Rainer Siegmund-Schultze

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We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on Z-lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources(More)
We present a quantum extension of a version of Sanov’s 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 exponential separating(More)
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(More)
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(More)
We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a multidimensional lattice setting. We use techniques of packings and coverings with multidimensional windows to(More)
In part I (math.PR/0406392) we proved for an arbitrary onedimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n). In higher dimensions we call a random walk ’polygonally recurrent’ (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive(More)
Abstract. We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n). Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P (|X+Y | ≤ 1) < 2P (|X−Y | ≤ 1) for independent identically distributed X, Y is used. In part II we(More)