Christoph Hirche

Divergences lie at the core of information-theoretic applications. A recently introduced family of f-divergences, defined via an integral representation, has exhibited remarkable properties --- for instance, for the study of contraction coefficients. However, many familiar properties of their classical analogous have remained elusive. In this work, we develop alternative representations of the quantum f-divergences by leveraging the recently established quantum layer-cake theorem. These new formulations enable us to establish several key properties, including monotonicity and connections to other divergences. As our main application, we show how these representations unify and streamline various proofs in quantum hypothesis testing, yielding tighter achievability bounds through conceptually simple arguments that apply across different error regimes.

The task of testing the validity of a hypothesis underlies numerous applications in quantum information theory. The most commonly investigated approach is that of gathering all the available (quantum) data and making a final decision based on a collective measurement. However, such offline strategies are often far from practical, both in the amount of data required as well as in the complexity of the required measurement. In some settings, when the goal is quick detection, offline algorithms are not applicable at all, as they can only make a decision once all samples are received. Sequential methods offer the use of online strategies, where samples are requested on a need-to-know basis, drastically reducing the number of required samples in order to guarantee the, task specific, associated performance criteria. While extensively investigated and applied in the classical setting, we know far less about the optimal performance of such online strategies when quantum data is available. In this joint submission we present major recent progress on sequential methods for the fundamental tasks of quantum state discrimination, channel discrimination and quickest change point detection. In summary, we provide a comprehensive picture of the optimal asymptotic performance of online strategies in these settings under different performance criteria.

We study unitary 2-designs and decoupling with random diagonal-unitaries. We first show that the alternate application of random diagonal-unitaries in the Pauli-Z and -X bases constitutes a unitary 2-design after a number of repetitions, implying that the process achieves decoupling. We then go on to show that even fewer repetitions are sufficient for achieving decoupling at the same rate as that with Haar random unitaries. These results imply that precise unitary 2-designs are not necessary for achieving the Haar decoupling rate, indicating the possibility to achieve the rate by random unitaries less uniform than 2-designs. We also provide a simple quantum circuit that implements a unitary 2-design and achieves decoupling, which is partitioned into a constant number of commuting parts.