The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of quantum Gaussian states

Quantum Gaussian systems are fundamental for modelling the modes of electromagnetic radiation in the quantum regime. Electromagnetic waves provide the most promising platform for quantum communication and quantum key distribution. In this field, entropic inequalities are a fundamental tool to determine the maximum achievable rates for the protocols for communication and key distribution. In the first part of this thesis, we prove the multimode conditional quantum Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum von Neumann conditional quantum entropy of the output of a generic bosonic quantum Gaussian channel among all the input states with given conditional quantum entropies. Bosonic quantum Gaussian channels are a fundamental element in quantum optics because they provide a model for the attenuation, amplification and noise of electromagnetic signals. Our proof of the multimode conditional quantum entropy power inequality is based on a new Stam inequality for the quantum conditional Fisher information and an important result which describes the universal asymptotic behaviour of the quantum conditional entropy under the time evolution generated by the heat semigroup. The study of entropic inequalities for generic bosonic quantum Gaussian channels can be useful to obtain a better description of the transformations that electromagnetic waves undergo during their transmission, and the new inequality we present in this work can have a strong impact in quantum information and quantum cryptography. In the second part of the thesis we apply the multimode conditional quantum Entropy Power Inequality to the study of the squashed entanglement of a family of bosonic Gaussian quantum channels. This application provides a justification for the relevance of the new entropy inequality we have proved in the first part of the thesis. The squashed entanglement of a quantum state is one of the two main entanglement measures in quantum communication theory and provides a very tight upper bound to the length of a shared secret key that can be generated by two parties holding many copies of the quantum state. Similarly, the squashed entanglement of a quantum channel provides an upper bound to the capacity of the channel to generate a secret key shared between the sender and receiver. We first prove a lower and an upper bound to the squashed entanglement of a broad family of quantum Gaussian states obtained by applying a generic Gaussian unitary transformation to a thermal state tensored with the vacuum state. It is well known that lower bounds to the squashed entanglement are difficult to find as they require an optimisation that is hardly analytically tractable. We overcome this difficulty with the multimode conditional quantum Entropy Power Inequality. For some choices of the unitary transformation, it can be proved that the lower bound matches the upper bound in the limit of infinite energy. We conjecture that in this limit the gap between lower and upper bound closes for any choice of the Gaussian unitary transformation. We have performed several numerical simulation, which are all in agreement with the conjecture. We then apply this result to determine a new lower bound for the maximum squashed entanglement achievable between a sender and a receiver of a family of bosonic Gaussian quantum channels. Again, particular choices of the channel make it possible to prove that the lower bound coincides with the upper bound, thus permitting the squashed entanglement of the analysed quantum channel to be determined exactly. In this thesis we deal with a general case and we provide an explicit expression for the upper bound