Effective Complexity and its Relation to Logical Depth

Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information content. In this paper, we give a precise formal definition of effective complexity and rigorous proofs of its basic properties. In particular, we show that incompressible binary strings are effectively simple, and we prove the existence of strings that have effective complexity close to their lengths. Furthermore, we show that effective complexity is related to Bennett's logical depth: If the effective complexity of a string $x$ exceeds a certain explicit threshold then that string must have astronomically large depth; otherwise, the depth can be arbitrarily small.Comment: 14 pages, 2 figure

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for $3\times 3$ matrices.Comment: 27 page

Der Shannon-McMillan-Satz und verwandte Resultate für ergodische Quantenspingittersysteme und Anwendung in der Quanteninformationstheorie

Wir zeigen, dass Quantenversionen des berühmten Shannon-McMillan Satzes und seiner auf Breiman zurückgehenden Verschärfung existieren. In der Ergodentheorie ist der SMB-Satz ein Grenzwertsatz für die dynamische Entropie. Diese ist im Fall klassischer Spingittersysteme gleich der Shannon-Entropierate. Wir betrachten Quantengittersysteme modelliert als C*-dynamische Systeme, wobei die Dynamik durch die Wirkung der Translationsgruppe auf einer quasilokalen C*-Algebra gegeben ist. Es stellt sich heraus, dass in diesem Kontext die von Neumann-Entropierate die Shannon-Entropierate verallgemeinert: Sie gibt die asymptotisch exponentielle Wachstumsrate der Dimension von Hilbertteilräumen an, die typisch sind bzgl. einem ergodischen Quantenzustand auf einer quasilokalen Algebra. Basierend auf dem Quanten-SM-Satz beweisen wir ein Quanten-Datenkompressionstheorem: Die von Neumann-Entropierate ergodischer Quanteninformationsquellen, modelliert durch 1-dimensionale Quantengittersysteme, ist die erreichbare untere Schranke an die Kompressionsrate asymptotisch zuverlässiger Block-Datenkompressionsschemen für diese Quelle.We show that there exist quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the SMB-theorem is a limit theorem for the dynamical entropy. This is equal to the Shannon entropy rate in the case of classical spin lattice systems. We consider quantum lattice systems modeled as C*-dynamical systems, where the dynamics is given by the action of the translation group on a quasi-local C*-algebra. It turns out that in this setting the von Neumann entropy rate generalizes the Shannnon entropy rate: It gives the asymptotically exponential growth rate of the dimension of Hilbert subspaces typical with respect to an ergodic quantum state on a quasi-local algebra. Based on the quantum SM-theorem we prove a quantum data compression theorem: The von Neumann entropy rate of an ergodic quantum information source -modeled by a 1-dimensional quantum lattice system- is the achievable lower bound on the compression rate of asymptotically reliably operating block data compression schemes for this source

Maximum likelihood type detectors for multiple quantum states

Non UBCUnreviewedAuthor affiliation: Max Planck Institute for Mathematics in the Sciences - LeipzigPostdoctora