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A Framework for Non-Asymptotic Quantum Information Theory

Marco Tomamichel

arXiv:1203.2142v2quant-phmath-ph

TL;DR

Non-asymptotic quantum information and cryptography need finite-size entropic tools beyond asymptotic limits. This thesis develops smooth entropies using purified distance, establishes their key properties and asymptotic convergence, and applies them to coding, randomness extraction, and quantum key distribution security.

  • Problem

    Finite-size quantum information and cryptography require an operational entropy framework that captures quantum side information and supports rigorous non-asymptotic analysis.

  • Method

    The thesis defines smooth min- and max-entropies using purified distance, then develops their properties, duality, uncertainty relations, and one-shot applications.

  • Results

    Smooth entropies converge to von Neumann entropy asymptotically and support strong converse source coding, randomness extraction, and information-theoretic quantum key distribution security.

  • Takeaways & Limitations

    The framework provides a reference for analyzing finite-blocklength quantum information tasks and quantum cryptography with smooth entropic tools.

  • Takeaways & Limitations

    Existing techniques establish the asymptotic limit but do not provide good finite-n convergence bounds, which depend on n, ε, and the quantum state.

Abstract

from arXiv · show

This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical Rényi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.

Abbreviations and Notation · Introduction

The section establishes notation and introduces quantum mechanics and information theory without assuming prior knowledge. It motivates non-asymptotic quantum information theory through elementary information-processing tasks and the growing relevance of quantum physics to technology and cryptography.

  • Abbreviations and Notation: Table 1 defines the abbreviations used throughout the thesis.
  • Abbreviations and Notation: Table 2 specifies notational conventions for quantum mechanics.
  • Abbreviations and Notation: Table 3 specifies notational conventions for mathematical expressions.
  • Introduction: The introduction begins with a philosophical explanation of quantum mechanics that assumes no prior knowledge.It justifies the thesis’s information-theoretic approach and previews relevant aspects of information and quantum information theory.
  • Introduction: The introduction presents non-asymptotic quantum information theory as a way to characterize elementary information-processing tasks.It also indicates that the chapter will discuss the importance of this approach before continuing beyond the supplied passage.
  • Introduction: The introduction frames quantum mechanics as experimentally verified, physically counterintuitive, and foundational to quantum cryptography.It restricts discussion to non-relativistic quantum mechanics and treats the physical system’s state as the central object of interest.
  • Introduction: Non-relativistic quantum mechanics underpins today’s semiconductor industry and is becoming increasingly relevant as information-processing devices are miniaturized.The passage emphasizes that understanding quantum physics is pivotal for engineering the next generation of information-processing devices.

1. INTRODUCTION

The thesis consolidates the smooth entropy framework for non-asymptotic quantum information theory, introducing purified distance, entropic asymptotic equipartition, and uncertainty relations. It establishes smooth entropies’ operational properties, including duality and convergence to von Neumann entropy in the i.i.d. limit.

  • Entropy properties: Smooth min- and max-entropies satisfy data-processing inequalities and chain rules, and converge to the von Neumann entropy in the i.i.d. limit.This convergence establishes the entropic form of the asymptotic equipartition property and connects one-shot resource usage with its asymptotic characterization.
  • Entropy properties: The smooth entropies obey a duality relation for tripartite quantum states, with equality when the joint state is pure.The relation connects min- and max-entropy and the tasks characterized by them.
  • Thesis goals: The thesis consolidates the smooth entropy framework for non-asymptotic information theory and introduces entropic asymptotic equipartition and uncertainty relations.Its primary focus is the properties of smooth entropies as a reference for non-asymptotic quantum information theory.
  • Core framework: The purified distance is introduced as a novel measure of distance between potentially incomplete quantum states.This distance supports the framework’s treatment of quantum states that may be unnormalized or incomplete.
  • Uncertainty relations: Entropic uncertainty relations bound uncertainty for two incompatible measurements using smooth and von Neumann entropies, enabling direct applications to quantum cryptography.These relations generalize the role of Heisenberg’s uncertainty principle in this framework.

