Hawking radiation and the quantum marginal problem

In 1974 Steven Hawking showed that black holes emit thermal radiation, which eventually causes them to evaporate. The problem of the fate of information in this process is known as the “black hole information paradox”. Two main types of resolution postulate either a fundamental loss of information in Nature — hence the breakdown of quantum mechanics — or some sort of new physics, e.g. quantum gravity, which guarantee the global preservation of unitarity. Here we explore the second possibility with the help of recent developments in continuous-variable quantum information. Concretely, we employ the solution to the Gaussian quantum marginal problem to show that the thermality of all individual Hawking modes is consistent with a global pure state of the radiation. Surprisingly, we find out that the mods of radiation of an astrophysical black hole are thermal until the very last burst. In contrast, the single-mode thermality of Hawking radiation originating from microscopic black holes, expected to evaporate through several quanta, is not excluded, though there are constraints on modes' frequencies. Our result paves the way towards a systematic study of multi-mode correlations in Hawking radiation.

Entanglement marginal problems

We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.

MICROPLASTICS RISK AT THE INTERFACE OF CIRCULAR ECONOMY, QUALITY AND FOOD SAFETY IN POLAND: A CASE STUDY

The purpose of this paper is to explore the scope of undertaken and planned actions related to microplastics risk in one of the biggest polish food sector company for better understanding the relations between quality management, food safety management and circular economy. To explore the planed and undertaken actions in relation to microplastics risk a case study was performed. It was performed in one of the leading polish organization producing and selling food products of an international range of operation. During the study representative of the leadership of the organization was interviewed as well as relevant instructions and records provided by the organization were analysed. Results indicate that the issue of microplastics for studied company is a marginal problem, that has not been given much attention so far. On the other hand, significant changes at the global level such as circular economy strategy lead to changes in the design of products (mainly packaging). To the best of the authors’ knowledge, this is the first study to examine the microplastics risk from the perspective of the management systems implemented in food sector organization with the circular economy context. This study enriches current knowledge about circular economy and its integration with other management systems in the microplastics elimination context.

From a ‘Marginal Problem’ to the ‘Supreme International Jurisdiction’

The first of two chapters to explore the theme protection, chapter 4 records the range of conflicting attitudes displayed by Jacob Robinson and Shabtai Rosenne towards the Genocide Convention during its drafting, with regard to and following its ratification, and at the International Court of Justice advisory proceedings on the question of reservations to the Convention. The chapter describes their early disinterest in and indifference towards the Genocide Convention as a ‘marginal problem’ on the United Nations agenda, but also the circumstances under which they came to acknowledge and appropriate the Convention’s Jewish paternity, exploit the opportunities it presented while, in private, recording their hostility towards Raphael Lemkin, its progenitor, as well as their derision of the Convention’s promise to protect Jewish existence.

Quantum marginal problem and incompatibility

One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.

Jordan products of quantum channels and their compatibility

Given two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

A complete hierarchy for the pure state marginal problem in quantum mechanics

Clarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are compatible with some pure global quantum state. As an application, we prove that the existence of multiparticle absolutely maximally entangled states for a given dimension is equivalent to the separability of an explicitly given two-party quantum state. Finally, we show that the existence of quantum codes with given parameters can also be interpreted as a marginal problem, hence, our complete hierarchy can also be used.