Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

k -uniform quantum states arising from orthogonal arrays

Mao-Sheng Li, Yan-Ling Wang

Published Apr 25, 2019• 34 citations
Publisher

Abstract

A pure quantum state of $N$ subsystems with local dimension $d$ is called a $k$-uniform state if every reduction to $k$ qudits is maximally mixed. Based on a special class of combinatorial design, namely irredundant orthogonal arrays, we show the existence of 2-uniform $N$-qudit states when $N\ensuremath{\ge}4$ and $d$ is a prime power other than 2. In addition, given any Hadamard matrix of order greater than 4, we demonstrate how it can be used to construct 3-uniform multiqubit states. In fact, we find that there exists some 3-uniform $N$-qubit when $N\ensuremath{\ge}8$ except case $N=9$. These give an answer to a question posed by Goyeneche et al. [Phys. Rev. A 90, 022316 (2014)]. Furthermore, starting from a minimal support $k$-uniform state, we show how to generate an orthogonal basis consisting of $k$-uniform states. At last, we derive a series of $(k\ensuremath{-}1)$-uniform $(N\ensuremath{-}1)$-qudit states from a single $k$-uniform $N$-qudit state. The $k$-uniform states are good candidates among the multipartite entangled states.

DOI
10.1103/physreva.99.042332

Related papers

Two and three-uniform states from irredundant orthogonal arrays

Shanqi Pang, Zhang Xiao, Lin Xiao +1 more

Abstract A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2-uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3-uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.

npj Quantum Information•Jun 17, 2019•53 citations

Genuinely multipartite entangled states and orthogonal arrays

Dardo Goyeneche, Karol Życzkowski

A pure quantum state of $N$ subsystems with $d$ levels each is called $k$-multipartite maximally entangled state, which we call a $k$-uniform state, if all its reductions to $k$ qudits are maximally mixed. These states form a natural generalization of $N$-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and $k$-uniform states and prove the existence of several classes of such states for $N$-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of $N>5$ qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of $k$-uniform states and classical and quantum error correction codes and provide a graph representation for such states.

arXiv:1404.3586•Physical Review A•Aug 15, 2014•157 citations

Constructions of $k$-uniform states from mixed orthogonal arrays

Fei Shi, Yi Shen, Lin Chen +1 more

We study $k$-uniform states in heterogeneous systems whose local dimensions are mixed. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and $k$-uniform states, we present two constructions of $2$-uniform states in heterogeneous systems. We also construct a family of $3$-uniform states in heterogeneous systems, which solves a question posed in [D. Goyeneche et al., Phys. Rev. A 94, 012346 (2016)]. We also show two methods of generating $(k-1)$-uniform states from $k$-uniform states. Some new results on the existence and nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of $k$-uniform states with minimum support. Moreover, we show that some $k$-uniform bases can not be distinguished by local operations and classical communications, and this shows quantum nonlocality with entanglement.

arXiv:2006.04086•Jun 7, 2020•7 citations

kkk-Uniform states and quantum information masking

Fei Shi, Mao-Sheng Li, Lin Chen +1 more

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. \textbf{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.

arXiv:2009.12497•Sep 26, 2020•3 citations

A new method for constructing entangled states via orthogonal arrays

Xiaoyuan Yu, Zha Xin-wei, CHE JunLing

A pure quantum state of $N$ subsystems with $d$ levels each is called $k$-multipartite maximally entangled state, which we call a $k$-uniform state, if all its reductions to $k$ qubits are maximally mixed. Recently, Goyeneche and Zyczkowski proposed a method of constructing $k$-uniform states via orthogonal arrays, by which, we will also propose a new method to construct quantum states with orthogonal arrays in this paper, which is the method of increasing phase parameters. We can find the quantum states constructed via our new method by using the same orthogonal array, for four-qubit quantum states, not only are they 1-uniform states, also may become maximally entangled states; for five-qubit quantum states, we find that it can make the original 1-uniform states into 2-uniform states. In particular, for four-qubit quantum states, we can construct a series of new forms of maximally entangled states via our new method.

Zhongguo kexue. Wulixue Lixue Tianwenxue•Sep 21, 2017•3 citations

Multipartite entangled states, symmetric matrices and error-correcting codes

Keqin Feng, Lingfei Jin, Chaoping Xing +1 more

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 the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there 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 in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.

arXiv:1511.07992•IEEE Transactions on Information Theory•Jan 1, 2017•22 citations
A pure quantum state of NNN subsystems with local dimension ddd is called a kkk-uniform state if every reduction to kkk qudits is maximally mixed. Based on a special class of combinatorial design, namely irredundant orthogonal arrays, we show the existence of 2-uniform NNN-qudit states when N\ensuremath≥4N\ensuremath{\ge}4N\ensuremath≥4 and ddd is a prime power other than 2. In addition, given any Hadamard matrix of order greater than 4, we demonstrate how it can be used to construct 3-uniform multiqubit states. In fact, we find that there exists some 3-uniform NNN-qubit when N\ensuremath≥8N\ensuremath{\ge}8N\ensuremath≥8 except case N=9N=9N=9. These give an answer to a question posed by Goyeneche et al. [Phys. Rev. A 90, 022316 (2014)]. Furthermore, starting from a minimal support kkk-uniform state, we show how to generate an orthogonal basis consisting of kkk-uniform states. At last, we derive a series of (k\ensuremath−1)(k\ensuremath{-}1)(k\ensuremath−1)-uniform (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1)-qudit states from a single kkk-uniform NNN-qudit state. The kkk-uniform states are good candidates among the multipartite entangled states.
Abstract A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2-uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3-uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qudits are maximally mixed. These states form a natural generalization of NNN-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and kkk-uniform states and prove the existence of several classes of such states for NNN-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5N>5N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of kkk-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
Constructions of kkk-uniform states from mixed orthogonal arrays
We study kkk-uniform states in heterogeneous systems whose local dimensions are mixed. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and kkk-uniform states, we present two constructions of 222-uniform states in heterogeneous systems. We also construct a family of 333-uniform states in heterogeneous systems, which solves a question posed in [D. Goyeneche et al., Phys. Rev. A 94, 012346 (2016)]. We also show two methods of generating (k−1)(k-1)(k−1)-uniform states from kkk-uniform states. Some new results on the existence and nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of kkk-uniform states with minimum support. Moreover, we show that some kkk-uniform bases can not be distinguished by local operations and classical communications, and this shows quantum nonlocality with entanglement.
A pure state of NNN parties with local dimension ddd is called a kkk-uniform state if all the reductions to kkk parties are maximally mixed. Based on the connections among kkk-uniform states, orthogonal arrays and linear codes, we give general constructions for kkk-uniform states. We show that when d≥4k−2d\geq 4k-2d≥4k−2 (resp. d≥2k−1d\geq 2k-1d≥2k−1) is a prime power, there exists a kkk-uniform state for any N≥2kN\geq 2kN≥2k (resp. 2k≤N≤d+12k\leq N\leq d+12k≤N≤d+1). Specially, we give the existence of 4,54,54,5-uniform states for almost every NNN-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. \textbf{120}, 230501 (2018)}] to kkk-uniform quantum information masking in multipartite systems, and we show that kkk-uniform states and quantum error-correcting codes can be used for kkk-uniform quantum information masking.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qubits are maximally mixed. Recently, Goyeneche and Zyczkowski proposed a method of constructing kkk-uniform states via orthogonal arrays, by which, we will also propose a new method to construct quantum states with orthogonal arrays in this paper, which is the method of increasing phase parameters. We can find the quantum states constructed via our new method by using the same orthogonal array, for four-qubit quantum states, not only are they 1-uniform states, also may become maximally entangled states; for five-qubit quantum states, we find that it can make the original 1-uniform states into 2-uniform states. In particular, for four-qubit quantum states, we can construct a series of new forms of maximally entangled states via our new method.
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 the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there 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 in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.