Coherence resource power of isocoherent states

We address the problem of comparing quantum states with the same amount of coherence in terms of their coherence resource power given by the preorder of incoherent operations. For any coherence measure, two states with null or maximum value of coherence are equivalent with respect to that preorder. This is no longer true for intermediate values of coherence when pure states of quantum systems with dimension greater than two are considered. In particular, we show that, for any value of coherence (except the extreme values, zero and the maximum), there are infinite incomparable pure states with that value of coherence. These results are not peculiarities of a given coherence measure, but common properties of every well-behaved coherence measure. Furthermore, we show that for qubit mixed states there exist coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states.


Scientific Reports
| (2022) 12:7329 | https://doi.org/10.1038/s41598-022-11300-x www.nature.com/scientificreports/ IO-incomparable pure states with that value of coherence, provided that the dimension of the quantum systems is greater than two. These results are not peculiarities of a given coherence measure, but common properties of every well-behaved coherence measure. Furthermore, for qubit mixed states, we show that there are coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states. In this way, our work complements other works [14][15][16][17][18][19][20] about ordering of quantum states with coherence. The paper is organized as follows. First, we review the main aspects of the resource theory of coherence, including the definition of incoherent states, incoherent operations and coherence measures, among other relevant results. Then, we provide our main results regarding the comparison of states with the same amount of coherence in terms of their coherence resource power. Finally, we summarize our results and discuss their roots and links with related problems. For the sake of readability all proofs and auxiliary results are given in the "Methods".

Preliminaries: resource theory of coherence, IO preorder and coherence measures
We will focus on quantum systems with finite dimension. The Hilbert space of the system is denoted by H and its dimension by d = dim H . The set of all quantum states of H is denoted by S (H) and the subset of pure states is denoted by P (H).
Without loss of generality we choose the computational basis B = {|i�} d−1 i=0 as the incoherent basis. In this way, incoherent states, are quantum states with diagonal density matrix in the computational basis. More precisely, ρ is an incoherent state if and only if ρ = d−1 i=0 i |i��i| , with i ≥ 0 and d−1 i=0 i = 1 . We denote the set of incoherent states as I.
The incoherent operations introduced in 3 , which have the property that coherence can not be created from an incoherent state, not even in a probabilistic way, are defined as follows. A completely positive trace-preserving map � : S (H) → S (H) is an incoherent operation (IO) if it admits a representation in terms of Kraus operators {K n } N n=1 such that K n ρK † n /Tr(K n ρK † n ) ∈ I for all 1 ≤ n ≤ N and ρ ∈ I. Interesting enough, incoherent operations induce a preorder among quantum states in terms of their coherence resource power:

