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On the Algorithmic Information Between Probabilities

Samuel Epstein

TL;DR

This work develops a unified framework for algorithmic information between probability measures, proving that the self-information between measures cannot increase under randomized processing across finite sequences, infinite sequences, and general $T_0$ second-countable topologies. It achieves this by mapping topological spaces to sequence spaces, employing computable random transitions, and establishing bounds via computable covers and averaging arguments. Key results include that Gaussian smoothing (convolution with a computable kernel) reduces self-information, and that averaged information across ensembles remains bounded, with strong implications for quantum measurements where most pure states produce negligible information. The framework broadens the scope of algorithmic information theory beyond computable objects to arbitrary measures and topologies, offering insights for signal processing and quantum information tasks.

Abstract

We extend algorithmic conservation inequalities to probability measures. The amount of self information of a probability measure cannot increase when submitted to randomized processing. This includes (potentially non-computable) measures over finite sequences, infinite sequences, and $T_0$, second countable topologies. One example is the convolution of signals over real numbers with probability kernels. Thus the smoothing of any signal due We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced.

On the Algorithmic Information Between Probabilities

TL;DR

This work develops a unified framework for algorithmic information between probability measures, proving that the self-information between measures cannot increase under randomized processing across finite sequences, infinite sequences, and general second-countable topologies. It achieves this by mapping topological spaces to sequence spaces, employing computable random transitions, and establishing bounds via computable covers and averaging arguments. Key results include that Gaussian smoothing (convolution with a computable kernel) reduces self-information, and that averaged information across ensembles remains bounded, with strong implications for quantum measurements where most pure states produce negligible information. The framework broadens the scope of algorithmic information theory beyond computable objects to arbitrary measures and topologies, offering insights for signal processing and quantum information tasks.

Abstract

We extend algorithmic conservation inequalities to probability measures. The amount of self information of a probability measure cannot increase when submitted to randomized processing. This includes (potentially non-computable) measures over finite sequences, infinite sequences, and , second countable topologies. One example is the convolution of signals over real numbers with probability kernels. Thus the smoothing of any signal due We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced.
Paper Structure (7 sections, 20 theorems, 18 equations, 1 figure)

This paper contains 7 sections, 20 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Probabilities over Sequences
  3. Probabilities Over General Spaces
  4. Information Transfer
  5. Computable Covers
  6. Averaged Information
  7. Quantum Mechanics

Key Result

Lemma 1

For partial recursive function $f:\{0,1\}^*\rightarrow\{0,1\}^*$, $\i(f(x):y)<^{+} \i(x:y)+{\mathbf K}(f)$.

Figures (1)

  • Figure 1: Information terms.

Theorems & Definitions (58)

  • Definition 1: Information, Discrete Semi-Measures
  • Lemma 1: Levin84
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Example 1
  • Definition 2: Channel
  • Example 2: Uniform Spread
  • Theorem 1
  • ...and 48 more