Universal communication part II: channels with memory

Consider communication over a channel whose probabilistic model is completely unknown vector-wise and is not assumed to be stationary. Communication over such channels is challenging because knowing the past does not indicate anything about the future. The existence of reliable feedback and common randomness is assumed. In a previous paper it was shown that the Shannon capacity cannot be attained, in general, if the channel is not known. An alternative notion of "capacity" was defined, as the maximum rate of reliable communication by any block-coding system used over consecutive blocks. This rate was shown to be achievable for the modulo-additive channel with an individual, unknown noise sequence, and not achievable for some channels with memory. In this paper this "capacity" is shown to be achievable for general channel models possibly including memory, as long as this memory fades with time. In other words, there exists a system with feedback and common randomness that, without knowledge of the channel, asymptotically performs as well as any block code, which may be designed knowing the channel. For non-fading memory channels a weaker type of "capacity" is shown to be achievable

Entanglement-assisted classical capacities of compound and arbitrarily varying quantum channels

We consider classical message transmission under entanglement assistance for compound memoryless and arbitrarily varying quantum channels. In both cases, we prove general coding theorems together with corresponding weak converse bounds. In this way, we obtain single-letter characterizations of the entanglement-assisted classical capacities for both channel models. Moreover, we show that the entanglement-assisted classical capacity does exhibit no strong converse property for some compound quantum channels for the average as well as the maximal error criterion. A strong converse to the entanglement-assisted classical capacities does hold for each arbitrarily varying quantum channel.Comment: Minor corrections, results unchanged, presentation updated, 21 pages, 0 figures, accepted for publication in Quant. Inf. Pro

Simultaneous transmission of classical and quantum information under channel uncertainty and jamming attacks

We derive universal codes for simultaneous transmission of classical messages and entanglement through quantum channels, possibly under attack of a malignant third party. These codes are robust to different kinds of channel uncertainty. To construct such universal codes, we invoke and generalize properties of random codes for classical and quantum message transmission through quantum channels. We show these codes to be optimal by giving a multi-letter characterization of regions corresponding to the capacity of compound quantum channels for simultaneously transmitting and generating entanglement with classical messages. Also, we give dichotomy statements in which we characterize the capacity of arbitrarily varying quantum channels for simultaneous transmission of classical messages and entanglement. These include cases where the malignant jammer present in the arbitrarily varying channel model is classical (chooses channel states of product form) and fully quantum (is capable of general attacks not necessarily of product form)

Resource cost results for one-way entanglement distillation and state merging of compound and arbitrarily varying quantum sources

We consider one-way quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelakovi\'c et. al. [Universal quantum state merging, J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum sources (AVQS). In the AVQS model, the source state is assumed to vary in an arbitrary manner for each source output due to environmental fluctuations or adversarial manipulation. We determine the one-way entanglement distillation capacity for AVQS, where we invoke the famous robustification and elimination techniques introduced by R. Ahlswede. Regarding quantum state merging for AVQS we show by example, that the robustification and elimination based approach generally leads to suboptimal entanglement as well as classical communication rates.Comment: Improved presentation. Close to the published version. Results unchanged. 25 pages, 0 figure

Resource Cost Results for Entanglement Distillation and State Merging under Source Uncertainties

We introduce one-way LOCC protocols for quantum state merging for compound sources, which have asymptotically optimal entanglement as well as classical communication resource costs. For the arbitrarily varying quantum source (AVQS) model, we determine the one-way entanglement distillation capacity, where we utilize the robustification and elimination techniques, well-known from classical as well as quantum channel coding under assumption of arbitrarily varying noise. Investigating quantum state merging for AVQS, we demonstrate by example, that the usual robustification procedure leads to suboptimal resource costs in this case.Comment: 5 pages, 0 figures. Accepted for presentation at the IEEE ISIT 2014 Honolulu. This is a conference version of arXiv:1401.606

Achieving the Empirical Capacity Using Feedback Part I: Memoryless Additive Models

We address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback, and show how rates corresponding to the empirical behavior of the channel can be attained, although no rate can be guaranteed in advance. First, we consider a discrete modulo-additive channel with alphabet $\mathcal{X}$, where the noise sequence $Z^n$ is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder's message, possibly in an adversarial fashion. Although the classical capacity of this channel is zero, we show that rates approaching the empirical capacity $\log|\mathcal{X}|-H_{emp}(Z^n)$ can be universally attained, where $H_{emp}(Z^n)$ is the empirical entropy of $Z^n$. For the more general setting where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality, we model the empirical channel actions as the modulo-addition of a realized noise sequence, and show that the same result applies if common randomness is available. The results are proved constructively, by providing a simple sequential transmission scheme approaching the empirical capacity. In part II of this work we demonstrate how even higher rates can be attained by using more elaborate models for channel actions, and by utilizing possible empirical dependencies in its behavior.Comment: Submitted to the IEEE Transactions on Information Theor

Universal superposition codes: capacity regions of compound quantum broadcast channel with confidential messages

We derive universal codes for transmission of broadcast and confidential messages over classical-quantum-quantum and fully quantum channels. These codes are robust to channel uncertainties considered in the compound model. To construct these codes we generalize random codes for transmission of public messages, to derive a universal superposition coding for the compound quantum broadcast channel. As an application, we give a multi-letter characterization of regions corresponding to the capacity of the compound quantum broadcast channel for transmitting broadcast and confidential messages simultaneously. This is done for two types of broadcast messages, one called public and the other common

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.Comment: 37 pages, 0 figures. Accepted for publication in the LNCS Volume in Memory of Rudolf Ahlswede. Includes a section on certain differences between classical and classical-quantum channels regarding their zero-error capacitie

Positivity, Discontinuity, Finite Resources and Nonzero Error for Arbitrarily Varying Quantum Channels

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.Comment: 19 pages, no figures. Corrected typos. Large parts of the introduction are rewritten, especially the historical part from the earlier verions is completely deleted. This version contains an additional theorem (Theorem 5) which summarizes our findings concerning the points of discontinuity of the deterministic message transmission capacit

A Characterization of the Minimal Average Data Rate that Guarantees a Given Closed-Loop Performance Level

This paper studies networked control systems closed over noiseless digital channels. By focusing on noisy LTI plants with scalar-valued control inputs and sensor outputs, we derive an absolute lower bound on the minimal average data rate that allows one to achieve a prescribed level of stationary performance under Gaussianity assumptions. We also present a simple coding scheme that allows one to achieve average data rates that are at most 1.254 bits away from the derived lower bound, while satisfying the performance constraint. Our results are given in terms of the solution to a stationary signal-to-noise ratio minimization problem and builds upon a recently proposed framework to deal with average data rate constraints in feedback systems. A numerical example is presented to illustrate our findings.Comment: Submitted to IEEE Transactions on Automatic Control on December 26, 201