Shan-Shan Yuwen¹, Lian-He Shao², Zheng-Jun Xi^,1,?1 College of Computer Science, Shaanxi Normal University, Xi'an 710062, China
2 School of Computer Science, Xi'an Polytechnic University, Xi'an 710048, China

Corresponding authors: ^?E-mail:xizhengjun@snnu.edu.cn

Received:2019-05-5Accepted:2019-06-18Online:2019-09-1

Fund supported:

*Supported by the National Natural Science Foundation of China under Grant.61671280 Supported by the National Natural Science Foundation of China under Grant.11771009 Supported by the National Natural Science Foundation of China under Grant.11847101 by the Natural Science Basic Research Plan in Shaanxi Province of China under Grant.2017KJXX-92 by the Fundamental Research Funds for the Central Universities under Grant.GK201902007 by the Funded Projects for the Academic Leaders and Academic Backbones, Shaanxi Normal University under Grant.16QNGG013

Abstract
We find tight upper bound on the coherence of a superposition of two states in terms of the coherence of the two states constituting the superposition with $l_{1}$-norm of coherence. Our upper bound is tighter than the one presented by Liu, et al. [Quantum Inf. Process. 15 (2016) 4209.] We also generalize the results to the case that the superposition is constituted with more than two states in high dimension, and we give the corresponding upper bounds.
Keywords： quantum coherence;superposition states;$l_1$-norm of coherence


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Shan-Shan Yuwen, Lian-He Shao, Zheng-Jun Xi. Coherence of Superposition States*. [J], 2019, 71(9): 1084-1088 doi:10.1088/0253-6102/71/9/1084

1 Introduction

Quantum coherence arises from quantum superposition, which plays a central role in applications of quantum physics and quantum information science, and also a common necessary condition for entanglement and other types of quantum correlations. Baumgratz, et al. proposed a theoretical framework for quantitative study of quantum coherence from the perspective of resource theory.^[1] Based on this, various ways have been presented to develop the resource-theoretic framework for understanding quantum coherence,^[2-11] we refer to two comprehensive reviews^[12-13] for more discussions about resource theory of coherence.

Although there are many discussions on coherence measure for quantum states, there are few studies on superposition states that what is the relationship between the superposition state and the superimposed states. In the field of entanglement, Linden, et al. have firstly studied the relationship between the entanglement of a superposition state and its arbitrary decomposition in Ref. [14]. Shortly after, for the same question, Gour gave a tight upper bound and a new lower bound on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition.^[15] Thereafter, many interesting results have been obtained.^[16-26] In quantum coherence theory, we are also interested in the relationship between the coherence of $|\Omega\rangle$ and that of $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, where $|\Omega\rangle=\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle$ with $|\alpha_1|^2+|\alpha_2|^2=1$. In particular, the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are maximum coherent states on qubit system, the coherence of superposition $|\Omega\rangle$ is possible to zero. We can raise a problem: given a superposition state, what is the relation between the coherence of it and those of the components in the superposition? Liu, et al. discussed the bounds of coherence of superpositions with two components using $l_1$-norm of coherence, where every component comes from the qubit system.^[27] Using three different measures, Yue, et al. gave the upper and lower bounds for coherence of the superposition states of two states in terms of the coherence of the two states being superposed.^[28]

In this paper we give a tight upper bound on the coherence of superposition states with two orthogonal states, which is tighter than the one presented in Ref. [27]. We then generalize the ideas raised in Ref. [28] to the coherence of superpositions with more than two $(n>2)$ components. We give the upper bounds using $l_{1}$-norm of coherence. This paper is organized as follows. In Sec. 2, we give the tight upper bound for the superposition of two states using $l_1$-norm of coherence. We then give the upper bounds for the superposition of more than two states using $l_1$-norm of coherence. We summarize our results and discuss further problems in Sec. 3.

