Two and three-uniform states from irredundant orthogonal arrays

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.


INTRODUCTION
Multi-particle entanglement is an essential component in describing the possible quantum advantages available to metrology or information processing. The identification of multipartite quantum states with the strongest possible quantum correlations is a crucial question in quantum information theory. 1 An important open issue concerns the construction of genuinely multipartite entangled states, 2 as these have been widely applied to quantum errorcorrecting codes (QECCs), 3,4 teleportation, 5-8 key distribution, 9 dense coding, and quantum computation. 10 For example, in the past few years, significant development has been made in the new area of quantum machine learning, where quantum information benefits from modern information-processing technologies. 11 Quantum entanglement as a resource has been used to experimentally demonstrate various modern quantum technologies.
A pure quantum state of N subsystems with d levels is said to be k-uniform if all of its reductions to k qudits are maximally mixed. 2 An orthogonal array OAðr; N; d n1 1 d n2 2 Á Á Á d n l l ; kÞ is an r × N matrix, having n i columns with d i levels, i = 1, 2, …, l, l is an integer, N ¼ P l i¼1 n i , and d i ≠ d j for i ≠ j, with the property that, in any r × k submatrix, all possible combinations of k symbols appear equally often as a row. The orthogonal array is called a mixed orthogonal array (MOA) if l ≥ 2. Otherwise, the array is called symmetrical. Additionally, the following Rao bounds hold in OA(r, N, d, k): OAs that achieve these bounds are called tight or saturated. 12 An OA(r, N, d, k) is said to be an irredundant orthogonal array (IrOA) if, in any r × (N −k ) subarray, all of its rows are different. 2 A link between an IrOA and a k-uniform state was established by Goyeneche et al., 2 i.e., every column and every row of the array correspond to a particular qudit and a linear term of the state, respectively. For simplicity, the normalization factors are omitted from this paper. We summarize the link as follows: Connection 1.1. Many efforts have attempted to find and identify k-uniform states. 2,4 Under the generalizations of the Meyer-Wallach measure defined in, 4 k-uniform states have the largest average entanglement between blocks of qudits and the remaining elements. There has also been some progress in the construction and characterization of k-uniform states. 1,2,11,[13][14][15][16][17][18] For example, Goyeneche et al. 2 constructed a 3-uniform state of six qubits and a 2uniform state of five qubits by the judicial insertion of some minus signs. The nonexistence of the 3-uniform states of seven qubits is proved in. 14 Using the above connection, Goyeneche et al. 2 constructed 2-uniform states for an arbitrary number of N ≥ 6 qubits using known Hadamard matrices. Yu et al. 19 constructed a 3-uniform state of eleven qubits via OAs. However, as stated in, 2 there are many open issues in multipartite quantum systems. Huber et al. 14 stated that it would be of great interest to settle the problem of whether N-qubit states exist in which all k-body reduced density are maximally mixed for k < N 2 The aim of the present study was to solve the open issues raised in 2 and the interesting question stated in 14 using IrOAs, the states related to which are useful and necessary in quantum error correction. The different uniformity of multipartite entangled states reflects different features. One motivation for searching for multipartite states of a higher uniformity is that k-uniform states of qudits offer advantages over k′-uniform states for any 0 < k′ < k. They are good for increasing the order of the density coding information rate from d k′ to d k . 20 It is interesting that these states form a natural extension of N-qudit Greenberger-Horne-Zeilinger states, which are 1-uniform. Moreover, a k-uniform state is also k′-uniform. As is often the case, [21][22][23] combinatorics can be useful to quantum information theory, and OAs are fundamental ingredients in the construction of other useful combinatorial objects. 12 The Hamming distance, difference schemes, and orthogonal partitions have been applied to many aspects of constructing OAs. [24][25][26][27] Recently, many new methods of constructing OAs of strength k, especially mixed OAs, have been presented, and many new classes of OAs have been obtained. [28][29][30][31][32][33] It is these new developments in OAs that suggest the possibility of constructing infinitely many new k-uniform states from IrOAs.
The following notation, concepts, and lemmas are used in this paper.
Let A T be the transposition of matrix A and (d) = (0, 1, …, d − 1) T . Let 0 r and 1 r denote the r × 1 vectors of 0s and 1s, respectively. If A = (a ij ) m×n and B = (b ij ) u×v with elements from a Galois field with binary operations (+ and ⋅), the Kronecker product A ⊗ B and the Kronecker sum A ⊕ B are respectively defined as . A matrix A can often be identified with a set of row vectors if necessary. Let x d e denote the least integer not less than x, and let A be an additive group of d elements.
Definition 1.1. 27 Suppose that a, b, c are three columns in an OA(r, N, 2, 2). Then, c is called the interaction column between a and b if c ≡ a + b or c ≡ 1 r + a + b (mod 2). Definition 1.2. 12 Let S l = {(v 1 , …, v l )|v i ∈ S, i = 1, 2, …, l}. The Hamming distance HD(u, v) between two vectors u = (u 1 , …, u l ), v = (v 1 , …, v l ) ∈ S l is defined as the number of positions in which they differ. The minimal distance of a matrix A, written MD(A), is defined to be the minimal Hamming distance between its distinct rows. HD (A) is used to represent all the values of the Hamming distances between two distinct rows of A. The matrix A is said to have constant Hamming distance if HD(A) is constant for any two distinct rows. Definition 1.3. 12 An r × c matrix D with elements from A is called a difference scheme if it has the property that, for all i and j with 1 ≤ i, j ≤ c, i ≠ j, among the vector differences between the ith and jth columns, every element of A appears equally often. Such a matrix is denoted as D(r, c, d).
For instance, if B r is an OA(r, c−1, 2, 2), then [0 r , B r ] is D(r, c, 2). A k , k ≥ 1, denotes the additive group of order d k consisting of all k-tuples of entries from A with the usual vector addition as the binary operation. Let 0 is a subgroup of A k of order d, and its cosets will be denoted by Definition 1.4. 12 An r × c matrix D based on A is called a difference scheme of strength k if, for every r × k submatrix, each set A k i , i = 0, 1, …, d k−1 −1, is represented equally often when the rows of the submatrix are viewed as elements of A k . Such a matrix is denoted by D k (r, c, d).
For k = 2, this definition is equivalent to Definition 1.3 In particular, when k 1 = 0, A i can be considered as an OA r u ; N; d n1   Lemma 1.14. An OA(r, N, d, k) with minimal distance w ≥ k + 1 implies that an IrOA(r, N′, d, k) exists for N − w + k + 1 ≤ N′ ≤ N.   ) is an OA(r′, N′ + N″, d, 2) and its minimal distance satisfies w ≥ min{w 1 + w 2 , N′}.
These lemmas will form the backbone of our main theorems and results. Especially, Lemmas 1.6, 1.15, and 1.18 play an essential role in finding infinite classes of uniform states. In the 'Discussion' section, we discuss the main results and some uniform states that could have fewer terms or qudits. In the "Methods" section, we describe the central idea of the construction method. We present some examples of 2 and 3-uniform states, summarize the constructed IrOAs, and present the proofs of some lemmas in the appendix (see Supplementary Information for a detailed description of the examples, IrOAs and proofs).

RESULTS
The main results are summarized in Table 1.
Very interestingly, not only can 2 and 3-uniform states of qubits be constructed using IrOAs, but the states of high-dimensional quantum systems (qudits) can also be obtained. We now address the open issue proposed by Goyeneche et al. 2 . There is a reciprocal link between k-uniform states of N qudits and ((N, 1, k + 1)) d QECCs. 3 The newly constructed uniform states might be applied in quantum information protocols, quantum secret sharing, 39 and holographic codes.