Definition 1
We say that the state ρ is more or equally coherent than σ if it is possible to transform the former into the latter by means of an incoherent operation. We denote this by ρ → IO σ.
(2) ψ = |�0|ψ�| 2 , . . . , |�d − 1|ψ�| 2 . www.nature.com/scientificreports/ where ↓ indicates that the entries of ψ and φ are sorted in non-increasing order, i.e., ψ As we say before, the majorization relation defines a preorder on the set of probability vectors d . If ψ φ or φ ψ , we say that the probability vectors are comparable. If both relation are satisfied, ψ and φ are equal up to a permutation. In this case we say that the probability vectors are equivalent. For dimensions greater than two 24 , there are cases in which neither ψ φ nor φ ψ are possible. When this is the case, we say that the probability vectors are incomparable.
Taking into account these definitions, we can state the following result that connects both preorders (see 7,8,[21][22][23]  In general, for mixed states, a finite number of conditions are not sufficient to fully characterize coherent transformations 25 . From a generalized notion of coherence vector, it can be obtained a necessary condition in terms of a majorization relation 13 . However, qubit transformations under incoherent operations are completely characterized 26,27 . Theorem 2 Let ρ and σ be two qubit states with Bloch vectors r 1 = (x 1 , y 1 , z 1 ) and r 2 = (x 2 , y 2 , z 2 ) , respectively. Then, ρ → IO σ if and only if with r 1 = x 2 1 + y 2 1 and r 2 = x 2 2 + y 2 2 .
In addition to the comparability notions between states, it is relevant to quantify the coherence amount of quantum states. In this paper, we mainly follow the axiomatic formulation for coherence measures. It can be shown that conditions 2 and 3 imply monotonicity under IO, that is, C(ρ) ≥ C(�(ρ)) for any incoherent operation and any state ρ . The relevance of condition 4 is discussed in 5 . In particular, this condition excludes some problematic cases as the one given in Ex. 4 of 28 .
An interesting result is that any coherence measure restricted to pure states can be expressed in terms of a real, symmetric and concave functions defined on d . More precisely, given the set we have the following result 7,8 .
Theorem 3 Let C be a coherence measure. Then, there exists a function f C ∈ F , such that the restriction of C to the set of pure states, denoted by C| P (H) , satisfies where ψ is the coherence vector of |ψ�.
By abuse of notation, we will use C(|ψ�) when evaluating the restriction of the measure C on a pure state, instead of C| P (H) (|ψ��ψ|) . In particular, we will focus on functions of F that are also strictly Schur-concave. Namely, a real function f defined on d is said to be Schur-concave, if f (ψ) ≥ f (φ) whenever ψ φ . If, in addition, f (ψ) > f (φ) whenever ψ φ and ψ = �φ , with a permutation matrix, then f is said to be strictly Schur-concave (see Def. A.1 in 24 ). We note that any function f ∈ F is Schur-concave, since they are symmetric and concave (see Prop. C.2 in 24 ), but they are not necessarily strictly Schur-concave. The condition that f C be www.nature.com/scientificreports/ strictly Schur-concave is not so restrictive, since the most used coherence measures satisfy it, including the relative entropy of coherence, the ℓ 1 -norm of coherence, and the coherence of formation. Now, we review some relevant coherence measures. The first one is the relative entropy of coherence C re , defined as where S(ρ||σ ) = Tr(ρ(ln ρ − ln σ )) . Alternatively, the relative of coherence can be expressed as i=0 �i|ρ|i�|i��i| and S(ρ) = −Tr(ρ ln ρ) is the von Neumann entropy. Accordingly, the associated function f C re ∈ F of the relative of coherence is the Shannon entropy, i.e., which is also strictly Schur-concave. The relative entropy of coherence has a particular operational interpretation. It coincides with the distillable coherence, that is, the maximal number of maximally coherent single-qubit states |ψ mcs 2 � which can be obtained per copy of a given state ρ by means of incoherent operations in the asymptotic limit 6 .
Another relevant coherence measure is given in terms of the off-diagonal elements of ρ . More precisely, the ℓ 1 -norm of coherence C ℓ 1 is defined as which is also strictly Schur-concave. We remark that the ℓ 1 -norm of coherence is useful for characterizing quantum interference and obtaining complementarity relations between coherence and path information in multipath interferometers [29][30][31] .
Finally, we recall that there are different ways to extend a coherence measure defined on pure states to mixed states 7,8,12,13 . The most common way is the convex roof method. For any f ∈ F the convex roof measure of coherence C cr f is given by 7,8 where +∞) , leads to the q-Tsallis coherence of formation C T q . Notice that f T q is strictly Schur-concave 33 and C T 1 (ρ) = lim q→1 C T q (ρ) = C CoF (ρ) , recovering the coherence of formation. This measure coincides with the coherence cost, that is, the minimal number of maximally coherent single-qubit states |ψ mcs 2 � required to produce a given state ρ by means of incoherent operations, in the asymptotic limit 6 .

Results
We are interesting in comparing states with the same amount of coherence. For a given coherence measure C and a non-negative number α , we introduce the set of isocoherence states E C,α as follows On the one hand, from the condition 1 of definition 2, it follows E C,0 = I . Thus, as a consequence of that all incoherent states are IO-equivalent (see Observation 1), the states belonging to E C,0 are all IO-equivalent. On the other hand, from condition 4 of definition 2, it follows that the set E C,M C , with M C = C(ρ mcs ) , is formed by maximally coherent states. Thus, the states belonging to E C,M C are all IO-equivalent. Therefore, as one might expect, all isocoherent states with an extreme value of coherence have the same coherence resource power. Moreover, this is also true for pure isocoherent states of qubit systems for any value of coherence.

