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.[-]

Among the set of all pure states living in a finite dimensional Hilbert space $\mathcal{H}_N$one distinguishes subsets of states satisfying some natural condition. One basis independent choice, consist in selecting the spin coherent states, corresponding to the $SU(2)$ group, or generalized, $SU(K)$ coherent states. Another often studied example is basis dependent, as states coherent with respect to a given basis are distinguished by the fact that the moduli of their off-diagonal elements (called 'coherences') are as large as possible. It is natural to define 'anti-coherent' states, which are maximally distant to the set of coherent states and to quantify the degree of coherence of a given state can by its distance to the set of anti-coherent states. For instance, the separable states of a system composed of two subsystems with $N$ levels are coherent with respect to the composite group $SU(N)\times SU(N)$, while in this setup, the anti-coherent states are maximally entangled.[-]

Conventional quantum field theory techniques do not work for extracting physical information from a background-independent quantum theory of gravity. A technique that works is Oeckl's boundary formalism, with semiclassical coherent states on the boundary. I illustrate how this technique has allowed us to compute the lifetime of a black hole in loop quantum gravity. This is an astrophysical relevant quantity that could have observational consequences.[-]

Ingrid Daubechies, James B. Duke Professor of Mathematics and Electrical and Computer Engineering at Duke University.

Baroness Ingrid Daubechies (In 2012 King Albert II of Belgium granted her the title of Baroness) is a Belgian physicist and mathematician. Between 2004 and 2011 she was the William R. Kenan, Jr. Professor in the mathematics and applied mathematics departments at Princeton University. She taught at Princeton for 16 years. In January 2011 she moved to Duke University as a professor in mathematics. She was the first woman to be president of the International Mathematical Union (2011–2014). She is best known for her work with wavelets in image compression.

Why she does mathematics, first mathematical memories, first encounter with mathematics, influences, research themes, wavelets theory, collaboration with Alex Grossman and Jean Morlet, first « Eurêka moment », etc.[-]

We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their duality structure, we introduce two natural partial inner product spaces and formulate a number of open questions.

Keywords: continuous frames - semi-frames - frame duality - reproducing pairs - partial inner product spaces[-]