Classical and quantum information:
Randomization and entanglement



Alexander Holevo
Moscow University

Randomization is known as a useful tool in classical statistics, in particular, in experimental design and in game theory. However, in Bayes approach the optimal decision rule can always be chosen nonrandomized (deterministic). Randomization means introduction of an independent source of randomness, so one could formulate the following principle:
Observation of a system in unknown state together with an ancillary independent system in a fixed state gives no more information about the unknown state than observation of the system alone.
In other words, introducing extra noise in observation cannot increase our knowledge about the state of observed system. Indeed, this is so, if by ``systems'' here one means ``classical systems'' and by ``noise'' -- a classical source of randomness (say, roulette). However, this principle looses its validity for quantum systems due to the new quality of entanglement which to some extent is similar to classical correlation but by no means can be reduced to it. Further, for two independent quantum systems there are entangled measurements which can bear more information than arithmetic sum of informations from these systems. This property of superadditivity of information has profound consequences for the theory of quantum communication channels and their capacities. We also discuss the famous ``additivity conjecture'' for the classical capacity as well as recently found deep connection between the secret classical and the quantum capacities of a channel.