Coherence measure in terms of the Tsallis relative α entropy

Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn’t increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative α entropy has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative α entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.


Result
The coherence and the Tsallis relative α entropy. The resource theory includes three ingredients: the free states, the resource states and the free operations 24,46 . For coherence, the free states are referred to as the incoherent states which are defined in a given fixed basis {|i〉} by the states with the density matrices in the diagonal form, i.e., δ δ = ∑ i i i i with δ ∑ = 1 i i for the positive δ i . All the states without the above diagonal form are the coherent states, i.e., the resource states. The quantum operations described by the Kraus operators {K n } with = † K K I n n are called as the incoherent operations and serve as the free operations for coherence, if δ ∈ † K K n n  for any incoherent δ. In this sense, the standard criteria of a good coherence quantifier C(ρ) for the state ρ can be rigorously rewritten as 18 (i) (Null) C(δ) = 0 for  δ ∈ ; (ii) (Strong monotonicity) for any state ρ and incoherent operations {K n }, ρ ρ ≥ ∑ C p C ( ) ( ) n n n with ρ = † p K K Tr n n n and ρ ρ = † K K p / n n n n ; (iii) (Convexity) For any ensemble i . In addition, ref. 18 also introduces the monotonicity (in contrast to the strong monotonicity) that requires ρ ρ ≥ ∑ C C p ( ) ( ) n n n . This actually can be automatically implied by (ii) and (iii). As mentioned in ref. 18 , the monotonicity is not laid in an important position compared with the strong monotonicity, because the measurement outcomes of {K n } can be well controlled (sub-selected) in practical experiments. In fact, the fundamental spirit of both the monotonicity and the strong monotonicity (or the resource theory) is to restrict that the coherence (resource) shouldn't be increased under the incoherent (free) operations, which is parallel with the resource theory of entanglement, namely, the average entanglement is not increased under the local operations and classical communication (LOCC). However, if for a quantum state ρ, there exists one incoherent operation {K n } such that where n denotes the measurement outcome with the probability ρ = † p K K Tr n n n , and the corresponding post-measurement state is ρ ρ = † K K n n n , this means that if we erase the information of the measurement outcomes, the coherence of the post-measurement state ρ′ is less than the coherence of the pre-measurement state, but if we keep the measurement information, the average coherence is increased. However, in the practical experiment, it is not necessary for us to erase any information. This means that the incoherent operation {K n } can increase the coherence, which violates the fundamental spirit of a resource theory. It is why we emphasize the strong monotonicity.
With the above criteria, any measure of distinguishability such as the (pseudo-) distance norm could induce a potential candidate for a coherence quantifier. But it has been shown that some candidates only satisfy the monotonicity rather than the strong monotonicity, so they are not ideal and could be only used in the limited cases. ref. 22 found that the coherence based on the Tsallis relative α entropy is also such a coherence quantifier without the strong monotonicity.
The Tsallis relative α entropy is a special case of the quantum f-divergences 22,47 . For two density matrices ρ and σ, it is defined as n n n n n n n n n n for the density matrices ρ n and σ n and the corresponding probability distribution p n .
Based on the Tsallis relative α entropy ρ σ α D ( ), the coherence in the fixed reference basis {|i〉} can be characterized by 22 However, it is shown that ρ α  C ( ) satisfies all the criteria for a good coherence measure but the strong monotonicity. Since ρ σ α→ D ( ) 1 reduces to the relative entropy ρ σ S( ) which has induced the good coherence measure, throughout the paper we are mainly interested in ∪ α ∈ (0, 1) (1, 2]. In addition, the Tsallis relative α entropy ρ σ α D ( ) can also be reformulated by a very useful function as Accordingly, the coherence ρ α  C ( ) can also be rewritten as j 1 1/ Based on Eq. (6) and the properties of ρ σ α D ( ) mentioned above, one can have the following observations for the function f α (ρ, σ) 22,47 .
The coherence measures based on the Tsallis relative α entropy. To proceed, we would like to present a very important lemma for the function f α (ρ, σ), which is the key to show our main result.  The proof is given in the Methods. Based on Lemma 1 and the preliminaries given in the previous section, we can present our main theorem as follows.

