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.

翻译：量子计算中的一个主要难点在于实现容错计算，以保护信息免受与环境的不良相互作用。稳定子码被引入作为一种在存储或应用计算时保护信息的手段，适用于局部维度固定的希尔伯特空间，即形如$({\mathbb C}^D)^{\otimes n}$的空间。若$D$为素数幂，则可考虑基于有限域的稳定子码\cite{KKKS2006}，这允许利用更深的数学结构来发展稳定子码。然而，子系统并无实际理由必须具有相同的局部维度。本文针对混合维度希尔伯特空间（即形如${\mathbb C}^{D_1} \otimes \cdots \otimes {\mathbb C}^{D_n}$的空间）引入了一种稳定子形式体系。更一般地，我们定义并证明了混合维度希尔伯特空间中量子纠错码的Singleton界。我们重新定义了这些希尔伯特空间中的纠缠度量，并遵循\cite{HESG2018}，将绝对最大纠缠态定义为最大化该纠缠度量的态。我们提供了在先前未知存在绝对最大纠缠态的维度空间中绝对最大纠缠态的具体实例。