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arXiv:2305.12769

Constructions of kkk-uniform states in heterogeneous systems

Keqin Feng, Lingfei Jin, Chaoping Xing, Yuan Chen

Published May 22, 2023• 0 citations
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Abstract

A pure quantum state of $n$ parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called $k$-uniform if all the reductions to $k$-parties are maximally mixed. The $n$ partite system is called homogenous if the local dimension $d_1=d_2=\cdots=d_n$, while it is called heterogeneous if the local dimension are not all equal. $k$-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing $k$-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the $k$-uniform states in heterogeneous systems for $k>3$. We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix $H$ meeting the condition that $H_{A\times \bar{A}}+H^T_{\bar{A}\times A}$ has full rank for any row index set $A$ of size $k$. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer $k$, one can construct $k$-uniform states of a heterogeneous system in many different Hilbert spaces.

DOI
10.48550/arxiv.2305.12769

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A pure quantum state of nnn parties associated with the Hilbert space \CCd1⊗\CCd2⊗⋯⊗\CCdn\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}\CCd1​⊗\CCd2​⊗⋯⊗\CCdn​ is called kkk-uniform if all the reductions to kkk-parties are maximally mixed. The nnn partite system is called homogenous if the local dimension d1=d2=⋯=dnd_1=d_2=\cdots=d_nd1​=d2​=⋯=dn​, while it is called heterogeneous if the local dimension are not all equal. kkk-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing kkk-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the kkk-uniform states in heterogeneous systems for k>3k>3k>3. We present two general methods to construct kkk-uniform states in the heterogeneous systems for general kkk. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new kkk-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix HHH meeting the condition that HA×Aˉ+HAˉ×ATH_{A\times \bar{A}}+H^T_{\bar{A}\times A}HA×Aˉ​+HAˉ×AT​ has full rank for any row index set AAA of size kkk. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer kkk, one can construct kkk-uniform states of a heterogeneous system in many different Hilbert spaces.
A pure quantum state ofnnnparties associated with the Hilbert spaceCd1⊗Cd2⊗⋯⊗Cdn\mathbb {C}^{d_{1}}\otimes \mathbb {C} ^{d_{2}}\otimes \cdots \otimes \mathbb {C} ^{d_{n}}Cd1​⊗Cd2​⊗⋯⊗Cdn​is calledkkk-uniform if all the reductions tokkk-parties are maximally mixed. Thennnpartite system is called homogenous if the local dimensionsd1=d2=⋯=dnd_{1}=d_{2}=\cdots =d_{n}d1​=d2​=⋯=dn​, while it is called heterogeneous if the local dimensions are not all equal.kkk-uniform sates play an important role in quantum information theory. There are much progress in characterizing and constructingkkk-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for thekkk-uniform states in heterogeneous systems fork>3k>3k>3. We present two general methods to constructkkk-uniform states in the heterogeneous systems for generalkkk. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many newkkk-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrixHHHmeeting the condition thatHA×Aˉ+HAˉ×ATH_{A\times \bar {A}}+H^{T}_{\bar {A}\times A}HA×Aˉ​+HAˉ×AT​has full rank for any row index setAAAof sizekkk. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integerkkk, one can constructkkk-uniform states of a heterogeneous system in many different Hilbert spaces.
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.
We studykkk-uniform states in heterogeneous systems whose local dimensions are not all the same. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays andkkk-uniform states, we present two constructions of 2-uniform states in heterogeneous systems. We also construct two families of 3-uniform states in heterogeneous systems, which solves a question posed in [D. Goyenecheet al., Phys. Rev. A94, 012346 (2016)]. We show two methods of generating(k−1)(k-1)(k−1)-uniform states fromkkk-uniform states. Some results on the nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting ofkkk-uniform states with the minimum support, and we show that somekkk-uniform bases are locally irreducible. Moreover, we connectkkk-uniform states with quantum information masking.
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.
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.
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 correction codes.To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes.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 the so-called maximum distance separable error correction codes.Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.