Preliminaries

The preliminaries establish finite-dimensional Hilbert-space quantum mechanics, introduce the thesis’s notation, and provide mathematical tools needed for later derivations.

  • Structure: The preliminaries comprise foundations of finite-dimensional Hilbert-space quantum mechanics, notation summarized in Tables 2 and 3, and mathematical tools for deriving subsequent results.Readers familiar with linear algebra used in quantum information theory can skip the first two sections.
  • Sources: The section’s foundations draw primarily on Bhatia’s Matrix Analysis and Positive Definite Matrices, with additional support from Watrous’s lecture notes and Nielsen and Chuang’s textbook.These sources are presented as resources for the introductory material.
  • Hilbert spaces: A Hilbert space is defined as a finite-dimensional complex vector space equipped with an inner product.The dual space consists of linear functionals from H to C.

2. PRELIMINARIES

This section establishes the Hilbert-space and operator-theoretic preliminaries used throughout the thesis, including inner products, metrics, bases, direct sums, tensor products, and operator properties.

  • Inner products and metrics: An inner product induces both a norm and a metric, while metrics generally satisfy positive-definiteness, symmetry, and the triangle inequality.The induced metric is verified to fulfill these conditions.
  • Bases and representations: Orthonormal bases satisfy ⟨ei|ej⟩=δij, contain d elements for Hilbert dimension d, and provide unique d × 1 ket and 1 × d bra representations.The Kronecker delta is 1 when i = j and 0 otherwise.
  • Direct sums and tensor products: Direct sums combine Hilbert spaces with dim {H ⊕H′}=d+d′, whereas tensor products have bases formed from tensor products of basis elements and dimension d·d′.Both constructions extend the inner product to the combined space.
  • Operators: Operators are linear maps represented by d′ × d matrices, with adjoints, transposes, kernels, supports, ranks, images, projectors, and generalized inverses defined for later use.The generalized inverse acts as the inverse on an operator’s support.
  • Isometries and unitaries: Unitary operators obey U†U=1 and U−1=U†, while partial isometries obey V†V=ΠV and have generalized inverse V†.An isometry is a partial isometry with full support on H.

The Purified Distance

The purified distance is introduced as a metric on sub-normalized quantum states, designed to support smooth entropies with natural invariance and data-processing properties. It addresses limitations of existing ε-ball definitions, especially dependence on the Hilbert-space representation.

  • Motivation: The purified distance was first proposed as a metric on the space of sub-normalized quantum states.Its usefulness is established through applications to smooth min- and max-entropies.
  • Motivation: Smooth entropies optimize an underlying entropy over states that are ε-close to ρ, producing ε-smooth quantities.The optimization may be a maximization or minimization, depending on the entropy.
  • Requirements: Existing definitions of ε-balls do not simultaneously satisfy the two properties considered important for smooth entropies in the quantum regime.These properties motivate introducing a more suitable notion of closeness.
  • Requirements: The definition of ε-smooth entropies should be independent of the Hilbert spaces used to represent ρ.This requirement concerns the representation of the quantum state rather than its underlying physical content.
  • Requirements: Embedding ρ into a larger Hilbert space before smoothing should leave the ε-smooth entropies unchanged.Larger spaces can otherwise provide extra dimensions orthogonal to ρ’s support, increasing optimization flexibility for some ε-balls.

3. THE PURIFIED DISTANCE

This section introduces the purified distance, a fidelity-based metric on sub-normalized quantum states, to define ε-closeness while enabling robust extension and purification properties. It establishes that the metric is compatible with operational trace-distance bounds, quantum operations, and Uhlmann-style purification results.

  • Definition and motivation: The purified distance is a metric on sub-normalized states, and ρ ≈ε τ denotes that their purified distance is at most ε.It is introduced specifically to include sub-normalized states in ε-balls.
  • Metric properties: When at least one state is normalized, the purified distance can be expressed in terms of fidelity and upper-bounds the trace distance.This preserves the operational interpretation of trace distance as an upper bound on distinguishing advantage.
  • Metric properties: Trace non-increasing CPMs preserve ε-closeness: ρ ≈ε τ implies E[ρ] ≈ε E[τ].This is the purified distance’s monotonicity under trace non-increasing completely positive maps.
  • Purifications and extensions: Every purification of ρ has a purification of τ at exactly the same purified distance, and extensions can likewise preserve that distance.The extension result also applies when an extension of ρ is fixed in advance.

