The Wehrl entropy has Gaussian optimizers

We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantum-classical channel that associates to a quantum state its Husimi Q representation is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to determine the p->q norms of the aforementioned quantum-classical channel in the two particular cases of one mode and p=q, and prove that they are achieved by thermal Gaussian states. The same equivalence permits to prove that the Husimi Q representation of a one-mode passive state (i.e. a state diagonal in the Fock basis with eigenvalues decreasing as the energy increases) majorizes the Husimi Q representation of any other one-mode state with the same spectrum, i.e. it maximizes any convex functional.Comment: Proof extended to multimode state

Uncertainty relations with quantum memory for the Wehrl entropy

We prove two new fundamental uncertainty relations with quantum memory for the Wehrl entropy. The first relation applies to the bipartite memory scenario. It determines the minimum conditional Wehrl entropy among all the quantum states with a given conditional von Neumann entropy and proves that this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The second relation applies to the tripartite memory scenario. It determines the minimum of the sum of the Wehrl entropy of a quantum state conditioned on the first memory quantum system with the Wehrl entropy of the same state conditioned on the second memory quantum system and proves that also this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The Wehrl entropy of a quantum state is the Shannon differential entropy of the outcome of a heterodyne measurement performed on the state. The heterodyne measurement is one of the main measurements in quantum optics and lies at the basis of one of the most promising protocols for quantum key distribution. These fundamental entropic uncertainty relations will be a valuable tool in quantum information and will, for example, find application in security proofs of quantum key distribution protocols in the asymptotic regime and in entanglement witnessing in quantum optics

New lower bounds to the output entropy of multi-mode quantum Gaussian channels

We prove that quantum thermal Gaussian input states minimize the output entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are entanglement breaking and of the multi-mode quantum Gaussian phase contravariant channels among all the input states with a given entropy. This is the first time that this property is proven for a multi-mode channel without restrictions on the input states. A striking consequence of this result is a new lower bound on the output entropy of all the multi-mode quantum Gaussian attenuators and amplifiers in terms of the input entropy. We apply this bound to determine new upper bounds to the communication rates in two different scenarios. The first is classical communication to two receivers with the quantum degraded Gaussian broadcast channel. The second is the simultaneous classical communication, quantum communication and entanglement generation or the simultaneous public classical communication, private classical communication and quantum key distribution with the Gaussian quantum-limited attenuator

Rational behaviour, Risk aversion, High stakes for society

Certain areas related to the topics under discussion here lie outside my field; for instance the evaluation of risk assessment and security deficiencies in the transport sector. What has convinced me of the importance of this subject are a few very general conclusions, indeed I would say, impressions, that I have drawn from the truly remarkable development of our powers to analyse the risk decision-making process over some years now.Risk, uncertainty, home security, expected utility, non-expected utility, OECD

Minimal perturbations approaching the edge of chaos in a Couette flow

This paper provides an investigation of the structure of the stable manifold of the lower branch steady state for the plane Couette flow. Minimal energy perturbations to the laminar state are computed, which approach within a prescribed tolerance the lower branch steady state in a finite time. For small times, such minimal-energy perturbations maintain at least one of the symmetries characterizing the lower branch state. For a sufficiently large time horizon, such symmetries are broken and the minimal-energy perturbations on the stable manifold are formed by localized asymmetrical vortical structures. These minimal-energy perturbations could be employed to develop a control procedure aiming at stabilizing the low-dissipation lower branch state

The conditional entropy power inequality for quantum additive noise channels

We prove the quantum conditional Entropy Power Inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional Entropy Power Inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional Entropy Power Inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogues of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on Entropy Power Inequalities.Comment: 26 pages; updated to match published versio

Defining Desire: Re(storyng) a 'fraudulent' marriage in 1901 Spain

In the second half of the 19th century, two Spanish primary school teachers were married despite the fact that their legal status as women rendered this union not only illegal but also publicly scandalous. In 2008 their story was resurrected in the form of a book based on an extensive review of educational, legal, and media archives. The Spanish press responded to the book’s publication by embedding the events within a more recent historical narrative around the struggle for gay marriage rights. In this article, we analyze the events in light of the understandings of sex, gender and sexuality that were available at the time, and then explore both the continuities and discontinuities with the modern interpretive framework that affords these women a lesbian identity, drawing upon Bennett’s notion of ‘‘lesbian-like’’ practices in eras where such identities were not yet conceptualized

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers

We determine the p->q norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the p->q norms of any quantum semigroup.Comment: Annales Henri Poincar\'e (2018

Experiments testing macroscopic quantum superpositions must be slow

We consider a thought experiment where the preparation of a macroscopically massive or charged particle in a quantum superposition and the associated dynamics of a distant test particle apparently allow for superluminal communication. We give a solution to the paradox which is based on the following fundamental principle: any local experiment, discriminating a coherent superposition from an incoherent statistical mixture, necessarily requires a minimum time proportional to the mass (or charge) of the system. For a charged particle, we consider two examples of such experiments, and show that they are both consistent with the previous limitation. In the first, the measurement requires to accelerate the charge, that can entangle with the emitted photons. In the second, the limitation can be ascribed to the quantum vacuum fluctuations of the electromagnetic field. On the other hand, when applied to massive particles our result provides an indirect evidence for the existence of gravitational vacuum fluctuations and for the possibility of entangling a particle with quantum gravitational radiation.Comment: 12 pages, 1 figur

Dual Labor Market and Endogenous Fluctuations

We study the influence of wage differential on the emergence of endogenous fluctuations. In this way, we introduce a dual labor market, based on the Shapiro-Stiglitz efficiency wage theory in an overlapping generations model. We show that wage inequality is a source of endogenous fluctuations. Indeed, a sufficiently strong wage differentialleads to the occurrence of cycles of period two and local indeterminacy. Moreover, in contrast to several existing contributions, these results depend neither on increasing returns to scale, nor on the degree of capital-labor substitution.Endogenous fluctuations ; dual labor market ; wage inequality ; efficiency wage