2 The Coherent Superposition of Pure Coherent States

We introduce some concepts about coherence measure that can be used for our main results.^{[1, 12]} Given a $d$-dimension Hilbert space $\mathcal{H}$ with a fixed orthogonal basis $\{|j\rangle\}_{j=1}^{d}$, we denote the set of all density operators acting on $\mathcal{H}$ by $\mathcal{D}(\mathcal{H})$. The density operators which are diagonal in this fixed basis are called incoherent, we denote the set of all incoherent states by $\mathcal{I}$, and $\mathcal{I}\subset\mathcal{D}(\mathcal{H})$. Any incoherent state $\delta$ is of the form

(1)$\delta=\sum_{j=1}^{d}\delta_j|j\rangle\langle j|,$
where $\delta_j$ are probability distribution. Any state which cannot be written as above form is defined as the coherent state, which means the coherence is basis-dependent. Baumgratz, et al. presented the axiomatic approach to quantify coherence in Ref. [1]. Subsequently, several authors generalized this result to different measures of coherence, and many interesting results are given. In this paper, we focus on the $l_1$-norm of coherence. For any quantum state $\rho$ on $\mathcal{H}$, the $l_{1}$-norm of coherence^[1] is defined as

(2)$\mathcal{C}_{l_1}(\rho):=\min_{\delta \in I}\parallel\rho-\delta\parallel_{l_{1}}\,. $
This quantifier has a more directly expression,^[1] and

(3)$\mathcal{C}_{l_{1}}(\rho)=\sum_{i\neq j}|\rho_{ij}|\,, $
where $\rho_{ij}$ is the matrix elements of density operator $\rho$ under the fixed basis $\{|j\rangle\}$.

We often say that a unit vector $|\psi\rangle$ is a pure state or $|\psi\rangle\langle\psi|$ is a pure state, and we use $\psi$ to denote pure state $|\psi\rangle$ in $\mathcal{D}(\mathcal{H})$. For any pure state $|\psi\rangle=\sum_ja_j|j\rangle$, one can easily check ^{[4, 29-30]}

(4)$\mathcal{C}_{l_{1}}(\psi)=\Bigl(\sum_{j}|a_{j}|\Bigr)^{2}-1 \leq d-1\,. $
In this paper, we are interested in the relation between the coherence of a superposition state and the coherence of the individual states in the superposition. Without of loss generality, we suppose that

(5)$|\Omega\rangle=\sum_{i=1}^k\alpha_i|\psi_i\rangle\,, $
where $2\leq k\leq d$, and $\sum_i|\alpha_i|^2=1$. Here, the states $\{|\psi_i\rangle\}$ are coherence states with the fixed basis $\{|j\rangle\}$, and

(6)$|\psi_i\rangle=\sum_{j=1}^d\mu_j^i|j\rangle\,, $
where $\sum_j|\mu_j^i|^2=1$.

2.1 The Superposition with Qubit System

Firstly, we consider the superposition state constituted with two individual states in qubit system.

Proposition 1 Given a state

(7) $|\Omega\rangle=\sum_{i=1}^2\alpha_i|\psi_{j}\rangle\,, $
where $|\psi_j\rangle=a_j|0\rangle+b_j|1\rangle$, and $\{|\psi_i\rangle\}$ are orthogonal, $\alpha_i$, $a_j$, and $b_j$ are real numbers and satisfy $\sum_{i=1}^2\alpha_i^2 =1$, $a_j^2+b_j^2=1$, $j=1,2$. The coherence of the superposition $|\Omega\rangle$ satisfies

(8) $\mathcal{C}_{l_1}(\Omega)<\alpha_1^2\mathcal{C}_{l_1}(\psi_1)+\alpha_2^2\mathcal{C}_{l_1}(\psi_2)+2|\alpha_1\alpha_2|\,. $
Proof Let us consider the states:

(9) $|\psi_{1}\rangle=a_1|0\rangle+b_1|1\rangle,\quad|\psi_{2}\rangle=a_2|0\rangle+b_2|1\rangle\,,\\ |\Omega\rangle=\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle=(\alpha_1a_1+\alpha_2a_2)|0\rangle\\ \quad\quad\,\,\,+(\alpha_1b_1+\alpha_2b_2)|1\rangle\,.$
From Eq. (4), we have

