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.
We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes, and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called
k-uniform, i.e., multipartite pure states such that every reduction to
k parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite
k-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.
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.
Huber et al. [Phys. Rev. Lett. 118 (2017) 200502] have proved that a seven-qubit state whose three-body marginal states are all maximally mixed does not exist. Here, we propose a method to build a maximally entangled state based on orthogonal arrays to construct maximally entangled seven-qubit states. Using this method, we not only determine that a seven-qubit state whose three-body marginals are all maximally mixed does not exist, but also find the condition for maximally entangled seven-qubit states. We consider that π ME = 19/140 is a condition for maximally entangled seven-qubit states. Furthermore, we derive three forms of maximally entangled seven-qubit states via orthogonal arrays.
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.