Quantum dynamical entropy, chaotic unitaries and complex Hadamard matrices

TL;DR

This paper introduces quantum dynamical entropies as invariants for unitary operators, characterizes chaotic unitaries via complex Hadamard matrices, and analyzes their properties and prevalence in low-dimensional quantum systems.

Contribution

It defines new quantum dynamical entropy measures, characterizes chaotic unitaries with necessary and sufficient conditions in low dimensions, and studies their distribution and average entropy.

Findings

Chaotic unitaries correspond to rescaled complex Hadamard matrices.

Necessary conditions for chaos involve trace and determinant relations.

Average dynamical entropy grows logarithmically with system dimension.

Abstract

We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies. They quantify the randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy or, equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.