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arXiv:1004.5551 · IEEE International Symposium on Information Theory (ISIT)

Entanglement transmission over arbitrarily varying quantum channels

Rudolf Ahlswede, Igor Bjelaković, Holger Boche, Janis Nötzel

Published Jun 1, 2010• 3 citations
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Abstract

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.

DOI
10.1109/isit.2010.5513541

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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 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.
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.
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.
As our main result we show that, in order to achieve the randomness assisted message - and entanglement transmission capacities of a finite arbitrarily varying quantum channel it is not necessary that sender and receiver share (asymptotically perfect) common randomness. Rather, it is sufficient that they each have access to an unlimited amount of uses of one part of a correlated bipartite source. This access might be restricted to an arbitrary small (nonzero) fraction per channel use, without changing the main result. We investigate the notion of common randomness. It turns out that this is a very costly resource - generically, it cannot be obtained just by local processing of a bipartite source. This result underlines the importance of our main result. Also, the asymptotic equivalence of the maximal- and average error criterion for classical message transmission over finite arbitrarily varying quantum channels is proven. At last, we prove a simplified symmetrizability condition for finite arbitrarily varying quantum channels.
As our main result show that in order to achieve the randomness assisted message and entanglement transmission capacities of a finite arbitrarily varying quantum channel it is not necessary that sender and receiver share (asymptotically perfect) common randomness. Rather, it is sufficient that they each have access to an unlimited amount of uses of one part of a correlated bipartite source. This access might be restricted to an arbitrary small (nonzero) fraction per channel use, without changing the main result. We investigate the notion of common randomness. It turns out that this is a very costly resource – generically, it cannot be obtained just by local processing of a bipartite source. This result underlines the importance of our main result. Also, the asymptotic equivalence of the maximal- and average error criterion for classical message transmission over finite arbitrarily varying quantum channels is proven. At last, we prove a simplified symmetrizability condition for finite arbitrarily varying quantum channels.
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.