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A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

Marco Tomamichel, Masahito Hayashi

arXiv:1208.1478v3quant-ph

TL;DR

The paper asks how quantum data compression with side information and randomness extraction can be characterized accurately beyond first-order asymptotics. It uses one-shot entropies and relates them to information-spectrum quantities, obtaining tight second-order i.i.d. characterizations and finite-blocklength bounds. The resulting hierarchy distinguishes operationally precise but computationally difficult quantities from spectrum-based quantities that are easier to evaluate asymptotically.

  • Problem

    Finite-blocklength quantum tasks require characterizations beyond earlier results that established only first-order convergence.

  • Method

    The paper characterizes both tasks with one-shot entropies and relates those quantities to quantum and classical information spectra.

  • Results

    Both tasks admit tight second-order i.i.d. asymptotics, while the associated Gaussian approximation is valid up to logarithmic terms.

  • Takeaways & Limitations

    One-shot entropies best describe operational quantities, whereas information-spectrum quantities are easier to calculate in the i.i.d. limit; the analysis also yields finite-blocklength bounds.

Abstract

from arXiv · show

We consider two fundamental tasks in quantum information theory, data compression with quantum side information as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. We show that these characterizations - in contrast to earlier results - enable us to derive tight second order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our techniques also naturally yields bounds on operational quantities for finite block lengths.

I. INTRODUCTION

The paper combines one-shot entropies with quantum and classical information-spectrum quantities to characterize compression and randomness extraction with quantum side information. This yields tight second-order i.i.d. asymptotics and numerically evaluable finite-blocklength bounds.

  • One-shot characterization: One-shot entropies provide general-source bounds for operational quantities, but are difficult to calculate for large systems.They can be evaluated numerically for small examples and asymptotically converge to conditional von Neumann entropy.
  • Information-spectrum hierarchy: The paper relates one-shot entropies to information-spectrum quantities that are easier to approximate in the i.i.d. setting.The classical information spectrum has a natural second-order expansion through the central limit theorem, whereas quantum extensions can be difficult because of non-commutativity.
  • Second-order asymptotics: Both direct and converse operational bounds converge to the same second-order expression for i.i.d. sources.This is reported as the first second-order expansion of an operational quantity involving quantum resources.
  • Finite block length: The resulting Gaussian approximation is valid up to logarithmic terms and is easy to evaluate for arbitrary block lengths and error parameters.The approximation mostly yields good estimates of the finite-blocklength quantities of interest.
  • Finite block length: The analysis gives numerically evaluable direct and converse bounds for finite block lengths.For randomness extraction with ε = 10^-6, the example has first-order rate H(X|B) ≈ 0.714, while the second-order term causes a 10% rate drop near n ≈ 1.8 · 10^4.

3) Tight One-Shot Characterization:

The paper bounds operational tasks using one-shot entropies, then relates these quantities to information-spectrum expressions and finite-blocklength asymptotics. These relations yield improved finite-n bounds and a hierarchy for quantum information tasks.

  • Tight One-Shot Characterization: The authors bound randomness extraction and source compression operational quantities using smooth min-entropy and conditional hypothesis-testing entropy.The smoothing parameter ε remains the operational error parameter, while η can be optimized.
  • Tight One-Shot Characterization: The resulting hierarchy ranges from operational quantities and one-shot entropies to quantum spectra, classical spectra, and second-order asymptotic expansions.Operational quantities give task-specific optimal performance, while higher classes provide increasingly macroscopic and calculable descriptions.
  • Tight One-Shot Characterization: One-shot entropies are computable through semidefinite optimization for small examples but become intractable at large i.i.d. block lengths.The optimization complexity scales exponentially with n.
  • Tight One-Shot Characterization: The one-shot quantities relate to quantum and classical information-spectrum quantities, with equivalence up to additive O(log n) terms in the i.i.d. setting.The classical spectrum is based on the corresponding Nussbaum-Szkoła distributions and supports evaluation of the second-order expansion.
  • Tight One-Shot Characterization: The analysis improves earlier finite-n convergence bounds whose √n second-order term was not tight.The improved bounds are reported as tight in the second-order regime.

