Quantum Entropies: Dynamics, Information and Complexity by Fabio Benatti

By Fabio Benatti

The top subject matter of the booklet is complexity in quantum dynamics. This factor is addressed through comparability with the classical ergodic, info and algorithmic complexity theories.

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

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Quantum Entropies: Dynamics, Information and Complexity

The prime subject of the publication is complexity in quantum dynamics. This factor is addressed by way of comparability with the classical ergodic, info and algorithmic complexity theories. Of specific significance is the concept of Kolmogorov-Sinai dynamical entropy and of its inequivalent quantum extensions formulated by way of Connes, Narnhofer and Thirring on one hand and Alicki and Fannes at the different.

Additional info for Quantum Entropies: Dynamics, Information and Complexity

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4 (von Neumann Ergodic Theorem). Let UT be the unitary Koopman operator acting on the Hilbert space H := L2μ (X ) of a dynamical triplet (X , T, μ), with T invertible. Let At : H → H be defined t−1 1 by At | ψ := U s | ψ , ψ ∈ H, and let P project onto the subspace K t s=0 T of vectors such that UT | ψ = | ψ . 1) of the sequence of operators At , P = s − limt→+∞ At . Proof: The subspace orthogonal to K is (UT − 1l)H; thus, for any ψ ∈ H, | ψ = P | ψ + (1l − P )| ψ = P | ψ + (UT − 1l)| φ , for some φ ∈ H.

Condition ii) expresses the fact that cylinder sets C[p,q] with p, q ∈ Z generate Σ, while in condition iii), C−n] = Tσ−j (C{0} ) n≥0 n≥0 j≥n denotes the largest σ-algebra, called tail of C{0} (Tail C{0} ) contained in all C−n] with n ≥ 0. [p,p+q] Cylinders in C−n] are of the form Ci(q+1) with p ≥ n , q ≥ 0; they become subsets of Tail C{0} when t → +∞. 3, one deduces that the characteristic functions of these atoms [0,q] go into the constant functions μ(Ci(q+1) ) 1l, asymptotically, whence condition iii).

4 (von Neumann Ergodic Theorem). Let UT be the unitary Koopman operator acting on the Hilbert space H := L2μ (X ) of a dynamical triplet (X , T, μ), with T invertible. Let At : H → H be defined t−1 1 by At | ψ := U s | ψ , ψ ∈ H, and let P project onto the subspace K t s=0 T of vectors such that UT | ψ = | ψ . 1) of the sequence of operators At , P = s − limt→+∞ At . Proof: The subspace orthogonal to K is (UT − 1l)H; thus, for any ψ ∈ H, | ψ = P | ψ + (1l − P )| ψ = P | ψ + (UT − 1l)| φ , for some φ ∈ H.

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