Optimal common resource in majorization-based resource theories

We address the problem of finding the optimal common resource for an arbitrary family of target states in quantum resource theories based on majorization, that is, theories whose conversion law between resources is determined by a majorization relationship, such as it happens with entanglement, coherence or purity. We provide a conclusive answer to this problem by proving that the majorization lattice is complete. The proof relies heavily on the more geometric construction provided by the Lorenz curves. Our framework includes the case of possibly non-denumerable sets of target states (i.e., targets sets described by continuous parameters). In addition, we show that a notion of approximate majorization, which has recently found application in quantum thermodynamics, is in close relation with the completeness of this lattice.


Introduction
Quantum resource theories (QRTs) are a very general and powerful framework for studying different phenomena in quantum theory from an operational point of view (see Ref. [1] for a recent review of the topic). Indeed, all QRTs are built from three basics components: free states, free operations and resources. These components are not independent among each other, and they are defined in a way that depends on the physical properties that one wants to describe. In general, for a given QRT, one defines the set of free sates F, formed by those states that can be generated without too much effort. Then, an operation E is said to be free, if it satisfies the condition of mapping free states into free states: ρ → free E(ρ) iff E(ρ) ∈ F ∀ ρ ∈ F. Thus, free operations can be interpreted as the ones that are easy to implement in the lab. Finally, quantum resources are defined as those states that do not belong to the set of free states (i.e., ρ / ∈ F). These states are the useful ones for doing the corresponding quantum tasks. Clearly, it is not possible to convert free states into resources by appealing to free operations alone. This is the reason why the term resource theory was coined. In fact, one of the main concerns of the QRTs is the characterization of transformations between resources by means of free operations. Here, we are focused on QRTs for which these transformations are fully characterized by a kind of majorization law between the resources. Precisely, we are interested in QRTs for which ρ → free σ is equivalent to x(ρ) x(σ) or x(σ) x(ρ), where x(ρ) and x(σ) are probability vectors associated to ρ and σ, respectively, and means a majorization relation (see e.g. [2] for an introduction to majorization theory). In addition to the characterization of the convertibility of free states by means of free operations [3][4][5][6][7][8], majorization theory has been applied to different problems in quantum information such as entanglement criteria [9,10], majorization uncertainty relations [11][12][13][14][15], quantum entropies [16][17][18] and quantum algorithms [19], among others [20][21][22][23][24].
Here, we aim to address the following problem. Let us suppose that one wants to have a set of target resources T . For obvious practical reasons, it is very useful to find a resource ρ, such that it can be converted by means of free operations to any other resource belonging to the target set, that is, ρ → free σ for all σ ∈ T . Clearly, the maximal resource (if it exists), by definition, has to perform this task for any QRT and target set. But a more interesting question is whether there exists a state that can carry out the same task, but using the least amount of resources as possible. More precisely, one aims to find a resource ρ ocr such that ρ ocr → free σ ∀σ ∈ T , and if ρ is another resource satisfying ρ → free σ ∀σ ∈ T , then either ρ → free ρ ocr or ρ free ρ ocr . If this state exists, we refer to it as the optimal common resource (ocr). In this work, we provide a solution for the problem of finding the optimal common resources for arbitrary target sets of all QRTs based on majorization. This problem was already posed and (partially) solved in Ref. [34], for possibly infinite (but denumerable) target sets of bipartite pure entangled states. Let us stress that our proposal is a twofold extension of that previous work. In the first place, we provide a unifying framework for arbitrary QRTs based on majorization, which includes not only entanglement resource theory, but also the important cases of coherence and purity resource theories. In the second place, we consider the most general case of possibly non-denumerable sets of target resources. This is a powerful extension of previous works, because it allows to apply this technique to target sets which are described by a continuous family of parameters. We provide the answer to this general problem by appealing to the completeness of the majorization lattice. This is an interesting order-theoretic property in itself that, as far as we know, no formal proof was provided in the literature up to now. Thus, we close this gap by giving a proof that relies on the geometrical properties of Lorenz curves associated to target sets.

Majorization lattice
Here, we introduce the majorization lattice and present one of the main results of this work, that is, the completeness property of the majorization lattice.
Let us consider probability vectors whose entries are sorted in non-increasing order, that is, vectors belonging to the set: Geometrically, this set is a convex polytope embedded in the d − 1-probability simplex.
Let us now introduce the notion of majorization between probability vectors (see, e.g. [2]).

