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In a Kitaev chain, a phase error \(\sigma^{\mathrm z}_j\) can occur where one state \(|1\rangle\) changes sign while \(|0\rangle\) relative to it does not:

`\begin{align} \sigma^{\mathrm z}_j \, |0\rangle ~&=~ |0\rangle \\ \sigma^{\mathrm z}_j \, |1\rangle ~&=~ -|1\rangle \end{align}`The phase flip error is represented by the Pauli-Z matrix \(\sigma^{\mathrm z}_j\), even if the Kitaev chain is not a spin system. According to the Wigner-Jordan transformation, a spin system is equivalent to a fermionic system (particle / no particle) like the Kitaev chain.