Quantum Dynamical Entropies and Gács Algorithmic Entropy

Abstract

Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov-Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following, we establish a similar relation between the Connes-Narnhofer-Thirring quantum dynamical entropy for the shift on quantum spin chains and the Gács algorithmic entropy. We further provide, for the same system, a weaker linkage between the latter algorithmic complexity and a different quantum dynamical entropy proposed by Alicki and Fannes.

Thermodynamic Properties and Decoherence of a Central Electron Spin of Atom Coupled to an Anti-Ferromagnetic Spin Bath

“…Thermodynamic properties of Quantum systems have become very attractive due to their potential applications in thermoelectric devices [1], tunneling and decoherence [2,3]. Thermodynamic properties of quantum systems now aid in the investigation of the dynamical entropy [4][5][6][7][8]. Recently, different definitions of specific heat are discussed [9] and the entropy for a quantum oscillator in an arbitrary heat bath at finite temperature is examined [10][11][12].…”

Gacs Algorithmic Complexity on Hilbert Spaces and Some of its Applications

“…In [6], a relation was established between the quantum algorithmic complexity as formulated in [21], that we shall refer to as Gacs complexity (entropy) in the following, and the quantum dynamical entropies of the shift automorphism on quantum spin chains as formulated by Connes, Narnhofer and Thirring (CNT-entropy) [1] and by Alicki and Fannes (AF-entropy) [2]. A quantum spin chain is a one-dimensional lattice with d-level quantum systems at each site and the lattice translations or shift-automorphisms are the simplest viii possible dynamics.…”

“…In [6], the extra term log d is given an informational interpretation in terms of the Gacs complexity per spin in the Alicki-Fannes construction. There, the limit rate is obtained starting from increasingly large, but finite-dimensional sub-chains and using the formulation in [21] that concerns arbitrary, but finite number of spins.…”

“…There, the limit rate is obtained starting from increasingly large, but finite-dimensional sub-chains and using the formulation in [21] that concerns arbitrary, but finite number of spins. As a consequence of the construction of the complexity rate from below, that is from finite dimensional subalgebras to the infinite dimensional spin-chain, a constraint had to be imposed in [6] on the growth of the classical complexity of finite-size density matrices; namely, that it be slower than the size of the sub-chain. Instead, in this thesis, we construct a Gacs complexity quantity starting directly from the infinite dimensional quantum spin chain.…”

“…Then, we introduce the two quantum dynamical entropies, CNT and AF, which are extensions of the classical KS-entropy, their relations and properties with applications to quantum spin chains. Most of the material in this chapter is taken from [6].…”