Proposition 1
For any function f ∈ F strictly Schur-concave, pure isocoherent states of qubit systems are IO-equivalent.
A natural question that arises from the previous observations is: In higher dimensional systems, do isocoherent pure states with an intermediate value of coherence have the same coherence resource power? In other words, for systems with d > 2 , we are asking if states of E C,α , with α ∈ (0, M C ) , are IO-equivalent.
We will show that this is not the case. More precisely, we will prove that for each value of coherence α in the interval (I C , M C ) , there are infinite IO-incomparable pure states with that amount of coherence, where I C = inf ψ∈ri(� d ) f C (ψ) with ri(� d ) the relative interior of the set d , i.e., Proposition 2 Let d > 2 and let C : S (H) → R be a coherence measure such that its restriction to pure states has an associated function f C ∈ F strictly Schur-concave. For any α ∈ (I C , M C ) , there are infinite pure states If a coherence measure also satisfies the continuity condition (5 ), we have that for any possible value of coherence (except the extreme cases zero and the maximal value M C ) there are an infinite number of IO-incomparable pure states with that amount of coherence. In particular, we remark that, Propositions 1 and 2, and Corollary 1 are valid for the relative entropy of coherence C re and the ℓ 1 -norm of coherence C ℓ 1 .
Finally, for the three-dimensional case, we provide an example of a family of IO-incomparable pure states with the same value of the relative entropy of coherence. For each a ∈ [0, 1] , we define the curve where u = (1/3, 1/3, 1/3) , v = (1, 0, 0) and w = (0, −1/3, 1/3) . For each a ∈ [0, 1] and t ∈ [0, 1] , ψ (a) (t) is a probability vector sorted in a non-increasing way. Moreover, it can be proved that different curves do not have equivalent probability vectors in common, except the extreme vectors u and v.
For some α ∈ (0, M C ) (with M C = ln 3 ) and for each a ∈ [0, 1] , we consider the intersection of the contour plot C r (|ψ�) = − 2 i=0 ψ i ln ψ i = α with the curve ψ (a) . We denote this intersection by ψ (a) (t * a ) , with t * a ∈ (0, 1) . In Fig. 1 Regarding qubit mixed states, we can distinguish two situations depending on the coherence measure. An arbitrary coherence measure for a qubit state ρ , with Bloch vector r = (x, y, z) , can be expressed in terms of r = x 2 + y 2 and z, i.e., C(ρ) = C(r, z) . For measures that do not depend on z and are strictly increasing on r, it is easy to see that there are no pair of incomparable isocoherent states. More precisely, let ρ and σ be two www.nature.com/scientificreports/ isocoherent states, with Bloch vectors r 1 = (x 1 , y 1 , z 1 ) and r 2 = (x 2 , y 2 , z 2 ) respectively. Then, since the coherence measure only depends on r, and it is strictly increasing, we have that r 1 = r 2 . Finally, from Theorem 2, we conclude that, if |z 1 | ≤ |z 2 | , then σ → IO ρ , and if |z 2 | ≤ |z 1 | , then ρ → IO σ . In other words, the isocoherent states ρ and σ are IO-comparable . In particular, the ℓ 1 -norm of coherence is an example of this kind of coherence measures: For coherence measures that also depend on z, there are examples of isocoherent states which are incomparable. For instance, for the relative entropy of coherence, which can be expressed as Fig. 2 an example of two incomparable and isocoherent states.

