$k$-uniform quantum states arising from 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 $N\ge 4$ 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 $N\ge 8$ 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 $(k-1)$-uniform $(N-1)$-qudit states from a single $k$-uniform $N$-qudit state. The $k$-uniform states are good candidates among the multipartite entangled states.

Figure 1

Main method to construct the $k$-uniform state in this paper.

Figure 2

Sketch map to show the results we obtained in Secs. 5 and 6.