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Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

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 Cr(ρ)=S(ρdiag)−S(ρ) and the other is the ℓ1-norm of coherence Cℓ1(ρ)=∑i≠j|ρij|. In this paper, we obtain a symmetry-like relation of relative entropy measure Cr(ρA1A2⋯An) of coherence for an n-partite quantum states ρA1A2⋯An, which gives lower and upper bounds for Cr(ρ). As application of our inequalities, we conclude that when each reduced states ρAi is pure, ρA1⋯An is incoherent if and only if the reduced states ρAi and trAiρA1⋯An(i=1,2,…,n) are all incoherent. Meanwhile, we discuss the conjecture that Cr(ρ)≤Cℓ1(ρ) for any state ρ, 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 η of a state ρ satisfying the conjecture with any incoherent state σ also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of log2, the reduced relative entropy measure of coherence C¯r(ρ)=−ρdiaglnρdiag+ρlnρ satisfies the inequality C¯r(ρ)≤Cℓ1(ρ) for any state ρ.


Introduction
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 [4]. 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. [15] 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 1 -norm of coherence and relative entropy of coherence [15], fidelity and trace norm distances for quantifying coherence [16], robustness of coherence [17], geometric measure of coherence [18], coherence of formation [19], relative quantum coherence [20], measuring coherence with entanglement concurrence [21], trace distance measure of coherence [22][23][24].
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 C r (ρ) and C 1 (ρ). In Section 5, we give our conclusions obtained in this paper.

Preliminaries
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 |ψ , |x and so on, and let B(H) be the C * -algebra consisting of all bounded linear operators on H. The adjoint operator of an operator T in B(H) is denoted by T † . The identity operator on H is denoted by I H , or simply, I. We use D(H) 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) e = {|e i } d i=1 for H, a state ρ of S is said to be incoherent with respect to (w.r.t.) the basis e if e i |ρ|e j = 0(i = j). Otherwise, it is said to be coherent w.r.t. e. Let I(e) be the set of all states of S that are incoherent w.r.t. e, that is, For every ρ ∈ D(H), we define Clearly, ρ e-diag ∈ I(e). By definition, a state ρ is incoherent w.r.t e if and only if ρ = ρ e-diag , i.e., it has a diagonal matrix representation w.r.t. e, i.e., e if and only if it can not be written as a diagonal matrix under this basis.
According to [42], a linear map E on the C * -algebra B(H) is a completely positive and trace preserving (CPTP) map if and only if there exists a set of operators K 1 , . . . , K m in B(H) (called Kraus operators of E ) with ∑ m n=1 K † n K n = I H such that A CPTP map E on B(H) is said to be an e-incoherent operation (IO) if it has Kraus operators K 1 , . . . , K m such that for all n = 1, 2, . . . , m, it holds that K n ρK n † ∈ tr K n ρK n † I(e), ∀ρ ∈ I(e).
In this case, we call {K n } m n=1 a set of e-incoherent Kraus operators of E . In order to measure coherence, Baumgratz et al. [15] presented the following four defining conditions for a coherence measure C e : (A 1 ) C e (ρ) ≥ 0, ∀ρ ∈ D(H); and C e (ρ) = 0 if and only if ρ ∈ I(e).
for any e-incoherent operation E and any state ρ ∈ D(H).
It was proved in [15] that the relative entropy C e r (ρ) and the 1 -norm measure C e 1 (ρ) of coherence satisfy these defining conditions, which are defined as follows: where S(ρ) = −tr(ρ log ρ) is the von Neumann entropy, and Notably, for a bipartite quantum system AB, the reference basis for H AB = H A ⊗ H B can be taken as a local basis: are the orthonormal bases for H A and H B , respectively. In this case, every ρ AB of AB has the following representation: Thus, a state ρ AB of the system AB is incoherent w.r.t. e AB if and only if ρ AB = ρ AB e AB -diag , i.e., Moreover, let ρ A := tr B (ρ AB ) and ρ B := tr A (ρ AB ). 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.

