Bounds for coherence of quantum superpositions in high dimension

Quantum coherence plays a major role in the promotion for quantum information processing and designing quantum technology. Since coherence is rooted in superposition principle, it is vital to understand the coherence change with respect to superpositions. Here we study the bounds for coherence of quantum superpositions in high dimension. We consider three most frequently used measures of coherence, i.e. the relative entropy of coherence, l1 norm of coherence and robustness of coherence. For a quantum state (an arbitrary dimension) and its arbitrary decomposition, we give the upper and lower bounds for coherence of the superposition state in terms of the coherence of the states being superposed.

Quantum coherence plays a major role in the promotion for quantum information processing and designing quantum technology. Since coherence is rooted in superposition principle, it is vital to understand the coherence change with respect to superpositions. Here we study the bounds for coherence of quantum superpositions in high dimension. We consider three most frequently used measures of coherence, i.e. the relative entropy of coherence, l 1 norm of coherence and robustness of coherence. For a quantum state (an arbitrary dimension) and its arbitrary decomposition, we give the upper and lower bounds for coherence of the superposition state in terms of the coherence of the states being superposed.
Quantum coherence is a fundamental feature of quantum mechanics. As one of the most crucial physical resources, it plays a primary role in quantum information processing 1-3 , computational task 4, 5 , quantum metrology 6,7 , thermodynamics [8][9][10] , and quantum biology 11,12 . The theory of coherence as a resource was first set forth in ref. 13, where the authors introduced a rigorous framework for the quantification of coherence and identified the computable measures of coherence. From resource-driven viewpoint, there is a growing number of work studying coherence including different coherence measures [14][15][16][17][18] , the properties of coherence 19,20 , the freezing phenomenon of coherence 21,22 , the relation among coherence, entanglement and quantum correlation [23][24][25] , and so on refs 26, 27. Despite coherence derives from the superposition of states, the coherence of a superposition state cannot be directly deduced from the coherence of the individual states being superposed. We illustrate it with the following examples: given a state 1 it is obvious that the coherence of |0〉, |1〉 is 0, while the coherence of |Ω 1 〉 reaches maximum value. Here, we consider coherence in the computational basis. In the following, we show an opposite example. Given a state 2 where | ± 〉 = | 〉 ± | 〉 ( 0 1 )/ 2. Interestingly, under computational basis the coherence of |+〉, |−〉 reaches maximum value while the coherence of their superposition |Ω 2 〉 is 0.
Our concern here is that: given an arbitrary state |Ω〉 and its arbitrary decomposition what is the relation between the coherence of |Ω〉 and the coherence of |Φ〉 and |Ψ〉? Because of the importance of coherence in quantum physics and superposition for coherence, the solutions for this problem will provide a theoretical foundation for potential applications of quantum resource and quantum information processing. Similar problem has been studied in the field of entanglement. In 2006, Linden et al. firstly studied the relation between the entanglement of |Ω〉 and the entanglement of |Φ〉 and |Ψ〉 28 , in which they gave the upper bounds on the entanglement of the superposition state in terms of the entanglement of the states being superposed using von Neumann entropy of the reduced state as a measure of entanglement. Thereafter, there are some related works. 1 Ref. 29 gave a tighter upper bound on the same question and also gave a lower bound. Refs 30, 31 studied the same problem considering concurrence (another measure of entanglement) and gave the corresponding upper and lower bound. However, this problem is still open in the field of coherence where only two specific cases have been discussed 32,33 .
In this work, we analyze the relation between the coherence of |Ω〉 and the coherence of its decomposition. We systematically study this problem considering three kinds of coherence measure, and give the corresponding tight upper and lower bounds. Our results can be used for estimating the coherence range of the superposition state. For example, given the coherence of two states, we do not even need to know what the state is, we can estimate the range of the coherence resource we can get from their superposition state. In addition, armed with these relationships on coherence of superpositions, we can easily monitor the coherence change in the quantum information processing, such as coherence distillation. Coherence is likewise a measure of information carrying ability. The more coherence, the more information can be carried in the states.
In our work, we focus on orthogonal version of problem (i.e. |Φ〉 and |Ψ〉 are orthogonal states). We can get the non-orthogonal version of the problem easily by the following decomposition, where |Φ〉 and |Φ ⊥ 〉 are orthogonal states.

