<![CDATA[ IEEE Transactions on Information Theory - new TOC ]]>
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TOC Alert for Publication# 18 2018June 21<![CDATA[Table of contents]]>647C1C4152<![CDATA[IEEE Transactions on Information Theory publication information]]>647C2C2112<![CDATA[Useful States and Entanglement Distillation]]>64746894708964<![CDATA[Communication Complexity of One-Shot Remote State Preparation]]>nonoblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate RSP (ARSP) in which the goal is to prepare an approximation of the desired quantum state. Jain (Quant. Inf. & Comp., 2006) showed that the worst-case communication complexity of ARSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this paper, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of nonasymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that $n$ bits cannot be communicated with less than $n$ transmitted bits in local operations and classical communication protocols. This strengthens a result due to Nayak and Salzman (J. ACM, 2006) and may be of independent interest.]]>64747094728638<![CDATA[Shorter Stabilizer Circuits via Bruhat Decomposition and Quantum Circuit Transformations]]>$(14n{-}4)$ implementation of stabilizer circuits over the gate library ${ mathrm{H}, mathrm{P}, mathrm{CNOT}}$ , executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth-$25n$ circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-)$^{m}$ into a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C- into a 9-stage decomposition of the form -C-P-C-P-H-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters.]]>64747294738479<![CDATA[Bounds on Information Combining With Quantum Side Information]]>et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to polar coding for binary-input classical-quantum channels, and show the following three results: 1) even non-stationary channels polarize under the polar transform; 2) the blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity; and 3) under the aforementione-
lower bound conjecture, a blocklength polynomial in the gap suffices.]]>647473947571279<![CDATA[Superadditivity of Quantum Relative Entropy for General States]]>$ {mathcal H}_{AB}= {mathcal H}_{A} otimes {mathcal H} _{B}$ , for every density operator $rho _{AB}$ , one has $ D(rho _{AB}||sigma _{A}otimes sigma _{B}) ge D(rho _{A}||sigma _{A})+ D(rho _{B}||sigma _{B})$ . In this paper, we provide an extension of this inequality for arbitrary density operators $sigma _{AB}$ . More specifically, we prove that $ alpha (sigma _{AB})cdot D(rho _{AB}||sigma _{AB}) ge D(rho _{A}||sigma _{A})+ D(rho _{B}||sigma _{B})$ holds for all bipartite states $rho _{AB}$ and $sigma _{AB}$ , where $alpha (sigma _{AB})= 1+2 |sigma _{A}^{-1/2} otimes sigma _{B}^{-1/2} , sigma _{AB} , sigma _{A}^{-1/2} otimes sigma _{B}^{-1/2} - {{mathsf{1}}}_{AB}|_infty $ .]]>64747584765259<![CDATA[Compression for Quantum Population Coding]]>$n$ quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with $f$ independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size $(f/2)log n$ , including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of $n$ identically prepared modes. In both cases, we show that $(f/2)log n$ is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.]]>64747664783969<![CDATA[Optimal Nonlinear Filtering of Quantum State]]>64747844791252<![CDATA[Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model]]>64747924818813<![CDATA[Energy Propagation in Deep Convolutional Neural Networks]]>$alpha $ )-curvelets, and shearlets) the feature map energy decays at least polynomially fast. For broad families of wavelets and Weyl-Heisenberg filters, the guaranteed decay rate is shown to be exponential. Moreover, we provide handy estimates of the number of layers needed to have at least $((1-varepsilon )cdot 100)%$ of the input signal energy be contained in the feature vector.]]>647481948421032<![CDATA[How Compressible Are Innovation Processes?]]