## Transmission of entanglement over compound - and arbitrarily varying channels

We investigate the following task: a sender tries to send one half of a maximally entangled state to a receiver. Available to him is some physical medium, like e.g. an optical fibre (simply called 'channel' in what follows), which is mathematically described by a completely positive map.

In the simplest scenario, this channel is memoryless and constant throughout the whole time of transmission. It is further assumed, that the channel can be used arbitrarily often - a paradigm that is being taken over from Shannon information theory to the quantum case.

In real world scenarios, however, these idealistic mathematical assumptions can never be met. There are different ways to model the limitations and uncertainties arising in these situations.

Two of them are given by models called compound- and aribtrarily varying channel, which can be described as follows:

The compound channel is still memoryless and constant. The additional constraints arising in a real-world application are accounted for through the assumption that neither sender nor receiver know explicitly, what channel exactly they are transmitting over.

Their knowledge is limited insofar as they only know that it belongs to a certain set (describing, for example, a maximal deviation of certain parameters in a mathematical description of an optical fibre which is guaranteed by the manufacturer).

The scenario can be modified by endowing either sender or receiver with complete channel knowlede.

The arbitrarily varying channel describes an even more complicated situation: The channel is allowed to change during transmission. Changes are, again, limited to a certain set of channels that is known to both sender and receiver. The only assumption remaining is the memoryless character of transmission.

Mathematical methods employed reach from linear algebra, including matrix inequalities, over (discrete) probability theory and representationtheory of groups to the geometry of highdimensional banach spaces.

References

- I. Bjelakovic, H. Boche, J. Nötzel, “Quantum capacity of a class of compound channels”, Phys. Rev. A 78, 042331, 2008
- Bjelakovic , I., Boche, H.: "Classical Capacities of Averaged and Compound Quantum Channels". IEEE Trans. Inf. Th. 57(7), 3360–3374, 2009
- "Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding" I. Bjelakovic, H. Boche, and J. Nötzel, Communications in Mathematical Physics 292, pp. 55-97, 2009, Preprint available at www.arxiv.org.
- "Quantum Capacity under adversarial quantum noise: arbitrarily varying quantum channels" R. Ahlswede, I. Bjelakovic, H. Boche, and J. Nötzel, submitted to Communications in Mathematical Physics