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Samuel Epstein

Publications and source records attributed to Samuel Epstein.

31 records · Page 2Linked to original sources
arXiv2023

A Quantum Outlier Theorem

In recent results, it has been proven that all sampling methods produce outliers. In this paper, we extend these results to quantum information theory. Projectors of large rank must contain pure quantum states in their images that are outlying states. Otherwise, the projectors are exotic, in that they have high mutual information with the halting sequence. Thus quantum coding schemes that use projections, such as Schumacher compression, must communicate using outlier quantum states.

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arXiv2023

On the Existence of Anomalies

The Independence Postulate (IP) is a finitary Church-Turing Thesis, saying mathematical sequences are independent from physical ones. IP implies the existence of anomalies.

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arXiv2023

A Quantum EL Theorem

In this paper, we prove a quantum version of the EL Theorem. It states that non-exotic projections of large rank must have simple quantum states in their images. A consequence to this is there is no way to communicate a quantum source with corresponding large enough von Neumann entropy without using simple quantum states.

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arXiv2022

All Sampling Methods Produce Outliers

Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P.

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arXiv2022

Outliers, Dynamics, and the Independence Postulate

We show that outliers occur almost surely in computable dynamics over infinite sequences. Ever greater outliers can be found as the number of visited states increases. We show the Independence Postulate explains how outliers are found in the physical world. We generalize the outliers theorem to uncomputable sampling methods.

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arXiv2022

On the Kolmogorov Complexity of Binary Classifiers

We provide tight upper and lower bounds on the expected minimum Kolmogorov complexity of binary classifiers that are consistent with labeled samples. The expected size is not more than complexity of the target concept plus the conditional entropy of the labels given the sample.

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arXiv2021

On the Algorithmic Content of Quantum Measurements

We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced. This is independent of the number of outcomes of the quantum measurement. Due to conservation inequalities, such random noise cannot be processed into coherent data.

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arXiv2021

On the Conditional Complexity of Sets of Strings

Given a set X of finite strings, one interesting question to ask is whether there exists a member of X which is simple conditional to all other members of X. Conditional simplicity is measured by low conditional Kolmogorov complexity. We prove the affirmative to this question for sets that have low mutual information with the halting sequence. There are two results with respect to this question. One is dependent on the maximum conditional complexity between two elements of X, the other is dependent on the maximum expected value of the conditional complexity of a member of X relative to each member of X.

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arXiv2021

Algorithmic No-Cloning Theorem

We introduce the notions of algorithmic mutual information and rarity of quantum states. These definitions enjoy conservation inequalities over unitary transformations and partial traces. We show that a large majority of pure states have minute self algorithmic information. We provide an algorithmic variant to the no-cloning theorem, by showing that only a small minority of quantum pure states can clone a non negligible amount of algorithmic information. We also provide a chain rule inequality for quantum algorithmic entropy. We show that rarity does not increase under POVM measurements.

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arXiv2014

Sets Have Simple Members

The combined Universal Probability M(D) of strings x in sets D is close to max M({x}) over x in D: their ~logs differ by at most D's information j=I(D:H) about the halting sequence H. Thus if all x have complexity K(x) >k, D carries >i bits of information on each its x where i+j ~ k. Note that there are no ways to generate D with significant I(D:H).

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arXiv2013

Randomness Conservation over Algorithms

Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation inequalities. We address this issue by proving tight bounds of randomness and information conservation with respect to recursively enumerable transformations, i.e. processing by algorithms. We also show conservation of randomness of finite strings with respect to enumerable distributions, i.e. semicomputable semi-measures.

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arXiv2011

An Information Theoretic Representation of Agent Dynamics as Set Intersections

We represent agents as sets of strings. Each string encodes a potential interaction with another agent or environment. We represent the total set of dynamics between two agents as the intersection of their respective strings, we prove complexity properties of player interactions using Algorithmic Information Theory. We show how the proposed construction is compatible with Universal Artificial Intelligence, in that the AIXI model can be seen as universal with respect to interaction.

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