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Update 07-stabilisers.Rmd
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diffeo-christopher authored Dec 21, 2024
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Expand Up @@ -182,7 +182,7 @@ The set of all stabilisers of a given state or given subspace form a group: if $
This group is called the **stabiliser group** $\mathcal{S}$ of the given state or subspace.

Using this language, we can rephrase the previous example by saying that the stabiliser group of the state $\ket{0}$ is $\{\id,Z\}=\langle Z\rangle$, the stabiliser group of the state $\ket{1}$ is $\{\id,-Z\}=\langle -Z\rangle$, the stabiliser group of the state $\ket{+}$ is $\{\id,X\}=\langle X\rangle$, and so on.
If we take the tensor product of a two states, with stabiliser groups $\mathcal{A}$ and $\mathcal{B}$ (respectively), then the resulting tensor product state has stabiliser group given by the cartesian product $\mathcal{A}\times\mathcal{B}$.
If we take the tensor product of two states, with stabiliser groups $\mathcal{A}$ and $\mathcal{B}$ (respectively), then the resulting tensor product state has stabiliser group given by the cartesian product $\mathcal{A}\times\mathcal{B}$.
For example, the state $\ket{1}\ket{+}$ is stabilised by the group
$$
\begin{aligned}
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