\\{|y_{i}\rangle,i=1,2\\}$ and
$\mathbb{Z}=\\{|z_{i}\rangle,i=1,2\\}$, we define the incompatibility vector
of $\mathbb{X}$, $\mathbb{Y}$, $\mathbb{Z}$ as
$\vec{\mathfrak{c}}=({\mathfrak{c}}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3})$,

$\displaystyle\vec{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3}):=(\max_{|x_{i}\rangle\in\mathbb{X},|y_{j}\rangle\in\mathbb{Y}}|\langle
x_{i}|y_{j}\rangle|^{2},\max_{|y_{i}\rangle\in\mathbb{Y},|z_{j}\rangle\in\mathbb{Z}}|\langle
y_{i}|z_{j}\rangle|^{2},\max_{|x_{i}\rangle\in\mathbb{X},|z_{j}\rangle\in\mathbb{Z}}|\langle
x_{i}|z_{j}\rangle|^{2})^{\downarrow},$ (17)

where the superscript ↓ means rearranging the entries in descending order. The
incompatibility vector
$\vec{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3})$
describes the commutativity of three orthnormal bases. It satisfies
$\frac{1}{2}\leq\mathfrak{c}_{i}\leq 1$ for $i=1,2,3$.
$\vec{\mathfrak{c}}=(1,1,1)$ if and only if three orthonormal bases are all
compatible. Based on the incompatibility vector $\vec{\mathfrak{c}}$, we
derive a quantum uncertainty relation of geometric coherence under three
measurement bases.

######  Theorem 3.

For any qubit state $\rho$ and three orthonormal bases
$\mathbb{X}=\\{|x_{i}\rangle,i=1,2\\}$, $\mathbb{Y}=\\{|y_{i}\rangle,i=1,2\\}$
and $\mathbb{Z}=\\{|z_{i}\rangle,i=1,2\\}$, suppose
$\vec{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3})$ is
the incompatibility vector of $\mathbb{X}$, $\mathbb{Y}$ and $\mathbb{Z}$.

  1. (1)

If
$1+\sqrt{\mathfrak{c}_{3}}<\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
then

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)+C_{g}^{\mathbb{Z}}(\rho)\geq
1-\frac{1}{2}[f(\sqrt{\mathfrak{c}_{1}})+f(\sqrt{\mathfrak{c}_{1}}-\sqrt{\mathfrak{c}_{3}})].$
(18)

  2. (2)

If
$1+\sqrt{\mathfrak{c}_{3}}\geq\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
then

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)+C_{g}^{\mathbb{Z}}(\rho)\geq\frac{3}{2}-\frac{1}{2}\max\\{1+f(\sqrt{\mathfrak{c}_{1}})+f(\sqrt{\mathfrak{c}_{2}}),$
$\displaystyle
f(\frac{1-\sqrt{\mathfrak{c}_{1}}-\sqrt{\mathfrak{c}_{2}}+\sqrt{\mathfrak{c}_{3}}}{2})+f(\frac{1-\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}-\sqrt{\mathfrak{c}_{3}}}{2})+f(\frac{1+\sqrt{\mathfrak{c}_{1}}-\sqrt{\mathfrak{c}_{2}}-\sqrt{\mathfrak{c}_{3}}}{2})\\},$

with $f(x)=\sqrt{1+4(2\mathcal{P}-1)(x^{2}-x)}$ and $\mathcal{P}$ the purity
of the quantum state $\rho$.