Min- and Max-Entropies

This section introduces quantum conditional min- and max-entropies, develops their expressions and properties, and motivates them as representatives of smoothed Rényi-entropy classes with operational interpretations. It also establishes duality, classical limits, guessing-probability interpretations, and continuity results.

  • Rényi-entropy representatives: Smooth Rényi entropies of order α > 1 are represented by the min-entropy Hmin(A)ρ = H∞(A)ρ, motivated by its guessing-probability interpretation.The min-entropy also has a simple quantum generalization.
  • Rényi-entropy representatives: Smooth Rényi entropies of order α < 1 are represented by the max-entropy Hmax(A)ρ = H1/2(A)ρ, whose choice is supported by duality of conditional min- and max-entropies.The surrounding discussion contrasts this choice with the order-0 Rényi entropy.
  • Expressions and formulations: The chapter’s main result is a collection of expressions for quantum conditional min- and max-entropies.These expressions include formulations as semi-definite programs and fidelity-based characterizations.
  • Classical limits and guessing: Classical conditional min-entropy equals the optimal probability of guessing X from Y, obtained by choosing the most probable x for each observed y.The observer’s optimal strategy is to guess the x with highest conditional probability given y.
  • Duality and operational structure: For a pure state, Hmin(A|B)ρ = −log ΦA|B(ρAB) and Hmax(A|B)ρ = log ΦA|C(ρAC), related by duality.The functional Φ’s properties provide the basis for further entropy bounds.
  • Properties: For normalized states, the min-entropy is no larger than the von Neumann entropy, which is no larger than the max-entropy, and both entropies are Lipschitz continuous in purified distance.Continuity of the max-entropy follows from min–max duality.

Smooth Entropies

Smooth entropies optimize min- and max-entropies over states close to the state under consideration, with purified distance proposed as the closeness metric. This choice supports key structural properties, including isometric invariance, duality, data processing inequalities, and chain rules.

  • Purified distance: Purified distance is proposed for defining the ε-ball because fidelity-based metrics permit extensions that are as close as their marginals by Uhlmann’s theorem.This property does not hold for the trace-distance metric used previously.
  • Properties: The resulting smooth entropies are invariant under isometries and satisfy a duality relation.These are among the useful properties enabled by measuring closeness with purified distance.
  • Properties: Smooth entropies also satisfy data processing inequalities and chain rules.These properties are established among various other properties of the smooth entropies.
  • Definition: Smooth entropies optimize the underlying min- and max-entropies over an ε-ball of states close to the state under consideration.The conditional smooth entropies for quantum states were first introduced by Renner.

5. SMOOTH ENTROPIES

The section defines smooth min- and max-entropies by optimizing over ε-close states using purified distance, and establishes their central structural properties. These include smooth duality, generalized data processing, and classical-register smoothing.

  • Duality: The smoothing framework extends the min- and max-entropy duality relation to ε-smooth entropies for pure tripartite states.This extension is identified as a central consequence of the purified-distance definition.
  • Data processing: Generalized data processing shows that conditioning on less side information can increase uncertainty by at most the permitted transformation-dependent amount.The result holds more generally for suitable trace-preserving completely positive maps applied to the conditioned system.
  • Classical smoothing: The smooth entropies retain a well-defined classical limit because the smoothing optimization may use valid classical states.The section also develops further properties for smooth entropies involving classical registers.
  • Definition and smoothing: Smooth min- and max-entropies optimize the corresponding unsmoothed entropies over states within an ε-ball around the original state.The purified-distance ball supplies the notion of ε-closeness used in the optimization.
  • Optimization properties: Normalized optimal smoothed states can be obtained after embedding the systems into sufficiently large Hilbert spaces.The embeddings address the fact that optimal states restricted to the original support are not necessarily normalized.
  • Classical smoothing: For classical registers, smoothing can be restricted to states classical on the same subsystems as the original state without loss of generality.This preserves the classical structure relevant to the smooth entropy optimization.

