Information Theory and Classical Error Correcting Codes

Ivan B. Djordjević

Emanuel H. Knill, Raymond Laflamme, Lorenza Viola

A measure of quality of an error-correcting code is the maximum number of errors that it is able to correct. We show that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions. Thus, e-error-correcting codes protect information without requiring the usual assumptions of independence. We prove the existence of large codes for both quantum and classical information. By viewing error-correcting codes as subsystems, we relate codes to irreducible representations of operator algebras and show that noiseless subsystems are infinite-distance error-correcting codes.

James L. Massey

Isaac Kremsky, Min-Hsiu Hsieh, Todd A. Brun

We present a general formalism for quantum-error-correcting codes that encode both classical and quantum information (the EACQ formalism). This formalism unifies the entanglement-assisted coding theory and the classical coding theory in the sense that, after the encoding, error-correcting, and decoding steps, the encoded quantum and classical information can be correctly recovered by the receiver. We formally define this kind of quantum codes using stabilizer language, and derive the appropriate error-correcting conditions. We give several examples to demonstrate the construction of such codes.

Ian F. Blake

Hsun‐Hsien Chang

This report surveys quantum error-correcting codes. As Preskill claimed, 21st century would be the golden age of quantum error correction. Quantum channels behave differently from classical channels, so researchers face difficulties in developing robust quantum codes. Fortunately, the classical error control methods have been well developed. If we can learn many lessons from classical coding theory, we can expedite the development of quantum codes. Scientists have discovered that quantum error correction shares many concepts with classical counterpart. Both quantum and classical coding schemes add redundancy to information to protect against noises. They also have similar conditions for error detectability and correctability.