###### Proof.

Since
$\vec{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3})$ is
the incompatibility vector of three orthonormal bases $\mathbb{X}$,
$\mathbb{Y}$ and $\mathbb{Z}$, without loss of generality, we suppose
$\mathfrak{c}_{1}=\max_{|x_{i}\rangle\in\mathbb{X},|y_{j}\rangle\in\mathbb{Y}}|\langle
x_{i}|y_{j}\rangle|^{2}=|\langle x_{1}|y_{1}\rangle|^{2}$,
$\mathfrak{c}_{2}=\max_{|x_{i}\rangle\in\mathbb{X},|z_{j}\rangle\in\mathbb{Z}}|\langle
x_{i}|z_{j}\rangle|^{2}=|\langle x_{1}|z_{1}\rangle|^{2}$, and
$\mathfrak{c}_{3}=\max_{|y_{i}\rangle\in\mathbb{Y},|z_{j}\rangle\in\mathbb{Z}}|\langle
y_{i}|z_{j}\rangle|^{2}$, and
$\mathfrak{c}_{1}\geq\mathfrak{c}_{2}\geq\mathfrak{c}_{3}$. Suppose
$\rho=p|\phi_{1}\rangle\langle\phi_{1}|+(1-p)|\phi_{2}\rangle\langle\phi_{2}|$
is the spectral decomposition of $\rho$, denote
$a=|\langle\phi_{1}|x_{1}\rangle|^{2}$, $b=|\langle\phi_{1}|y_{1}\rangle|^{2}$
and $n=|\langle\phi_{1}|z_{1}\rangle|^{2}$, then using Eq. (3), we have

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)+C_{g}^{\mathbb{Y}}(\rho)+C_{g}^{\mathbb{Z}}(\rho)$
$\displaystyle=$
$\displaystyle\frac{3}{2}-\frac{1}{2}[\sqrt{1+4(2\mathcal{P}-1)(a^{2}-a)}+\sqrt{1+4(2\mathcal{P}-1)(b^{2}-b)}+\sqrt{1+4(2\mathcal{P}-1)(n^{2}-n)}]$
$\displaystyle=$ $\displaystyle\frac{3}{2}-\frac{1}{2}[f(a)+f(b)+f(n)].$

Now we consider the maximum of $G(a,b,n)=f(a)+f(b)+f(n)$. By the symmetry of
the function $G$ and Lemma 1, we only need to consider the following
optimization problem:

$\displaystyle\max\quad G(a,b,n)=f(a)+f(b)+f(n)$ (19)
$\displaystyle\begin{array}[]{r@{\quad}r@{}l@{\quad}l}s.t.&&a+b\in[1-\sqrt{\mathfrak{c_{1}}},1+\sqrt{\mathfrak{c_{1}}}]\\\
&&a+n\in[1-\sqrt{\mathfrak{c_{2}}},1+\sqrt{\mathfrak{c_{2}}}]\\\
&&b+n\in[1-\sqrt{\mathfrak{c_{3}}},1+\sqrt{\mathfrak{c_{3}}}]\\\
&&b-a\in[-\sqrt{1-\mathfrak{c_{1}}},\sqrt{1-\mathfrak{c_{1}}}]\\\
&&n-a\in[-\sqrt{1-\mathfrak{c_{2}}},\sqrt{1-\mathfrak{c_{2}}}]\\\
&&a,b,n\in[0,\frac{1}{2}],\end{array}$

where we have extended the range of $b+n$ from $[1-\sqrt{|\langle
y_{1}|z_{1}\rangle|^{2}},1+\sqrt{|\langle y_{1}|z_{1}\rangle|^{2}}]$ to
$[1-\sqrt{\mathfrak{c_{3}}},1+\sqrt{\mathfrak{c_{3}}}]$ and remove the
restriction of $n-b$ for simplicity.

First if $\mathfrak{c}_{1}\in[0.5,0.75]$, then we divide the discussion of the
optimization problem in Eq. (19) into two cases.

(1) When
$1+\sqrt{\mathfrak{c_{3}}}<\sqrt{\mathfrak{c_{1}}}+\sqrt{\mathfrak{c_{2}}}$,
the feasible domain is a convex polyhedron in the first subfigure in Fig. 3.
Since $G(a,b,n)$ is a convex function of $a,b,n$, its maximum over this
feasible domain must be achieved at the extreme points, which are the vertices
of the polyhedron
$C_{1}(1-\sqrt{\mathfrak{c_{1}}},0,1-\sqrt{\mathfrak{c_{3}}})$,
$C_{2}(0,1-\sqrt{\mathfrak{c_{1}}},\sqrt{\mathfrak{c_{1}}}-\sqrt{\mathfrak{c_{3}}})$,
$C_{5}(0,\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}},1-\sqrt{\mathfrak{c_{2}}})$,
$C_{6}(1-\sqrt{\mathfrak{c_{2}}},1-\sqrt{\mathfrak{c_{3}}},0)$ and the
vertices

