A major difficulty in quantum computation is the ability to implement fault-tolerant computations, protecting information against undesired interactions with the environment. Stabilizer 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 ${\left({\mathbb{C}}^D\right)}^{\otimes n}$. If $D$ is a prime power then one can consider stabiliser codes over finite fields [A. Ketkar et al., IEEE Trans. Inf. Theory 52, 4892 (2006)], which allows a deeper mathematical structure to be used to develop stabilizer codes. However, there is no practical reason that the subsystems should have the same local dimension and in this article we introduce a stabilizer 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 [F. Huber et al., J. Phys. A: Math. Theor. 51, 175301 (2018)] and define absolutely maximally entangled states as states which maximize this entanglement measure. We provide examples of absolutely maximally entangled states in spaces of dimensions not previously known to have absolutely maximally entangled states.

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• Quantum error correction
• Quantum information theory