We prove that the average error capacity Cqof a quantum arbitrarily varying channel (QAVC) equals 0 or else the random code capacity lowbarC (Ahlswede's dichotomy). We also establish a necessary and sufficient condition for Cq> 0
We consider a model of communication via a fully quantum jammer channel with quantum jammer, quantum sender and quantum receiver, which we dub quantum arbitrarily varying channel (QAVC). Restricting to finite dimensional user and jammer systems, we show, using permutation symmetry and a de Finetti reduction, how the random coding capacity (classical and quantum) of the QAVC is reduced to the capacity of a naturally associated compound channel, which is obtained by restricting the jammer to i.i.d. input states. Furthermore, we demonstrate that the shared randomness required is at most logarithmic in the block length, via a quantum version of the “elimination of of correlation” using a random matrix tail bound. This implies a dichotomy theorem: either the classical capacity of the QAVC is zero, and then also the quantum capacity is zero, or each capacity equals its random coding variant.
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.
We derive a regularized formula for the common randomness assisted entanglement transmission capacity of finite arbitrarily varying quantum channels (AVQC's). For finite AVQC's with positive capacity for classical message transmission we show, by derandomization through classical forward communication, that the random capacity for entanglement transmission equals the deterministic capacity for entanglement transmission. This is a quantum version of the famous Ahlswede dichotomy. In the infinite case, we derive a similar result for certain classes of AVQC's. At last, we give two possible definitions of symmetrizability of an AVQC.
We establish the Ahlswede Dichotomy for arbitrarily varying classical-quantum wiretap channels, i.e., either the deterministic secrecy capacity of an arbitrarily varying classical-quantum wiretap channel is zero, or it equals its randomness assisted secrecy capacity. We analyze the secrecy capacity of arbitrarily varying classical-quantum wiretap channels when the sender and the receiver use various resources. It turns out that having randomness, common randomness, and correlation as resources are very helpful for achieving a positive deterministic secrecy capacity of arbitrarily varying classical-quantum wiretap channels. We prove the phenomenon “super-activation” for arbitrarily varying classical-quantum wiretap channels, i.e., two arbitrarily varying classical-quantum wiretap channels, both with zero deterministic secrecy capacity, if used together allow perfect secure transmission.
This work contains results concerning transmission of entanglement and subspaces as well as generation of entanglement in the limit of arbitrary many uses of compound- and arbitrarily varying quantum channels (CQC, AVQC).
In both cases, the channel is described by a set of memoryless channels. Only forward communication between one sender and one receiver is allowed. A code is said to be „good“ only, if it is „good“ for every channel out of the set.
Both settings describe a scenario, in which sender and receiver have only limited channel knowledge.
For different amounts of information about the channel available to sender or receiver, coding theorems are proven for the CQC.
For the AVQC, both deterministic and randomised coding schemes are considered. Coding theorems are proven, as well as a quantum analogue of the Ahlswede-dichotomy.
The connection to zero-error capacities of stationary memoryless quantum channels is investigated.
The notion of symmetrisability is defined and used for both classes of channels.