By Fabio Benatti

The top subject of the booklet is complexity in quantum dynamics. This factor is addressed by way of comparability with the classical ergodic, details and algorithmic complexity theories.

Of specific significance is the suggestion of Kolmogorov-Sinai dynamical entropy and of its inequivalent quantum extensions formulated via Connes, Narnhofer and Thirring on one hand and Alicki and Fannes at the different. Their connections with extensions to quantum structures of Kolmogorov-Chaitin-Solomonoff algorithmic complexity concept is usually provided. The technical instruments hired are these of the algebraic method of quantum statistical mechanics which bargains a unifying view of classical and quantum dynamical platforms. Proofs and examples are supplied with a view to make the presentation self consistent.

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As the measured values depend on the system state, it follows that measuring g and then f yields the same results as measuring f and then g. Also, it is practically convenient to approximate certain observables in the algebra by means of other observables that are in a certain sense close to them; we shall thus assume these commutative algebras of observables to be endowed with topologies and to be closed with respect to them; in particular, we shall consider algebras of observables where converging sequences of functions do converge to observables in the algebra.

1. 72). 6 they are K-systems. 2. 3). As a partition of T2 , choose the Cartesian product C of the partitions of the 1-dimensional torus T into atoms C1 = {0 ≤ θ < π}, C2 = {π ≤ θ < 2π}. Because of ergodicity, the trajectories of the end points of the atoms Ci × Cj fill T2 densely and the intersections of their images T k (Ci × Cj ) under the dynamics become finer and finer and approximate better and better the Borel σ-algebras of T2 . Actually, this already occurs if one restricts to T −j (C) with j ≥ 0, +∞ namely j=0 T −j (C) = Σ.

64) S0 ∈ Tail (P), then S0 belongs to P−n] := j≥n T −j (P) for all n and thus to the σ-algebra Σn∞ (P), generated by P−n] . 64) which then yields μ(S0 ∩ S0 ) = μ(S0 )2 and, in turn, Tail (P) = N . 64) a finite partition P 9 and consider the σ-subalgebras P−n] ⊆ Σ generated by the infinite refinements −k (P). 50)) are such that, for any A0 ∈ Σ and B ∈ P−n] , μ(A0 ∩ B)(x) − μ(A0 )μ(B) ≤ B dμ (x) μ(A0 |P−n] )(x) − μ(A0 ) . e. when n → ∞, whence K-mixing follows from Lebesgue dominated convergence theorem.

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