Analog repetition code

Description

An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number of \(n\) modes.

There are two variants, a bit- and a phase-flip code, whose encoding for \(n=3\) is \begin{align} |\overline{x}_{\text{bit}}\rangle&\rightarrow|x,x,x\rangle\tag*{(1)}\\ |\overline{x}_{\text{phase}}\rangle&\rightarrow \int dx_{1}dx_{2}dx_{3}\delta(x_{1}+x_{2}+x_{3}-x)|x_{1},x_{2},x_{3}\rangle~. \tag*{(2)}\end{align}

Nullifiers for the bit-flip analog repetition code are differences \(\hat{x}_{j+1} - \hat{x}_{j}\). Bit-flip codewords can be superposed to yield the logical momentum basis of analog GHZ states \begin{align} |\overline{p}\rangle=\int dx e^{ipx}|x\rangle^{\otimes n}~, \tag*{(3)}\end{align} a bosonic version of GHZ states. At \(p=0\), the above is an analog stabilizer state nullified by the bit-flip nullifiers and the total momentum operator \(\hat{p}_1+\hat{p}_2+\cdots+\hat{p}_n\) [1]. For \(n=2\), this state is known as an EPR pair [2], an infinitely squeezed version of the two-mode squeezed (TMS) a.k.a. twin-beam state.