Quantum Capacity under Adversarial Quantum Noise: Arbitrarily Varying Quantum Channels

Abstract

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 entanglementassisted zero-error capacities of quantum channels are generically zero and are discontinuous at every positivity point.

Arbitrarily small amounts of correlation for arbitrarily varying quantum channels

“…2, it is conjectured in Ref. 4 that A det (I) = A random (I) holds for every AVQC I. To complete the list of models and capacities that one could investigate in this matter we point out that no such example has been given for classical message transmission over AVQCs, although one should expect that the case C random (I) > C det (I) occurs.…”

“…Note that we can safely assume C det (I) = 0, since otherwise we already achieve above numbers (in case of entanglement transmission, this is stated and proven as Theorem 5 in Ref. 4, for message transmission it is obvious for the expert but yet unproven), even without the use of common randomness. …”

“…The techniques necessary to cope with arbitrary sets have been developed in Ref. 4. They are based on the case |S| < ∞.…”

“…4 cannot be checked that easily, especially not since it might be unstable with respect to small variations of the set I. By the results of Ahlswede et al, 4 symmetrizability of an AVQC is equivalent to C det (I) = 0, hence a simpler version of the symmmetrizability conditions seems desirable. This simplified version is the fourth result of this paper.…”

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

“…Observe however that what we treated here is not an "arbitarily varying quantum channel" in any sense previously considered [3,9], going beyond the model in [36], too.…”