Communications in Mathematical Physics

, Volume 317, Issue 1, pp 103-156

First online:

Quantum Capacity under Adversarial Quantum Noise: Arbitrarily Varying Quantum Channels

  • Rudolf AhlswedeAffiliated withFakultät für Mathematik, Universität Bielefeld
  • , Igor BjelakovićAffiliated withTheoretische Informationstechnik, Technische Universität München
  • , Holger BocheAffiliated withLehrstuhl für Theoretische Informationstechnik, Technische Universität München
  • , Janis NötzelAffiliated withTheoretische Informationstechnik, Technische Universität München Email author 

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We investigate entanglement transmission over an unknown channel in the presence of a third party (called the adversary), which is enabled to choose the channel from a given set of memoryless but non-stationary channels without informing the legitimate sender and receiver about the particular choice that he made. This channel model is called an arbitrarily varying quantum channel (AVQC). We derive a quantum version of Ahlswede’s dichotomy for classical arbitrarily varying channels. This includes a regularized formula for the common randomness-assisted capacity for entanglement transmission of an AVQC. Quite surprisingly and in contrast to the classical analog of the problem involving the maximal and average error probability, we find that the capacity for entanglement transmission of an AVQC always equals its strong subspace transmission capacity.

These results are accompanied by different notions of symmetrizability (zero-capacity conditions) as well as by conditions for an AVQC to have a capacity described by a single-letter formula. In the final part of the paper the capacity of the erasure-AVQC is computed and some light shed on the connection between AVQCs and zero-error capacities. Additionally, we show by entirely elementary and operational arguments motivated by the theory of AVQCs that the quantum, classical, and entanglement-assisted zero-error capacities of quantum channels are generically zero and are discontinuous at every positivity point.