Optimal common resource in majorization-based resource theories

Abstract

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 appealing to the completeness property of the majorization lattice. We give a proof of this property that relies heavily on the more geometric construction provided by the Lorenz curves, which allows to explicitly obtain the corresponding infimum and supremum. Our framework includes the case of possibly nondenumerable 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. Finally, we provide some examples of optimal common resources within the resource theory of quantum coherence.

Extremal elements of a sublattice of the majorization lattice and approximate majorization

“…Clearly, the majorization lattice is a bounded lattice, with top and bottom elements e 1 and e d , respectively. Moreover, it turns out that the majorization lattice is indeed complete [20][21][22]. In other words, the infimum and supremum exist for every family of probability vectors in ∆ ↓ d .…”

“…For completeness, in Appendix B we recall the explicit algorithm to compute the upper envelope given in Ref. [22]. When the set P ⊆ ∆ ↓ d is a convex polytope, the corresponding infimum and supremum can be computed as the infimum and supremum of the set of vertices, vert(P), as explained in the following Lemma.…”

Generalized coherence vector applied to coherence transformations and quantifiers

“…The majorization relation is preorder relation on the set ∆ d and a partial order on the set ∆ ↓ d . Moreover, the set ∆ ↓ d together with the majorization relation is a complete lattice 1 [12,13], and the algorithms to obtain the supremum and infimum can be found in [13][14][15]. In particular, the supremum of a set U ⊆ ∆ ↓ d , denoted as U, can be computed as follows.…”

“…Let L U be the upper envelope 2 of the polygonal curve given by the linear interpolation of the set of points {(j, S j )} 0≤j≤d , where S j is the supremum of s j (u) for all u ∈ U and s j (u) = j−1 i=0 u i , with the convention S 0 = 0. As it is shown in [13], L U is the Lorenz curve 3 associated to the probability vector U.…”

“…This characterization is given in terms of the majorization relation [11] between the coherence vectors of the pure states. Motivated by this fact, we propose a generalization of the coherence vector to general quantum states, and we advance on the characterization of the state conversion by means of incoherent operations by appealing to the majorization lattice theory [12][13][14][15]. More precisely, given a pure state decomposition of a quantum state, we define the coherence vector of the decomposition in terms of the coherence vectors of the pure states.…”