Tensor network decompositions for absolutely maximally entangled states

Abstract

Absolutely maximally entangled (AME) states of $k$ qudits (also known as perfect tensors) are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension $D$. We find that certain AME states with $k=6$ can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension $D=5$ and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for $k=8$, where we find similar tensor network decompositions with six 4-leg tensors.

21 pages, v2: a new Section in the Appendix, and some minor modifications

References in corpus (4)

• Maximally multipartite entangled states
• Genuinely multipartite entangled states and orthogonal arrays
• All Maximally Entangled Four Qubits States
• Maximally Entangled States of Four Nonbinary Particles

Cited by in corpus (6)

• Exactly solvable many-body dynamics from space-time duality
• Operator dynamics and entanglement in space-time dual Hadamard lattices
• Construction of perfect tensors using biunimodular vectors
• Matrix-product unitaries: Beyond quantum cellular automata
• Geometric constructions of generalized dual-unitary circuits from biunitarity
• Absolutely maximally entangled pure states of multipartite quantum systems