Related papers: Improved direct sum theorem in classical communica…

This paper has been withdrawn due to an error in one of the equations in the extended portion.

This paper has been withdrawn by the author due to a crucial error in equation.

An improved estimate is given for $|\theta(x) -x|$, where $\theta(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the…

This paper has been withdrawn because Proposition 2.2 (c) is false. This invalids the main results of section 2 and 3. We thank A. Canonaco for pointing us the error.

This paper has been withdrawn due to a critical error near equation (71). This error causes the entire argument of the paper to collapse. Emmanuel Candes of Stanford discovered the error, and has suggested a correct analysis, which will be…

This paper has been withdrawn by the authors, due a crucial error in the proof of the main theorem.

This paper has been withdrawn by the author due to a crucial sign error in equation 2.

This paper has been withdrawn by the authors due to a mistake in the proof of the chief result. In particular Theorem 1.3 is correct, while Theorem 1.1 and Theorem 1.2 hold with \mu>0 and a suitable restriction on the exponent p. The proof…

This paper has been withdrawn by the authors due to crucial error on assumption in Eq. 2. We cannot assume v_R and v_L to be equal or even have the same sign.

A process of preparation, transmission and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However no model for n qubits with a…

This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking.

This paper has been withdrawn by the author, due to obsolete reference [4], insufficient discussion in sec. 4, and major conceptual error in sec. 5.

Major mistake. The paper has been withdrawn.

On [3, p. 199] one says "We mention parenthetically that the proof of [99, Lemma 41.3] is incorrect, and we do not know whether it, [99, Theorem 41.5] and [99, Theorem 41.6] are true". The previously cited reference [99] is our reference…

This paper has been withdrawn because the result turns out to be trivial.

The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information [1]. This capacity can be expressed using the mutual information between input…

This paper has been withdrawn by the author due to a crucial sign error in equation 1

This paper has been withdrawn.

The paper has been withdrawn because the proof of part (b) of the main theorem is incomplete.

The manuscript has been withdrawn because it has a fatal flaw caused by a misinterpretation of the Copeland-Erdos theorem.