Related papers: Two Simple Proofs of M\"uller's Theorem
Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical…
The incompressibility method is a counting argument in the framework of algorithmic complexity that permits discovering properties that are satisfied by most objects of a class. This paper gives a preliminary insight into Kolmogorov's…
As illustrated by Schrodingers cat, there are often several macroscopically different versions of reality simultaneously existing in the wave function. On the face of it, this would seem to imply that an observer could perceive a…
Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and,…
A formula for the capacity of a quantum channel for transmitting private classical information is derived. This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward…
The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space C, but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a…
Classical machine learning theory and theory of quantum computations are among of the most rapidly developing scientific areas in our days. In recent years, researchers investigated if quantum computing can help to improve classical machine…
In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a…
This article is about the proof of the celebrated KAM theorem as sketched out in \cite{KOL} Kolmogorov's original presentation to the ICM. The proof presented here has been detailed as an effort to clarify if Kolmogorov's argument can be…
Consider a fixed universe of $N=2^n$ elements and the uniform distribution over elements of some subset of size $K$. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset.…
Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only…
We show that there are infinitely many binary strings z, such that the sum of the on-line decision complexity of predicting the even bits of z given the previous uneven bits, and the decision complexity of predicting the uneven bits given…
Intuitively, the more powerful a theory is, the greater the variety and quantity of ideas can be expressed through its formal language. Therefore, when comparing two theories concerning the same subject, it seems only reasonable to compare…
By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a "quantum probability theory". As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum…
Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the…
Kolmogorov suggested to measure quality of a statistical hypothesis $P$ for a data $x$ by two parameters: Kolmogorov complexity $C(P)$ of the hypothesis and the probability $P(x)$ of $x$ with respect to $P$. P. G\'acs, J. Tromp, P.M.B.…
The power of quantum computers is still somewhat speculative. While they are certainly faster than classical ones at some tasks, the class of problems they can efficiently solve has not been mapped definitively onto known classical…
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
After reviewing unnormalized and normalized information distances based on incomputable notions of Kolmogorov complexity, we discuss how Kolmogorov complexity can be approximated by data compression algorithms. We argue that optimal…
Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity…