We determine the secrecy capacities of arbitrarily varying quantum channels (AVQCs). Both secrecy capacities with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. As an application of our result for secret message transmission over AVQCs, we determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon “superactivation” for secrecy capacities of arbitrarily varying quantum channels, i.e., two quantum channels both with zero secrecy capacity, which, if used together, allow secure transmission with positive capacity. We give therewith an answer to the question “When is the secrecy capacity a continuous function of the system parameters?,” which has been listed as an open problem in quantum information problem page of the Institut für Theoretische Physik (ITP) Hannover. We also discuss the relations between the entanglement distillation capacity, the entanglement generating capacity, and the strong subspace transmission capacity for AVQCs. Ahlswede et al. made in 2013 the conjecture that the entanglement generating capacity of an AVQC is equal to its entanglement generating capacity under shared randomness assisted quantum coding. We demonstrate that the validity of this conjecture implies that the entanglement generating capacity, the entanglement distillation capacity, and the strong subspace transmission capacity of an AVQC are continuous functions of the system parameters. Consequently, under the premise of this conjecture, the secrecy capacities of an AVQC differ significantly from the general quantum capacities.

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