The Quantum Asymptotic Equipartition Property

The chapter presents the asymptotic equipartition property as the basis for using Shannon entropy in information-theoretic tasks and extends it fully to quantum outputs with quantum side information. In the classical i.i.d. setting, sufficiently long sequences lie with probability greater than 1 − ε in a typical set whose members satisfy an entropy-based probability bound.

  • Quantum generalization: The chapter proves a fully quantum generalization of the AEP in which both the experiment’s output and its side information are quantum.The result expands on previous work cited as [TCR09].
  • Classical AEP: The classical AEP states that long i.i.d. random-variable sequences are overwhelmingly likely to belong to a typical set of nearly equally likely outputs.For any 0 < ε < 1 and µ > 0, the probability of the typical set exceeds 1 − ε for sufficiently large n.
  • Classical AEP: The typical set A_n contains length-n sequences satisfying the entropy-based probability condition n log P_Xn(x^n) < H(X) + µ.The joint probability factors into the product of the probabilities of the independent events x_1, …, x_n.

6. THE QUANTUM ASYMPTOTIC EQUIPARTITION PROPERTY

The chapter establishes a fully quantum asymptotic equipartition property: smooth conditional min- and max-entropies converge to conditional von Neumann entropy for i.i.d. quantum states. Its finite-copy bounds scale with conditional entropies rather than system dimensions and support strong converse statements.

  • Fully quantum AEP: The fully quantum AEP applies when both A and B are general quantum systems.It provides a quantum generalization of the entropic classical AEP, avoiding reliance on conditional probabilities for quantum side information.
  • Finite-copy bounds: The convergence-rate term δ scales with Hmax(A|B) and Hmax(A|R), rather than H0(A).Here R purifies ρAB; the conditional entropies measure correlations and do not depend on subsystem Hilbert-space dimensions.
  • Fully quantum AEP: Smooth conditional min- and max-entropies converge to the von Neumann entropy in the asymptotic limit of many copies.The result holds for every 0 < ε < 1.
  • Extensions and consequences: Converse bounds for large smoothing parameters imply strong converse statements for information-theoretic tasks characterized by smooth entropies.The chapter notes that this extension is important for strong converse results, including applications discussed later.
  • Extensions and consequences: The chapter proves the AEP for relative entropies, with conditional entropies as special cases.The results extend earlier work to relative entropies and a more general class of operators, including settings with non-normalized smoothed states.

Uncertainty Relations for Smooth Entropies

This section establishes entropic uncertainty relations that lower-bound the uncertainty of two incompatible measurements given side information, using smooth min- and max-entropies and von Neumann entropy.

  • Entropic uncertainty relations: Entropic uncertainty relations lower-bound the uncertainty of outcomes from two incompatible measurements given side information through conditional entropies.The chapter proves several such relations using smooth min- and max-entropies, as well as von Neumann entropy, as uncertainty measures.
  • Motivation: The standard-deviation uncertainty relation can be trivial for eigenstates and depends on the premeasurement state, which may be unknown or adversarially prepared.The passage also criticizes standard deviation because it conflates uncertainty about measurement-outcome values with uncertainty about differing outcomes.

7. UNCERTAINTY RELATIONS FOR SMOOTH ENTROPIES

This section extends entropic uncertainty relations from classical side information to smooth min- and max-entropies with quantum side information. It develops generalized and effective-overlap bounds, connects them to von Neumann entropy through the asymptotic equipartition property, and establishes tight bipartite formulations.

