Quantum divergences, channel discrimination and strong data-processing

Information-theoretic divergences (also called relative entropies) serve as parent quantities for a plethora of operationally relevant quantities both in classical and quantum information theory. They can be used to express entropy (relevant quantity for data-compression), mutual information (relevant quantity in classical data transmission and channel simulation) and quantify distinguishability of states in discrimination tasks. Understanding mathematical properties of these divergences often immediately translates to making progress on related operational questions. While divergences in classical information theory have been long studied and are essentially fully classified and understood, in the quantum setting, due to different operator orderings, there are infinitely many more possibilities, and new useful variants keep getting introduced, which open up new pathways in research. By building on mathematical tools and frameworks provided by recent advances in quantum information theory, we want to further investigate properties of quantum divergences, and specifically focus on these two areas of questions:

1. Entropic quantities for quantum channels: Divergences serve as relevant distinguishability measures in discrimination tasks, and while the theory of state discrimination is very well understood, many questions regarding discrimination of channels (i.e. physical transformations between states) are still open. These are again often tied to mathematical properties of the underlying channel divergences which involve an optimization over input states and are hence trickier to deal with. One important question here is to find out when (and how specifically) discrimination strategies in which channels are used sequentially, and in which inputs to successive channel uses, are chosen based on previous channel outputs can give an advantage over simpler strategies, in which channels are used in parallel. The latter is related to a "chain-rule" property of the channel divergence. We want to study this in detail with the aim of fully understanding how parallel and sequential strategies are related and how there can sometimes be an advantage of one over the other.

2. Strong data processing: One of the defining (and arguably most important) properties of a quantum divergence is the data processing inequality: distinguishability of a pair of states can only decrease if they are subjected to processing operation (i.e. the action of a quantum channel). Strong data processing involves finding conditions under which distinguishability strictly decreases, and ideally also bound this decrease. Recently several lower bounds have been obtained on the difference of the divergences, which depend on whether the action of the channel can be approximately reversed. This has turned out to be very useful in many subsequent applications. It is still an open question if and to what degree these bounds are tight, with some indication that they are often not, which we would like to investigate. Instead of bounding the difference, we also want to look at bounding the quotient of the divergences before and after application of a channel, optimized over input states. These so-called contraction coefficients are closely related to very practical questions of rapid mixing (i.e. exponentially fast convergence of an open quantum system towards its equilibrium state), especially relevant in the context of quantum dynamical semigroups (i.e. quantum evolution of a system interacting with a memoryless environment). While classically these contraction coefficients are well understood, quantum-mechanically there are still many open questions, and already finding simple sufficient conditions for a contraction coefficient to be less than one would be a major step forward for many applications.