$\begin{array}[]{rcl}&&D_{1}(1-\sqrt{\mathfrak{c_{1}}},0,\frac{1}{2}),\ \
D_{2}(0,1-\sqrt{\mathfrak{c_{1}}},\frac{1}{2}),\ \
D_{3}(0,\frac{1}{2},\frac{1}{2}),\ \
D_{4}(\frac{1}{2},\frac{1}{2},\frac{1}{2}),\\\
&&D_{5}(\frac{1}{2},0,\frac{1}{2}),\ \
D_{6}(0,\frac{1}{2},1-\sqrt{\mathfrak{c_{2}}}),\ \
D_{7}(1-\sqrt{\mathfrak{c_{2}}},\frac{1}{2},0),\ \
D_{8}(\frac{1}{2},\frac{1}{2},0),\\\
&&D_{9}(\frac{1}{2},1-\sqrt{\mathfrak{c_{3}}},0),\ \
D_{10}(\frac{1}{2},0,1-\sqrt{\mathfrak{c_{3}}}).\end{array}$ (20)

By calculating and comparing these values, we get $\max
G(a,b,n)=\max\\{G(C_{2}),G(C_{5})\\}$. Let
$H(x)=f(0)+f(x)+f(1-\sqrt{\mathfrak{c_{3}}}-x)$,
$x\in[0,1-\sqrt{\mathfrak{c_{3}}}]$, one can verify that $H(x)$ is a convex
function and symmetric about $x=\frac{1-\sqrt{\mathfrak{c_{3}}}}{2}$. Since
$|1-\sqrt{\mathfrak{c_{1}}}-\frac{1-\sqrt{\mathfrak{c_{3}}}}{2}|\geq|\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}}-\frac{1-\sqrt{\mathfrak{c_{3}}}}{2}|$,
the value of $C_{5}$ is no more than the value of $C_{2}$, therefore the
maximum of function $G$ is definitely

$\displaystyle\max
G(a,b,n)=G(0,1-\sqrt{\mathfrak{c}_{1}},\sqrt{\mathfrak{c}_{1}}-\sqrt{\mathfrak{c}_{3}}).$
(21)

(2) When
$1+\sqrt{\mathfrak{c}_{3}}\geq\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
the feasible domain is a convex polyhedron in the second subfigure in Fig. 3.
Similarly, to get the maximum of function $G$ in this polyhedron, we need to
calculate the values of the vertices
$C_{1}(1-\sqrt{\mathfrak{c_{1}}},0,1-\sqrt{\mathfrak{c_{3}}})$,
$C_{3}(0,1-\sqrt{\mathfrak{c_{1}}},1-\sqrt{\mathfrak{c_{2}}})$,
$C_{6}(1-\sqrt{\mathfrak{c_{2}}},1-\sqrt{\mathfrak{c_{3}}},0)$,
$C_{7}(\frac{1-\sqrt{\mathfrak{c_{1}}}-\sqrt{\mathfrak{c_{2}}}+\sqrt{\mathfrak{c_{3}}}}{2},\frac{1-\sqrt{\mathfrak{c_{1}}}+\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}}}{2},\frac{1+\sqrt{\mathfrak{c_{1}}}-\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}}}{2})$
and the vertices $\\{D_{i},i=1,...,10\\}$ in Eq. (20). By comparing the
values, we get

$\displaystyle\max
G(a,b,n)=\max\\{G(0,1-\sqrt{\mathfrak{c_{1}}},1-\sqrt{\mathfrak{c_{2}}}),G(\frac{1-\sqrt{\mathfrak{c_{1}}}-\sqrt{\mathfrak{c_{2}}}+\sqrt{\mathfrak{c_{3}}}}{2},$
$\displaystyle\frac{1-\sqrt{\mathfrak{c_{1}}}+\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}}}{2},\frac{1+\sqrt{\mathfrak{c_{1}}}-\sqrt{\mathfrak{c_{2}}}-\sqrt{\mathfrak{c_{3}}}}{2})\\}.$
(22)