Lower and Upper Bounds for the Relative Entropy of Coherence
Xi et al. [30] proved that for any bipartite quantum state ρ AB , the relative entropy of coherence obeys some uncertainty-like relation by using the properties of relative entropy, which reads where Afterwards, Liu et al. [31] proved that any tripartite pure state ρ ABC satisfies where e ABC := e A ⊗ e B ⊗ e C , e AB := e A ⊗ e B , e AC := e A ⊗ e C , ρ AB = tr C ρ ABC and ρ AC = tr B ρ ABC , provided that for some 0 ≤ λ ≤ 1. Combining Equations (6) and (7), the following inequality was derived in [31]: for a pure state ρ ABC satisfying the condition (8).
The aim of this section is to establish lower and upper bounds of C r (ρ A 1 A 2 ···A n ) for a general n-partite state ρ A 1 A 2 ···A n . To do this, we use ρ X diag and C r (ρ X ) to denote ρ X e X -diag and C e X r (ρ X ), respectively.
First, for a bipartite ρ AB of the system AB, we know from Equation (5) and the subadditivity of von Neumann entropy that and so Combing this with Equation (6), we have Second, for a tripartite quantum state ρ ABC , according to the super-additivity inequality (6), we have By finding the sums of two sides of the inequalities above, we obtain On the other hand, using definition (1) yields that Combining Equations (11) and (12) gives As a generalization of inequalities (10) and (13), we can prove the following inequalities (14) for any n-partite state ρ A 1 ···A n of the system H A 1 A 2 ···A n = H A 1 ⊗ H A 2 ⊗ · · · ⊗ H A n , which give lower and upper bounds for the relative entropy of coherence. To do this, we let e A k = {|e k i k } d k i k =1 be an orthogonal basis for the Hilbert space H A k (k = 1, 2, . . . , n), and let which is an orthogonal basis for the Hilbert space H A 1 A 2 ···A n . Thus, becomes an orthogonal basis for the Hilbert space With these notations, we have the following.

Theorem 1. For any state ρ
where ρ A i denotes the reduced state of ρ A 1 ···A n on the subsystem A i .

Proof.
To prove that the first inequality in Equation (14) holds, we know from Equation (6) that . . .
and consequently, 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 n = 2 and any bipartite state. Secondly, we assume the second inequality in (14) holds for n = N − 1 and any N − 1-partite state. Then for any N-partite state ρ A 1 ···A N , we have By using Equation (6), we know that C r (tr X η) ≤ C r (η). Thus, Combining the fact that we get that 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.

Corollary 1.
Let ρ A 1 ···A n be a state of the system H A 1 A 2 ···A n = H A 1 ⊗ H A 2 ⊗ · · · ⊗ H A n . If ρ A 1 ···A n is incoherent, then the reduced states ρ A i and tr A i ρ A 1 ···A n (i = 1, 2, . . . , n) are all incoherent. The converse is true if each reduced states ρ A i is pure.

Corollary 2.
Let ρ A 1 ···A n be a state of the system H A 1 A 2 ···A n = H A 1 ⊗ H A 2 ⊗ · · · ⊗ H A n such that the reduced states ρ A i (i = 1, 2, . . . , n) are pure and incoherent. Then It is remarkable that the equalities in Equation (14) may hold in some cases. For example, when d 1 = d 2 = · · · = d n = d and the maximally coherent state due to the fact that C r (ρ A j ) = log 2 d for j = 1, 2, . . . , n, and Moreover, the second inequality in Equation (14) also becomes equality when n = 2. This shows that the inequalities in Equation (14) are tight and can not be improved.

Conclusions
In this paper, we have established lower and upper bounds for relative entropy of coherence C r (ρ A 1 A 2 ···A n ) for an n-partite quantum states ρ A 1 A 2 ···A n . As application of our inequalities, we have found that when each reduced states ρ A i is pure, ρ A 1 ···A n is incoherent if and only if the reduced states ρ A i and tr A i ρ A 1 ···A n (i = 1, 2, . . . , n) are all incoherent. Moreover, we have discussed the conjecture that C r (ρ) ≤ C 1 (ρ) for any state ρ and observed that every mixture η of a state ρ satisfying the conjecture with any incoherent state σ also satisfies the conjecture. We have also proved that when the von Neumann entropy is defined by the natural logarithm ln instead of log 2 , the reduced relative entropy measure of coherenceC r (ρ) = −ρ diag ln ρ diag + ρ ln ρ satisfies the inequalityC r (ρ) ≤ C 1 (ρ) for any state ρ.