Minimal decomposition entropy and optimal representations of absolutely maximally entangled states

TL;DR

This paper introduces a minimal decomposition entropy measure to analyze and classify absolutely maximally entangled states, providing efficient algorithms and revealing their optimal representations and entanglement properties across various quantum systems.

Contribution

It develops an efficient algorithm for computing minimal decomposition entropy and applies it to classify AME states, revealing their sparser representations and entanglement characteristics.

Findings

AME states have lower minimal entropy for q=2, indicating sparser forms.

AME states show higher entanglement at q=∞, related to geometric measures.

The algorithm simplifies AME states into more manageable representations.

Abstract

Understanding and classifying multipartite entanglement is fundamental to quantum information processing. This work focuses on absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. To analyze and classify AME states, we employ the minimal decomposition entropy, defined as the minimum R\'{e}nyi entropy $S_q$ associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing properties of AME states. We develop an efficient algorithm for computing the minimal decomposition entropy for finite $q>1$ and compare AME and Haar-random states for \( q = 2 \) and \( q = \infty \) in qubit, qutrit, and ququad systems. For \( q = 2 \), AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For \( q = \infty \) -- related to the geometric measure of entanglement -- AME states exhibit higher entanglement. The algorithm also simplifies known AME states into sparser representations, aiding in distinguishing genuinely quantum AME states from those constructible from classical combinatorial designs. Our results advance the classification of AME states and demonstrate the utility of minimal decomposition entropy as both a local unitary invariant and a tool for state simplification.