On the nonlinearity of quantum dynamical entropy

TL;DR

This paper demonstrates that quantum dynamical entropy is nonlinear in time, contrasting with classical entropy, and provides exact calculations for specific quantum systems like the Hadamard walk.

Contribution

It reveals the nonlinearity of quantum dynamical entropy and computes exact values for the Hadamard walk with different measurement schemes.

Findings

Quantum dynamical entropy is nonlinear in time.

Classical Kolmogorov-Sinai entropy is linear in time.

Exact entropy values are computed for the Hadamard walk.

Abstract

Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Luders-von Neumann instruments and partitions.