Discussions
In this work, we have considered the subset of quantum states formed by those states with a fixed value of coherence for a given coherence measure. We have analyzed its coherence resource power in terms of the preorder induced by the incoherent operations.
First, we have observed that, as one might expect, isocoherent states with an extreme value of coherence have the same coherence resource power in terms of the incoherent preorder. Second, we have shown that pure isocoherent states of qubit systems with arbitrary value of coherence are IO-equivalent (Proposition 1).
Third, we have proved that, in higher dimensional systems ( d > 2 ), pure isocoherent states are not necessarily IO-equivalent. In particular, for any value of coherence, we have shown that there are infinite IO-incomparable pure states with that value of coherence (Proposition 2 and Corollary 1). The essence of these results is that, in general, the coherence measures do not fully preserve the preorder structure of the quantum states induced by incoherent operations. Indeed, coherence measures map the set of quantum states to the positive real numbers, which is a total order set. In this way, the quantum states go from being pre-ordered by IO to being totally ordered by the coherence measure. Our results Proposition 2 and Corollary 1 arise as a consequence of this discrepancy. Another related consequence is the fact that different coherence measures induce different total orders on the set of quantum states, as it was observed in 14,[16][17][18] . In this way, our work complements these studies about ordering quantum states with coherence.
Regarding qubit mixed states, we have distinguished two situations depending on the coherence measure. For measures that do not depend on z and are strictly increasing on r, we have shown that there are no pair of incomparable isocoherent states. In particular, the ℓ 1 -norm of coherence is an example of this kind of coherence measures. For coherence measures that also depend on z, we have shown that there are examples of isocoherent mixed states which are incomparable. In particular, we have considered the case of the relative entropy of coherence.
Finally, we remark that we have focused on the resource theory of coherence to illustrate these observations due to its topicality and practical relevance 1 . However, as our proofs are posed in a wider and simpler context based on majorization theory 24 , the results can be easily extended to any majorization-based quantum resource theory 34 , such as entanglement theory 35 and nonuniformity 36,37 . In fact, the observations made for the entanglement entropy in 38 , namely there are infinite incomparable bipartite pure states with a fixed value of entanglement entropy, can be extend to any entanglement monotone 39 by exploiting similar majorization arguments to those given in our proofs. The reason behind this generality is again that the preorder induced by the free operations of the resource theory and the total order induced by the measures are not isomorphic.

Methods
First, we provide a proof of the following intuitive result: all incoherent states have the same coherence resource power. Indeed, for any quantum resource theory that admits a tensor product structure, like the quantum coherence resource theory considered here, it is valid that the free states are interconvertible by means of the free operations of the theory 2 . For the sake of completeness, we provide a directly proof of the Observation 1, giving an explicit incoherent operation that allows to transform an arbitrary incoherent state into another arbitrary one.
Both results imply that, for all ρ ∈ I , |0��0| ↔ IO ρ . Then, we can conclude that all incoherent states are IO-equivalent.
For the proof of Proposition 2 we need the following lemma.
Lemma 1 Let C : S (H) → R be an coherence measure and f C its associated function.

Proof
(i) Since f C is a concave function on d , −f C is a convex function on the same domain. We consider the following extension of f C over all R d : The function g C is convex on R d , therefore it is continuous on ri(dom g) (see Th. 10.1 in 40 ). Since ri(dom g C ) = ri(� d ) , g C is continuous on ri(� d ) . Finally, we conclude that f C is continuous on ri(� d ). Then, since f C is a bounded Schur-concave function, {f C (v n )} n∈N >1 is a bounded and monotonic decreasing sequence. Therefore, there exists L = lim n→∞ f C (v n ) , and L ≤ f C (v n ) for all n > 1.
On the other hand, for all ψ ∈ ri(� d ) , I C ≤ f C (ψ) . In particular, I C ≤ f C (v n ) for all n ∈ N . This implies, I C ≤ lim n→∞ f C (v n ).
Summing up, I C = lim n→∞ f C (v n ) . In particular, if f C is continuous on d , we have I C = lim n→∞ f C (v n ) = f C (1, 0, . . . , 0) = 0.

Proof of Proposition 2.
Proof On the one hand, M C = C(ρ mcs ) = f C (u) , with u = (1/d, . . . , 1/d) . On the other hand, due to Lemma 1, I C = lim n→∞ f C (v n ) . Since, for all n ∈ N , u v n , but v n u , and f C is strictly Schur-concave, we have f C (v n ) < f C (u) . Therefore, I C < M C .
Let α ∈ (I C , M C ) . Since α > I C = lim n→∞ f C (v n ) and {f C (v n )} n∈N >1 is monotonic decreasing, there is some n α > 1 , such that f C (v n α ) < c . Therefore, f C (v n α ) < α < f C (u). Now, we construct a family of probability vectors, such that the value of f C on these vectors is equal to α . Notice that the entries of ψ (a) (t) are Then, for t ∈ [0, 1] and a ∈ [0, 1/d] , all the entries are greater than zero, and d−1 i=0 ψ (a) i (t) = 1 . In other words, for each a ∈ [0, 1/d] , the curve ψ (a) (t) is formed by probability vectors. In addition, ψ (a) (t) ∈ ri(� d ) for t ∈ [0, 1] and a ∈ [0, 1/d].
Next, we show that different curves do not have equivalent probability vectors in common, except u and v n α . First, we observe that their entries are decreasingly ordered: