Positivity, Discontinuity, Finite Resources and Nonzero Error for
Arbitrarily Varying Quantum Channels
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by
H. Boche, J. Noetzel
2014
Abstract
This work is motivated by a quite general question: Under which circumstances
are the capacities of information transmission systems continuous? The research
is explicitly carried out on arbitrarily varying quantum channels (AVQCs). We
give an explicit example that answers the recent question whether the
transmission of messages over AVQCs can benefit from distribution of randomness
between the legitimate sender and receiver in the affirmative. The specific
class of channels introduced in that example is then extended to show that the
deterministic capacity does have discontinuity points, while that behaviour is,
at the same time, not generic: We show that it is continuous around its
positivity points. This is in stark contrast to the randomness-assisted
capacity, which is always continuous in the channel. Our results imply that the
deterministic message transmission capacity of an AVQC can be discontinuous
only in points where it is zero, while the randomness assisted capacity is
nonzero. Apart from the zero-error capacities, this is the first result that
shows a discontinuity of a capacity for a large class of quantum channels. The
continuity of the respective capacity for memoryless quantum channels had,
among others, been listed as an open problem on the problem page of the ITP
Hannover for about six years before it was proven to be continuous. We also
quantify the interplay between the distribution of finite amounts of randomness
between the legitimate sender and receiver, the (nonzero) decoding error with
respect to the average error criterion that can be achieved over a finite
number of channel uses and the number of messages that can be sent. This part
of our results also applies to entanglement- and strong subspace transmission.
In addition, we give a new sufficient criterion for the entanglement
transmission capacity with randomness assistance to vanish.
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