Error-correcting codes and absolutely maximally entangled states for mixed dimensional Hilbert spaces

TL;DR

This paper extends quantum error correction and entanglement theory to mixed-dimensional Hilbert spaces, introducing a stabiliser formalism, proving a Singleton bound, and identifying new absolutely maximally entangled states.

Contribution

It develops a stabiliser code framework for mixed-dimensional spaces, establishes a Singleton bound, and finds new examples of maximally entangled states in previously unexplored dimensions.

Findings

Introduced stabiliser formalism for mixed-dimensional Hilbert spaces

Proved a Singleton bound for quantum codes in these spaces

Discovered new absolutely maximally entangled states in novel dimensions

Abstract

A major difficulty in quantum computation is the ability to implement fault tolerant computations, protecting information against undesired interactions with the environment. Stabiliser codes were introduced as a means to protect information when storing or applying computations in Hilbert spaces where the local dimension is fixed, i.e. in Hilbert spaces of the form $({\mathbb C}^D)^{\otimes n}$. If $D$ is a prime power then one can consider stabiliser codes over finite fields \cite{KKKS2006}, which allows a deeper mathematical structure to be used to develop stabiliser codes. However, there is no practical reason that the subsystems should have the same local dimension and in this article we introduce a stabiliser formalism for mixed dimensional Hilbert spaces, i.e. of the form ${\mathbb C}^{D_1} \otimes \cdots \otimes {\mathbb C}^{D_n}$. More generally, we define and prove a Singleton bound for quantum error-correcting codes of mixed dimensional Hilbert spaces. We redefine entanglement measures for these Hilbert spaces and follow \cite{HESG2018} and define absolutely maximally entangled states as states which maximise this entanglement measure. We provide examples of absolutely maximally entangled states in spaces of dimensions not previously known to have absolutely maximally entangled states.