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arXiv:1712.05702 · Journal of Mathematical Physics

Secret message transmission over quantum channels under adversarial quantum noise: Secrecy capacity and super-activation

Holger Boche, Minglai Cai, Janis Nötzel, Christian Deppe

Published Jun 1, 2019• 3 citations
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Abstract

We determine the secrecy capacities of arbitrarily varying quantum channels (AVQCs). Both secrecy capacities with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. As an application of our result for secret message transmission over AVQCs, we determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon “superactivation” for secrecy capacities of arbitrarily varying quantum channels, i.e., two quantum channels both with zero secrecy capacity, which, if used together, allow secure transmission with positive capacity. We give therewith an answer to the question “When is the secrecy capacity a continuous function of the system parameters?,” which has been listed as an open problem in quantum information problem page of the Institut für Theoretische Physik (ITP) Hannover. We also discuss the relations between the entanglement distillation capacity, the entanglement generating capacity, and the strong subspace transmission capacity for AVQCs. Ahlswede et al. made in 2013 the conjecture that the entanglement generating capacity of an AVQC is equal to its entanglement generating capacity under shared randomness assisted quantum coding. We demonstrate that the validity of this conjecture implies that the entanglement generating capacity, the entanglement distillation capacity, and the strong subspace transmission capacity of an AVQC are continuous functions of the system parameters. Consequently, under the premise of this conjecture, the secrecy capacities of an AVQC differ significantly from the general quantum capacities.

DOI
10.1063/1.5019461

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We determine the secrecy capacities of arbitrarily varying quantum channels (AVQCs). Both secrecy capacities with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. As an application of our result for secret message transmission over AVQCs, we determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon “superactivation” for secrecy capacities of arbitrarily varying quantum channels, i.e., two quantum channels both with zero secrecy capacity, which, if used together, allow secure transmission with positive capacity. We give therewith an answer to the question “When is the secrecy capacity a continuous function of the system parameters?,” which has been listed as an open problem in quantum information problem page of the Institut für Theoretische Physik (ITP) Hannover. We also discuss the relations between the entanglement distillation capacity, the entanglement generating capacity, and the strong subspace transmission capacity for AVQCs. Ahlswede et al. made in 2013 the conjecture that the entanglement generating capacity of an AVQC is equal to its entanglement generating capacity under shared randomness assisted quantum coding. We demonstrate that the validity of this conjecture implies that the entanglement generating capacity, the entanglement distillation capacity, and the strong subspace transmission capacity of an AVQC are continuous functions of the system parameters. Consequently, under the premise of this conjecture, the secrecy capacities of an AVQC differ significantly from the general quantum capacities.
We determine the secrecy capacities of AVQCs (arbitrarily varying quantum channels). Both secrecy capacity with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. We determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon “super-activation” for secrecy capacities of AVQCs, i.e., two quantum channels both with zero secrecy capacity, which, if used together, allow secure transmission with positive capacity. We also discuss the relations between the entanglement distillation capacity, the entanglement generating capacity, and the strong subspace transmission capacity for AVQCs.
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 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 finite 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 assistance by 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 unassisted capacity does have discontinuity points, while it is known that the randomness-assisted capacity is always continuous in the channel. We characterize the discontinuity points and prove that the unassisted capacity is always continuous around its positivity points. After having established shared randomness as an important resource, we quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) probability of a decoding error with respect to the average error criterion and the number of messages that can be sent over a finite number of channel uses. We relate our results to the entanglement transmission capacities of finite AVQCs, where the role of shared randomness is not yet well understood, and give a new sufficient criterion for the entanglement transmission capacity with randomness assistance to vanish.
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.
We analyze arbitrarily varying classical-quantum wiretap channels. These channels are subject to two attacks at the same time: one passive (eavesdropping), and one active (jamming). We progress on previous works [5] and [6] by introducing a reduced class of allowed codes that fulfills a more stringent secrecy requirement than earlier definitions. In addition, we prove that non-symmetrizability of the legal link is sufficient for equality of the deterministic and the common randomness assisted secrecy capacities. At last, we focus on analytic properties of both secrecy capacities: We completely characterize their discontinuity points, and their super-activation properties.