B. The Smooth Entropy Framework

The smooth entropy framework defines distance-based quantum entropies and develops their operational and optimization properties. It also establishes data processing, isometric invariance, and structured optimal solutions for hypothesis-testing entropies.

  • The Smooth Entropy Framework: Smooth entropies optimize min- or max-entropy quantities over sub-normalized states within purified distance ε of the original state.The framework uses balls of states defined by purified-distance proximity.
  • The Smooth Entropy Framework: Purified distance supports Uhlmann extension and monotonicity under trace-nonincreasing completely positive maps.These properties allow nearby marginal states to be extended and processed while preserving the distance bound.
  • The Smooth Entropy Framework: Smooth min-entropy is invariant under isometries and monotone under classical functions applied to a classical register.The proof constructs extensions and uses pinching maps.
  • The Smooth Entropy Framework: Hypothesis-testing entropy satisfies data processing under a sub-unital map on A and a trace-preserving map on B.The proof maps feasible primal and dual solutions through the channels.
  • The Smooth Entropy Framework: When the state is preserved by suitable channels, optimal primal and dual SDP operators can inherit that structure.For states classical on X and Y, corresponding optimal operators can also be chosen classical on those registers.

III. ONE-SHOT CHARACTERIZATION OF RANDOMNESS EXTRACTION

Randomness extraction uses randomized seeded hashing to produce an output that is close to uniform and independent of quantum side information. Its maximal extractable length is characterized by smooth min-entropy through direct and converse bounds.

  • III. ONE-SHOT CHARACTERIZATION OF RANDOMNESS EXTRACTION: A randomized extraction protocol consists of a seed, an output set, seed probabilities, and seed-dependent hash functions from X to Z.The protocol applies the selected hash function to the classical register X.
  • III. ONE-SHOT CHARACTERIZATION OF RANDOMNESS EXTRACTION: The protocol output is modeled as a trace-preserving completely positive map acting on the source and seed registers.This map produces the final state τZBS.
  • III. ONE-SHOT CHARACTERIZATION OF RANDOMNESS EXTRACTION: Extractable randomness is measured by the largest output length for which Z is close to uniform and independent of B and the seed S.Security minimizes purified distance to a fully mixed output tensored with side-information and seed states.
  • III. ONE-SHOT CHARACTERIZATION OF RANDOMNESS EXTRACTION: The operational quantity ℓε is characterized by smooth min-entropy through direct and converse bounds for 0 < η ≤ ε < 1.The direct and converse parts are proved separately.

A. Proof of Converse

The section establishes operational characterizations for randomized source compression with quantum side information, using hypothesis testing entropy and separate direct and converse arguments.

  • Protocol definition: Randomized compression protocols use a seed, codebook, encoder functions, and decoder POVMs to recover X from the compressed register and quantum side information.The encoding applies a seed-dependent function to X, while decoding measures B using the seed and compressed message.
  • Operational quantity: The operational quantity mε is the minimum compression length required to achieve an error probability at most ε.The final protocol state is obtained by applying the decoding map to the encoded state.
  • Characterization: Theorem 9 characterizes mε using the hypothesis testing entropy for CQ states with 0 < η ≤ ε < 1.The proof separates the direct and converse inequalities.

A. Proof of Converse

The section develops links among one-shot entropies, quantum information-spectrum quantities, and classical Nussbaum–Szkoła distributions, while proving the needed entropy relations.

  • Converse strategy: The converse relies on monotonicity of smooth min-entropy under classical functions and adapts earlier arguments.The proof is formulated by contradiction for arbitrary protocols.
  • Direct construction: Two-universal hashing and pretty good measurements provide the operational direct construction for source compression.The encoder family bounds collisions, while the decoder uses measurements conditioned on the seed and message.
  • Quantum spectrum: The entropic quantum information spectrum is introduced to connect one-shot entropy quantities with information-spectrum analysis.The asymptotic analysis is reduced to relating one-shot entropies to this spectrum quantity.
  • Entropy relations: The resulting relations bound hypothesis-testing and smooth entropies through the classical information spectrum of Pρ,σ and Qρ,σ.The classical spectrum is connected to the commuting-state formulation and hypothesis testing.
  • Classical reduction: Nussbaum–Szkoła distributions preserve the first two moments of the log-likelihood ratio corresponding to log ρ − log σ.This moment correspondence supports reducing quantum asymptotic analysis to classical distributions.

