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Asymptotic Equipartition Theorems in von Neumann algebras

Omar Fawzi, Li Gao, Mizanur Rahaman

TL;DR

The paper extends the AEP framework to general von Neumann algebras, proving a second-order AEP for the smooth max-relative entropy of i.i.d. states and an analogous result for quantum channels under suitable assumptions. It develops a robust chain rule for sandwiched Rényi divergences across channels, introduces an amortized divergence tool, and demonstrates a sequential (entropic) accumulation bound for compositions of channels. A central technical advance is the Haagerup reduction, enabling Type III algebras to be treated via semifinite approximations, which unifies finite-dimensional, infinite-dimensional, classical-quantum, and quantum-field-theoretic settings. The results have broad implications for quantum hypothesis testing, channel discrimination, and information-processing tasks in the algebraic setting, including potential extensions to quantum field theory and quantum gravity models modeled by von Neumann algebras.

Abstract

The Asymptotic Equipartition Property (AEP) in information theory establishes that independent and identically distributed (i.i.d.) states behave in a way that is similar to uniform states. In particular, with appropriate smoothing, for such states both the min and the max relative entropy asymptotically coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max relative entropy can be upper bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.

Asymptotic Equipartition Theorems in von Neumann algebras

TL;DR

The paper extends the AEP framework to general von Neumann algebras, proving a second-order AEP for the smooth max-relative entropy of i.i.d. states and an analogous result for quantum channels under suitable assumptions. It develops a robust chain rule for sandwiched Rényi divergences across channels, introduces an amortized divergence tool, and demonstrates a sequential (entropic) accumulation bound for compositions of channels. A central technical advance is the Haagerup reduction, enabling Type III algebras to be treated via semifinite approximations, which unifies finite-dimensional, infinite-dimensional, classical-quantum, and quantum-field-theoretic settings. The results have broad implications for quantum hypothesis testing, channel discrimination, and information-processing tasks in the algebraic setting, including potential extensions to quantum field theory and quantum gravity models modeled by von Neumann algebras.

Abstract

The Asymptotic Equipartition Property (AEP) in information theory establishes that independent and identically distributed (i.i.d.) states behave in a way that is similar to uniform states. In particular, with appropriate smoothing, for such states both the min and the max relative entropy asymptotically coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max relative entropy can be upper bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.
Paper Structure (21 sections, 27 theorems, 217 equations)

This paper contains 21 sections, 27 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Preliminary on von Neumann algebras
  4. Examples
  5. Classical systems:
  6. Quantum systems:
  7. Composite systems and tensor product:
  8. Fermionic systems:
  9. Type III algebras
  10. Relative Entropies
  11. Relative entropies on general von Neumann algebras
  12. Quantum channels
  13. Haagerup Reduction
  14. $D_{\max}$ AEP for states
  15. $D_{\max}$ AEP for channels
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let ${\mathcal{M}}$ be a von Neumann algebra, $\rho$ be a normal state on ${\mathcal{M}}$ and $\sigma$ a normal positive linear functional. Assume that $D(\rho \| \sigma) < \infty, V(\rho \| \sigma) < \infty$, and $T(\rho \| \sigma) < \infty$ are all finite. Then for any $\varepsilon \in (0,1)$ and where the $O(\cdot)$ hides constants that only depend on $\rho$ and $\sigma$ and $\varepsilon$.

Theorems & Definitions (50)

  • Theorem 1.1: $D_{\max}$ AEP for states
  • Theorem 1.2: $D_{\max}$ AEP for channels
  • Theorem 1.3: Chain rule for quantum channels
  • Theorem 1.4: Relative EAT
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1: AEP for states
  • ...and 40 more