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 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.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed.These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes.To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes.We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from the so-called maximum distance separable error correction codes.Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
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 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.