Results
Relative entropy of coherence. A well-defined and frequently used coherence measure is the relative entropy of coherence, which is proposed and studied in ref. 13. With a particular entropic formula, the relative entropy of coherence has some clear physical meanings, such as it is equal to the optimal distillation rate for standard coherence distillation 34 , and can also be interpreted as the minimal amount of noise required for fully decohering the state 27,35 . In this section, we study the relationship between the coherence of two orthogonal states and the coherence of its decomposition using the relative entropy of coherence. Given a particular basis re d where ρ is density operator and ρ d denotes the state obtained from ρ by deleting all off-diagonal elements under the particular basis, and S(ρ) is von Neumann entropy of ρ. In the case of a pure state |φ〉, its relative entropy of coherence can be expressed as In the following, we will give the bounds with respect to relative entropy of coherence. The proof of this Theorem is in Methods.
where n is the dimension of Hilbert space.
In this case, we can see that the upper bound in Theorem 1 is tight. Considering p = |α| 2 , the first bound in Eq.
Scientific RepoRts | 7: 4006 | DOI:10.1038/s41598-017-03885-5 Now, we move to analyze the lower bounds. By constructing a special state and measuring it with an incoherent operation, we can get the lower bound for coherence of superpositions considering relative entropy of coherence as stated in following theorem. Its detailed proof is in Methods. for 0 < p < 1, and

Theorem 2. Given two orthogonal states |Φ〉, |Ψ〉, and two complex number α, β satisfying
l 1 norm coherence. Another measure of coherence, l 1 norm, is defined with the off-diagonal elements of the considered quantum state. This definition is intuitive for the measure of coherence and satisfies the necessary properties presented in ref. 13.
It can also be expressed as We consider l 1 norm of coherence measure and obtain corresponding bounds for coherence of superpositions. See Methods for the proof of the following Theorem.
Remark 3: An interesting symmetric inequality can be deduced from the second line in Eq. (14) as follows: This inequality may have applications in other analysis of coherence by considering superposition effects.
Remark 4: Here, we want to emphasize that in Theorem 3 the dimension n is an arbitrary positive integer and our result covers the two dimension case mentioned in ref. 33. Now, we compare the values of two expressions in Theorem 3. The only difference between them locates at the last term. One is 2|αβ| (n − 1), and the other is αβ . Thus, we only need to compare the values of n − 1 and . In Figure 2, we plot function as independent variables whose domains are[0, 10]. Since n − 1 is an integer, it is easy for us to compare the value of Φ + Ψ + C C ( ( ) 1)( ( ) 1) l l 1 1 and coordinate axis scale which can be recognized as n − 1. If n − 1 is smaller than the value of Φ + Ψ + C C ( ( ) 1)( ( ) 1) l l 1 1 , then the upper bound will be α β αβ is the tighter upper bound. Next, we focus on discussing lower bound. Through using the absolute value inequality, we can obtain the lower bound for l 1 norm of coherence of superpositions as illustrated in the following theorem. Its proof can be found in Methods. Theorem 4. Given two orthogonal states |Φ〉, |Ψ〉, and two complex number α, β satisfying α β Φ + Ψ = 1, the coherence of the superposition where n is the dimension of Hilbert space.
Remark 5: An interesting symmetric inequality can be deduced from the second line in Eq. (16) as follows: Same as the Eq. (15), this inequality may have applications in other analysis of coherence by considering superposition effects. The only difference between the first line and second line in Eq. (16) is also in the last term, which are −2|αβ| (n − 1) and − αβ respectively. This comparison is dimension dependent as shown in Fig. 2. Note that since a measure of coherence is nonnegative, we compare the expressions in Eq. (16) and choose the maximum value as the lower bound.