Figure 3: The geometry of the feasible domain in the optimization problem in
Eq. (19) for $\mathfrak{c}_{1}\in[0.5,0.75]$. When
$1+\sqrt{\mathfrak{c}_{3}}<\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
the feasible domain is in shape of the subfigure (a). When
$1+\sqrt{\mathfrak{c}_{3}}\geq\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
the feasible domain is in shape of the subfigure (b). Both polyhedra are
formed by cutting three triangular prisms from a cube with length
$\frac{1}{2}$. We label these cross-sectional planes as i@, ii@ and iii@
respectively. The intersection points of the cross-sectional planes of i@ and
iii@ are $C_{1}$ and $C_{2}$, the intersection points of the cross-sectional
planes of i@ and ii@ are $C_{3}$ and $C_{4}$, the intersection points of the
cross-sectional planes of ii@ and iii@ are $C_{5}$ and $C_{6}$. The vertex
$C_{7}$ in subfigure (b) is the intersection point of i@, ii@ and<EMAIL_ADDRESS>$D_{i}$ ($1\leq i\leq 10$) are other vertices of the polyhedra.

Second, if $\mathfrak{c}_{1}\in(0.75,1]$, then the feasible domain in the
optimization problem in Eq. (19) is a smaller polyhedron contained in Fig. 3.
So the maximum reached in the first case is an upper bound for this case,
which completes the proof.

Theorem 3 provides a quantum uncertainty relation for the geometric coherence
under three measurement bases, in terms of the purity of quantum states and
the incompatibility vector of three bases. Similarly, this quantum uncertainty
relation is also state-independent for pure state. Next we illustrate the
lower bound of the geometric coherence by an explicit example.

######  Example 3.

For the maximally coherent mixed states in Eq. (5) in qubit systems, we fix
$\mathbb{X}=\\{\frac{1}{\sqrt{5}}(-2|0\rangle+|1\rangle),\frac{1}{\sqrt{5}}(|0\rangle+2|1\rangle)\\}$,
$\mathbb{Y}=\\{\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle),\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\\}$
and $\mathbb{Z}=\\{|0\rangle,|1\rangle\\}$. By calculation we get the
incompatibility vector of $\mathbb{X}$, $\mathbb{Y}$, $\mathbb{Z}$ is
$\vec{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}_{3})=(\frac{9}{10},\frac{4}{5},\frac{1}{2})$.
Since
$1+\sqrt{\mathfrak{c}_{3}}<\sqrt{\mathfrak{c}_{1}}+\sqrt{\mathfrak{c}_{2}}$,
by Eq. (18) we have the lower bound of
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})+C_{g}^{\mathbb{Z}}(\rho_{m})$
is

$\displaystyle
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})+C_{g}^{\mathbb{Z}}(\rho_{m})\geq
1-\frac{1}{2}\left[\sqrt{1+\frac{6(3-\sqrt{10})q^{2}}{5}}+\right.$
$\displaystyle\left.\sqrt{1+\frac{2(14-6\sqrt{5}-3\sqrt{10}+5\sqrt{2})q^{2}}{5}}\,\right].$
(23)

In fact the geometric coherence is

$\displaystyle
C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})+C_{g}^{\mathbb{Z}}(\rho_{m})=\frac{1}{2}(1-\sqrt{1-\frac{9}{25}q^{2}})+\frac{1}{2}(1-\sqrt{1-q^{2}}),$
(24)

by Eq. (3). The geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})+C_{g}^{\mathbb{Z}}(\rho_{m})$
and its lower bound in Eq. (23) are plotted in Fig. 4.

Figure 4: The evaluation of the geometric coherence
$C_{g}^{\mathbb{X}}(\rho_{m})+C_{g}^{\mathbb{Y}}(\rho_{m})+C_{g}^{\mathbb{Z}}(\rho_{m})$.
The green dot-dashed line is exact value in Eq. (24) and the red solid line is
the lower bound in Eq. (23).

##  V The geometric coherence and quantum state discrimination

In this section we show the application of the geometric coherence in quantum
state discrimination. First, let us recall the process of the ambiguous
quantum state discrimination with von Neumann measurement. Suppose the sender
draws at some states from $\\{\rho_{i}\\}$ with probabilities $\\{p_{i}\\}$
and sends them to the receiver. Then the receiver tries to determine which
state he has received by a von Neumann measurement $\\{\Pi_{i}\\}$ satisfying
$\sum_{i}\Pi_{i}=I$. He declares the state is $\rho_{j}$ if the measurement
result is $j$. Since the probability to get the result $j$ is
$p(j|i)=tr(\Pi_{j}\rho_{i})$ provided the state is $\rho_{i}$, so the maximal
success probability to discriminate this set of states $\\{p_{i},\rho_{i}\\}$
is $\max_{\Pi_{i}}\sum_{i}p_{i}tr(\Pi_{i}\rho_{i})$, where the maximum is over
all von Neumann measurement $\\{\Pi_{i}\\}$. Correspondingly, the minimum
error probability is