>$k$ -term approximation. In this paper, we use the entropy measure to study the compressibility of continuous-domain innovation processes (alternatively known as white noise). Specifically, we define such a measure as the entropy limit of the doubly quantized (time and amplitude) process. This provides a tool to compare the compressibility of various innovation processes. It also allows us to identify an analogue of the concept of “entropy dimension” which was originally defined by Rényi for random variables. Particular attention is given to stable and impulsive Poisson innovation processes. Here, our results recognize Poisson innovations as the more compressible ones with an entropy measure far below that of stable innovations. While this result departs from the previous knowledge regarding the compressibility of impulsive Poisson laws compared with continuous fat-tailed distributions, our entropy measure ranks $alpha$ -stable innovations according to their tail.]]>64748434871722<![CDATA[Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization]]>$sqrt {2}$ when the number of clusters is large. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information-theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured “hard but detectable” regime for community detection in sparse graphs.]]>64748724894499<![CDATA[The Shortest Possible Return Time of <inline-formula> <tex-math notation="LaTeX">$beta$ </tex-math></inline-formula>-Mixing Processes]]>$n$ -string. We study the shortest possible return time (or shortest return path) of the string over all the realizations of process starting from this string. For a $beta $ -mixing process having complete grammar, and for each size $n$ of the strings, we approximate the distribution of this short return (properly re-scaled) by a non-degenerated distribution. Under mild conditions on the $beta $ coefficients, we prove the existence of the limit of this distribution to a non-degenerated distribution. We also prove that ergodicity is not enough to guaranty this convergence. Finally, we present a connection between the shortest return and the Shannon entropy, showing that maximum of the re-scaled variables grow as the matching function of Wyner and Ziv.]]>64748954906382<![CDATA[Asymptotic Theory for Estimators of High-Order Statistics of Stationary Processes]]>64749074922345<![CDATA[Neyman–Pearson Test for Zero-Rate Multiterminal Hypothesis Testing]]>64749234939653<![CDATA[Feedback Capacity and Coding for the BIBO Channel With a No-Repeated-Ones Input Constraint]]>$(1,infty )$ -RLL input constraint, that is, the input sequence contains no consecutive ones. These results are obtained via explicit solution of an equivalent dynamic programming optimization problem. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The posterior matching scheme for our input-constrained setting is shown to achieve capacity using two new ideas: history bits, which captures the memory embedded in our setting, and message-interval splitting, which eases the analysis of the scheme. Additionally, in the special case of an S-channel, we give a very simple zero-error coding scheme that is shown to achieve capacity. For the input-constrained binary symmetric channel, we show using our capacity formula that feedback increases capacity when the cross-over probability is small.]]>647494049611116<![CDATA[Information Structures for Feedback Capacity of Channels With Memory and Transmission Cost: Stochastic Optimal Control and Variational Equalities]]>${mathbf{P}}_{B_{i}|B^{i-1}, A_{i}}$ and ${mathbf{P}}_{B_{i}|B_{i-M}^{i-1}, A_{i}}$ , where $M$ is the memory of the channel, $B^{i} stackrel {triangle }{=}{B^{-1},B_{0}, ldots, B_{i}}$ are the channel outputs, and $A^{i} stackrel {triangle }{=}{A_{0}, A_{1}, ldots, A_{i}}$ , are the channel inputs, for $i=0, ldots, n$ . The characterizations of FTFI capacity are obtained by first identifying the information structures of the optimal channel input conditional distributions ${mathscr P}_{[0, n]} stackrel {triangle }{=}big { {mathbf{P}}_{A_{i}|A^{i-1}, B^{i-1}}: i=0, ldots, nbig }$ , which maximize directed information $C_{A^{n} rightarrow B^{n}}^{FB} stackrel {triangle }{=}sup _{mathscr P_{[0, n]} } I(A^{n} rightarrow B^{n}), hspace {.2in} I(A^{n} rightarrow B^{n}) stackrel {triangle }{=}sum _{i=0}^{n} I(A^{i};B_{i}|B^{i-1})$ . The main theorem states that, for any channel with memory $M$ , the optimal channel input conditional distributions occur in the subset satisfying conditional independence $stackrel {circ }{mathscr P}_{[0, n]} stackrel {triangle }{=}big { {mathbf{P}}_{A_{i}|A^{i-1}, B^{i-1}}= {math-