Definition 1.
For given x, y ∈ ∆ ↓ d , it is said that x majorizes y, denoted as x y, if and only if, Notice that d i=1 x k = d i=1 y k is trivially satisfied, because x and y are probability vectors (so we can discard this condition from the definition of majorization).
The intuitive idea of majorization is that a probability distribution majorizes another one, whenever the former is more concentrated than the latter. In this sense, majorization provides a quantification of the notion of non-uniformity. To fix ideas, let us observe that any probability vector x ∈ ∆ ↓ d trivially satisfies the majorization relations: where rankx is the number of positives entries of x, and e d and u d are the extreme d-dimensional probability vectors in the sense of maximum non-uniformity (e d ) and minimum non-uniformity (u d , i.e. the uniform probability vector), respectively. Let us remark that there are several equivalent definitions of majorization that connect it with the notions of double stochastic matrices, Schur-concave functions and entropies, among others (see e.g. [2]).
Here we are interested in the order-theoretic properties of majorization. Indeed, it can be shown that the set ∆ ↓ d together with the majorization relation is a partially ordered set (POSET, see e.g. [35] for an introduction to order theory). This means that that, for every x, y, z ∈ ∆ ↓ d one has (i) reflexivity: x x, (ii) antisymmetry: x y and y x, then x = y, and (iii) transitivity: x y and y z, then x z.
Notice that if one leaves the constraint that the entries of the probability vectors are sorted in nonincreasing order, then condition (ii) is not valid in general. Instead of this, a weaker version holds, where x and y differ only by a permutation of its entries. In such case, majorization gives a preorder because condition (i) and (iii) remain valid.
In general, majorization does not yields a total order for probability vectors belonging to ∆ ↓ d . This is because there exist x, y ∈ ∆ ↓ d such that x y and y x for any d > 2. In this situation, we say that the probability vectors are incomparable. For instance, it is straightforward to check that There is a visual way to address majorization that consists in appealing to the notion of Lorenz curve [36]. More precisely, for a given (with the convention (0, 0) for k = 0). Then, the Lorenz curve of x, say L x (ω) with ω ∈ [0, d], is obtained by the linear interpolation of these points. At the end, one obtains a non-decreasing and concave polygonal curve from (0, 0) to (d, 1). In this way, given two Lorenz curves of x and y, if the Lorenz curve of x is greater than one of y, it implies that x majorizes y, and viceversa. On the other hand, if the Lorenz curves intersect at least at one point (in addition to the points (0, 0) and x u 4 and e 4 y u 4 , but x y and y x. However, in such case, one can easily realize that there are infinite Lorenz curves below the ones of x and y, and among of all them, there is one which is the greatest one. In the same vein, there are infinitely many Lorenz curves above those of x and y, and there is one which is the lowest one. These intuitions can be formalized and allow to formulate a notion of infimum and supremum in the general case [25]. Consequently, the definition of majorization lattice is introduced as follows: Definition 2. The quadruple L = ∆ ↓ d , , e d , u d defines a bounded lattice order structure, where e d is the top element, u d is the bottom element and for all x, y ∈ ∆ ↓ d the infimum x ∧ y and the supremum x ∨ y are expressed as in [25] (or see below).
Precisely, the components of the infimum are given by iteration of the formula for k = 1, . . . , d and the convention that summations with the upper index smaller than the lower index are equal to zero. For the supremum, one has to proceed in two steps. First, one has to calculate the probability vector, say z, with components given by In general, this vector does not belong to ∆ ↓ d , because its components are not in a decreasing order. If it is the case that z ∈ ∆ ↓ d , then z = x ∨ y. Otherwise, one has to apply the flatness process (see [25,Lemma 3]) in order to get the supremum, as follows. For a probability vector w = [w 1 , . . . , w d ] t , let j be the smallest integer in [2, d] such that w j > w j−1 and let k be the greatest integer in [1, j − 1] such that with w 0 > 1. Then, a flatness probability vector w ′ is given by Then, the supremum is obtained in no more than d − 1 iterations, by iteratively applying the above transformations with the input probability vector z given by (4), until one obtains a probability vector in ∆ ↓ d . Let us consider a finite set of probability vectors, that is, By appealing to the algebraic properties of the definition of lattice, it is straightforward to show that the infimum and the supremum of P always exist, and are given by However, if one considers an arbitrary set of probability vectors (which could be infinite), the lattice properties are not strong enough to guarantee the existence of infimum and supremum. If the infimum and supremum exist for arbitrary families, the lattice is said to be complete. Here, we show that the majorization lattice is indeed complete. Proposition 1. Let P an arbitrary set of probability vectors such that P ⊆ ∆ ↓ d . Then, there exist the infimum x inf ≡ P and the supremum x sup ≡ P of P. In addition, the components of the x inf are given by where , . . . , d} and S 0 (x) ≡ 0. On the other hand, to obtain the components of the x sup , we have first to define the probability vector with components given byx Then, we compute the upper envelope of the polygonal given by the linear interpolation of the points {(k, S k (x))} d k=0 , sayL(ω), by using the Algorithm 1. Finally, the components of the supremum are given by: The proof of Proposition 1 is given in A. Clearly, when the set is given by two probability vectors in ∆ ↓ d , that is P = {x, y}, the calculus of infimum and supremum of the Proposition 1 reduces to the procedure given in Ref. [25] (see Eqs. (3)-(6)).

