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.

翻译：与希尔伯特空间 $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ 相关联的 $n$ 方纯量子态称为 $k$-均匀态，若其所有 $k$ 方约化态均为最大混合态。当各局部维度 $d_1=d_2=\cdots=d_n$ 时，该 $n$ 方系统称为同质系统；反之，若局部维度不完全相等，则称为异质系统。$k$-均匀态在量子信息论中具有重要作用。目前，关于同质系统中 $k$-均匀态的表征与构造已取得诸多进展，但异质系统中纠缠特性研究远较同质情形更具挑战性。对于 $k>3$ 的异质系统，目前仅有极少数已知结果。我们提出两种一般性方法，可用于构造一般 $k$ 值下异质系统中的 $k$-均匀态。第一种构造方法源于纠错码，通过建立无冗余混合正交阵列与纠错码之间的关联得以实现。该方法可构造大量新型 $k$-均匀态，其中每个子系统的局部维度均可为素数幂。第二种构造方法源于满足特定条件的矩阵 $H$：对于任意大小为 $k$ 的行指标集 $A$，矩阵 $H_{A\times \bar{A}}+H^T_{\bar{A}\times A}$ 满秩。该矩阵构造法可为局部维度提供更灵活的选择，即在满足特定约束条件下，局部维度可为任意整数（不必为素数幂）。我们的构造方法表明：对于任意正整数 $k$，均可在多种不同的希尔伯特空间中构建异质系统的 $k$-均匀态。