(10) $\mathcal{C}_{l_1}(|\psi_1\rangle)=2|a_1||b_1|,\quad\mathcal{C}_{l_1}(|\psi_2\rangle)=2|a_2||b_2|\,,\\ \mathcal{C}_{l_1}(|\Omega\rangle)=2(|\alpha_1a_1+\alpha_2a_2|)(|\alpha_1b_1+\alpha_2b_2|)\,. $
Then we will have

(11) $\mathcal{C}_{l_1}(|\Omega\rangle)=2|\alpha_1^{2}a_1b_1+a_1b_2\alpha_1\alpha_2+a_2b_1\alpha_1\alpha_2+\alpha_2^2a_2b_2|\\ \leq\alpha_1^{2}\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^{2}\mathcal{C}_{l_{1}}(\psi_{2})+|2(a_1b_2+a_2b_1)\alpha_1\alpha_2|. $
Because $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthogonal, we can get

(12) $a_1a_2+b_1b_2=0\,.$
Combining $a_j^2+b_j^2=1$ with Eq. (12), we have

(13) $a_1^2+a_2^2=1\,.$
Thus, we have

(14) $(a_1b_2+a_2b_1)^2=a_1^2b_2^2+a_2^2b_1^2+2a_1a_2b_1b_2\\ =a_1^2(1-a_2^2)+a_2^2(1-a_1^2)-2a_1^2a_2^2\\ =1-4a_1^2(1-a_1^2)\\ =(1-2a_1^2)^2\,. $
Since $a_1$ is a real number and can not be $0$ and $1$, we have $|1-2a_1^2|<1$. Then we can get our upper bound

(15) $\mathcal{C}_{l_1}(|\Omega\rangle)\leq\alpha_1^{2}\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^{2}\mathcal{C}_{l_{1}}(\psi_{2})+2|(1-2a_1^2)\alpha_1\alpha_2|\\ <\alpha_1^{2}\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^{2}\mathcal{C}_{l_{1}}(\psi_{2})+2|\alpha_1\alpha_2|. $
The coefficient $|\alpha_1\alpha_2|$ can be considered as the relative coherence, because we find that the $|\alpha_1\alpha_2|$ is only relative to the states $|\psi_1\rangle$ and $|\psi_2\rangle$, from Eq. (15) we can see that the coherence of $|\Omega\rangle$ is also related to the internal coherence of the states $|\psi_1\rangle$ and $|\psi_2\rangle$. And for the same question in Ref. [27], the authors did not consider the nature of orthogonality, that is why our upper bound is tighter than the one they presented. Now we consider two examples.

Example 1 Consider the states

(16) $|\psi_1\rangle=\sqrt{\frac{2}{3}}|0\rangle+\sqrt{\frac{1}{3}}|1\rangle\,,\\ |\psi_2\rangle=-\sqrt{\frac{1}{3}}|0\rangle+\sqrt{\frac{2}{3}}|1\rangle\,,\\ |\Omega\rangle=\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle\\ =\Big(\sqrt{\frac{2}{3}}\alpha_1-\sqrt{\frac{1}{3}}\alpha_2\Big)|0\rangle+\Big(\sqrt{\frac{1}{3}}\alpha_1+\sqrt{\frac{2}{3}}\alpha_2\Big)|1\rangle. $
It is clearly that $\mathcal{C}_{l_1}(\psi_1)=\mathcal{C}_{l_1}(\psi_2)=({2\sqrt{2}})/{3}$. The coherence of the superposition $|\Omega\rangle$ satisfies

(17) $\mathcal{C}_{l_1}(\Omega)\leq\alpha_1^2\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^2\mathcal{C}_{l_{1}}(\psi_{2})+\frac{2}{3}|\alpha_1\alpha_2|\\ <\alpha_1^2\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^2\mathcal{C}_{l_{1}}(\psi_{2})+2|\alpha_1\alpha_2|. $
Example 2 Consider the states