Using difference schemes and orthogonal partitions, we propose a recursive construction method for OAs. By exploring the Hamming distances of OAs, especially newly constructed OAs, we explicitly construct a finite set of IrOA(r, N, d, k) for n ≤ N ≤ m. The recursive nature of the method allows us to generate infinite classes of IrOAs (see Table 1). Relying on Connection 1.1, we prove the existence of the uniform states in Table 1. Some examples are presented.
First, we describe the construction of 3-uniform states from IrOAs. Construction of 3-uniform states of N ≥ 8 qubits By using the minimal distance and other properties of OAs, we can obtain an IrOA(r, N, 2, 3) and 3-uniform states of N qubits for N = 8 and N ≥ 10. We also find a 3-uniform state of 9 qubits by adding some unimodular complex numbers in Appendix A (see Supplementary Example 4). Hence, we have 3-uniform states of N qubits for every N ≥ 8.
Construction of 3-uniform states of N qudits with d ≥ 6 levels Starting from the difference schemes D 3 (d 2 , d, d) and D 3 (d 2 , 4, d) respectively and exploring the minimal distances of OAs, we will construct an IrOA(r, N, d, 3) and the 3-uniform states of N qudits for every N ≥ 6 with any prime power d > 6 levels, and an IrOA(r, N, d, 3) and the 3-uniform states of N qudits for N = 8 and every N ≥ 12 with any non-prime-power d ≥ 6 levels. Using similar arguments, we can construct 3-uniform states with any non-prime-power d ≥ 6 levels. The existence of maximally entangled states lies at the intersection of quantum theory and discrete mathematics, and determining their mathematical structure is extremely technical. The characterization and classification of such entanglement have now become issues of great interest. The constructed states with higher dimensions offer an increase in the channel capacity for quantum communication, and provide a higher error rate tolerance and enhanced security in quantum key distribution. 41 High-dimensional entanglement offers great potential for applications in quantum information, particularly in quantum communications.
However, the states of higher dimensions are significantly different from the trivial extension of those of lower dimensions. A key problem in the construction of 3-uniform states of a nonprime-power dimension is the shortage of proper mathematical tools. For example, in non-prime-power dimensions the Galois fields do not exist. Therefore, studying these states is even more challenging than in the case of prime-power dimensions. In practice, the available experimental data are usually quite limited, such that devising effective ways for characterizing highdimensional entanglement is very challenging. Interestingly, the states of any dimensions can be consistently obtained using the proposed methods.
It is interesting that 3-uniform states are a natural extension of 1-uniform states. They are useful for increasing the order of density coding information rate from d 1 to d 3 . 20 Moreover, a 3-uniform state is also 2-uniform.
We continue to use our method for 2-uniform states. New results of construction of 2-uniform states of N ≥ 5 qubits For N ≥ 6, there exist 2-uniform states of N qubits that can be constructed from two infinite classes of Hadamard matrices based on the Hadamard matrices H 2 and H 12 of orders 2 and 12. 2 However, we simplify the construction of these 2-uniform states using only the known class of orthogonal arrays OA(2 n , 2 n −1, 2, 2) with n ≥ 3. Theorem 2.9. A saturated orthogonal array OA(r, r−1, 2, 2) for r ≥ 8 implies the existence of an IrOA(r, N, 2, 2) and 2-uniform states of N qubits for r 2 þ 2 N r À 1. Proof. It follows from Lemma 1.2 that the OA(r, r−1, 2, 2) has constant Hamming distance r 2 . Then we have an IrOA(r, N, 2, 2) for r 2 þ 2 N r À 1 from Lemma 1.14. By Connection 1.1, we can obtain 2-uniform states of N qubits for r 2 þ 2 N r À 1.
High-dimensional entanglement is different from two-level entanglement. Furthermore, general quantum states could be multilevel. They break the generic classical constraints and provide a new perspective for quantum mechanics.
Construction of 2-uniform states of N ≥ 4 qudits with a prime power d ≥ 4 levels For a prime power d ≥ 4, we can obtain an IrOA(r, N, d, 2) and 2uniform states of N qudits for every N ≥ 4 from the property and minimal distances of OAs and some special difference schemes.
