Complexity Measure: A Quantum Information Approach

Abstract

In the past decades, all of the efforts at quantifying systems complexity with a general tool has usually relied on using Shannon's classical information framework to address the disorder of the system through the Boltzmann–Gibbs–Shannon entropy, or one of its extensions. However, in recent years, there were some attempts to tackle the quantification of algorithmic complexities in quantum systems based on the Kolmogorov algorithmic complexity, obtaining some discrepant results against the classical approach. Therefore, an approach to the complexity measure is proposed here, using the quantum information formalism, taking advantage of the generality of the classical-based complexities, and being capable of expressing these systems' complexity on other framework than its algorithmic counterparts. To do so, the Shiner–Davison–Landsberg (SDL) complexity framework is considered jointly with linear entropy for the density operators representing the analyzed systems formalism along with the tangle for the entanglement measure. The proposed measure is then applied in a family of maximally entangled mixed state.

LMC and SDL Complexity Measures: A Tool to Explore Time Series

“…Since the early 2000s, the idea of adapting LMC and SDL to dynamical systems was successfully applied to different types of time evolution problems: bird songs [13], neural plasticity [14], interactions between species in ecological systems [2], physiognomies of landscapes [15], economic series [16], spread depression [17], and quantum information [18].…”

Dynamic Complexity Measures: Definition and Calculation

“…Since the early 2000's, the idea of adapting LMC and SDL to dynamical systems was successfully applied to different types of time evolution problems: sleep-awake cycle [21], bird songs [24], neural plasticity [10,14], interactions between species in ecological systems [1,19], physiognomies of landscapes [9], economical series [12,20], spread depression [18] and quantum information [2,17].…”

Tree-size complexity of multiqubit states

“…We believe that an explicit construction will be very useful for further studies on the topic of complex quantum states. Among the several complexity measures proposed [16,[19][20][21][22][23], we focus on the tree size of a quantum state, introduced by Aaronson in an attempt to give a more rigorous foundation to the debate on the possibility of large-scale quantum computing versus a hypothetical breakdown of quantum mechanics [16]. This measure of complexity is motivated by the work of Raz, who showed that any multilinear formula for the determinant and permanent of a matrix must be superpolynomial in size [24].…”