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.

如果最大程度地减少对k个量子态的所有归约，则N个子系统的纯量子态（每个子系统具有d个能级）被称为k-均匀的。在整个这项工作中，仅考虑从正交阵列（OAs）获得的均匀状态。OA的汉明距离特别适用于量子信息理论。通过使用差分方案和正交分区，我们构建了一系列无穷类非冗余正交阵列（IrOAs），然后回答的是否存在的3均匀状态下的开放性问题的Ñ量子位和2-均匀状态Ñ qutrits，以及是否 高N值时qudits的3均匀状态（d > 2）可以显式构造。事实上，我们得到3-均匀状态为任意数量的Ñ  ≥8倍的量子位和2-均匀状态Ñ qutrits每Ñ  ≥4中。另外，我们提供的3均匀状态的明确的结构Ñ  ≥8个qutrits，ñ  = 6和ñ  ≥8个ququarts和ququints，ñ  ≥6个qudits具有d为任何素功率电平d > 6，和ñ  = 8和ñ  ≥12个qudits具有d为非黄金功率电平d ≥6.此外，我们描述了用于具有qudits的2均匀状态的明确的结构方案d  ≥4个级别。的2均匀状态的存在的证明Ñ  ≥6个量子位是通过使用一类的OA的简化。使用OAs的相互作用列特性，从IrOA（32，10，2，3）和IrOA（32，11，2，3）获得了两个特殊的3均匀状态。