[Paper Review] Multipartite entangled states, symmetric matrices and error-correcting codes

Keqin Feng, Lingfei Jin+2 more|arXiv (Cornell University)|Nov 25, 2015

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper presents two novel constructions for $k$-uniform multipartite entangled quantum states using symmetric matrices and classical error-correcting codes. It proves the existence of $n/2$-uniform $n$-qudit states for even $n$ and sufficiently large prime $d$, establishes $k = \Omega(n)$ uniformity, and resolves an open question by constructing $3$-uniform states for all $n \geq 8$. The symmetric matrix method outperforms the code-based approach in achieving higher $k$-uniformity.

ABSTRACT

A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit are maximally mixed. We investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ levels. We provide one construction via symmetric matrices and the second one through classical error-correcting codes. There are three main results arising from our constructions. Firstly, we show that for any given even $n\ge 2$, there always exists an $n/2$-uniform $n$-qudit quantum state of level $p$ for sufficiently large prime $p$. Secondly, both constructions show that their exist $k$-uniform $n$-qudit pure quantum states such that $k$ is proportional to $n$, i.e., $k=Ω(n)$ although the construction from symmetric matrices outperforms the one by error-correcting codes. Thirdly, our symmetric matrix construction provides a positive answer to the open question in \cite{DA} on whether there exists $3$-uniform $n$-qudit pure quantum state for all $n\ge 8$. In fact, we can further prove that, for every $k$, there exists a constant $M_k$ such that there exists a $k$-uniform $n$-qudit quantum state for all $n\ge M_k$. In addition, by using concatenation of algebraic geometry codes, we give an explicit construction of $k$-uniform quantum state when $k$ tends to infinity.

Motivation & Objective

• To resolve the open question of whether $3$-uniform $n$-qudit pure quantum states exist for all $n \geq 8$.
• To construct $k$-uniform $n$-qudit quantum states with $k$ proportional to $n$, i.e., $k = \Omega(n)$, for arbitrary $n$.
• To establish the existence of $k$-uniform states for all $n \geq M_k$ for any fixed $k$, with $M_k$ a constant depending on $k$.
• To provide explicit constructions using symmetric matrices and classical error-correcting codes, particularly algebraic geometry codes, for asymptotic $k \to \infty$.
• To compare the performance of symmetric matrix and code-based constructions in achieving high $k$-uniformity.

Proposed method

• Define $k$-uniform quantum states via a map from $\mathbb{Z}_d^n$ to $\mathbb{C}$, providing an equivalent characterization for constructing such states.
• Construct $k$-uniform states using symmetric matrices over finite fields, leveraging their algebraic structure to ensure maximal mixed reductions.
• Construct $k$-uniform states via classical error-correcting codes with large minimum distance and dual distance, exploiting the duality between quantum codes and orthogonal arrays.
• Use concatenation of algebraic geometry codes to achieve $k$-uniform states when $k$ tends to infinity, enabling asymptotic constructions.
• Prove that for any even $n \geq 2$, there exists an $n/2$-uniform $n$-qudit state over level $p$ for sufficiently large prime $p$.
• Demonstrate that the symmetric matrix construction yields higher $k$-uniformity than the code-based method, especially for large $n$.

Experimental results

Research questions

• RQ1 Does there exist a $3$-uniform $n$-qudit pure quantum state for all $n \geq 8$?
• RQ2 Can $k$-uniform $n$-qudit states be constructed such that $k$ is proportional to $n$, i.e., $k = \Omega(n)$?
• RQ3 For any fixed $k$, does there exist a constant $M_k$ such that $k$-uniform $n$-qudit states exist for all $n \geq M_k$?
• RQ4 Can explicit constructions of $k$-uniform states be achieved when $k$ tends to infinity?
• RQ5 How do the symmetric matrix and error-correcting code constructions compare in terms of achievable $k$-uniformity?

Key findings

• For any even $n \geq 2$, there exists an $n/2$-uniform $n$-qudit quantum state over level $p$ for sufficiently large prime $p$.
• Both constructions yield $k$-uniform states with $k = \Omega(n)$, demonstrating that high uniformity is achievable for large $n$.
• The symmetric matrix construction provides a positive answer to the open problem in DA regarding the existence of $3$-uniform $n$-qudit states for all $n \geq 8$.
• For every $k$, there exists a constant $M_k$ such that $k$-uniform $n$-qudit states exist for all $n \geq M_k$.
• By concatenating algebraic geometry codes, the paper gives an explicit construction of $k$-uniform quantum states when $k$ tends to infinity.
• The symmetric matrix method outperforms the error-correcting code method in achieving higher $k$-uniformity, especially for large $n$.