Construction of 2-uniform states of N qudits with a non-primepower d ≥ 4 levels By computing the minimal distance of symmetrical OAs constructed from the orthogonal partition methods in, 32 we derive the following results.
When d > 6 ðd 6 2 ðmod 4ÞÞ is not a prime power, we can obtain an IrOA(r, N, d, 2) and 2-uniform states of N qudits for N ≥ 4.
We constructed 3-uniform states before 2-uniform states because we needed to use some results from the former case. That the newly constructed 3-uniform states are 2-uniform follows immediately from the strength of their corresponding IrOAs. We hope that our method for 2-and 3-uniform states will also contribute to demonstrating the existence of k-uniform states for k ≥ 4 in the homogeneous case.

DISCUSSION
The present work not only answers the question proposed by Goyeneche et al. 2 and provides a positive answer to the interesting problem stated by Huber et al., 14 but also presents more general results. In Tables 2 and 3, for any given d ≥ 2, we summarize the precise existence of the states for every value of N. In the tables, "a" denotes that a state can be constructed from an IrOA, whereas "b" represents that a state can be constructed with some judicially inserted minus signs. The symbol "-" means that such a state cannot exist by Lemma 1.11, whereas "?" indicates that we have not yet solved the problem. Tables 2 and 3 indicate that for any given d ≥ 2 (prime power or non-prime-power), we exhaustively solve the problem regarding an explicit construction of the 2 and 3-uniform states of every N qudits from IrOAs, except for at most five values of N. They are closely related to QECCs, multi-unitary permutation matrices, and mutually orthogonal quantum Latin squares and cubes. 1 Furthermore, for the k-uniform states with symbol "?" in Tables 2 and 3, we consider the following five cases: (1) The nonexistence of 3-uniform states of seven qubits and 2uniform states of four qubits has already been proved. 2,13 In Theorem 3.1, we prove the nonexistence of IrOA(r, 7, 2, 3) and IrOA(r, 4, 2, 2). (2) The existence of 3-uniform states of six qubits and 3uniform states of N qutrits for N = 6, 7 has been presented in, 44 whereas the nonexistence of IrOA(r, 6, 2, 3), IrOA(r, 6, 3, 3), and IrOA(r, 7, 3, 3) has been proved in Theorem 3.1. It is tempting to believe that some of the states obtained in this paper will be good for experimental purposes and help to quantify 46 the level of entanglement in some multipartite system. The states are also ideal candidates for quantum information protocols and quantum secret sharing. It is expected that remarkable progress will be made in the field of QECCs by application of the results presented herein. As stated in, 2 there are many open issues to solve regarding the construction and characterization of entanglement in multipartite quantum systems. The results presented in this paper will establish a foundation for solving other open problems, such as the construction of k-uniform states of N qudits (d ≥ 2) for k ≥ 4, including the problem stated by Huber et al., 14 and heterogeneous multipartite systems, 17 since the proposed construction methods can be suitable for IrOAs of any strength k ≥ 4 and irredundant mixed orthogonal arrays (IrMOAs). 17,32,33 In the construction process, we often encounter the problem that some uniform states could have fewer terms or qudits. If the Hadamard conjecture is considered, Theorems 2.9 and 2.10 state that the number of terms in many 2-uniform states could be reduced. Moreover, the following three theorems indicate that some 2-uniform states can have fewer terms. Furthermore, from Lemmas 1.13 and 1.14, there exist some 3-uniform states with fewer terms than those obtained by Theorem 2.1 (for instance, the two 3-uniform states obtained from IrOA (40,16,2,3) and IrOA(48, 16, 2, 3) by Lemmas 1.13, 1.14, and Theorem 2.1, respectively). Additionally, the states obtained by Theorem 3.5 have fewer qubits.  