  • Smooth-entropy uncertainty relations: The smooth-entropy uncertainty relation extends the von Neumann-entropy bound to incompatible measurements on a system with quantum side information.The lower bound increases when incompatible measurements are applied, generalizing the earlier tripartite relation.
  • Effective overlap: A tighter bound uses an effective overlap that depends on the marginal state before measurement as well as both POVMs.This state-dependent quantity improves on the state-independent overlap and also yields generalized relations for smooth min- and max-entropies.
  • Generalized uncertainty relation: The generalized uncertainty relation bounds uncertainty about two incompatible measurements conditioned on quantum side information and an additional projective measurement.The theorem provides a broad formulation from which the uncertainty relations in Section 7.4 follow as corollaries.
  • Same-state relations: The framework also relates smooth min- and max-entropies of the same post-measurement state, rather than only entropies associated with different measurements.This result gives another uncertainty relation within the smooth-entropy framework.
  • Asymptotic relations: The asymptotic equipartition property yields corresponding uncertainty relations for von Neumann entropy in the limit of many copies.The resulting state-dependent bound is H(X|B)ρ + H(Y|C)ρ ≥ log 1 c∗(ρA, X, Y).
  • Tightness: The bipartite uncertainty relation is tight because uncertainty can be produced in two separate steps without weakening the bound.For rank-1 projective measurements, one auxiliary system contains a copy of the outcome and the second entropy vanishes.

Applications

The chapter presents three applications of the smooth entropy framework. It shows how one-shot characterizations yield finite-blocklength bounds and strong converses, with source compression treated in detail.

  • Applications: Three example applications use results developed in Chapters 3–7 of the smooth entropy framework.The applications are source compression, randomness extraction, and a further application introduced in Section 8.3.
  • Source compression: One-shot source-compression characterizations yield direct and converse finite-blocklength bounds and strong converse statements.This illustrates how one-shot characterizations can determine resource-usage bounds for finite block lengths and the i.i.d. limit.
  • Source compression: For source compression with quantum side information, the thesis gives a strong converse for information reconciliation.The derivation uses purified-distance properties, data-processing inequalities, and the asymptotic equipartition property.
  • Source compression: The resulting bounds also cover classical side information and no side information, encompassing Shannon’s theorem and the Slepian–Wolf and quantum extensions.The source-compression results therefore apply to classical source-compression tasks as well.

8. APPLICATIONS

The applications demonstrate that smooth entropies characterize one-shot source compression, randomness extraction, and quantum-key-distribution security. They yield strong converses, near-optimal extraction bounds, and an intuitive security proof based on entropic uncertainty relations.

  • Source compression with quantum side information: One-shot information reconciliation asks for the minimum message length mε needed for Bob to reconstruct classical Z from quantum side information B with error at most ε.The protocols are non-interactive and one-way from Alice to Bob.
  • Source compression with quantum side information: Theorem 8.1 provides an improved converse bound, requiring every protocol with error at most ε to satisfy a smooth max-entropy lower bound on transmitted message length.The improvement addresses the weakness of the earlier converse when ε approaches 1.
  • Source compression with quantum side information: Asymptotically, the minimum communication rate converges to H(Z|B)ρ, and transmitting less than this Shannon limit causes the maximal success probability to drop exponentially in n.This exponential decrease is identified as the strong converse.
  • Randomness extraction: The extractable uniform and independent randomness is characterized by the smooth min-entropy, and the corresponding protocols are essentially optimal.The protocol families operate on any state with sufficiently high min-entropy, without using other state properties.
  • Quantum key distribution: Smooth-entropy uncertainty relations provide a concise and intuitive security proof for quantum key distribution.The proof uses quantum correlations to establish both secrecy and correctness, unlike the unrestricted classical case.
  • Quantum key distribution: If Alice’s and Bob’s classical correlations are sufficiently good, meaning the corresponding smooth max-entropies are small, the protocol safely extracts a secret key.Correctness is controlled by Bob’s estimation of Alice’s string, while secrecy follows from the smooth min-entropy bound against the eavesdropper.

Conclusions and Outlook

The thesis consolidates the smooth entropy framework for non-asymptotic quantum information theory and introduces additions including the entropic asymptotic equipartition property and uncertainty relations. It aims to serve as a reference for research in non-asymptotic quantum information theory and quantum cryptography.

  • Conclusions and Outlook: The thesis consolidates the smooth entropy framework and introduces the entropic asymptotic equipartition property and various uncertainty relations.These additions are presented as important extensions to the framework.
  • Conclusions and Outlook: Smooth entropies have become a standard tool for analyzing finite-key security in quantum key distribution, with entropic uncertainty relations simplifying some protocol analyses.The formalism also supports work on entropically secure encryption and randomness extraction.
  • Conclusions and Outlook: Decoupling is a fully quantum generalization of randomness extraction characterized by smooth entropies and applicable to direct bounds for many information-theoretic tasks.The thesis also shows decoupling is possible with approximate two-designs, suggesting efficient realization in nature.
  • Conclusions and Outlook: The smooth entropy formalism has been used to investigate various quantum channel capacities and converses.This places the framework within a broader range of non-asymptotic quantum information applications.

9. CONCLUSIONS AND OUTLOOK

The thesis improves the smooth entropy framework with finite-smoothing converse bounds, generalized uncertainty relations, and a complete set of chain rules. It also identifies open questions and opportunities for broader applications in quantum information theory.

  • Contributions and outlook: The improved entropic asymptotic equipartition property provides a converse bound for finite smoothing and may support strong converse statements for quantum information tasks, including channel capacities.The arguments also apply to classical theory, while comparisons with existing converse bounds remain open.
  • Open questions: Whether the effective overlap is always smaller than the overlap remains open for general POVMs.The conjecture was stated in Chapter 7, Eq. (7.15).
  • Contributions and outlook: The generalized entropic uncertainty relation expresses its lower bound using an effective overlap, which can be bounded through the maximal CHSH value attainable with the same measurement setup and a second party.The passage characterizes the CHSH value as a measure of non-locality in correlations produced by two parties.
  • Contributions and outlook: A complete set of chain rules for smooth entropies supplies an important missing link and expands the framework’s potential range of applicability.This addition is presented as a recent development in the thesis’s outlook.

Appendix A … B. PROPERTIES OF QUASI-ENTROPIES

Appendix A develops technical lemmas for tensor spaces, coherent classical states, and entropy relations, while Appendix B establishes structural and monotonicity properties of quasi-entropies and relative Rényi entropies.

  • A.1 Two Lemmas for Tensor Spaces: Trace-preserving CPMs leave the marginal state on an untouched Hilbert space unchanged, while trace-nonincreasing maps give the corresponding trace inequality.The result follows from an operator inequality and the fact that the marginal difference is positive with vanishing trace in the trace-preserving case.
  • A. VARIOUS LEMMAS: A rank bound derived from the Schmidt decomposition supplies an operator inequality used among the appendix’s general tensor-space lemmas.The proof reduces to rank-one operators and uses tr(Λ) ≤ dim H′.
  • A.2 Entropies of Coherent Classical States: For coherent classical states, measuring X′ provides candidate states that relate conditional min-entropies conditioned on X′B and on XX′B.The inequalities follow by selecting optimizing states and applying the trace-preserving measurement M_X′.
  • A.3 Selected Relations between Entropies: For pure states, a projector Π_AC and truncated state ˜ρ can be chosen within purified-distance ε while lower-bounding a relative min-entropy using H_min(A|B)_ρ.The construction uses a dual projector Π_B, cuts off large eigenvalues, and controls the purified distance through the projector choice.
  • A.3 Selected Relations between Entropies: The projector construction extends to smooth min-entropy by producing a state within B_{ε+2ε′}(ρ) through a filtered approximation and purified-distance triangle inequality.The proof combines an ε′-close purification, projection monotonicity, and a filtering operator F_B.
  • B.1 Properties of Quasi-Entropies: All f-quasi-entropies are covariant under isometries, with zero eigenvalues introduced by the isometry contributing no terms under the stated finite-f(0) condition.The isometry preserves eigenvalues and scalar products, while the added orthogonal zero-eigenspace does not contribute.
  • B. PROPERTIES OF QUASI-ENTROPIES: For operator-concave f, quasi-entropies are monotone under trace-preserving CPMs, extending partial-trace monotonicity through isometric dilations and continuity for noninvertible arguments.The extension is highlighted as necessary because an isometry can map an invertible B to a generally noninvertible E(B).
  • B.2 Properties of the Rényi Entropy: Relative Rényi entropies decrease with α and satisfy data processing under trace-preserving CPMs, while their conditional versions admit a duality relation for pure tripartite states.The monotonicity statement applies for α ≥ β ≥ 0, and the CPM monotonicity lemma covers α ∈ [0, 2].
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