Janis Nötzel

Quantum capacity under adversarial quantum noise: arbitrarily varying quantum channels

AbstractWe 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.

The arbitrarily varying wiretap channel - secret randomness, stability and super-activation

We study the arbitrarily varying wiretap channel (AVWC) under average error criterion when external common randomness (CR) can be used between the legitimate parties. We consider three scenarios: In the first one the CR is known to the eavesdropper, in the second it is not known to her and in the third there is no CR available. For the second scenario, we prove a complete coding theorem. For the third scenario it is known that the capacity function is discontinuous. We prove that it is nonetheless stable in the sense of being continuous around its positivity points. We characterize the points of discontinuity in terms of continuous functions. We then give a complete characterization of those pairs of AVWCs whose capacity can be super-activated in the unassisted third case - in terms of the capacity function describing the first case.

Arbitrarily varying and compound classical-quantum channels and a note on quantum zero-error capacities

We consider compound as well as arbitrarily varying classical-quantum channel models. For classical-quantum compound channels, we give an elementary proof of the direct part of the coding theorem. A weak converse under average error criterion to this statement is also established. We use this result together with the robustification and elimination technique developed by Ahlswede in order to give an alternative proof of the direct part of the coding theorem for a finite classical-quantum arbitrarily varying channels with the criterion of success being average error probability. Moreover we provide a proof of the strong converse to the random coding capacity in this setting. The notion of symmetrizability for the maximal error probability is defined and it is shown to be both necessary and sufficient for the capacity for message transmission with maximal error probability criterion to equal zero. Finally, it is shown that the connection between zero-error capacity and certain arbitrarily varying channels is, just like in the case of quantum channels, only partially valid for classical-quantum channels.

Quantum capacity of a class of compound channels

In this paper we address the issue of universal or robust communication over quantum channels. Specifically, we consider a memoryless communication scenario with channel uncertainty which is an analog of the compound channel in classical information theory. We determine the quantum capacity of finite compound channels and arbitrary compound channels with an informed decoder. Our approach in the finite case is based on the observation that perfect channel knowledge at the decoder does not increase the capacity of finite quantum compound channels. As a consequence, we obtain a coding theorem for finite quantum averaged channels, the simplest class of channels with long-term memory. The extension of these results to quantum compound channels with uninformed encoder and decoder and infinitely many constituents remains an open problem.

A Channel Under Simultaneous Jamming and Eavesdropping Attack—Correlated Random Coding Capacities Under Strong Secrecy Criteria

We give a complete characterization of the correlated random coding secrecy capacity of arbitrarily varying wiretap channels (AVWCs). We apply two alternative strong secrecy criteria, which both lead to the same multi-letter formula. The difference of these criteria lies in the treatment of correlated randomness; they coincide in the case of uncorrelated codes. On the basis of the derived formula, we show that the correlated random coding secrecy capacity is continuous as a function of the AVWC, in contrast to the discontinuous uncorrelated coding secrecy capacity. In the proof of the secrecy capacity formula for correlated random codes, we apply an auxiliary channel, which is compound from the sender to the intended receiver and arbitrarily varying from the sender to the eavesdropper.We give a complete characterization of the secrecy capacity of arbitrarily varying wiretap channels (AVWCs) with correlated random coding. We apply two alternative strong secrecy criteria, which both lead to the same formula and which coincide in the case of deterministic codes. On the basis of the derived formula, we show that the correlated random coding secrecy capacity is continuous as a function of the AVWC. We show that the deterministic coding secrecy capacity of the AVWC either equals 0 or the correlated random coding secrecy capacity. For the case that only a weak secrecy criterion is applied, we derive a complete characterization of the corresponding secrecy capacity for deterministic codes. In the proof of the secrecy capacity formula for correlated random codes, we apply an auxiliary channel which is compound from the sender to the intended receiver and varies arbitrarily from the sender to the eavesdropper.

Arbitrarily small amounts of correlation for arbitrarily varying quantum 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 simplifed symmetrizability condition for finite arbitrarily varying quantum channels.

Erratum to: Entanglement Transmission and Generation under Channel Uncertainty: Universal Quantum Channel Coding

We determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty. In the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels, so that they are forced to use quantum codes that are reliable for the whole set of channels. This is precisely the quantum analog of the compound channel coding problem. We determine the entanglement transmission and entanglement-generating capacities of compound quantum channels and show that they are equal. Moreover, we investigate two variants of that basic scenario, namely the cases of informed decoder or informed encoder, and derive corresponding capacity results.

The Arbitrarily Varying Wiretap Channel—Secret Randomness, Stability, and Super-Activation

We define the common randomness-assisted capacity of an arbitrarily varying wiretap channel (AVWC) when the eavesdropper is kept ignorant about the common randomness. We prove a multi-letter capacity formula for this model. We prove that, if enough common randomness is used, the capacity formula can be given a single-shot form again. We then consider the opposite extremal case, where no common randomness is available, and derive the capacity. It is known that the capacity of the system can be discontinuous under these circumstances. We prove here that it is still stable in the sense that it is continuous around its positivity points. We further prove that discontinuities can only arise if the legal link is symmetrizable and characterize the points where it is positive. These results shed new light on the design principles of communication systems with embedded security features. At last, we investigate the effect of super-activation of the message transmission capacity of AVWCs under the average error criterion. We give a complete characterization of those AVWCs that may be super-activated. The effect is thereby also related to the (conjectured) super-activation of the common randomness assisted capacity of AVWCs with an eavesdropper that gets to know the common randomness. Super-activation is based on the idea of wasting a few bits of non-secret messages in order to enable provably secret transmission of a large bulk of data, a concept that may prove to be of further importance in the design of communication systems. In this paper, we provide further insight into this phenomenon by providing a class of codes that is capacity achieving and does not convey any information to the eavesdropper.

Positivity, discontinuity, finite resources, and nonzero error for arbitrarily varying quantum channels

We give an explicit example that answers the question whether the transmission of messages over arbitrarily varying quantum channels 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 in fact continuous around its positivity points. This is in stark contrast to the randomness-assisted capacity, which is continuous in the channel. We then 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. These results also apply to entanglement and strong subspace transmission.

Entanglement transmission capacity of compound channels

We determine the optimal achievable rate at which entanglement can be reliably transmitted when the memoryless channel used during transmission is unknown both to sender and receiver. To be more precise, we assume that both of them only know that the channel belongs to a given set of channels. Thus, they have to use encoding and decoding schemes that work well for the whole set.