B. Useful Inequalities for Relative Entropies

This section proves inequalities relating hypothesis-testing, smooth, and information-spectrum relative entropies through pinching and classical probability representations.

  • Proposition 13: Proposition 13 gives inequalities relating several relative entropies for 0 < ε < 1 and 0 < δ < ε.The bounds use ν(σ), the number of distinct eigenvalues of σ.
  • Pinching: Pinching Eσ(ρ) commutes with σ, enabling a common eigenbasis and classical treatment of the resulting operators.The pinching projects onto eigenspaces associated with σ’s distinct eigenvalues.
  • Classical representation: The information-spectrum relative entropy of the pinched state can be expressed through probability events involving logarithmic likelihood ratios.This uses the induced classical distributions and threshold events.
  • Proof ingredients: The proofs combine Markov’s inequality, monotonicity, joint convexity, and spectral decompositions to establish the entropy inequalities.The arguments also use the relation between pinching and projective measurements.

C. One-Shot Entropies and the Information Spectrum

The section replaces difficult one-shot quantities with information-spectrum quantities while controlling the resulting bounds through spectral parameters.

  • Spectral parameters: The parameter θ(σ) is controlled by the smaller of the distinct-eigenvalue count and a quantity derived from the eigenvalue range.The proof first establishes bounds using ν(σ), then replaces it with 2⌈λ(σ)⌉ and takes the minimum.
  • Theorem 14: Theorem 14 states bounds relating one-shot entropies to classical information-spectrum quantities for 0 < δ < min{ε, 1 − ε}.The theorem uses θ(σ), defined from the spectrum of σ, together with the Nussbaum–Szkoła distributions.
  • Proof structure: The proof partitions the smoothing parameter δ among intermediate inequalities and optimizes over the resulting choices.The partition satisfies δ1 + δ2 + δ3 = δ.
  • Resulting bounds: A representative bound contains the classical spectrum term, −log δ, log ν, and −log(1 − ε) as finite-block corrections.The relation is stated explicitly after combining earlier inequalities.
  • Refinement: Replacing ν(σ) by 2⌈λ(σ)⌉ extends the inequalities to a potentially smaller spectral correction parameter.The construction modifies σ so that the number of relevant eigenvalues is bounded by the chosen integer parameter.

VI. ASYMPTOTIC ANALYSIS

The analysis connects information-spectrum quantities to i.i.d. quantum states through Nussbaum-Szkoła distributions and derives asymptotic expansions using normal approximation and finite-n bounds.

  • Information-spectrum reduction: The Nussbaum-Szkoła distributions of i.i.d. states inherit an i.i.d. structure, reducing the relevant information-spectrum analysis to averages of independent likelihood-ratio variables.The variables are defined from log Pρ,σ − log Qρ,σ, and their average converges toward a normal distribution.
  • Normal approximation: The central limit theorem and Berry-Esseen theorem provide the normal approximation used to analyze the information spectrum and control its finite-n deviation.The argument assumes a finite third absolute centered moment for the likelihood-ratio variable.
  • Asymptotic expansion: For fixed ε and perturbations proportional to 1/√n, continuous differentiability of the Gaussian inverse yields the relevant large-n asymptotic expansion.The expansion is applied after combining the information-spectrum expression with the preceding bounds.
  • Finite-n control: The auxiliary spectral quantity θ(σ^n) grows at most linearly in n when λ(σ) is finite, allowing δ = 1/√n in the finite-n estimates.This choice supports the subsequent asymptotic control of the bounds.

B. Asymptotic Behavior of Operational Quantities

The paper derives i.i.d. asymptotic characterizations for source compression and randomness extraction, then extends the hierarchy to information-spectrum and operational quantities under spectral conditions.

  • Operational tasks: Source compression with quantum side information and randomness extraction are analyzed from one-shot characterizations of their operational quantities.The source-compression quantity is mε, while the extraction quantity is ℓε.
  • Asymptotic expansions: Corollaries 15 and 16 give i.i.d. asymptotic expansions for both tasks for every CQ state and fixed 0 < ε < 1.The expansions are obtained by combining one-shot bounds with the information-spectrum analysis and choosing auxiliary parameters of order 1/√n.
  • Randomness extraction: The extraction analysis requires a separate treatment because optimization over σB prevents directly bounding θ(σB) for the optimal state.The paper instead uses an additional relation together with Theorem 9 to derive the needed bounds.
  • Finite block lengths: Theorem 17 supplies computable finite-block-length upper and lower bounds, with an auxiliary parameter ξ that can be optimized numerically.Choosing ξ, ξ0, and ξ1 proportional to 1/√n recovers the asymptotic statement.
  • Information-spectrum hierarchy: Relative-entropy information-spectrum quantities provide entropic versions of the quantum information spectrum through the paper’s asymptotic limit relations.The same hierarchy connects operational quantities, one-shot entropies, and information-spectrum quantities.
  • General sequence results: The relations between one-shot and information-spectrum quantities extend to hypothesis testing and, under eigenvalue-growth conditions, to Nussbaum-Szkoła classical distributions.The conditions require the number of distinct eigenvalues or the logarithm of the minimum eigenvalue to grow at most polynomially in n.

IX. CONCLUSION AND DISCUSSION

The conclusion argues that task-specific one-shot entropies are needed for tight second-order analysis and that these quantities also clarify asymptotic information-spectrum behavior.

  • Main result: The paper recovers tight second-order asymptotics for source compression and randomness extraction, improving earlier characterizations limited to first-order convergence.The result applies to both tasks using quantum side information.
  • Second-order precision: First-order tightness does not determine second-order tightness because the one-shot quantities behave differently beyond the leading rate.The first-order analysis is independent of the error or security parameter ε.
  • Task dependence: The hypothesis-testing entropy gives the tight one-shot characterization for source compression, whereas smooth min-entropy serves that role for randomness extraction under purified-distance secrecy.The task-dependent choice indicates that no single one-shot entropy is expected to characterize every relevant task correctly at second order.
  • Information-spectrum consequence: The asymptotic information spectrum, including direct and converse parts, can be expressed as an appropriate limit of the corresponding one-shot quantity.This connects detailed one-shot analysis with asymptotic information-spectrum behavior.

APPENDIX A EXAMPLE OF FINITE BLOCK LENGTH ANALYSIS: EAVESDROPPING ON PAULI CHANNEL

The appendix demonstrates finite-block-length randomness-extraction analysis for a Pauli-channel example by reducing the relevant quantities to binomial information-spectrum expressions.

  • Bounding procedure: The extractable randomness ℓε(X^n|B^n) is bounded using the classical information spectrum with auxiliary parameters ξ1 and ξ2.The resulting direct and converse bounds are evaluated through the relations developed earlier.
  • Classical reduction: The likelihood-ratio random variables can be rescaled into Bernoulli trials, so the information-spectrum calculation reduces to a binomial cumulative distribution function.The remaining optimization is evaluated numerically.
  • Asymptotic framework: The appendix places source compression and randomness extraction within the information-spectrum framework for general sequences of information sources.It defines corresponding asymptotic operational quantities and asymptotic quantum conditional entropies.
  • Asymptotic relations: The resulting asymptotic quantities are related to smooth conditional min-entropy and conditional information-spectrum expressions under polynomial eigenvalue-growth conditions.In the commutative case, the equations hold without those eigenvalue conditions.
  • Scope: The information-spectrum equations remain valid for 0 ≤ ε < 1, and the eigenvalue assumptions can be removed in the commutative case.These scope statements delimit the generality of the appendix’s asymptotic relations.
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