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.
We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for
n-qubit states, we express the condition that all
k-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all
k vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group
Pn that is isomorphic to
Z2n. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed
k for fixed
n. In particular, the ratio
k/n can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed
k-qubit reductions with
k>1. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though
k>1 is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.
Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on the properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.
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.
parties are distinguished by the property that for any splitting at least one partial trace is maximally mixed. Due to maximal possible correlations between any two selected subsystems these states have numerous applications in various fields of quantum information processing including multi-user teleportation, quantum error correction and secret sharing. We present an updated survey of various techniques to generate such strongly entangled states, including those going beyond the standard construction of graph and stabilizer states. Our contribution includes, in particular, analysis of the degree of entanglement of reduced states obtained by partial trace of AME projectors, states obtained by a symmetric superposition of GHZ states, an orthogonal frequency square representation of the 'golden' AME state and an updated summary of the number of local unitary equivalence classes.
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.