$\displaystyle
P_{e}(\\{p_{i},\rho_{i}\\})=1-\max_{\Pi_{i}}\sum_{i}p_{i}tr(\Pi_{i}\rho_{i}).$
(25)

Ref. Chunhe Xiong  shows the geometric coherence of a quantum state amounts to
the minimum error probability to discriminate a set of pure states via von
Neumann measurement.

######  Lemma 2.

Chunhe Xiong  For any given orthonormal basis $\mathbb{X}=\\{|x_{i}\rangle\\}$
in a $d$-dimensional system and any quantum state $\rho$, the geometric
coherence of $\rho$ is equal to the minimum error probability to discriminate
the pure-state ensemble $\\{\eta_{i},|\psi_{i}\rangle\\}$ with von Neumann
measurement

$\displaystyle
C_{g}^{\mathbb{X}}(\rho)=P_{e}(\\{\eta_{i},|\psi_{i}\rangle\\}),$ (26)

with $\eta_{i}=\langle x_{i}|\rho|x_{i}\rangle$ and
$|\psi_{i}\rangle=\eta_{i}^{-1/2}\sqrt{\rho}|x_{i}\rangle$.

In fact, for any given quantum state $\rho$, the ensemble
$\\{\eta_{i},|\psi_{i}\rangle\\}$ with $\eta_{i}=\langle
x_{i}|\rho|x_{i}\rangle$ and
$|\psi_{i}\rangle=\eta_{i}^{-1/2}\sqrt{\rho}|x_{i}\rangle$ is a pure-state
ensemble of $\rho$, that is,
$\rho=\sum_{i}\eta_{i}|\psi_{i}\rangle\langle\psi_{i}|$. So the minimum error
probability to discriminate this pure-state ensemble corresponds to the
geometric coherence of the associated mixed state. In light of the relation
between the geometric coherence and purity in Theorem 1, the minimum error
probability to discriminate any pure-state ensemble and the purity are
connected.

######  Corollary 2.

For any qubit systems, the minimum error probability to discriminate any pure-
state ensemble $\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2}$ by von Neumann
measurement is bounded from above as

$\displaystyle
P_{e}(\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2})\leq\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}},$
(27)

with $\mathcal{P}$ the purity of the associated quantum state
$\rho=\sum_{i=1}^{2}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$.

###### Proof.

Suppose $\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2}$ is any pure-state ensemble in
the qubit system, and
$\rho=\sum_{i=1}^{2}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ is the quantum
state associated to it. We further suppose
$\rho=\sum_{j=1}^{2}\lambda_{j}|\phi_{j}\rangle\langle\phi_{j}|$ is the
spectral decomposition of $\rho$. Then the two pure-state decompositions of
$\rho$ are related as
$\sqrt{p_{i}}|\psi_{i}\rangle=\sum_{j}\sqrt{\lambda_{j}}U_{ij}|\phi_{j}\rangle$,
$\forall i$, for some unitary operator $U=(U_{ij})$ M.A. . Since the
eigenvectors of $\rho$ compose an orthonormal basis in the qubit system and
$U=(U_{ij})$ is unitary, so there exist another orthonormal basis denoted as
$\mathbb{X}=\\{|x_{i}\rangle\\}$ such that
$U_{ij}=\langle\phi_{j}|x_{i}\rangle$. Under the selected orthonormal basis
$\mathbb{X}$, we have $\sqrt{p_{i}}|\psi_{i}\rangle=\sqrt{\rho}|x_{i}\rangle$,
$\forall i$. Therefore the pure-state ensemble
$\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2}$ satisfies
$|\psi_{i}\rangle=p_{i}^{-1/2}\sqrt{\rho}|x_{i}\rangle$ and $p_{i}=\langle
x_{i}|\rho|x_{i}\rangle$. By Lemma 2 and Theorem 1, we have
$P_{e}(\\{\eta_{i},|\psi_{i}\rangle\\})=C_{g}^{\mathbb{X}}(\rho)\leq\frac{1}{2}-\frac{1}{\sqrt{2}}\sqrt{1-\mathcal{P}}$,
which completes the proof.