Robustness of coherence.
As a quantifier of the advantage enabled by a quantum state in phase discrimination task, the robustness of coherence is defined and proved to be a full measure for the framework proposed in ref. 13. Robustness of coherence is shown to be an observable related to the notion of coherence witness 17,18 . Given a fixed basis where n is the dimension of Hilbert space. The definition of robustness of coherence of a state ρ is as following 17 : is the convex set of density operators acting on a n-dimensional Hilbert space, and  is the set of all incoherent states. Notice that the robustness of coherence of a pure state |φ〉 satisfies 18 , From this equation, we can see the equivalence between robustness of coherence and l 1 norm of coherence for pure states. The bounds are same as in Theorem 3 and Theorem 4.
Coherence of superpositions for two states from orthogonal subspaces. Quantum states from orthogonal subspaces play an important role in quantum information and coding 36 . Here we consider the bounds for this special case. That is, the decomposition states (|Φ〉 and |Ψ〉) come from orthogonal subspaces. . Then the coherence of the superposition state |Ω〉 has the following bounds: (I) for relative entropy of coherence re re re 2 2 2 2 where |α| 2 + |β| 2 = 1 and For the superposition of two states from two orthogonal subspaces, the relative entropy coherence is the sum of three terms: the average of the coherence of two states being superposed, the binary entropy of probability |α| 2 . Instead of bounds, this is an accurate expression as shown in Eq. (20). The maximum increase for coherence is bounded as following: Scientific RepoRts | 7: 4006 | DOI:10.1038/s41598-017-03885-5 re re re 2 2 The bounds for l 1 norm coherence of superposition considering orthogonal subspaces in Corollary 1 is tighter compared with the general case in Theorem 3 and 4. Specifically, the expression given in the first line of Eq. (21) is lowered and the expression given in Eq. (22) is raised.

Discussion
Using the relative entropy of coherence, l 1 norm of coherence and robustness of coherence, we give the upper and lower bounds for coherence of the superposition. Bounds for coherence of superpositions of multiple terms can be easily found by generalizing our methods. Superposition is the root of both coherence and entanglement. Coherence is a property of an arbitrary quantum state, while entanglement is a property of a bipartite or multipartite state. In this sense, it is straightforward to understand that coherence is a more general quantum property than entanglement 20 . Entanglement of the superposition is closely linked to the entanglement of these two superposed states 28,29,31 . In this work, we have shown that the coherence of the superposition is intimately related to the coherence of these two superposed states. However, it is still unclear that the difference between entanglement of superpositions and coherence of superpositions. As shown in Tables 1 and 2, there exist strong similarities between coherence and entanglement for the analogy problem.
From the expressions in Table 2, we can see that the upper bound and lower bound for coherence are symmetric about the statistical average coherence of the two superposed orthogonal states, which is defined as , and the green line as shown in Fig. 2(a). While the bounds for entanglement are asymmetric around the statistical average coherence of the two superposed orthogonal states defined as , and the green line as shown in Fig. 2(b). Here, we consider the case where our lower bound is valid, i.e. positive. In the following, we give an intuitive comparison between coherence and entanglement for Table 2 by taking a simple example. We focus on fluctuating ranges of l 1 norm coherence and concurrence for the superposition of two orthogonal states. Here in order to make the discussion meaningful we consider the superposition of two bipartite orthogonal states.
Example 2: Let |Ψ〉 and |Φ〉 be two orthogonal states, defined by 00  11  2 and 00 2 01 10 11 5 (24) Their coherence and entanglement are depicted in Fig. 3. Evidently, the fluctuation of coherence is smoother than entanglement. The trend of change for coherence and entanglement with respect to the parameter |α| 2 is not positive correlated with each other. Furthermore, there are two inflections points in the change of exact value of coherence while only one for entanglement considering the change of |α| 2 in the case of example 2. Also in this upper bound for partial entropy of entanglement 29 upper bound for relative entropy of coherence lower bound for partial entropy of entanglement 29 lower bound for relative entropy of coherence  Table 1. Comparison between bounds on partial entropy of entanglement and relative entropy coherence for the superposition of two orthogonal states ( is an entanglement measure of pure state φ AB 28 ).
upper bound for concurrence 31 upper bound for l 1 norm of coherence   Table 2. Comparison between bounds on concurrence and l 1 norm coherence for the superposition of two orthogonal states (Here E C is concurrence which is a measure of entanglement 31,38 ).
example we can see that when entanglement of the superposition state disappears, the coherence still exists. From these observations, we can see that coherence is a kind of resource easier accessed compared with entanglement. In some information processing tasks, such as coherence distillation, we are more concerned with the coherence of the state, rather than the state itself. When we only know the coherence value of |Φ〉 and |Ψ〉, and do not know what the specific state is, we can use our results to estimate the upper and lower bound of the superposition, then we can know that the least amount and the most amount of coherence we can distill from the superposition state.
Another interesting and challenging problem related to our work is the effects of superposition on the coherence of superposition of mixed states, which can be used to analyze the effects of de-coherence resulted from the interactions with environment.  and with probability 1 − q, the state becomes . This measurement is an incoherent operation and C re is a coherence monotone 13 , we can get the following inequality