f{P}}_{A_{i}|B_{i-M}^{i-1}}: i=0, ldots, nbig }$ , and the characterization of FTFI capacity is given by $C_{A^{n} rightarrow B^{n}}^{FB, M} stackrel {triangle }{=}sup _{ stackrel {circ }{mathscr P}_{[0, n]} } sum _{i=0}^{n} I(A_{i}; B_{i}|B_{i-M}^{i-1})$ . Similar conclusions are derived for problems with average cost constraints of the form $frac {1}{n+1} {mathbf{E}}Big {c_{0,n}(A^{n}, B^{n-1})Big } leq kappa,,, kappa geq 0$ , for specific functions $c_{0,n}(a^{n},b^{n-1})$ . The feedback capacity is addressed by investigating $lim _{n longrightarrow infty } frac {1}{n+1}C_{A^{n} rightarrow B^{n}}^{FB, M} $ . The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and often a variational equality of directed information, to derive upper bounds on $I(A^{n} rightarrow B^{n})$ , which are achievable over specific subsets of channel input conditional distributions ${mathscr P}_{[0, n]}$ , which are characterized by conditional independence. The main results illustrate a direct analogy, in terms of conditional independence, of the characterizations of FTFI capacity and Shannon’s capacity formulae of memoryless channels. An example is presented to illustrate the role of optimal channel input process in the derivations of the direct and converse coding theorems.]]>64749624992777<![CDATA[Coding Theorem and Converse for Abstract Channels With Time Structure and Memory]]>$psi$ -mixing output memory condition used by Kadota and Wyner is quite restrictive and excludes important channel models, in particular for the class of Gaussian channels. In fact, it is proved that for Gaussian (e.g., fading or additive noise) channels, the $psi$ -mixing condition is equivalent to finite output memory. Furthermore, it is demonstrated that the measurability requirement of Kadota and Wyner is not satisfied for relevant continuous-time channel models such as linear filters, whereas the condition used in this paper is satisfied for these models. Moreover, a weak converse is derived for all stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a general and flexible way, including amplitude and average power constraints as special case. Formulated in rigorous mathematical terms complete, explicit, and transparent proofs are presented. As a side product a gap in the proof of Kadota and Wyner – illustrated by a counterexample – is closed by providing a corrected proof of a lemma on the monotonicity of some sequence of normalized mutual information qu-
ntities. An operational perspective is taken, and an abstract framework is established, which allows to treat discrete- and continuous-time channels with (possibly infinite input and output) memory and arbitrary alphabets simultaneously in a unified way.]]>64749935016625<![CDATA[On the Capacity Region of the Parallel Degraded Broadcast Channel With Three Receivers and Three-Degraded Message Sets]]>64750175041401<![CDATA[Capacity Regions of Two-Receiver Broadcast Erasure Channels With Feedback and Memory]]>$L$ acknowledgments and its achievable rate region approaches the capacity region exponentially fast in $L$ . The second algorithm is a backpressure-like algorithm that -
erforms optimally in the long run.]]>647504250692447<![CDATA[Exact Random Coding Exponents and Universal Decoders for the Asymmetric Broadcast Channel]]>64750705086798<![CDATA[Empirical and Strong Coordination via Soft Covering With Polar Codes]]>explicit and low-complexity coding schemes that achieve the capacity regions of both empirical coordination and strong coordination for sequences of actions taking value in an alphabet of prime cardinality. Our results improve previously known polar coding schemes, which (i) were restricted to uniform distributions and to actions obtained via binary symmetric channels for strong coordination, (ii) required a non-negligible amount of common randomness for empirical coordination, and (iii) assumed that the simulation of discrete memoryless channels could be perfectly implemented. As a by-product of our results, we obtain a polar coding scheme that achieves channel resolvability for an arbitrary discrete memoryless channel whose input alphabet has prime cardinality.]]