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Cambridge University Press eBooks

Majorization, across the (Quantum) Universe

Guido Bellomo, G. M. Bosyk

Published Apr 11, 2019• 6 citations

Abstract

In how many ways can one represent a given quantum mixed state as a mixture of pure states? Why (and in which sense) are separable states more disordered globally than locally? Is it possible to transform a given pure state into another by means of local operations and classical communication? How should an adequate formulation of the uncertainty principle be? All these questions, as dissimilar as they may seem, share one element in common: They can be answered by appealing to the notion of majorization partial order. Majorization is nowadays a well-established and powerful mathematical tool with many and different applications in several disciplines, such as economics, biology, and physics, among others. Indeed, the seminal idea of this concept had already been glimpsed by Lorenz (1905) while studying the inequality of wealth distribution and developing the representation of the (nowadays called) Lorenz curves. Moreover, the famous Gini coefficient (Gini 1912), widely accepted as a legitimate quantifier of income distribution inequality, is merely a ratio between graphical areas defined by a Lorenz curve. Other key contributions to the subject were those by Muirhead (1903), Dalton (1920), Schur (1923), and Hardy, Littlewood, and Pólya (1929). The name “majorization,” though, appears first in the prominent book by Hardy, Littlewood, and Pólya (1934).

DOI: 10.1017/9781108562218.018

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In how many ways can one represent a given quantum mixed state as a mixture of pure states? Why (and in which sense) are separable states more disordered globally than locally? Is it possible to transform a given pure state into another by means of local operations and classical communication? How should an adequate formulation of the uncertainty principle be? All these questions, as dissimilar as they may seem, share one element in common: They can be answered by appealing to the notion of majorization partial order. Majorization is nowadays a well-established and powerful mathematical tool with many and different applications in several disciplines, such as economics, biology, and physics, among others. Indeed, the seminal idea of this concept had already been glimpsed by Lorenz (1905) while studying the inequality of wealth distribution and developing the representation of the (nowadays called) Lorenz curves. Moreover, the famous Gini coefficient (Gini 1912), widely accepted as a legitimate quantifier of income distribution inequality, is merely a ratio between graphical areas defined by a Lorenz curve. Other key contributions to the subject were those by Muirhead (1903), Dalton (1920), Schur (1923), and Hardy, Littlewood, and Pólya (1929). The name “majorization,” though, appears first in the prominent book by Hardy, Littlewood, and Pólya (1934).

The question of how to characterize disorder arises in a wide variety of settings, including information theory, economics, and quantum thermodynamics. The theory of majorization provides an elegant answer, playing a central role in these fields. However, the existing majorization framework is inapplicable to functions that both take negative values and are defined on infinite spaces. Yet such functions, in the form of quasiprobability distributions, are ubiquitous in fields such as quantum optics, signal analysis, and bosonic quantum computation. Here we develop a notion of majorization that is applicable to such functions, proving that it admits four equivalent characterizations that naturally reduce to the finite case, thereby generalizing a seminal theorem by Hardy, Littlewood, and Pólya. Moreover, we extend this equivalence to the setting where majorization is considered relative to an arbitrary positive distribution. We give several applications of our results in the context of quantum resource theories. These include deriving families of resource monotones and constraining quantum state conversions. We analytically and numerically study examples using the Wigner and Husimi functions, which feature prominently in quantum optics. Our results provide a comprehensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces. Quasiprobability distributions, crucial in quantum optics and computation, challenge existing majorization frameworks due to their negative values and infinite domains. Here, the authors establish a novel majorization concept for such functions, offering four equivalent characterizations and extending applications to quantum resource theories, enhancing our understanding of quantum state transformations.

Lorenz values and the Gini index are popular quantities in Mathematical Economics, and are used here in the context of quantum systems with finite-dimensional Hilbert space. They quantify the uncertainty in the probability distribution related to an orthonormal basis. It is shown that Lorenz values are superadditive functions and the Gini indices are subadditive functions. The supremum over all density matrices of the sum of the two Gini indices with respect to position and momentum states is used to define an uncertainty coefficient which quantifies the uncertainty in the quantum system. It is shown that the uncertainty coefficient is positive, and an upper bound for it is given. Various examples demonstrate these ideas.

So far we have dealt with quantum systems in a well-defined state, namely, systems that can be fully described by a single state vector https://www.w3.org/1998/Math/MathML" display="inline"> | ψ ⟩ . Such states are called pure . Here we will learn how to describe situations where knowing the state vector is not enough because the system is under the influence of another, typically larger, system which we know little about. This kind of interaction degrades the “quantumness” of the system we are interested to describe. The mathematics used to described this phenomenon is also suitable for establishing the degree of entanglement present in a quantum system, being it in a pure state or not.

Abstract Employing the lattice theory on majorization, the universal quantum uncertainty relation for any number of observables and general measurement is investigated. It is found that 1) the least bounds of the universal uncertainty relations can only be properly defined in the lattice theory; 2) contrary to variance and entropy, the metric induced by the majorization lattice implies an intrinsic structure of the quantum uncertainty; and 3) the lattice theory correlates the optimization of uncertainty relation with the entanglement transformation under local quantum operation and classical communication. Interestingly, the optimality of the universal uncertainty relation found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society.