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It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call the complex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space., SCOPUS: ar.j, info:eu-repo/semantics/published

We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson’s theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson’s theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space., SCOPUS: ar.j, info:eu-repo/semantics/published

This thesis is centered on a novel approach to quantum uncertainty based on applying the theory of continuous majorization to quantum phase-space distributions. Majorization theory is a powerful mathematical framework that is aimed at comparing distributions with respect to intrinsic disorder. It is particularly significant in the sense that establishing a majorization relation between two distributions amounts to proving that every (Shur-concave) measure of disorder will categorize one distribution as more ordered than the other. Although this is less known, the distributions here do not need to be normalized nor positive for majorization theory to apply, so the latter even extends beyond probability distributions. Further, a majorization relation can rigorously be defined for both discrete and continuous distributions over a finite-size domain, as well as for (discrete and continuous) distributions that are positive over an infinite-size domain.The central thrust of this thesis is to characterize quantum uncertainty in phase space by applying the tools of majorization theory to the Wigner function, which is the most common (quasi)distribution that embodies a quantum state in phase space. Wigner functions are in general positive and negative, putting them beyond the reach of most information-theoretical measures but perfect candidates for the theory of majorization. We start our manuscript with a succinct overview of the basics of quantum optics in phase space, which are a prerequisite for the characterization of disorder in phase space. This gives us the occasion to present a secondary achievement of the thesis consisting in establishing a resource theory for local Gaussian work extraction, which exploits the symplectic formalism within quantum thermodynamics. In this context, work can be defined as the difference between the trace and symplectic trace of the covariance matrix of the state, and it displays a number of interesting properties. Back to our primary in, Cette thèse est centrée sur une nouvelle approche de l'incertitude quantique basée sur l'application de la théorie de la majorisation continue aux distributions de l'espace des phases quantique. La théorie de la majorisation est un cadre mathématique puissant qui vise à comparer des distributions par rapport à leur désordre intrinsèque. Elle est particulièrement importante dans le sens où établir une relation de majorisation entre deux distributions revient à prouver que toute mesure (Shur-concave) du désordre classera une distribution comme plus ordonnée que l'autre. Bien que cela soit moins connu, les distributions ici n'ont pas besoin d'être normalisées ni positives pour que la théorie de la majorisation s'applique, de sorte que cette dernière s'étend même au-delà des distributions de probabilité. De plus, une relation de majorisation peut être rigoureusement définie pour les distributions discrètes et continues sur un domaine de taille finie, ainsi que pour les distributions (discrètes et continues) qui sont positives sur un domaine de taille infinie. L'idée maîtresse de cette thèse est de caractériser l'incertitude quantique dans l'espace des phases en appliquant les outils de la théorie de la majorisation à la fonction de Wigner, qui est la (quasi-)distribution la plus commune qui incarne un état quantique dans l'espace des phases. Les fonctions de Wigner sont en général positives et négatives, ce qui les place hors de portée de la plupart des mesures de la théorie de l'information mais en fait des candidats parfaits pour la théorie de la majorisation. Nous commençons notre manuscrit par un aperçu succinct des bases de l'optique quantique dans l'espace de phase, qui sont une condition préalable à la caractérisation du désordre dans l'espace des phases. Cela nous donne l'occasion de présenter une réalisation secondaire de la thèse consistant à établir une théorie des ressources pour l'extraction du travail gaussien local, qui exploite le formalisme symplectique, Doctorat en Sciences de l'ingénieur et technologie, info:eu-repo/semantics/nonPublished

Quantum thermodynamics can be cast as a resource theory by considering free access to a heat bath, thereby viewing the Gibbs state at a fixed temperature as a free state and hence any other state as a resource. Here, we consider a multipartite scenario where several parties attempt at extracting work locally, each having access to a local heat bath (possibly with a different temperature), assisted with an energy-preserving global unitary. As a specific model, we analyze a collection of harmonic oscillators or a multimode bosonic system. Focusing on the Gaussian paradigm, we construct a reasonable resource theory of local activity for a multimode bosonic system, where we identify as free any state that is obtained from a product of thermal states (possibly at different temperatures) acted upon by any linear-optics (passive Gaussian) transformation. The associated free operations are then all linear-optics transformations supplemented with tensoring and partial tracing. We show that the local Gaussian extractable work (if each party applies a Gaussian unitary, assisted with linear optics) is zero if and only if the covariance matrix of the system is that of a free state. Further, we develop a resource theory of local Gaussian extractable work, defined as the difference between the trace and symplectic trace of the covariance matrix of the system. We prove that it is a resource monotone that cannot increase under free operations. We also provide examples illustrating the distillation of local activity and local Gaussian extractable work., SCOPUS: ar.j, info:eu-repo/semantics/published