The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

Coherent states have been long known for their applications in quantum optics and atomic physics. In recent years, a number of new applications have emerged in the area of quantum information theory. In this talk I will highlight two such applications. The first is the comparison between classical and quantum strategies to process information. Byproducts of this comparison are benchmarks that can be used to certify quantum advantages in realistic experiments, fundamental relations between quantum copy machines and precision measurements, and theoretical tools for security proofs in quantum cryptography. The second application is the simulation of unitary gates in quantum networks. Here the task is to simulate a given set of unitary gates using gates in another set, a general problem that includes as special cases the simulation of charge conjugate dynamics and the emulation of an unknown unitary gate. The problem turns out to have useful connections with the ultimate precision limits of quantum metrology.[-]

Combining the relativistic speed limit on transmitting information with linearity and unitarity of quantum mechanics leads to a relativistic extension of the no-cloning principle called spacetime replication of quantum information. We introduce continuous-variable spacetime-replication protocols, expressed in a Gaussian-state basis, that build on novel homologically constructed continuous-variable quantum error correcting codes. Compared to qubit encoding, our continuous-variable solution requires half as many shares per encoded system. We show an explicit construction for the five-mode case and how it can be implemented experimentally. As well we analyze the ramifications of finite squeezing on the protocol.[-]