No invariant perfect qubit codes

TL;DR

This paper investigates the existence of SU(2) invariant perfect tensors (IPTs) in quantum information and quantum gravity, establishing constraints and proving non-existence results for certain qubit encodings, including a no-go theorem.

Contribution

It introduces a systematic approach to determine the existence of invariant perfect tensors and proves that no invariant perfect qubit codes exist for valence 6, with additional proofs for valence 4.

Findings

No invariant perfect qubit encoding of valence 6 exists.

Provided two alternative proofs for the non-existence of 4-valent qubit IPTs.

Established constraints on the layout of invariant perfect tensors.

Abstract

Perfect tensors describe highly entangled quantum states that have attracted particular attention in the fields of quantum information theory and quantum gravity. In loop quantum gravity, the natural question arises whether SU(2) invariant tensors, which are fundamental ingredients of the basis states of spacetime, can also be perfect. In this work, we present a number of general constraints for the layout of such invariant perfect tensors (IPTs) and further describe a systematic and constructive approach to check the existence of an IPT of given valence. We apply our algorithm to show that no qubit encoding of valence 6 can be described by an IPT and close a gap to prove a no-go theorem for invariant perfect qubit encodings. We also provide two alternative proofs for the non-existence of 4-valent qubit IPTs which has been shown in [1,2].