Stabilizer codes and absolutely maximally entangled states for mixed-dimensional systems

Abstract

A major difficulty in quantum computation is the ability to implement fault tolerant computations, protecting information against undesired interactions with the environment. The theory of stabiliser codes has been developed over recent years which protects information when storing or applying computations in Hilbert spaces where the local dimension is fixed, i.e. in Hilbert spaces of the form (CD)⊗n. If D is a prime power then one can consider stabiliser codes over finite fields [KKKS06], which allows a deeper mathematical structure to be used to develop stabiliser codes. However, there is no practical reason that the subsystems should be required to have the same local dimension and in this work, we introduce a stabiliser formalism for mixed dimension Hilbert spaces, i.e. of the form CD1 ⊗ · · · ⊗ CDn. We redefine entanglement measures for these Hilbert spaces and follow [HESG18] to define absolutely maximally entangled states as states which maximize this entanglement measure, and give an example of such a state on a mixed dimension Hilbert space.