>64750875100615<![CDATA[On the Capacity of Write-Constrained Memories]]>64751015109778<![CDATA[Strong Secrecy for Interference Channels Based on Channel Resolvability]]>64751105130633<![CDATA[Autocorrelation Function for Dispersion-Free Fiber Channels With Distributed Amplification]]>647513151552577<![CDATA[Codes in the Space of Multisets—Coding for Permutation Channels With Impairments]]>64751565169531<![CDATA[An Improvement of the Asymptotic Elias Bound for Non-Binary Codes]]>64751705178377<![CDATA[Using the Difference of Syndromes to Decode Quadratic Residue Codes]]>647517951901390<![CDATA[Weak Flip Codes and their Optimality on the Binary Erasure Channel]]>nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to Shannon et al. and which was shown to achieve the error exponent with a fixed number of codewords $mathsf {M}$ , can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the $r$ -wise Hamming distance is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum $r$ -wise Hamming distance satisfies a generalized$r$ -wise Plotkin bound. The $r$ -wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the $r$ -wise Plotkin bound with equality for all $r$ . In the second part of this paper, these insights are applied to a binary erasure channel with an arbitrary erasure probability 0 < $delta$ < 1. An exact formula for the average error probability of an arbitrary (linear or nonlinear) code using maximum likelihood decoding is derived and shown to be expressible using only the $r$ -wise Hamming distance structure of the code. For a number of codewords $mathsf {M}$ satisfying $mathsf {M}le 4$ and an arbitrary finite blocklength $n$ , the globally optimal codes (in the sense of minimizing the average error probability) are found. For $mathsf {M}$ = 5 or $mathsf {M}$ = 6 and an arbitrary finite blocklength $n$ , the optimal codes are conjectured. For larger $mathsf {M}$ , observations regarding the optimal design are presented, e.g., that good codes have a large $r$ -wise Hamming distance structure for all $r$ . Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes.]]>64751915218976<![CDATA[Infinity-Norm Permutation Covering Codes From Cyclic Groups]]>$ell _infty $ -metric. We provide a general code construction, which combines short building-block codes into a single long code. We focus on cyclic transitive groups as building blocks, determining their exact covering radius, and showing a linear-time algorithm for finding a covering codeword. When used in the general construction, we show that the resulting covering code asymptotically out-performs the best known code while maintaining linear-time decoding. We also bound the covering radius of relabeled cyclic transitive groups under conjugation, showing that the covering radius is quite robust. While relabeling cannot reduce the covering radius by much, the downside is that we prove the covering radius cannot be increased by more than 1 when using relabeling.]]>647521952304492<![CDATA[Random Ensembles of Lattices From Generalized Reductions]]>64752315239355<![CDATA[Defect Tolerance: Fundamental Limits and Examples]]>647524052601216<![CDATA[Sign-Compute-Resolve for Tree Splitting Random Access]]>i.e., compute, the sum of the packets that were transmitted by the individual users. For each user, the packet consists of the user’s signature, as well as the data that the user wants to communicate. As long as no more than $K$ users collide, their identities can be recovered from the sum of their signatures. This framework for creating and transmitting packets can be used as a fundamental building block in random access algorithms, since it helps to deal efficiently with the uncertainty of the set of contending terminals. In this paper, we show how to apply the framework in conjunction with a tree-splitting algorithm, which is required to deal with the case that more than $K$ users collide. We demonstrate that our approach achieves throughput that tends to 1 rapidly as $K$ increases. We also present results on net data-rate of the system, showing the impact of the overheads of the constituent elements of the proposed protocol. We compare the performance of our scheme with an upper bound that is obtained under the assumption that the active users are a priori known. Also, we consider an upper bound on the net data-rate for any PLNC-based strategy in which one linear equation per slot is decoded. We show that already at modest packet lengths, the net data-rate of our scheme becomes close to the second upper bound, i.e., the overhead of the contention resolution algorithm and the signature codes vanishes.]]