Optimal common resource
Now, we are ready to apply the above Proposition to the problem of finding the optimal common resource in QRTs based on majorization.
In the first place, we have to distinguish between two possible cases of QRTs based on majorization. We call direct majorization-based QRTs to those QRTs such that ρ → free σ iff x(ρ) x(σ), whereas we call reversed majorization-based QRTs to those that reverse the majorization relation (that is, ρ → free σ iff x(σ) x(ρ)). Notice that entanglement and coherence are of the former type, whereas purity is of the latter one (see Tab. 1).
Let us consider an arbitrary set of target resources T . We show now that the problem of finding the optimal common resource of a QRT based on majorization, can be reduced to an application of the completeness of the majorization lattice. Indeed, by directly applying Proposition 1, one finds that the optimal common resource for direct majorization-based QRTs is the supremum of the target set, whereas for reversed majorization-based QRTs it is the infimum. As we have already stressed in the Introduction, this is a twofold extension of the proposal of Ref. [34].

Infimum and supremum over convex polytopes
Let us illustrate the meaning and relevance of the infimum and supremum discussed above with an interesting example. First, let us note that if P ⊆ ∆ ↓ d is a convex polytope, then the corresponding infimum and supremum can be computed as the infimum and supremum of the set of vertices, vert(P). Lemma 1. Let P be a convex polytope contained in ∆ ↓ d , and vert(P) the set of vertices, vert(P) = {v n } N n=1 . Then, the infimum x inf ≡ P and the supremum x sup ≡ P of P are given by the infimum and supremum elements of vert(P), namely The proof of Lemma 1 is given in B. Notice that, although the problem is reduced to the calculation of the infimum and supremum among the extreme points of the convex polytope, x inf and x sup do not necessarily belong to it (see e.g., Fig. 2.(a)). However, we will see an interesting example where the infimum and supremum do belong to the given convex polytope (see e.g., Fig. 2.(b)). Let us consider the ℓ 1 -norm ǫ-ball centered in |x i | denotes the ℓ 1 -norm of a probability vector. Let us first note that {x ′ ∈ R d : ||x ′ − x 0 || 1 ≤ ǫ} is a convex polytope (see Ref. [18]). Then, B ǫ (x 0 ) is also a convex polytope, because it is the intersection of that convex polytope with ∆ ↓ d . Therefore, by applying Lemma 1, B ǫ (x 0 ) and B ǫ (x 0 ) reduces to finding the infimum and supremum of the vertices of B ǫ (x 0 ). Interestingly enough, our contribution in this paper can be posed in strong connection with the notion of approximate majorization [18,37], which has recently found application in quantum thermodynamics [38]. More precisely, the steepest ǫ-approximation,x 0(ǫ) ∈ B ǫ (x 0 ), and the flattest ǫapproximation, x 0(ǫ) ∈ B ǫ (x 0 ), of x 0 given in [18,37] satisfy that,x 0(ǫ) x x 0(ǫ) for all x ∈ B ǫ (x 0 ). Using the definition of infimum and supremum of a given family, it follows thatx 0(ǫ) = x sup and x 0(ǫ) = x inf , although the algorithms to obtain them are different to the ones presented here. Thus, we see that the notion of approximate majorization is in strong connection with the property of completeness of the majorization lattice. Furthermore, we have shown that it can be reduced to the application of the algorithm of infimum and supremum to the set of vertices of B ǫ (x 0 ).

