We study
k-uniform states in heterogeneous systems whose local dimensions are mixed. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and
k-uniform states, we present two constructions of
2-uniform states in heterogeneous systems. We also construct a family of
3-uniform states in heterogeneous systems, which solves a question posed in [D. Goyeneche et al., Phys. Rev. A 94, 012346 (2016)]. We also show two methods of generating
(k−1)-uniform states from
k-uniform states. Some new results on the existence and nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of
k-uniform states with minimum support. Moreover, we show that some
k-uniform bases can not be distinguished by local operations and classical communications, and this shows quantum nonlocality with entanglement.
We study
k-uniform states in heterogeneous systems whose local dimensions are not all the same. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and
k-uniform states, we present two constructions of 2-uniform states in heterogeneous systems. We also construct two families of 3-uniform states in heterogeneous systems, which solves a question posed in [D. Goyenecheet al., Phys. Rev. A94, 012346 (2016)]. We show two methods of generating
(k−1)-uniform states from
k-uniform states. Some results on the nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of
k-uniform states with the minimum support, and we show that some
k-uniform bases are locally irreducible. Moreover, we connect
k-uniform states with quantum information masking.
Abstract Goyeneche et al [2018 Phys. Rev. A 97 062326] introduced several classes of quantum combinatorial designs, namely quantum Latin squares, quantum Latin cubes, and the notion of orthogonality on them. They also showed that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Moreover, they established a relationship between quantum combinatorial designs and a remarkable class of entangled states called k -uniform states, i.e. multipartite pure states such that every reduction to k parties is maximally mixed. In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays of size d 2 and d 3 , respectively, and thus naturally provide two- and three-uniform states.
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.
Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.
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.