[Paper Review] $k$-Uniform states and quantum information masking

Fei Shi, Mao-Sheng Li+2 more|arXiv (Cornell University)|Sep 26, 2020

Quantum Computing Algorithms and Architecture42 references4 citations

TL;DR

This paper presents general constructions of $k$-uniform quantum states using linear codes, orthogonal arrays, and algebraic geometry codes, proving the existence of $k$-uniform states for $d \geq 4k-2$ (prime power) and $N \geq 2k$, and for $d \geq 2k-1$ when $2k \leq N \leq d+1$. It further introduces $k$-uniform quantum information masking, showing that $k$-uniform states and quantum error-correcting codes enable strong masking protocols where information is hidden from any $k$ subsystems.

ABSTRACT

A pure state of $N$ parties with local dimension $d$ is called a $k$-uniform state if all the reductions to $k$ parties are maximally mixed. Based on the connections among $k$-uniform states, orthogonal arrays and linear codes, we give general constructions for $k$-uniform states. We show that when $d\geq 4k-2$ (resp. $d\geq 2k-1$) is a prime power, there exists a $k$-uniform state for any $N\geq 2k$ (resp. $2k\leq N\leq d+1$). Specially, we give the existence of $4,5$-uniform states for almost every $N$-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. extbf{120}, 230501 (2018)}] to $k$-uniform quantum information masking in multipartite systems, and we show that $k$-uniform states and quantum error-correcting codes can be used for $k$-uniform quantum information masking.

Motivation & Objective

• To establish general constructions of $k$-uniform states for $k \geq 4$ using connections with orthogonal arrays and linear codes.
• To resolve the long-standing open problem of existence conditions for $k$-uniform states, particularly for $k \geq 4$, in multipartite qudit systems.
• To generalize quantum information masking to $k$-uniform masking in multipartite systems, ensuring information is hidden from any $k$ subsystems.
• To demonstrate that $k$-uniform states and quantum error-correcting codes can be used to implement strong masking protocols.

Proposed method

• Leverages the equivalence between $k$-uniform states and orthogonal arrays (OAs) to construct states via linear codes over finite fields.
• Applies algebraic geometry codes to construct $k$-uniform states when $d$ is a prime power, using curves with many rational points and genus constraints.
• Uses duality and minimum distance properties of linear codes to ensure that all $k$-party reductions are maximally mixed.
• Employs Lemma 15 to verify that the Schmidt decomposition of a state yields a maximally mixed reduced density matrix when the basis states are orthonormal.
• Applies Lemma 16 to link $k$-uniform states to quantum error-correcting codes (QECCs), showing that $k$-uniform spaces correspond to $((N,K,k+1))_d$ QECCs.
• Combines theoretical bounds (e.g., Rains’ bound) with explicit constructions (e.g., self-dual codes, AME tables) to verify existence across $N$ and $d$.

Experimental results

Research questions

• RQ1 Under what conditions does a $k$-uniform state exist for $N$ qudits with local dimension $d$ when $k \geq 4$?
• RQ2 Can $k$-uniform states be systematically constructed using linear codes and orthogonal arrays?
• RQ3 Can quantum information masking be generalized to $k$-uniform masking in multipartite systems, such that information is hidden from any $k$ subsystems?
• RQ4 What is the relationship between $k$-uniform states and quantum error-correcting codes in enabling strong masking protocols?

Key findings

• For $d \geq 4k-2$ and $d$ a prime power, a $k$-uniform state exists for all $N \geq 2k$, providing a broad existence condition.
• For $d \geq 2k-1$ and $d$ a prime power, a $k$-uniform state exists when $2k \leq N \leq d+1$, extending known existence ranges.
• The paper proves the existence of $4$- and $5$-uniform states for almost every $N$-qudit system when $d$ is a prime power.
• Explicit constructions of $5$-uniform states are provided for $d \geq 18$ not a prime power and $N \geq 18$, using algebraic geometry codes.
• The existence of $5$-uniform states in $(\mathbb{C}^d)^{\otimes N}$ is confirmed for all $d \geq 2$ and $N \geq 18$, using duality and code concatenation.
• The paper establishes that $k$-uniform states and $((N,K,k+1))_d$ quantum error-correcting codes are equivalent, enabling $k$-uniform masking of one-qudit states.