Salman Beigi

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.

Abstract A correlation measure of a bipartite or multipartite source is said to satisfy the tensorization property if evaluating it on independent copies of the source, we get the same quantity as we evaluate it on a single copy. Such measures of correlation naturally appear in the study of some source coding and channel coding problems, as well as quantum nonlocality. This tutorial begins with some motivating examples that elucidate the importance of the tensorization property. Then two main measures of correlation with the tensorization property, namely the maximal correlation and the hypercontractivity ribbon are introduced. Finally some applications of these correlation measures, particularly in the study of quantum nonlocality in various settings are discussed.

Feynman [1] remarked that the spin–statistics theorem is one of the few principles in physics that can be simply stated yet whose proof requires the full machinery of relativistic quantum field theory. A central implication of this theorem is that fermions cannot be composite bosons. This naturally raises the question: can this fact admit an elementary, non-relativistic proof? Bell’s theorem [2] provides a paradigm for such elementary arguments: under the minimal assumption of causality, it rules out classical (local hidden-variable) explanations of quantum correlations. In particular, it shows that quantum systems such as qubits — carried, for instance, by bosons — cannot be simulated by classical bits, and that bosonic correlations cannot arise from compositions of classical particles. Inspired by this framework, we introduce a fermionic thought experiment whose outcome shows that fermions cannot be composite bosons through reasoning analogous to Bell’s theorem. The thought experiment is formulated in the setting of distributed quantum networks and draws on concepts from distributed computing. Within this framework, we prove the existence of fermionic correlations that admit no local hidden-qubit model and are strictly stronger than Bell nonlocal correlations achievable with qubits. This shows that standard quantum information theory is insufficient to represent information carried by indistinguishable fermions in distributed settings. The assumptions remain minimal: causality is preserved, and the distributed parties have no knowledge of the network topology. Our result therefore provides an information-theoretic, non-relativistic proof of the fundamental fermion–boson distinction implied by the spin–statistics theorem. References: [1] R. P. Feynman, The Character of Physical Law, MIT Press (1965). [2] J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics (1964).