Related papers: Computability of the Zero-Error capacity with Kolm…
We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a…
One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which…
Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender…
We establish a strong converse bound for the private classical capacity of anti-degradable quantum channels. Specifically, we prove that this capacity is zero whenever the error $\epsilon > 0$ and privacy parameter $\delta > 0$ satisfy the…
The capacity of finite state channels (FSCs) with feedback has been shown to be a limit of a sequence of multi-letter expressions. Despite many efforts, a closed-form single-letter capacity characterization is unknown to date. In this…
It was shown [T.S. Cubitt et al., IEEE Trans. Inform. Theory 57, 8114 (2011)] that there exist quantum channels where a single use cannot transmit classical information perfectly yet two uses can. This phenomenon is called the…
In networked control, a fundamental problem is to determine the smallest capacity of a communication channel between a dynamical system and a controller above which a prescribed control objective can be achieved. Often, a preliminary task…
We study various super-activation effects in the following zero-error communication scenario: One sender wants to send classical or quantum information through a noisy quantum channel to one receiver with zero probability of error. First we…
In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there…
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
Most communication channels are subjected to noise. One of the goals of Information Theory is to add redundancy in the transmission of information so that the information is transmitted reliably and the amount of information transmitted…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
The adaptive zero-error capacity of discrete memoryless channels (DMC) with noiseless feedback has been shown to be positive whenever there exists at least one channel output "disprover", i.e. a channel output that cannot be reached from at…
We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite…
Zero-Error communication investigates communication without any error. By defining channels without probabilities, results from Elias can be used to completely characterize which channel can simulate which other channels. We introduce the…
The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
By the seminal paper of Claude Shannon \cite{Shannon48}, the computation of the capacity of a discrete memoryless channel has been considered as one of the most important and fundamental problems in Information Theory. Nearly 50 years ago,…
A fully general strong converse for channel coding states that when the rate of sending classical information exceeds the capacity of a quantum channel, the probability of correctly decoding goes to zero exponentially in the number of…