A pure quantum state of [Formula: see text] subsystems with [Formula: see text] levels is called [Formula: see text]-uniform state if all its reductions to [Formula: see text] qudits are maximally mixed. We construct 3-uniform states for an arbitrary number of [Formula: see text] via orthogonal arrays and the method of adding minus signs mathematically.
A pure quantum state of [Formula: see text] subsystems with [Formula: see text] levels is called [Formula: see text]-uniform state if all its reductions to [Formula: see text] qudits are maximally mixed. We construct 3-uniform states for an arbitrary number of [Formula: see text] via orthogonal arrays and the method of adding minus signs mathematically.
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 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.
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.
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.
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.
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≥4k−2 (resp. d≥2k−1) is a prime power, there exists a k-uniform state for any N≥2k (resp. 2k≤N≤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.