42 Then, the minimal distance of the OA is 3. Therefore, it is an IrOA(200, 6, 10, 2). By exchanging the 16th and 17th rows of the D(40, 6, 10) in, 42 we obtain the difference scheme D′(40, 6, 10) and an OA(400, 7, 10, 2) = [D′(40, 6, 10) ⊕ (10),(10) ⊕ 0 40 ] with minimal distance 3. Hence, this is IrOA(400, 7, 10, 2). Utilizing Lemmas 1.6 and 1.7 and the case u = v in Lemma 1.15, we can construct OA(2000, 10, 10, 2) with minimal distance 4 from the OA(200, 5, 10, 2) = D(20, 5, 10) ⊕ (10) with minimal distance 2. Then we obtain an IrOA(2000, 10, 10, 2) and an IrOA(2000, 9, 10, 2). However, from Theorem 2.16, we can only obtain IrOA(10 3 , 6, 10, 2), IrOA(10 4 , 7, 10, 2), IrOA (10 4 , 9, 10, 2), and IrOA(10 4 , 10, 10, 2). Thus, the theorem holds. Theorem 3.3. For N = 6, 9, 10, there are 2-uniform states of N qudits with 14 levels that have fewer terms than those given by Theorem 2.16. Proof. By exchanging the 26th and 27th rows in the D (28,5,14) from, 42 we obtain the difference scheme D′(28, 5, 14) and an  Remark 2. The following theorem and Theorems 2.10 and 2.11 indicate that we can find more IrOA(n, N, 2, 2) than that suggested by Theorem 2.9 using the construction methods for OA(n, n − 1, 2, 2). Therefore, we can consider the lowest value of N such that kuniform states exist for fixed r, d, and k.
Theorem 3.5. If an OA(r, r − 1, 2, 2) is obtained from an MOA ðr; l þ u; d 1 1 Á Á Á d 1 l 2 u ; 2Þ by using the expansive replacement method and OA(d i , d i − 1, 2, 2), where d i ≥ 4 for i = 1, …, l, then for any 1 ≤ j ≤ l, there exists an OA(r, r−1 − [(d 1 − 1) + ⋯+ (d j −1)], 2, 2) with a minimal distance that is at least 1 2 ½r À ðd 1 þ d 2 þ Á Á Á þ d j Þ. Proof. It follows from Lemma 1.2 that the Hamming distances of the OA(r, r −1, 2, 2), OA(d i , d i − 1, 2, 2) (i = 1, …, l) are r 2 , di 2 , respectively, since they are saturated. For any 1 ≤ j ≤ l, deleting the corresponding d 1 − 1 + d 2 − 1 + ⋯ + d j − 1 columns obtained by the expansive replacement method in 12  In the future, we will study the optimal problems of the uniform states from IrOAs. At first, we will investigate the optimal problems of IrOAs, as they play an important role in uniform states. For some given N, d, k, we consider an IrOA(r, N, d, k) in which r attains its minimum, while we can also consider the lowest value of N such that an OA(r, N, d, k) exists for fixed r, d, k. For some given r, N, d, we consider an IrOA(r, N, d, k) in which k attains its maximum, and we can also consider the highest value of d such that an OA(r, N, d, k) exists for fixed r, N, k.

METHODS
By means of the construction methods of OAs, we can investigate the properties of the required IrOAs (see "Introduction" section). Using difference schemes, orthogonal partitions, and the Hamming distance, we creatively construct several infinite series of IrOAs. From these IrOAs, we then obtain the 2 and 3-uniform states of almost every N qudits for any given number of levels d ≥ 2 by means of the link between an IrOA and a kuniform state, which was established by Goyeneche et al. 2 In particular, a key construction of uniform states of non-prime-power dimensions is due to the orthogonal partition method for recursively constructing OAs, which is new development in combinatorial designs (see 'Results' section). Starting with some special difference schemes and IrOAs, or exploring the expansive replacement method, we can then obtain some uniform states having fewer terms or qudits (see 'Discussion' section).

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.