Concluding remarks
In this paper we gave a solution for the problem of finding an optimal common resource for an arbitrary family of target states of a given a QRT based on majorization like entanglement, coherence or purity (see Tab. 1). Our method relies on the completeness properties of the majorization lattice. Indeed, by appealing to the geometrical properties of Lorenz curves, we rigorously proved the completeness of the majorization lattice (Proposition 1). Furthermore, we provided concrete algorithms for computing the infimum and supremum of an arbitrary family of states. Our contribution improves previous works (e.g. [34]), in the sense that our method works for target sets of arbitrary cardinality (i.e., we allow for possibly non-denumerable families of states).
In addition, we showed that the notion of approximate majorization is in strong connection with the property of completeness of the majorization lattice [18,37]. Indeed, the flattest and steepest approximations are nothing more than the infimum and supremum of the corresponding set, respectively, and they can be calculated only from their vertices (Lemma 1).

A Proof of Proposition 1
In this section we show that for an arbitrary set P of probability vectors (whose components are arranged in non-increasing order), there exists an infimum x inf ≡ P and a supremum x sup ≡ P, with respect to the majorization relation. Furthermore, we provide the algorithms to obtain them.
Let us first introduce some notations and definitions. Let us define the partial sum of the first k components of a given vector x as S k (x) ≡ k i=1 x i with the convention S 0 (x) ≡ 0. Now, let us consider the set formed by all partial sums up to k that come from probability vectors in P, that is, S k = {S k (x) : x ∈ P} and its infimumS k ≡ inf S k and supremumS k ≡ sup S k . Notice that, for each k = 0, . . . d, bothS k andS k exist, since each S k is a set of real numbers bounded from below by k d and above by 1. Finally, let us consider the probability vectorsx = [S 1 ,S 2 −S 1 , . . . , . Let us prove that from these probability vectors one can obtain the infimum and the supremum, respectively.

Infimum
Let us now prove thatx = x inf . To prove that, we appeal to the description of majorization in terms of Lorenz curves. First we show that the curve Lx(ω) with ω ∈ [0, d], formed by the linear interpolation of the points {(k,S k )} d k=0 (notice thatS 0 = 0 andS d = 1) is a Lorenz curve. This is equivalent to prove thatx ∈ ∆ ↓ d . We proceed in two steps: (a) Lx(ω) is non-decreasing i.e., Lx(k) ≤ The proofs of both points are given by reductio ad absurdum.
Let us proceed with the proof of (a) Lx(k) ≤ Lx(k + 1) for all k ∈ {0, . . . , d − 1}. Let us assume that there exists k ′ such that Lx(k ′ ) > Lx(k ′ + 1). By construction, there exists a sequence, say Let us pick one of them, say i 0 . On the other hand, by definition ofx, one has . But this is in contradiction with the fact that L x i 0 (k) ≤ L x i 0 (k + 1) for all k ∈ {0, . . . , d − 1}, which is true by definition of Lorenz curve. Then, (a) holds. Now, we proceed with the proof of (b): , which is true by definition of Lorenz curve. Then, (b) holds.
Up to now, we have proved that Lx(ω) is a Lorenz curve that, by construction, satisfies Lx(ω) ≤ L x (ω) ∀ω ∈ [0, d] and ∀x ∈ P. In other words, we obtain thatx ∈ ∆ ↓ d and x x ∀x ∈ P. It remains to be proved that for any x ′ ∈ ∆ ↓ d such that x x ′ ∀x ∈ P, one hasx x ′ . In order to do this, we appeal again to the reductio ad absurdum and the notion of Lorenz curve. Let us assume that there exist x ′ such that x x ′ ∀x ∈ P, butx x ′ . This happens if at least one partial sum of x ′ is greater than the one of thex, say the k ′ partial sum. In other words, . But, by hypothesis, one has L x ′ (k) ≤ L x i 0 (k) for all k ∈ {0, . . . , d}, which is in contradiction with the previous inequality. Thus, there does not exist such x ′ . Therefore,x = x inf .

