We investigate the maximum purity that can be achieved by k-uniform mixed states of N parties. Such N-party states have the property that all their k-party reduced states are maximally mixed. A scheme to construct explicitly k-uniform states using a set of specific N-qubit Pauli matrices is proposed. We provide several different examples of such states and demonstrate that in some cases the state corresponds to a particular orthogonal array. The obtained states, despite being mixed, reveal strong non-classical properties such as genuine multipartite entanglement or violation of Bell inequalities.
We investigate the maximum purity that can be achieved by
k-uniform mixed states of
N parties. Such
N-party states have the property that all their
k-party reduced states are maximally mixed. A scheme to construct explicitly
k-uniform states using a set of specific
N-qubit Pauli matrices is proposed. We provide several different examples of such states and demonstrate that in some cases the state corresponds to a particular orthogonal array. The obtained states, despite being mixed, reveal strong nonclassical properties such as genuine multipartite entanglement or violation of Bell inequalities.
A pure quantum state of
n parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called
k-uniform if all the reductions to
k-parties are maximally mixed. The
n partite system is called homogenous if the local dimension
d1=d2=⋯=dn, while it is called heterogeneous if the local dimension are not all equal.
k-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing
k-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the
k-uniform states in heterogeneous systems for
k>3. We present two general methods to construct
k-uniform states in the heterogeneous systems for general
k. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new
k-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix
H meeting the condition that
HA×Aˉ+HAˉ×AT has full rank for any row index set
A of size
k. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer
k, one can construct
k-uniform states of a heterogeneous system in many different Hilbert spaces.
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 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.
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.