Niset-Andersen-Cerf code[1] 


Coherent-state c-q code encoding two-mode coherent states \(\{|\alpha\rangle, |\beta\rangle\}\) into four modes such that the complex values \((\alpha,\beta)\) are recoverable after a single-mode erasure. There are two variations of the storage procedure: a deterministic protocol that offers recovery against a single mode erasure, and a probabalistic that can protect against multiple errors with post selection. This code is effectively protecting classical information stored in \((\alpha,\beta)\) using quantum operations.


The deterministic protocol protects against a single erasure error on a known mode. This recovers one state perfectly and the other state with fidelity \(F = \frac{1}{1 + e^{-2 r}}\) for an initial EPR pair squeezed with variance \(e^{-2r}\). The probabalistic protocol utilizes post-selection to protect against multiple erasures with state-dependent fidelity.


After an EPR pair preparation, use 2 continuous CNOT and 2 continuous inverse CNOT gates to entangle a bosonic EPR pair with initial states \(|\alpha \rangle\) and \(|\beta \rangle\).Alternate optical encoder using a two-mode squeezed vacuum state and two balanced beam splitters to mix the input coherent states with the EPR pair.


Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [1].


Realized in Ref. [2] in an optical system with 3 beam-splitters. The fidelity peaked around \(0.6\) for deterministic approach, and around \(0.77\) for the probabilistic approach (with a 25% chance of error).



  • Quadrature-amplitude modulation (QAM) code — The Niset-Andersen-Cerf code encodes two coherent states at a time with arbitrary complex values, making it analogous to a two-point QAM code. The code does not encode any quantum information since superpositions of the coherent states are not stored. However, analysis of the code is done via a quantum treatment.
  • Homological bosonic code — The Niset-Andersen-Cerf code can be viewed as a scheme to replicate quantum information in multiple regions [3].


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
M. Lassen et al., “Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code”, Nature Photonics 4, 700 (2010) arXiv:1006.3941 DOI
P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI
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Zoo Code ID: niset_andersen_cerf

Cite as:
“Niset-Andersen-Cerf code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_niset_andersen_cerf, title={Niset-Andersen-Cerf code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Niset-Andersen-Cerf code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.