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Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence and the other is the -norm of coherence . In this paper, we obtain a symmetry-like relation of relative entropy measure of coherence for an n-partite quantum states , which gives lower and upper bounds for . As application of our inequalities, we conclude that when each reduced states is pure, is incoherent if and only if the reduced states and are all incoherent. Meanwhile, we discuss the conjecture that for any state , which was proved to be valid for any mixed qubit state and any pure state, and open for a general state. We observe that every mixture of a state satisfying the conjecture with any incoherent state also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of , the reduced relative entropy measure of coherence satisfies the inequality for any state .
Quantum computing utilizes the superposition and entanglement of quantum states to operate and process information. Its most significant advantage lies in the parallelism of operations [1,2,3]. To achieve efficient parallel computing in quantum computers, quantum coherence is essentially used. Quantum coherence arising from quantum superposition plays a central role in quantum mechanics and so becomes an important physical resource in quantum information and quantum computation . It also plays an important role in a wide variety of research fields, such as quantum biology [5,6,7,8,9,10], nanoscale physics [11,12], and quantum metrology [13,14].
In 2014, Baumgratz et al.  proposed a framework to quantify coherence. In their seminal work, conditions that a suitable measure of coherence should satisfy have been put forward, including nonnegativity, the monotonicity under incoherent completely positive and trace preserving operations, the monotonicity under selective incoherent operations on average and the convexity under mixing of states. By introducing such a rigorous theoretical framework, a mass of properties and operations of quantification of coherence were discussed. Moreover, based on that framework, many coherence measures have been found, such as -norm of coherence and relative entropy of coherence , fidelity and trace norm distances for quantifying coherence , robustness of coherence , geometric measure of coherence , coherence of formation , relative quantum coherence , measuring coherence with entanglement concurrence , trace distance measure of coherence [22,23,24].
In addition, some research related to quantum coherence have been developed, including quantum coherence and quantum correlations [25,26,27,28,29,30], an uncertainly-like relation about coherence , distribution of quantum coherence in multipartite systems , quantum coherence over the noisy quantum channels , maximally coherent mixed states , ordering states with coherence measures , coherence and path information , complementarity relations for quantum coherence , converting coherence to quantum correlations  and logarithmic coherence , quantum coherence and geometric quantum discord . Recently, Guo and Cao  discussed the question of creating quantum correlation from a coherent state via incoherent quantum operations and obtained explicit interrelations among incoherent operations (IOs), maximally incoherent operations, genuinely incoherent operations and coherence breaking operations.
In this paper, we discuss some inequalities on the measures of quantum coherence. The organization of this paper is as follows: In Section 2, we recall the framework of coherence measure and basic properties of quantum coherence. In Section 3, we establish lower and upper bounds for the relative entropy measure of coherence in a multipartite system. In Section 4, we discuss the relation between and . In Section 5, we give our conclusions obtained in this paper.
In this section, we give a review of some fundamental notions about quantification of coherence, such as incoherence states, incoherence operations, and measures of coherence.
Let H be a d-dimensional Hilbert space, whose elements are denoted by the Dirac notations and so on, and let be the -algebra consisting of all bounded linear operators on H. The adjoint operator of an operator T in is denoted by . The identity operator on H is denoted by , or simply, I. We use to denote the set of all density operators (positive and trace-1 operators) on H, whose elements are said to be the states of the quantum system S described by H. Fixed an orthonormal basis (ONB) for H, a state of S is said to be incoherent with respect to (w.r.t.) the basis e if . Otherwise, it is said to be coherent w.r.t. e. Let be the set of all states of S that are incoherent w.r.t. e, that is,
For every , we define
Clearly, By definition, a state is incoherent w.r.t e if and only if , i.e., it has a diagonal matrix representation w.r.t. e, i.e.,
where are eigenvalues of and it is coherent w.r.t. e if and only if it can not be written as a diagonal matrix under this basis.
According to , a linear map on the -algebra is a completely positive and trace preserving (CPTP) map if and only if there exists a set of operators in (called Kraus operators of ) with such that
A CPTP map on is said to be an e-incoherent operation (IO) if it has Kraus operators such that for all , it holds that
In this case, we call a set of e-incoherent Kraus operators of .
