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.
Realizations
Quantum teleportation [3], secret sharing [4], and superdense coding [5] protocols have been realized with analog GHZ states states for \(n=2,3\).Notes
Analog GHZ states are useful for quantum teleportation [6].Cousin
- Niset-Andersen-Cerf code— EPR pairs are used in an encoding of the Niset-Andersen-Cerf code [7].
Primary Hierarchy
References
- [1]
- P. van Loock, “Quantum Communication with Continuous Variables”, Fortschritte der Physik 50, 1177 (2002) DOI
- [2]
- A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, Physical Review 47, 777 (1935) DOI
- [3]
- H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables”, Nature 431, 430 (2004) DOI
- [4]
- A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite Quantum State Sharing”, Physical Review Letters 92, (2004) arXiv:quant-ph/0311015 DOI
- [5]
- J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental Demonstration of Tripartite Entanglement and Controlled Dense Coding for Continuous Variables”, Physical Review Letters 90, (2003) arXiv:quant-ph/0210132 DOI
- [6]
- P. van Loock and S. L. Braunstein, “Multipartite Entanglement for Continuous Variables: A Quantum Teleportation Network”, Physical Review Letters 84, 3482 (2000) arXiv:quant-ph/9906021 DOI
- [7]
- J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008) arXiv:0710.4858 DOI
Page edit log
- Victor V. Albert (2025-01-03) — most recent
Cite as:
“Analog repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/analog_repetition