Corollary 2 shows the minimum error probability to discriminate any pure-state
ensemble $\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2}$ is up bounded by the purity
of the corresponding mixed states. Different from Lemma 2, the minimum error
probability is just relied on the purity and independent of the reference
basis. Eq. (27) can be rewritten in form of the complementarity relation as

$\displaystyle
P_{e}(\\{p_{i},|\psi_{i}\rangle\\})+\frac{1}{{2}}\sqrt{S_{L}(\rho)}\leq\frac{1}{2},$
(28)

which imposes a limit on the amount of the minimum error probability for any
pure-state ensemble $\\{p_{i},|\psi_{i}\rangle\\}_{i=1}^{2}$ with given
mixedness.

######  Example 4.

For any orthonormal basis $\mathbb{X}=\\{|x_{i}\rangle\\}$ in qubit system,
let

$\displaystyle\begin{array}[]{rcl}|\psi_{1}\rangle=\cos\theta|x_{1}\rangle+\sin\theta|x_{2}\rangle,\\\
|\psi_{2}\rangle=\sin\theta|x_{1}\rangle+\cos\theta|x_{2}\rangle,\end{array}$
(31)

and $p_{1}=p_{2}=1/2$ with $\theta\in[-\frac{\pi}{4},\frac{\pi}{4}]$. Now we
consider the discrimination of the ensemble $\\{p_{i},|\psi_{i}\rangle\\}$ by
von Neumann measurement. On one hand, the minimum error probability to
discriminate the ensemble $\\{p_{i},|\psi_{i}\rangle\\}$ is
$P_{e}(\\{p_{i},|\psi_{i}\rangle\\})=\sin^{2}\theta$ and the optimal von
Neumann measurement $\\{\Pi_{i}\\}$ is $\Pi_{1}=|x_{1}\rangle\langle x_{1}|$
and $\Pi_{2}=|x_{2}\rangle\langle x_{2}|$ by Ref. Helstrom . On the other
hand, the quantum state with respect to this ensemble
$\\{p_{i},|\psi_{i}\rangle\\}$ is

$\rho=\frac{1}{{2}}(|x_{1}\rangle\langle x_{1}|+|x_{2}\rangle\langle
x_{2}|)+\sin\theta\cos\theta(|x_{1}\rangle\langle x_{2}|+|x_{2}\rangle\langle
x_{1}|)$ (32)

with purity $\mathcal{P}=\frac{1}{2}+2\sin^{2}\theta\cos^{2}\theta$. In this
case, the minimum error probability reaches the upper bound in Corollary 2.

##  VI Conclusions

To summarize, we study the trade-off relations of the geometric coherence in
qubit systems. We first show the complementarity relation between the
geometric coherence and the purity of quantum states. Then we derive the
quantum uncertainty relations of the geometric coherence on two and three
general measurement bases respectively, in terms of the purity of the quantum
states and the incompatibility of the bases. At last, we obtain the
complementarity relation between the minimum error probability for
discriminating a pure-states ensemble and the mixedness of quantum states.
These trade-off relations not only provide the limit to the amount of quantum
coherence, but also reveal the intimate relations between different
quantities. Since the geometric coherence is important both geometrically and
operationally, we hope this work is helpful for the characterization and
application of the geometric coherence.

Generalizations of the results to general high dimensional systems and more
orthonormal bases are natural extensions of this work. On one hand, for
general $m$ orthonormal bases, the quantum uncertainty relation of geometric
coherence depends on the incompatibility vector of dimensional
$\frac{m(m-1)}{2}$. Although it is not feasible to depict the feasible domain
as Fig. 1 or Fig. 3, a lower bound may be derived by convex analysis. On the
other hand, for general high dimensional systems, the incompatibility of two
orthonormal bases may be not enough to get a good lower bound. One may need to
make use of more parameters such as the second largest value of their inner
products.

##  VII Acknowledgement

This work is supported by the National Natural Science Foundation of China
under grant Nos. 12171044.

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