A fundamental tool to prove inner bounds in classical network information theory is the so-called conditional joint typicality lemma'. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments fail spectacularly in the quantum setting. This bottleneck shows up in the fact that so-called simultaneous decoders', as opposed to successive cancellation decoders', are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called simutaneous smoothing' theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so, we develop two novel tools in quantum information theory which may be of independent interest. The first tool is a simple geometric idea called tilting', which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing', is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm by much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.
The bottleneck of simultaneous decoding was first pointed out by @cite_14 in their paper on the quantum interference channel. Subsequently, Dutil @cite_13 pointed out a related bottleneck of simultaneous smoothing', which was further discussed by Drescher and Fawzi @cite_10 . Simultaneous decoders have been recently used in several papers e.g. @cite_1 , but the inner bounds obtained there were suboptimal compared to known inner bounds when restricted to the asymptotic iid setting.
@cite_1 @cite_10 @cite_14 @cite_13
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Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates. This paper provides new results on computing simultaneous sparse approximations of multichannel signals over redundant dictionaries using two greedy algorithms. The first one, p-thresholding, selects the S atoms that have the largest p-correlation while the second one, p-simultaneous matching pursuit (p-SOMP), is a generalisation of an algorithm studied by Tropp in (Signal Process. 86:572–588, 2006). We first provide exact recovery conditions as well as worst case analyses of all algorithms. The results, expressed using the standard cumulative coherence, are very reminiscent of the single channel case and, in particular, impose stringent restrictions on the dictionary. For the additive white Gaussian noise channel with average codeword power constraint, sparse superposition codes are developed. These codes are based on the statistical high-dimensional regression framework. In a previous paper, we investigated decoding using the optimal maximum-likelihood decoding scheme. Here, a fast decoding algorithm, called the adaptive successive decoder, is developed. For any rate R less than the capacity C, communication is shown to be reliable with nearly exponentially small error probability. Specifically, for blocklength n, it is shown that the error probability is exponentially small in n logn. Recently, it was observed that spatially-coupled LDPC code ensembles approach the Shannon capacity for a class of binary-input memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived in [1]. In particular, it was shown that the belief propagation (BP) threshold of the spatially coupled codes is equal to the maximum a posteriori (MAP) decoding threshold of the underlying constituent codes. In this sense, the BP threshold is saturated to its maximum value. Moreover, it has been empirically observed that the same phenomena also occurs when transmitting over more general classes of BMS channels. In this paper, we show that the effect of spatial coupling is not restricted to the realm of channel coding. The effect of coupling also manifests itself in compressed sensing. Specifically, we show that spatially-coupled measurement matrices have an improved sparsity to sampling threshold for reconstruction algorithms based on verification decoding. For BP-based reconstruction algorithms, this phenomenon is also tested empirically via simulation. At the block lengths accesible via simulation, the effect is rather small but, based on the threshold analysis, we believe this warrants further study.