Genuinely multipartite entangled states and orthogonal arrays

Abstract

A pure quantum state of N subsystems with d levels each is called k-multipartite maximally entangled state, written k-uniform, if all its reductions to k qudits are maximally mixed. These states form a natural generalization of N-qudits GHZ 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 new 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. Additionally, single vectors of another class of 2-uniform states are one-to-one related to maximal sets of mutually unbiased bases. Furthermore, we establish links between existence of k-uniform states, classical and quantum error correction codes and provide a novel graph representation for such states.

24 pages, 7 figures. Comments are very welcome!

References in corpus (12)

• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Maximally multipartite entangled states
• Multi-Qubit Systems: Highly Entangled States and Entanglement Distribution
• Multiqubit symmetric states with high geometric entanglement
• All Maximally Entangled Four Qubits States
• Entanglement monotones and maximally entangled states in multipartite qubit systems
• Exploring pure quantum states with maximally mixed reductions
• On polynomial invariants of several qubits
• On maximal entanglement between two pairs in four-qubit pure states
• Normal Forms and Tensor Ranks of Pure States of Four Qubits
• Duality and the geometric measure of entanglement of general multiqubit W states
• Some Entanglement Features of Highly Entangled Multiqubit States

Cited by in corpus (8)

• Tensor Representation of Spin States
• Multiqubit symmetric states with maximally mixed one-qubit reductions
• A brief introduction to multipartite entanglement
• Constructions of $k$-uniform states from mixed orthogonal arrays
• Quantum $k$-uniform states from quantum orthogonal arrays
• A note on the Bloch representation of absolutely maximally entangled states
• Purity and construction of arbitrary dimensional $k$-uniform mixed states
• Planar k-Uniform States: a Generalization of Planar Maximally Entangled States