Supremum
Notice that, according to lattice theory, the arbitrary supremum can be expressed in terms of the arbitrary infimum, and vice versa [40]. This means that our proof of the existence of the infimum for an arbitrary set P of probability vectors (whose components are arranged in non-increasing order), automatically implies the existence of its supremum x sup = {x ′ ∈ ∆ ↓ d : x ′ x ∀x ∈ P}. With this observation we finish our proof that the majorization lattice is complete. Notice that the mere proof of the existence of a supremum, does not guarantee the existence of an algorithm to compute it. Thus, in the sequel, we focus our efforts in providing such an algorithm.
Consider the polygonal curve Lx(ω), with ω ∈ [0, d], formed by the linear interpolation of the points (k,S k ) d k=0 (notice thatS 0 = 0 andS d = 1). By construction, Lx(ω) is non-decreasing and Algorithm 1 Upper envelope input: x ∈ R d output: coordinates of the upper envelope of the polygonal curve joining ⊲ Finds position of the last maximum slope K ← append{K, k} i ← k ⊲ Updates i end while return {(k, S k (x))} k∈K ⊲ Coordinates of the upper envelope end procedure satisfies that Lx(ω) ≥ L x (ω), ∀ω ∈ [0, d] and ∀x ∈ P. But, alike Lx(ω), Lx(ω) is not necessarily a Lorenz curve. Thus, it cannot be used to construct the (ordered) probability vector associated to the supremum of the given family. Instead, let us show that the upper envelope of Lx(ω), that is, L(ω) ≡ inf{g(ω) : g is concave and g(ω) ≥ Lx(ω) ∀ω ∈ [0, d]} (see e.g., [39,Def Our method to obtain the supremum x sup has three steps: first, we calculatex; second, we compute the upper envelope of Lx(ω),L(ω); third, we compute the elements of x sup as the components of the probability vector associated to the Lorenz curveL(ω). The first and last steps are straightforward. We also provide the Algorithm 1 to find the upper envelope of a polygonal curve with coordinates {(k, S k (x))} d k=0 . Notice that for a given probability vectorx ∈ R d , the output of the Algorithm 1 is a set of points {(k, S k (x))} k∈K . It is clear that the linear interpolation of these points is a Lorenz curve, say L x up (ω), which has associated some probability vector x up ∈ ∆ ↓ d . Let us show that L x up (ω) is equal to the upper envelope of Lx(ω). To see that, take two consecutive indices, k i , k i+1 ∈ K. By construction, Lx(ω) = L x up (ω) for ω = k i and ω = k i+1 . For ω ∈ [k i , k i+1 ], L x up (ω) is the linear interpolation and so one has two possibilities: either k i+1 = k i + 1 and Lx(ω) = L x up (ω) for all ω ∈ [k i , k i+1 ], or k i+1 > k i + 1 and Lx(ω) < L x up (ω) for some integer ω ∈ (k i , k i+1 ). In both cases, since the interpolation is linear, there is no concave curve such that L x (ω) ≥ Lx(ω) and L x (ω) < L x up (ω) for all ω ∈ (k i , k i+1 ). Since this is case for any k i ∈ K, L x up (ω), we necessarily obtain the upper envelope of the polygonal curve joining {(k, S k (x))} d k=0 . Then, we have proved that L x up (ω) =L(ω). This last equality implies in turn that (a)L(ω) it is a Lorenz curve. As a consequence, by construction of L(ω), we also have that x up = [L(1),L(2) −L(1), . . . ,L(i) −L(i − 1), . . . ,L(d) −L(d − 1)] satisfies that x up x ∀x ∈ P. In addition, we have that ∄ x ′ ∈ ∆ ↓ d such that L x ′ (ω) ≥ Lx(ω) ∀ω ∈ [0, d] and x up x ′ . Therefore, (b) holds and x up = x sup .

B Proof of Lemma 1
We prove now that x inf ≡ P = vert(P) ≡ v inf and x sup ≡ P = vert(P) ≡ v sup , that is to say that infimum and supremum can be computed among the set of vertices of the convex polytope. Let x be an arbitrary probability vector in P ⊆ ∆ ↓ d . Since P is a convex polytope, x can be written as a convex combination of the vertices, x = N n=1 p n v n , with v n ∈ vert(P ), p n ≥ 0 and N n=1 p n = 1. For arbitrary k, the k-partial sum of x gives where we have used that, by definition, v n v inf , ∀v n ∈ vert(P). On the other hand, since vert(P) ⊆ P and given that x inf ≡ P, we know by definition of infimum that v inf x inf must hold. Hence, using (11), x v inf x inf , ∀x ∈ P.
Therefore, by definition of infimum, one has x inf = v inf . Analogously, for the supremum one obtains that and the desired result follows as before, by definition of supremum, x sup = v sup .