(18) $|\psi_1\rangle=\sqrt{\frac{1}{2}}|0\rangle+\sqrt{\frac{1}{2}}|1\rangle\,,\\ |\psi_2\rangle=\sqrt{\frac{1}{2}}|0\rangle-\sqrt{\frac{1}{2}}|1\rangle\,,\\ |\Omega\rangle=\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle\\ =\Big(\sqrt{\frac{1}{2}}\alpha_1+\sqrt{\frac{1}{2}}\alpha_2\Big)|0\rangle+\Big(\sqrt{\frac{1}{2}}\alpha_1-\sqrt{\frac{1}{2}}\alpha_2\Big)|1\rangle. $
It is clearly that $\mathcal{C}_{l_1}(\psi_1)=\mathcal{C}_{l_1}(\psi_2)=1$. The coherence of the superposition $|\Omega\rangle$ satisfies

(19)$\mathcal{C}_{l_1}(\Omega)=2\Big(|\sqrt{\frac{1}{2}}\alpha_1+\sqrt{\frac{1}{2}}\alpha_2|\Big)\Big(|\sqrt{\frac{1}{2}}\alpha_1-\sqrt{\frac{1}{2}}\alpha_2|\Big)\\ <\alpha_1^2\mathcal{C}_{l_{1}}(\psi_{1})+\alpha_2^2\mathcal{C}_{l_{1}}(\psi_{2})+2|\alpha_1\alpha_2|. $
From Proposition 1, we can easily get the maximum increase of the coherence of superposition states.

Corollary 1 Given states $|\psi_1\rangle$ and $|\psi_2\rangle$ orthogonal, the maximum increase of coherence satisfies

(20) $\mathcal{C}_{l_1}(|\Omega\rangle)-\alpha_1^{2}\mathcal{C}_{l_{1}}(\psi_{1})-\alpha_2^{2}\mathcal{C}_{l_{1}}(\psi_{2})<1\,. $
From the Example 1, we can see that the increase of the coherence of $|\Omega\rangle$ is $({2}/{3})|\alpha_1\alpha_2|$. And this is for the superposition of orthogonal states. Now we consider the general case, which means the two superimposed states are non-orthogonal, we can also get an upper bound. We believe that the upper bound of the coherence of the superposition states with two non-orthogonal states is consistent with the upper bound of the coherence of the superposition states with two orthogonal states proposed by Liu, et al.^[27]

Proposition 2 Given a state $|\Omega\rangle=\sum_{i=1}^2\alpha_i|\psi_{j}\rangle$, where $|\psi_j\rangle=a_j|0\rangle+b_j|1\rangle$, $\alpha_i$, $a_j$, and $b_j$ are real numbers and satisfy $\sum_{i=1}^2\alpha_i^2 =1$, $a_j^2+b_j^2=1$, $j=1,2$. The coherence of the superposition $|\Omega\rangle$ satisfies

(21) $\mathcal{C}_{l_1}(\Omega)\leq\alpha_1^2\mathcal{C}_{l_1}(\psi_1)+\alpha_2^2\mathcal{C}_{l_1}(\psi_2)+2|\alpha_1\alpha_2|\,. $
From Proposition 2, we can easily get the maximum increase of the coherence of superposition states.

Corollary 2 Given arbitrary states $|\psi_1\rangle$ and $|\psi_2\rangle$, the maximum increase of coherence satisfies}

(22) $\mathcal{C}_{l_1}(|\Omega\rangle)-\alpha_1^{2}\mathcal{C}_{l_{1}}(\psi_{1})-\alpha_2^{2}\mathcal{C}_{l_{1}}(\psi_{2})\leq1.$

2.2 The Superposition with more than Two States

The above upper bounds are for the states with qubit system. Yue, et al. proposed upper and lower bounds for coherence of the superposition with two states in $d$-dimensional system. We now present the upper bound for the superposition of more than two states using $l_1$-norm of coherence.

Proposition 3 Given a state $|\Omega\rangle=\sum_{i=1}^3\alpha_i|\psi_{j}\rangle$, where $\{|\psi_{j}\rangle\}$ are orthogonal, $\alpha_i$ are real numbers and satisfy $\sum_{i=1}^3|\alpha_i|^2 =1$. The coherence of the superposition $|\Omega\rangle$ satisfies}

(23) $\mathcal{C}_{l_{1}}(\Omega)\leq\min\left\{f,g\right\}\,,$
where

(24) $f=\sum_{i=1}^3|\alpha_i|^2\mathcal{C}_{l_1}(\psi_i)+2\sum_{i\neq j}|\alpha_i\alpha_j|(d-1)\,,\\ g=\sum_{i=1}^3|\alpha_i|^2\mathcal{C}_{l_1}(\psi_i)\\ \quad\quad+2\sum_{i\neq j}|\alpha_i\alpha_j|\sqrt{(\mathcal{C}_{l_{1}}({\psi_{i}})+1)(\mathcal{C}_{l_{1}}(\psi_{j})+1)}\,.$
Proof Without loss of generality, let

(25) $|\psi_{1}\rangle=\sum_{j=1}^{d}a_{j}|j\rangle, \quad|\psi_{2}\rangle=\sum_{j=1}^{d}b_{j}|j\rangle, \quad|\psi_{3}\rangle=\sum_{j=1}^{d}c_{j}|j\rangle, $
where $a_{j}$, $b_{j}$, and $c_{j}$ are complex numbers and satisfy $\sum_{j=1}|a_{j}|^{2}=\sum_{j=1}|b_{j}|^{2}=\sum_{j=1}|c_{j}|^{2}=1$. For superposition state $|\Omega\rangle$, we have

(26) $\mathcal{C}_{l_{1}}(\Omega)=\sum_{i\neq j}|(\alpha_1 a_{i}+\alpha_2 b_{i}+\alpha_3 c_{i}) (\alpha_1 a_{j}+\alpha_2 b_{j}+\alpha_3 c_{j})| \\ =\sum_{i\neq j}|\alpha_1^{2}a_{i}a_{j}+\alpha_2^{2}b_{i}b_{j}+\alpha_3^{2}c_{i}c_{j} +2\alpha_1\alpha_2 a_{i}b_{j}+2\alpha_1\alpha_3 a_{i}c_{j} +2\alpha_2\alpha_3 b_{i}c_{j}| \\ \leq|\alpha_1|^{2}\sum_{i \neq j}|a_{i}a_{j}|+|\alpha_2|^{2}\sum_{i \neq j} |b_{i}b_{j}|+|\alpha_3|^{2}\sum_{i\neq j}|c_{i}c_{j}| +2|\alpha_1\alpha_2|\sum_{i\neq j}|a_{i}b_{j}|+2|\alpha_1\alpha_3|\sum_{i\neq j} |a_{i}c_{j}|+2|\alpha_2\alpha_3|\sum_{i\neq j}|b_{i}c_{j}| \\ =|\alpha_1|^{2}\mathcal{C}_{l_{1}}(\psi_{1})+|\alpha_2|^{2}\mathcal{C}_{l_{1}}(\psi_{2}) +|\alpha_3|^{2}\mathcal{C}_{l_{1}}(\psi_{3}) +2|\alpha_1\alpha_2|\sum_{i\neq j}|a_{i}b_{j}|+2|\alpha_1\alpha_3|\sum_{i\neq j}|a_{i}c_{j}| +2|\alpha_2\alpha_3|\sum_{i\neq j}|b_{i}c_{j}| \\ \leq\sum_{i=1}^3|\alpha_i|^2\mathcal{C}_{l_1}(\psi_i)+2(|\alpha_1\alpha_2|+|\alpha_1\alpha_3| +|\alpha_2\alpha_3|)(d-1), $
where the first inequality comes from the absolute value inequality, the second inequality comes from successive application of the mean inequality.

Next we will prove the second upper bound in Eq. (23). Using the relation (4), we have

(27) $\mathcal{C}_{l_{1}}(\Omega)=\Bigl(\sum_{i=1}|\alpha_1 a_{i}+\alpha_2 b_{i}+\alpha_3 c_{i}|\Bigr)^{2} -1\leq\Bigl(\sum_{i}|\alpha_1 a_{i}|+\sum_{i}|\alpha_2 b_{i}|+\sum_{i}|\alpha_3 c_{i}|\Bigr)^{2}-1\\ =|\alpha_1|^{2}\Bigl(\sum_{i}|a_{i}|\Bigr)^{2}+|\alpha_2|^{2}\Bigl(\sum_{i}|b_{i}|\Bigr)^{2}+|\alpha_3|^{2} \Bigl(\sum_{i}|c_{i}|\Bigr)^{2}+ 2|\alpha_1\alpha_2|\Bigl|\Bigl(\sum_{i}|a_{i}|\Bigr)\Bigl(\sum_{i}|b_{i}|\Bigr)\Bigr|\\\ \quad+2|\alpha_1\alpha_3|\Bigl|\Bigl(\sum_{i}|a_{i}|\Bigr)\Bigl(\sum_{i}|c_{i}|\Bigr)\Bigr|+ 2|\alpha_2\alpha_3|\Bigl|\Bigl(\sum_{i}|b_{i}|\Bigr)\Bigl(\sum_{i}|c_{i}|\Bigr)\Bigr|-1\\ =|\alpha_1|^{2}\mathcal{C}_{l_{1}}(\psi_{1})+|\alpha_2|^{2}\mathcal{C}_{l_{1}}(\psi_{2})+|\alpha_3|^{2} \mathcal{C}_{l_{1}}(\psi_{3})+2|\alpha_1\alpha_2|\Bigl|\Bigl(\sum_{i}|a_{i}|\Bigr)\Bigl(\sum_{i}|b_{i}|\Bigr)\Bigr|\\ \quad+2|\alpha_1\alpha_3|\Bigl|\Bigl(\sum_{i}|a_{i}|\Bigr)\Bigl(\sum_{i}|c_{i}|\Bigr)\Bigr|+2|\alpha_2\alpha_3|\Bigl| \Bigl(\sum_{i}|b_{i}|\Bigr) \Bigl(\sum_{i}|c_{i}|\Bigr)\Bigr|\\ =\sum_{i=1}^3|\alpha_i|^2\mathcal{C}_{l_1}(\psi_i)+2\sum_{i\neq j}|\alpha_i\alpha_j| \sqrt{(\mathcal{C}_{l_{1}}({\psi_{i}}) + 1)(\mathcal{C}_{l_{1}}(\psi_{j}) + 1)}\,, $
where the inequality is due to the absolute value inequality.

Note that the relationship between the coherence of two orthogonal states and the coherence of their combination using $l_{1}$-norm of coherence is given in Ref. [28]. Our result focuses on the coherence of superpositions with more than two components, and our result can also be generalized to the state which is superposed by $n$ orthogonal states.

Corollary 3 Given states $\{|\psi_{i}\rangle\}$ mutually orthogonal, the upper bound of coherence of the superposition state $|\Omega\rangle = \sum_{i=1}^{n}\alpha_{i}|\psi_{i}\rangle$ with $\sum|\alpha_{i}|^{2}=1$ is given as

(28) $\mathcal{C}_{l_{1}}(\Omega)\leq\min\left\{f,g,d-1\right\}\,,$
where

(29) $f=\sum_{i=1}^{n}|\alpha_{i}|^{2}\mathcal{C}_{l_{1}}(\psi_{i})+2\sum_{i\neq j}|\alpha_{i}\alpha_{j}| (d-1)\,,\quad g=\sum_{i=1}^{n}|\alpha_{i}|^{2}\mathcal{C}_{l_{1}}(\psi_{i})+2\sum_{i\neq j}|\alpha_{i}\alpha_{j} |\sqrt{(\mathcal{C}_{l_{1}}(\psi_{i})+1)(\mathcal{C}_{l_{1}}(\psi_{j})+1)}\,. $
Remark 1

(i) It is obviously that our results can be reduced from $n$ components to two components.

(ii) We find that for the general states, while satisfying $\sum_{i=1}^3|\alpha_i|^2 =1$, the larger the value of $\alpha_1$, the smaller the values of $\alpha_2$ and $\alpha_3$, the tighter the upper bound we get. For example, we suppose that $\alpha_1=\sqrt{{99}/{100}}$, $\alpha_2=\alpha_3=\sqrt{{1}/{200}}$, we have

(30) $\mathcal{C}_{l_{1}}\leq\min\left\{f,g\right\}\,,$
where

(31) $f=0.99 \mathcal{C}_{l_{1}}(\psi_{1})+0.005 \mathcal{C}_{l_{1}}(\psi_{2}) +0.005 \mathcal{C}_{l_{1}}(\psi_{3}) +0.29(d-1)\,,\\ g=0.99 \mathcal{C}_{l_{1}}(\psi_{1})+0.005 \mathcal{C}_{l_{1}}(\psi_{2}) +0.005 \mathcal{C}_{l_{1}}(\psi_{3})+2\sum_{i\neq j}|\alpha_{i}\alpha_{j}| \sqrt{(\mathcal{C}_{l_{1}}(\psi_{i})+1)(\mathcal{C}_{l_{1}}(\psi_{j})+1)}\,. $
(iii) For the maximum coherent state, this upper bound only makes sense in special cases. We suppose that $\{|\psi_i\rangle\}$ are maximum coherent states, with Proposition 3, we have

(32) $\mathcal{C}_{l_{1}}(\Omega)\leq\min\Bigl\{(d-1)\Bigl(1+2\sum_{i\neq j}|\alpha_i\alpha_j|\Bigr),d\Bigl(1+2\sum_{i\neq j}|\alpha_i\alpha_j|\Bigr)-1\Bigr\}.$
For the special case that $\alpha_1\rightarrow1$, $\alpha_2,\alpha_3\rightarrow0$, we can get a meaningful upper bound $\mathcal{C}_{l_{1}}(\Omega)\leq d-1$, but for another special case that $\alpha_1=\alpha_2=\alpha_3={1}/{\sqrt{3}}$, we get a meaningless upper bound $\mathcal{C}_{l_{1}}(\Omega)\leq \{3(d-1),3d-1\}$.

We find that in Ref. [28], the same problem also exists. When $|\psi_1\rangle$, $|\psi_2\rangle$ are maximum coherent states, $\alpha_1=\alpha_2={1}/{\sqrt{2}}$, the upper bound we get is

(33) $\mathcal{C}_{l_{1}}(\Omega)\leq\min\left\{2(d-1), 2d-1\right\}\,.$
The reason for this is the shrinking of inequations in Eqs. (26) and (27), and we do not have better solutions now.

(iv) We suppose that $|\psi_1\rangle$ is a maximum coherent state, and for other states $\{|\psi_i\rangle\}$, they tend to be incoherent. For this case, we have

(34) $\mathcal{C}_{l_{1}}(\Omega)\leq\min \Bigl\{(d-1)\Bigl(|\alpha_1|^2+2\sum_{i\neq j}|\alpha_i\alpha_j|\Bigr), |\alpha_1|^2(d-1)+2\sqrt{d}(|\alpha_1\alpha_2|+|\alpha_1\alpha_3|) +2|\alpha_2\alpha_3|\Bigr\}, $
when $\alpha_1\rightarrow1$, $\alpha_2,\alpha_3\rightarrow0$, we have $\mathcal{C}_{l_{1}}(\Omega)\leq|\alpha_1|^2(d-1)$.

3 Conclusion

In this paper, we have presented a tighter upper bound on the coherence of a superposition of two states in terms of the coherence of the two states constituting the superposition with $l_{1}$-norm of coherence. We have established the relationship between the coherence of the superposition of three states and the coherence of the three states constituting the superposition with $l_{1}$-norm of coherence. We also generalized the result of the superposition of three states to the superposition of $n$ states. In the calculation we found that it is more difficult for the superposition of more than two states. The bounds we get is not very tight in some cases. For the future work we hope to find new methods to make the upper bounds we get more tighter.

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