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

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.

💡 Research Summary

The paper tackles two intertwined problems in quantum information theory: (i) extending the stabiliser formalism to Hilbert spaces whose subsystems have heterogeneous local dimensions, and (ii) defining and characterising absolutely maximally entangled (AME) states in such mixed‑dimensional spaces.

First, the authors observe that all existing stabiliser codes assume a uniform local dimension $D$, i.e. a space $(\mathbb C^D)^{\otimes n}$. While this matches many theoretical models, real quantum hardware often contains qubits, qutrits, or higher‑dimensional qudits in the same processor. To address this, they define a general stabiliser code as the common +1 eigenspace of an arbitrary abelian subgroup $S$ of unitary operators on $\mathcal H=\bigotimes_{i=1}^n\mathbb C^{D_i}$. Crucially, $S$ is not required to be a subgroup of a Pauli‑Weyl group or any “nice error basis”; it merely needs to consist of permutations of the computational basis up to phase factors. Theorem 1 shows that the dimension of the code space is \