Theorem 1
The coherence of a quantum state ρ can be measured by Proof . At first, one can note that the function x α is a monotonically increasing function on x, so Eq. (10) obviously holds for positive x due to Eq. (6).
Null. Since the original Tsallis entropy defined by Eq. (2) can unambiguously distinguish a coherent state from the incoherent one. Eq. (2) implies that is sufficient and necessary condition for incoherent states. Thus the zero C α (ρ) is also a sufficient and necessary condition for incoherent state ρ.
Convexity. From ref. 48 , one can learn that the function g(A) = Tr(XA p X † ) s is convex in positive matrix A for p ∈ [1, 2] and ≥ s p 1 , and concave in A for p ∈ (0, 1] and ≤ ≤ s 1 p 1 . Now let's assume A = ρ, = X j j and p = α and = α s 1 , thus one has is convex in density matrix ρ for α ∈ [1, 2] and = α s 1 , and concave in ρ for α ∈ (0, 1] and = α s 1 . Here the subscript α and the superscript j in α g j specifies the particular choice. So it is easy to find that ρ ∑   n . Therefore, one can immediately find that where we use the function x 1/α is monotonically increasing on x. According to Eqs (12) and (15), we obtain In addition, the Hölder inequality 49 implies that for α ∈ (0, 1),  Maximal coherence and several typical quantifiers. Next, we will show that the maximal coherence can be achieved by the maximally coherent states. At first, we assume α ∈ (0, 1). Based on the eigen-decomposition of a d-dimensional state ρ ρ λ ψ ψ = ∑ : k k k k with λ k and ψ k representing the eigenvalue and eigenvectors, we have Similarly, for α ∈ (1, 2], the function x 1/α is concave, which leads to that Eq. (19) with the inverse inequality sign holds. The inequality can also saturate for ρ m . The corresponding coherence can be found to have the same form as C 0<α<1 (ρ m ). In other words, actually defines a family of coherence measures related to the Tsallis relative α entropy. This family includes several typical coherence measures. As mentioned above, the most prominent coherence measure belonging to this family is the coherence in terms of relative entropy, i.e., C 1 (ρ) = S(ρ).
One can also find that with ||·|| 2 denoting l 2 norm. So the l 2 norm has been revived for coherence measure by considering the square root of the density matrices. This is much like the quantification of quantum correlation proposed in ref. 50 . In addition, C 1/2 (ρ) can also be rewritten as Finally, one can also see that which is a simple function of the density matrix.
Applications. As applications, we would like to compare our coherence measure with other analytic coherence measures, that is, the measure based on l 1 norm, the relative entropy and the skew information. Let's consider a decoherence process where a bipartite maximally entangled state ψ | 〉 = |+ +〉 + |− −〉 ( ) Thus one can easily find that the coherence based on the l 1 norm can be given by , and the coherence based on our Tsallis relative α entropy can be given by ρ γ In particular, it is shown that C α (ρ(γ)) for α → 1 corresponds to the coherence based on the relative entropy defined by R(ρ(γ)) = S(I  ρ(γ)) − S(ρ(γ)) with  meaning the Hadamard product of matrices and C 1/2 (ρ(γ)) corresponds to the skew information 53 . In order to explicitly show the difference between the various coherence measures, we plot the coherence of the state ρ(γ) for C l 1 and C α (ρ(γ)) for various α in Fig. 1.

Conclusion
We establish a family of coherence measures that are closely related to the Tsallis relative α entropy. We prove that these coherence measures satisfy all the required criteria for a satisfactory coherence measure especially including the strong monotonicity. We also show this family of coherence measures includes several typical coherence measures such as the coherences measure based on von Neumann entropy, skew information and so on. Additionally, we show how to validate the l 2 norm as a coherence measure. In addition, one can find that our current coherence measure can be easily related to the original Tsallis relative α entropy in Theorem 1, thus our The solid line corresponds to C l 1 and the dashed line corresponds to C 1/2 which corresponds to the coherence in terms of skew information. The 'diamond' line, the '+' line and the dash-dotted line, respectively correspond to C 2/3 , C 3/2 and C 2 . In particular, the line marked by 'o' corresponds to C α→1 and the dot line corresponds to the coherence based on relative entropy R(ρ(γ)), which shows the perfect consistency. current coherence measure has many potential applications or connections in both thermo-statistics and the information theory, since the Tsallis relative α entropy lays the foundation to the non-extensive thermo-statistics and have important applications in the information theory 44,45 . This could require the further investigation. Finally, we would like to emphasize that the convexity and the strong monotonicity could be two key points which couldn't easily be compatible with each other to some extent. Fortunately, ref. 48 provides the important knowledge to harmonize both points in this paper. This work builds the bridge between the Tsallis relative α entropy and the strong monotonicity and provides the important alternative quantifiers for the coherence quantification. This could shed new light on the strong monotonicity of other candidates for coherence measure.

Methods
Proof of Lemma 1 Any TPCP map can be realized by a unitary operation on a composite system followed by a local projective measurement 54