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 ν-lattices: the en-tropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information… (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 give a self-contained, new proof of the monotonicity of the quantum relative entropy which seems to be natural from the point of view of quantum information theory. It is based on the quantum version of Stein's lemma which provides an operational interpretation of the quantum relative entropy.
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)
—We define an algorithm that parses multidimen-sional arrays sequentially into mainly unrepeated but nested multidimensional sub-arrays of increasing size, and show that the resulting sub-block pointer encoder compresses almost every realization of any finite-alphabet ergodic process on Z d ≥0 to the entropy, in the limit.
30. A. Hone. Diophantine nonintegrability of a third order recurrence with the Laurent property. two discrete integrability criteria: singularity confinement and algebraic entropy. (87j:05003) 45. T. Takenawa. A geometric approach to singularity confinement and algebraic entropy. Algebraic entropy and the space of initial values for discrete dynamical… (More)