An intuitive proof of the data processing inequality

Normand J. Beaudry; Renato Renner

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An intuitive proof of the data processing inequality

Normand J. Beaudry, Renato Renner

arXiv:1107.0740v2Jul 4, 2011quant-ph

TL;DR

The paper addresses how to prove the data processing inequality for quantum information measures in an intuitive and self-contained way. It proves the inequality for smooth min-entropy, specializes that result to von Neumann entropy using the QAEP, and supplies a shorter QAEP proof. The resulting construction gives a self-contained proof of the von Neumann entropy DPI.

Problem
The paper seeks an intuitive, self-contained proof of the data processing inequality for von Neumann entropy.
Method
The proof establishes DPI for smooth min-entropy and specializes it to von Neumann entropy through the quantum asymptotic equipartition property.
Results
The smooth min-entropy DPI directly reduces to the DPI for von Neumann entropy through the QAEP.
Takeaways & Limitations
The paper provides a new perspective that decomposes the von Neumann entropy DPI proof into a fundamental smooth min-entropy result followed by specialization.
Takeaways & Limitations
The von Neumann entropy DPI proof applies only to finite-dimensional systems, while the smooth min-entropy proof also applies to infinite-dimensional systems.

Abstract

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The data processing inequality (DPI) is a fundamental feature of information theory. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. When the smooth min-entropy is used as the relevant information measure, then the DPI follows immediately from the definition of the entropy. The DPI for the von Neumann entropy is then obtained by specializing the DPI for the smooth min-entropy by using the quantum asymptotic equipartition property (QAEP). We provide a new, simplified proof of the QAEP and therefore obtain a self-contained proof of the DPI for the von Neumann entropy.

1 Introduction

The paper develops an intuitive, self-contained route to the data processing inequality (DPI), starting from smooth min-entropy and specializing to von Neumann entropy through the QAEP.

The DPI states that local data processing cannot increase the information content of a quantum system.
Conditional von Neumann entropy is a widely used entropy measure for which the DPI holds and is equivalent to strong subadditivity.
The paper first proves the DPI for smooth min-entropy using the partial trace and then specializes it to von Neumann entropy through the QAEP.
The paper presents a shorter proof of the QAEP to obtain a self-contained proof of the von Neumann entropy DPI.

2 Smooth Min-Entropy

Smooth min-entropy is introduced as a quantum-information measure with operational relevance in one-shot settings, and the QAEP connects it to von Neumann entropy.

Smooth min-entropy is relevant for measuring quantum information and characterizes one-shot tasks such as data compression.
The one-shot setting makes no assumptions that relevant states have product form.
The QAEP expresses von Neumann entropy as an averaged smooth entropy under additional assumptions.
Smooth min-entropy is defined using nearby subnormalized states within an ε-ball and a normalized auxiliary state on B.

3 Data processing inequality

The paper proves the data processing inequality first for smooth min-entropy, then specializes it to von Neumann entropy through the QAEP. The resulting proof applies the partial trace argument and gives a self-contained finite-dimensional von Neumann entropy DPI.

3.2 Specialized Data Processing Inequality: The von Neumann entropy proof is restricted to finite-dimensional systems, whereas the smooth min-entropy proof also covers infinite dimensions.This is an explicit scope distinction between the two theorem proofs.
3.1 General Data Processing Inequality: The smooth min-entropy DPI follows from applying a partial trace to the defining operator inequality and preserving the smoothing condition.Tracing out system C yields a valid candidate for the corresponding smoothed optimization on AB.
3.1 General Data Processing Inequality: The smooth min-entropy DPI applies to the general tri-partite state setting and provides the starting point for the von Neumann entropy proof.The theorem is presented for states on HABC.
3.2 Specialized Data Processing Inequality: The QAEP specializes the smooth min-entropy DPI to the von Neumann entropy by considering many i.i.d. copies of a state.The specialization uses the limit of many identical copies, called the QAEP.
3.2 Specialized Data Processing Inequality: The resulting von Neumann entropy DPI applies to any CPTP map on system BC after representing the map as a unitary followed by a partial trace.Entropy invariance under unitaries reduces the general operation to the partial-trace case.
3.2 Specialized Data Processing Inequality: The paper supplies a shorter alternative proof of the QAEP to make the von Neumann entropy DPI self-contained.The proof omits analysis of the specialization's convergence rate.

4 Quantum Asymptotic Equipartition Property

The QAEP connects conditional smooth min-entropy of many i.i.d. copies with conditional von Neumann entropy. The paper proves this by establishing matching asymptotic upper and lower bounds using smooth-entropy techniques.

