Steeve Zozor - On the lossless quantum coding with exponential penalization

This talk is devoted to the lossless quantum data coding problem. To this end, we appeal to an encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. In the standard situation, studied for instance by Schumacher et al., Müller et al., Bostroem et al., or Hayashi, the goal is to minimize the arithmetic average length of the quantum codewords. Here, following the Campbell’s approach in the classical settings, we are interested in the minimization of an exponential type average, in order to penalize the large codeword lengths. We show that, similarly to the classical context, the exponential average length of the optimal code is related to the quantum Rényi entropy of the source, the von Neumann case being the particular case corresponding to the usual linear penalization. Moreover, using an exponential penalization, it also appears that the usual average length is then linked to both the Rényi and the von Neumann entropies.