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Two Simple Proofs of Müller's Theorem

Samuel Epstein

TL;DR

This work addresses whether quantum mechanics can improve the compression of classical information and affirms that it cannot, by proving Müller's theorem: $C(x)=^+\mathbf{Hbvl}(\ket{x}\bra{x})$. It provides two simple, self-contained proofs: a first proof leveraging a universal lower computable semi-density operator to relate $K$, Hbvl, and algorithmic probability, and a second, self-contained construction that derives the classical complexity bound directly from the quantum complexity via finite-precision witnesses and an extraction procedure. The paper also introduces new bounds on quantum Kolmogorov complexity with error and rigorously formalizes the indeterminate-length quantum state framework with a concrete approximation mechanism for universal quantum Turing machines. Together, these results establish the invariance of the informational content of classical data across physical models and clarify the limited or no advantage offered by quantum information processing for classical data compression in this setting.

Abstract

Due to Müller's theorem, the Kolmogorov complexity of a string was shown to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to using quantum mechanics to compress classical information. The quantitative amount of information in classical sources is invariant to the physical model used. These consequences make this theorem arguably the most important result in the intersection of algorithmic information theory and physics. The original proof is quite extensive. This paper contains two simple proofs of this theorem. This paper also contains new bounds for quantum Kolmogorov complexity with error.

Two Simple Proofs of Müller's Theorem

TL;DR

This work addresses whether quantum mechanics can improve the compression of classical information and affirms that it cannot, by proving Müller's theorem: . It provides two simple, self-contained proofs: a first proof leveraging a universal lower computable semi-density operator to relate , Hbvl, and algorithmic probability, and a second, self-contained construction that derives the classical complexity bound directly from the quantum complexity via finite-precision witnesses and an extraction procedure. The paper also introduces new bounds on quantum Kolmogorov complexity with error and rigorously formalizes the indeterminate-length quantum state framework with a concrete approximation mechanism for universal quantum Turing machines. Together, these results establish the invariance of the informational content of classical data across physical models and clarify the limited or no advantage offered by quantum information processing for classical data compression in this setting.

Abstract

Due to Müller's theorem, the Kolmogorov complexity of a string was shown to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to using quantum mechanics to compress classical information. The quantitative amount of information in classical sources is invariant to the physical model used. These consequences make this theorem arguably the most important result in the intersection of algorithmic information theory and physics. The original proof is quite extensive. This paper contains two simple proofs of this theorem. This paper also contains new bounds for quantum Kolmogorov complexity with error.
Paper Structure (15 sections, 11 theorems, 18 equations)

This paper contains 15 sections, 11 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. BvL Complexity
  4. An Elementary Approximation of $\mathfrak{U}_0$
  5. Proof.
  6. First Proof
  7. Proof.
  8. Proof.
  9. Proof.
  10. Proof.
  11. Proof.
  12. Second Proof
  13. Proof.
  14. Proof.
  15. The Multiverse

Key Result

Theorem 1

There is quantum Turing machine $\mathfrak{U}$ such that for every QTM $M$ and mixed state $\sigma$ for which $M(\sigma)$ is defined, there is mixed state $\sigma'$ such that for every $\delta\in\mathbb{Q}_{>0}$ where $\|\sigma'\|<^+ \|\sigma\|+{\mathbf K}(M,\delta$.

Theorems & Definitions (17)

  • Definition 1: Indeterminate Length Quantum States
  • Definition 2: Trace Distance and Fidelityt of Quantum States
  • Theorem 1: Muller08
  • Definition 3
  • Theorem 2: Muller08
  • Definition 4: BvL Complexity
  • Remark 1
  • Remark 2
  • Theorem 3: Muller07Muller08
  • Lemma 1
  • ...and 7 more