In this article, we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.

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.

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B p (x) with p ≥ 1 formed by the probability vectors whose p -norm distance to the center x is less than or equal to a radius . Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls B p (x) with 1 < p < ∞. On the other hand, we show that B ∞ (x) is a complete sublattice of the majorization lattice. As a consequence, this ball has also extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.

Li and Luo [Phys. Rev. A 78 (2008), 024303] discovered a remarkable relation between discord and entanglement. It establishes that all separable states can be obtained via reduction of a classicaly-correlated state 'living' in a space of larger dimension. Starting from this result, we discuss here an optimal classical extension of separable states and explore this notion for low-dimensional systems. We find that the larger the dimension of the classical extension, the larger the discord in the original separable state. Further, we analyze separable states of maximum discord in C 2 ⊗ C 2 and their associated classical extensions showing that, from the reduction of a classical state in (C 2 ⊗ C 3 ) ⊗ C 2 , one can obtain a separable state of maximum discord in C 2 ⊗ C

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.

We study the problem of deterministic transformations of an initial pure entangled quantum state, |ψ , into a target pure entangled quantum state, |φ , by using local operations and classical communication (LOCC). A celebrated result of Nielsen [Phys. Rev. Lett. 83, 436 (1999)] gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation ψ ≺ φ holds, where ψ and φ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an approximate entanglement transformation. Vidal et. al [Phys. Rev. A 62, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state |χ opt most approximate to |φ in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the majorization lattice. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between ψ and φ. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.

In [2014 Eur.J.Phys. D 68 1], Hou, Huang, and Cheng present, using Tsallis' entropy, possible generalizations of the quantum discord measure, finding original results. As for the mutual informations and discord, we show here that these two types of quantifiers can take negative values. In the two qubits instance we further determine in which regions they are non-negative. Additionally, we study alternative generalizations on the basis of Renyi entropies.On an interesting recent paper, Hou et al. [1] introduce generalizations for two quantifiers: mutual information and quantum discords, which they use for the study of quantum correlations in two qubits systems. It is conventionally agreed that the mutual information (MI) quantifies total correlations in bipartite systems. Given a system described by the state ρ ab , with subsystems a and b, the MI readswhere ρ a := Tr b ρ ab y ρ b := Tr a ρ ab are reduced states associated to our subsystems. S(·) is * Electronic address: