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.

arXiv comment24 pages, 7 figures. Comments are very welcome!

References in corpus (35)

• Quantum entanglement
• Three qubits can be entangled in two inequivalent ways
• Distributed Entanglement
• On mutually unbiased bases
• Four qubits can be entangled in nine different ways
• Induced measures in the space of mixed quantum states
• Quantum error-correcting codes associated with graphs
• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Bell inequality, Bell states and maximally entangled states for n qubits
• All Teleportation and Dense Coding Schemes
• Absolute Maximal Entanglement and Quantum Secret Sharing
• How entangled can two couples get?
• Maximally multipartite entangled states
• Multi-Qubit Systems: Highly Entangled States and Entanglement Distribution
• Non-local properties of multi-particle density matrices
• Multiqubit symmetric states with high geometric entanglement
• All Maximally Entangled Four Qubits States
• Exploring pure quantum states with maximally mixed reductions
• Entanglement monotones and maximally entangled states in multipartite qubit systems
• Mutually Unbiased Bases and Finite Projective Planes
• On polynomial invariants of several qubits
• Matrix models as solvable glass models
• Generalized Criterion of Maximally Multi-Qubit Entanglement
• Random graph states, maximal flow and Fuss-Catalan distributions
• On maximal entanglement between two pairs in four-qubit pure states
• Typical entanglement in multi-qubit systems
• Multipartite Entanglement and Frustration
• Absolutely Maximally Entangled Qudit Graph States
• Classical Statistical Mechanics Approach to Multipartite Entanglement
• The invariant-comb approach and its relation to the balancedness of multipartite entangled states
• Normal Forms and Tensor Ranks of Pure States of Four Qubits
• Duality and the geometric measure of entanglement of general multiqubit W states
• Location of quantum information in additive graph codes
• Proof of the Gour-Wallach conjecture
• Some Entanglement Features of Highly Entangled Multiqubit States

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