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Gaussian entropic inequalities and uncertainty relations for communication and secure quantum key distribution

Periodic Reporting for period 1 - GENIUS (Gaussian entropic inequalities and uncertainty relations for communication and secure quantum key distribution)

Reporting period: 2018-04-01 to 2020-03-31

Information and communication technologies are essential to modern society as the internet is pervading all aspects of our lives. Quantum mechanics imposes fundamental limits to the communication rates. The first objective of GENIUS was to determine these limits. Given the amount of sensitive information sent through the internet, secure communications are essential to our society. To fulfill this need, the EU is investing in quantum key distribution with the 1G€ Quantum Technology Flagship. Quantum key distribution is a technique to generate a secret key shared between two parties. The essential feature of quantum mechanics is that any measurement of a system unavoidably disturbs it. Quantum key distribution exploits this feature to detect any attempt of intercepting the communication and guarantee the security of the key. The second objective of GENIUS was to determine the maximum rates for quantum key distribution achievable by the forthcoming generation of quantum key distribution devices and to prove their perfect security.
The information sent through internet is encoded in light traveling through optical fibers. Optical fibers also provide the most promising platform for the forthcoming generation of devices for quantum key distribution. Quantum Gaussian channels provide the mathematical model for the propagation of light through optical fibers in the quantum regime. The entropy of a physical system quantifies its information content or the uncertainty in its state, and its conditional entropy quantifies its uncertainty from the point of view of a third party who has interacted with the system to get information on it. The maximum rates for communication and for quantum key distribution are determined by the properties of the entropy and of the conditional entropy of the output of the communication channel. GENIUS has achieved its goals by proving the following properties of the entropy:
- I have determined the minimum entropy of the output of a large family of quantum Gaussian channels. I have then exploited this result to prove new properties of the entropy of the output of all quantum Gaussian channels. This result allowed me to prove new limits to the maximum communication rates achievable in communication to multiple receivers and to the maximum rates achievable for simultaneous communication and quantum key distribution.
- I have determined the minimum conditional entropy of the outcome of the heterodyne measurement, which is at the basis of the most promising protocol for quantum key distribution. This result provides a guarantee on the uncertainty of the key from the point of view of any possible party who wishes to intercept it, and therefore contributes to prove the perfect security of the protocol.
- I have determined the minimum conditional entropy of the output of the quantum Gaussian channels that model the noise on optical fibers. This fundamental result allowed me to determine how much quantum correlations can help to increase the communication rates through noisy optical fibers.
- I have proven the Entropy Power Inequality for classical random variables in the presence of quantum conditioning. This is a new entropic inequality with applications in determining the help of quantum correlations in the task of distributed source coding, in which multiple senders communicate their share of information.
- I have determined the maximum amount of quantum correlations that an optical fibre is able to generate. This result implies limits on the maximum achievable rates for quantum key distribution.
Moreover, I have determined the optimal probes in quantum illumination, a new protocol that exploits quantum correlations to increase the sensitivity in the detection of the presence of objects, with applications ranging from radar technology to medical sensors.

I have presented the results of the project in five talks at leading international conferences attended by all the main experts of the field and in two seminars at the Massachusetts Institute of Technology and at the Institute of Science and Technology Austria, respectively. Moreover, I have organized at the University of Copenhagen a masterclass on the topic of the project, which attracted almost 100 participants from both academia and industry.
The results of the project constitute a fundamental contribution to the determination of the maximum communication rates for communication to multiple receivers and to the determination of the increase of the communication rates that can be achieved exploiting quantum correlations. The project has thus contributed to the achievement of high-rate communications, which are fundamental for the European society and have been declared as a priority by the EU with the “Digital Single Market Strategy”. Furthermore, the results of the project contribute to the proof of the security of a protocol to generate shared secret keys and to the determination of the maximum rates achievable for simultaneous communication and generation of a shared secret key through optical fibres. Therefore, the project has contributed to the achievement of secure communications, which are a need of pivotal importance for the European society and have been declared as a priority by the EU with the “Network and Information Security Directive”.
The results of the project will be precious for both applied and theoretical researchers in quantum communication. The theoretical researchers will have at disposal a new set of entropic inequalities to determine the maximum rates for communication and key distribution in further new scenarios, while the applied researchers will be able to benchmark their protocols and proposed devices for quantum communication with the rates determined by this project.