4 Quantum Asymptotic Equipartition Property: The lower bound uses a chain rule to reduce conditional smooth min-entropy to a difference of non-conditional smooth entropies.Non-conditional i.i.d. limits then supply the needed asymptotic behavior.
4 Quantum Asymptotic Equipartition Property: The upper bound relates smooth min-entropy to the von Neumann entropy of a nearby state and uses continuity of von Neumann entropy.The argument is applied to ρ⊗n before normalization and taking limits.
4 Quantum Asymptotic Equipartition Property: The QAEP is proved by upper and lower bounding the asymptotic conditional smooth min-entropy of i.i.d. states.The limits take ϵ → 0 and n → ∞.
4 Quantum Asymptotic Equipartition Property: The proof requires non-conditional smooth min-entropy and smooth 0th-order Rényi entropy as intermediate quantities.Their definitions use optimization over states within a purified-distance ball.
4 Quantum Asymptotic Equipartition Property: For trivial conditioning and ϵ = 0, conditional smooth min-entropy reduces to ordinary min-entropy.This reduction links the conditional construction to the non-conditional quantities used in the proof.

5 General Properties of Smooth Entropies

The paper develops entropy and distance bounds needed for the QAEP, including smooth Rényi-entropy estimates, purified-distance constructions, and asymptotic links to von Neumann entropy.

5 General Properties of Smooth Entropies: The QAEP proof bounds smooth min-entropy and smooth 0th-order Rényi entropy to control the limits ϵ → 0 and n → ∞.These bounds rely on basic properties of von Neumann entropy and distance measures.
5 General Properties of Smooth Entropies: The conditional smooth-entropy bounds use purification, support projectors, triangle inequalities, and monotonicity of purified distance under completely positive trace-nonincreasing maps.These steps produce nearby states suitable for the conditional optimization.
5 General Properties of Smooth Entropies: The non-conditional QAEP is obtained from bounds relating smooth entropies to Rényi entropies and then taking the i.i.d. limit.The paper presents this as an alternative proof to typical-projector or law-of-large-numbers arguments.
5 General Properties of Smooth Entropies: A nearby state is constructed by retaining eigenvectors of ρ while truncating eigenvalues above a selected threshold.The construction controls fidelity and purified distance while bounding the min-entropy.
5 General Properties of Smooth Entropies: The Rényi-entropy argument reaches von Neumann entropy by taking the order parameter α to 1 from above.The paper invokes the limit limα→1 Hα(A)ρ = H(A)ρ.
5 General Properties of Smooth Entropies: The conditional von Neumann entropy is expressed using quantum relative entropy and the optimizing smooth-entropy parameter.The proof begins from the conditional entropy definition for subnormalized states.

Appendix A: Known Distance Properties

The appendix records distance properties used in the QAEP proof, especially monotonicity of purified distance and a spectral relation for fidelity.

Appendix A: Known Distance Properties: Purified distance does not increase under completely positive trace-nonincreasing maps.This property supports applying partial traces and related maps while preserving smoothing neighborhoods.
Appendix A: Known Distance Properties: The appendix relates purified distance to fidelity for states whose eigenvalues are arranged in non-increasing order.The proof establishes the needed fidelity comparison under a spectral construction.

Appendix B: Known Entropic Properties

The appendix proves the conditional entropy limit for an almost i.i.d. state by first treating trivial B and then extending the argument to non-trivial B. The proof constructs an auxiliary state and controls entropy changes using purified and trace-distance bounds.

Lemma B.1: Lemma B.1 analyzes the conditional von Neumann entropy of σn close to an i.i.d. state ρ⊗n.The lemma assumes σn ∈ Bε(ρ⊗n).
Distance bounds: The closeness of σn to ρ⊗n implies closeness of their B marginals, while purified distance is bounded below by trace distance.These distance relations support the entropy comparison used in the proof.
Trivial B: The proof first reduces the argument to the case where system B is trivial, so conditional entropy becomes ordinary entropy.For trivial B, H(An|Bn)σn = H(An)σn and H(A|B)ρ = H(A)ρ.
Auxiliary-state construction: An auxiliary state σ̃n is introduced on an enlarged space to compare the entropy of σn with the trivial-B case.The construction uses a one-dimensional auxiliary space H1.
Entropy comparison: The entropy comparison includes the correction η(1 − Trσn), with η(x) := −x log x and d = dim(HA).The correction is bounded by 1/2 when 0 ≤ 1 − TrσnA ≤ 1.
Non-trivial B: For non-trivial B, the proof combines the intermediate entropy bound with the stated relation and the definition of conditional von Neumann entropy.This completes the lemma.