>647526152761201<![CDATA[Initialization Algorithms for Convolutional Network Coding]]>64752775295576<![CDATA[Streaming Codes for Multiplicative-Matrix Channels With Burst Rank Loss]]>647529653111135<![CDATA[Information-Theoretic Characterization of MIMO Systems With Multiple Rayleigh Scattering]]>et al., we derive another important metric, namely, the sum rate, under linear receive processing and independent stream decoding. In particular, we characterize the performance of the minimum mean squared error receiver in closed form, and that of the zero forcing receiver by resorting to bounding techniques. The bulk of the work relies on results about finite-dimensional random matrix products, a number of which are novel and detailed in the Appendices. The analysis, validated through numerical results, highlights the severe degradation in the performance of linear receivers due to multi-fold scattering. It also unveils the performance trend of multiple scattering MIMO channels as a function of the number of antennas and the number of scattering stages.]]>64753125325887<![CDATA[Approximate Capacity Region of the Two-User Gaussian Interference Channel With Noisy Channel-Output Feedback]]>647532653581996<![CDATA[Degrees of Freedom of Cache-Aided Wireless Interference Networks]]>647535953801496<![CDATA[Optimal Link Scheduling for Age Minimization in Wireless Systems]]>647538153941121<![CDATA[Asymptotically Optimal Pilot Allocation Over Markovian Fading Channels]]>exploitation, or assign a pilot to a user with outdated CSI for exploration. As we show, the arising pilot allocation problem is a restless bandit problem and thus its optimal solution is out of reach. In this paper, we propose an approximation based on the Lagrangian relaxation method, which provides a low-complexity Whittle index policy. We prove this policy to be asymptotically optimal in the many users regime (when the number of users in the system and the available pilots for channel sensing grow large). We evaluate the performance of Whittle’s index policy in various scenarios and illustrate its remarkably good performance for small number of users, where it is not guaranteed to be optimal.]]>64753955418968<![CDATA[Asymptotically Optimal Optical Orthogonal Signature Pattern Codes]]>$r$ -simple matrices to present a recursive construction for OOSPCs. These constructions yield new families of asymptotically optimal OOSPCs.]]>64754195431540<![CDATA[Full Characterization of Generalized Bent Functions as (Semi)-Bent Spaces, Their Dual, and the Gray Image]]>${mathbb F}_{2}^{n}$ to ${mathbb Z}_{2^{k}}$ which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when $n$ is even. Then, depending on the parity of $n$ , it is shown that the Gray image of a gbent function is $(k-1)$ or $(k-2)$ plateaued, which generalizes previous results for $k$ = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of $n$ , a gbent function is a $(k-1)$ -dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from ${mathbb F}_{2}^{n}$ to ${mathbb Z}_{2^{k}}$ , called ${mathbb Z}_{q}$ -bent functions (which are necessarily gbent), which essentially gives rise to relativ-
difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not ${mathbb Z}_{q}$ -bent.]]>64754325440279<![CDATA[Further Results on Generalized Bent Functions and Their Complete Characterization]]>$p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon’s $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh–Hadamard transform of their component functions, and we generalize these results to the case of odd prime.]]>64754415452301<![CDATA[Counterexample to the Vector Generalization of Costa’s Entropy Power Inequality, and Partial Resolution]]>et al. In particular, the claimed inequality can fail if the matrix-valued parameter in the convex combination does not commute with the covariance of the additive Gaussian noise. Conversely, the inequality holds if these two matrices commute.]]>64754535454177<![CDATA[Corrections to “Second-Order Asymptotics of Conversions of Distributions and Entangled States Based on Rayleigh-Normal Probability Distributions”]]>[2], Fig. 1 and Fig. 4 are based on incorrect numerical calculation. The correct plots of [2, Fig. 1] are given in [1, Fig. 3] as follows.]]>64754555455253<![CDATA[[Blank page]]]>647B5456B54564<![CDATA[IEEE Transactions on Information Theory information for authors]]>647C3C3100