In order to measure coherence, Baumgratz et al.  presented the following four defining conditions for a coherence measure :
if and only if
for any e-incoherent operation and any state .
for any e-incoherent operation with a set of e-incoherent Kraus operators and any state where with
for any ensemble
It was proved in  that the relative entropy and the -norm measure of coherence satisfy these defining conditions, which are defined as follows:
where is the von Neumann entropy, and
Notably, for a bipartite quantum system , the reference basis for can be taken as a local basis:
where and are the orthonormal bases for and , respectively. In this case, every of has the following representation:
Thus, a state of the system is incoherent w.r.t. if and only if , i.e.,
Moreover, let and . Then from Equations (3) and (4), we get that
In next section, we derive some inequalities, which give lower and upper bounds for the relative entropy of coherence of multi-partite states.
3. Lower and Upper Bounds for the Relative Entropy of Coherence
Xi et al.  proved that for any bipartite quantum state , the relative entropy of coherence obeys some uncertainty-like relation by using the properties of relative entropy, which reads
Afterwards, Liu et al.  proved that any tripartite pure state satisfies
where and , provided that
for some Combining Equations (6) and (7), the following inequality was derived in :
As a generalization of inequalities (10) and (13), we can prove the following inequalities (14) for any n-partite state of the system , which give lower and upper bounds for the relative entropy of coherence. To do this, we let be an orthogonal basis for the Hilbert space , and let
which is an orthogonal basis for the Hilbert space . Thus,
becomes an orthogonal basis for the Hilbert space . With these notations, we have the following.
For any state of the system , it holds that
where denotes the reduced state of on the subsystem .
To prove that the first inequality in Equation (14) holds, we know from Equation (6) that
Next, let us prove that the second inequality in (14) holds by using mathematical induction. Firstly, we know from Equation (10) that the desired inequality holds for and any bipartite state. Secondly, we assume the second inequality in (14) holds for and any -partite state. Then for any N-partite state , we have
Thus, the validity of the second inequality in Equation (14) is proved. The proof is completed. □
As immediate application of Theorem 1, we have the following corollaries.
Let be a state of the system . If is incoherent, then the reduced states and are all incoherent. The converse is true if each reduced states is pure.
Let be a state of the system such that the reduced states are pure and incoherent. Then
It is remarkable that the equalities in Equation (14) may hold in some cases. For example, when and
the maximally coherent state
due to the fact that for , and
Moreover, the second inequality in Equation (14) also becomes equality when . This shows that the inequalities in Equation (14) are tight and can not be improved.
4. The Relation between and
In this section, we discuss the relation between and . Rana et al. found that the inequality
holds for any mixed qubit state (, Proposition 1) and any pure state (, Proposition 3). Moreover, they conjectured that the inequality (17) holds for all states . It was also proved (, Proposition 6) that inequality (17) holds for any state of the form provided that is an incoherent state w.r.t. the reference basis. As an extension of this result, we have the following.
Let ρ be a state of S satisfying Equation (17) and let σ be any incoherent state of S. Then every mixture of ρ and σ satisfies (17).
The convexity of implies that
The proof is completed. □
Rana et al. proved in (, Proposition 6) that for arbitrary state of a d-dimensional system, it holds that
So, for all . Thus, if we redefine the von Neumann entropy as , then the resulted relative entropy of coherence reads
This leads to the following inequality:
In this paper, we have established lower and upper bounds for relative entropy of coherence for an n-partite quantum states . As application of our inequalities, we have found that when each reduced states is pure, is incoherent if and only if the reduced states and are all incoherent. Moreover, we have discussed the conjecture that for any state and observed that every mixture of a state satisfying the conjecture with any incoherent state also satisfies the conjecture. We have also proved that when the von Neumann entropy is defined by the natural logarithm ln instead of , the reduced relative entropy measure of coherence satisfies the inequality for any state .
Conceptualization, C.Z., Z.G. and H.C.; methodology, C.Z., Z.G. and H.C.; validation, C.Z., Z.G. and H.C.; formal analysis, C.Z. and H.C.; investigation, C.Z., Z.G. and H.C.; writing-riginal draft preparation, C.Z.; writing-review and editing, H.C.; visualization, Z.G.; supervision, H.C.; project administration, funding acquisition, Z.G. and H.C. All authors have read and agreed to the published version of the manuscript.
This work was supported by the National Natural Science Foundation of China (Nos. 11871318, 11771009) and the Fundamental Research Funds for the Central Universities (GK201903001).
Conflicts of Interest
The authors declare no conflict of interest.
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