Tensor network decompositions for absolutely maximally entangled states

Balázs Pozsgay1 and Ian M. Wanless2

1MTA-ELTE ``Momentum'' Integrable Quantum Dynamics Research Group, Department of Theoretical Physics, ELTE Eötvös Loránd University, Hungary
2School of Mathematics, Monash University, Australia

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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.

Entanglement is the distinguishing feature of quantum states, which makes them truly quantum mechanical. Starting from a product state, entanglement can be created by the action of unitary gates. In this work we consider quantum states that have maximal entanglement for all possible bi-partitions. We show, that contrary to general expectations, in certain cases these states can be created by a small number of two-site unitary gates. Equivalently, the states can be decomposed into a tensor network with few elementary tensors. This interpretation is also counter-intuitive, because it is well known that tensor networks describe states with low entanglement. Therefore, our results provide counter-examples to the general expectations, with concrete examples at small system sizes.

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Cited by

[1] Márton Mestyán, Balázs Pozsgay, and Ian M. Wanless, "Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states", SciPost Physics 16 1, 010 (2024).

[2] Suhail Ahmad Rather, "Construction of perfect tensors using biunimodular vectors", arXiv:2309.01504, (2023).

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