Proof of the
re re re re 2 Since Γ ≥ C ( ) 0 re , we can get the following inequality Due to |α| 2 + |β| 2 = 1, and we can obtain . In sub-figure (b), the black dotted line is the actual values of concurrence, the blue dashed line is upper bound on concurrence, the red solid line is lower bound on concurrence, and the green line represents the statistical average entanglement of the superposed states, . Note that coherence and entanglement are both non-negative. Through substituting Eq. (29) into Eq. (28), and we can obtain 2 Finally, substitute Eq. (30) into Eq. (26), and the result can be proved re re re 2 2 2 2 with 0 < p < 1. □ (2) The other bound in Eq. (7) is obtained through getting a tighter lower bound of Γ C ( ) re . Firstly, we introduce two lemmas on which the proof is based. 37 ) Suppose ρ and σ are density matrices such that the trace distance is given by T,

Lemma 1. (Fannes-Audenaert inequality
where d is the dimension of the Hilbert space, and

Lemma 2.
Given two orthogonal states |Φ〉, |Ψ〉, and two complex number α, β satisfying α β Then we can get the diagonal matrices of |Ω〉〈Ω| and Γ Γ by deleting all off-diagonal elements as follow   where the first inequality is due to fundamental inequality, and the last equality is due to the □ We can get the following inequality from lemma 1 and 2 Furthermore, we can get re re and through shifting one term in LHS we can get Substitute Eq. (42) into Eq. (25), then we can get

SA S A S
A , where we used an ancillary system A. The coherence of χ′ can be expressed as . We measure ancillary system A with Kraus operators . This measurement is an incoherent operation and C re is a coherence monotone 13 , we can get re re re re 1 2 2 Since Γ ≥ C ( ) 0 re 2 , we can obtain re re re 1 2 Through the following setting like in ref. 29, we can obtain Γ = Φ ω ω p t e p t e cos 1 sin 0 (47) i i 1 2 The parameters α and β satisfying Eqs (46) and (47) are as follows  . Let α β Ω = Φ + Ψ , and Ω = 1.
(1) Firstly, we prove the first upper bound in Eq. (14). From the definition of l 1 norm of coherence Eq. (12). We have  where the first inequality is due to absolute value inequality. Successive application of the mean inequality, we will get which is the first line in Eq. (14).
(2) Now, we prove the other upper bound in Eq. (14). From the definition of l 1 norm of coherence Eq. (13), we have  where the first inequality is due to absolute value inequality. Further, we have  . Let α β Ω = Φ + Ψ , and Ω = 1.
(1) From the definition of l 1 norm of coherence Eq. (12), we have where the first inequality is due to absolute value inequality. Further, we find (1) According to the conditions of Corollary Thus, we can get Eq. (20) in terms of Eq. (6).
(2) Let m 1 Since the lower bound Eq. (16) presented in Theorem 4 also works for this case, we can get the lower bound for the coherence of superposition of two states from two orthogonal subspaces as follows: