Rainer Siegmund-Schultze

A Quantum Version of Sanov's Theorem

We present a quantum version of Sanovs theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.

Typical Support and Sanov Large Deviations of Correlated States

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.

An Ergodic Theorem for the Quantum Relative Entropy

Abstract:We prove the ergodic version of the quantum Stein’s lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.

Threshold-based epidemic dynamics in systems with memory

In this article we analyze an epidemic dynamics model (SI) where we assume that there are k susceptible states, that is a node would require multiple

The Shannon-McMillan theorem for ergodic quantum lattice systems

(k)

Quantum Stein's lemma revisited, inequalities for quantum entropies, and a concavity theorem of Lieb

contacts before it gets infected. In specific, we provide a theoretical framework for studying diffusion rate in complete graphs and d -regular trees with extensions to dense random graphs. We observe that irrespective of the topology, the diffusion process could be divided into two distinct phases: i) the initial phase, where the diffusion process is slow, followed by ii) the residual phase where the diffusion rate increases manifold. In fact, the initial phase acts as an indicator for the total diffusion time in dense graphs. The most remarkable lesson from this investigation is that such a diffusion process could be controlled and even contained if acted upon within its initial phase.