[Paper Review] Constructing new k-uniform and absolutely maximally entangled states

Zahra Raissi, Adam Teixidó-Bonfill+2 more|arXiv (Cornell University)|Oct 28, 2019

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a novel systematic method to construct k-uniform and absolutely maximally entangled quantum states beyond those derived from maximum distance separable (MDS) codes. The approach yields new classes of states that are inequivalent to MDS-based constructions and resolves open questions by explicitly constructing previously unknown absolutely maximally entangled states for the first time.

ABSTRACT

Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.

Motivation & Objective

• To develop a systematic construction method for k-uniform quantum states beyond the known MDS code-based approach.
• To address the open problem of constructing absolutely maximally entangled states that are not equivalent to those from MDS codes.
• To demonstrate the existence of new classes of k-uniform states with distinct entanglement properties.

Proposed method

• The method leverages a class of classical error-correcting codes different from MDS codes to generate quantum states with desired uniformity properties.
• It ensures that all reductions to any k parties result in maximally mixed states, satisfying the k-uniform condition.
• The construction is systematic and generalizable, enabling the derivation of multiple new examples across various n and q.
• The approach uses algebraic and combinatorial techniques to verify k-uniformity and entanglement structure.
• It establishes a framework to identify and generate absolutely maximally entangled states through state reduction analysis.
• The method is proven to produce states not equivalent to any MDS-based construction, confirming their novelty.

Experimental results

Research questions

• RQ1 Can k-uniform quantum states be systematically constructed outside the MDS code framework?
• RQ2 Are there absolutely maximally entangled states that are not equivalent to those derived from MDS codes?
• RQ3 What is the structural difference between MDS-based and non-MDS-based k-uniform states?
• RQ4 Can the existence of previously unknown absolutely maximally entangled states be rigorously established?
• RQ5 What are the necessary and sufficient conditions for a quantum state to be both k-uniform and absolutely maximally entangled?

Key findings

• The proposed method successfully generates new k-uniform states that are not equivalent to any constructed from MDS codes.
• Several examples of absolutely maximally entangled states are explicitly constructed, resolving an open problem regarding their existence.
• The constructed states exhibit distinct entanglement properties compared to MDS-based counterparts, confirming their non-equivalence.
• The method provides a general framework for generating k-uniform states beyond the limitations of MDS codes.
• The results demonstrate that the class of k-uniform and absolutely maximally entangled states is richer than previously known through MDS-based constructions.