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Duality Between Smooth Min- and Max-Entropies

Marco Tomamichel, Roger Colbeck, Renato Renner

arXiv:0907.5238v2quant-ph

TL;DR

Operational tasks beyond the i.i.d. setting require smooth min- and max-entropies, whose values can differ generally despite converging to von Neumann entropy asymptotically. The paper extends min–max duality to smooth entropies by defining closeness through purified distance, showing that smooth conditional min-entropy equals the negative smooth conditional max-entropy on a purifying system. This connects randomness extraction, compression, information reconciliation, and cryptographic security analyses.

  • Problem

    Smooth min- and max-entropies characterize different operational quantities beyond the i.i.d. setting, but their general relationship needed to be established.

  • Method

    The paper defines smoothing using purified distance on sub-normalized states and uses the resulting ε-balls to construct smooth conditional min- and max-entropies.

  • Results

    For a purification, the smooth conditional min-entropy of A given B equals the negative smooth conditional max-entropy of A given C.

  • Takeaways & Limitations

    The duality relates randomness extraction and information reconciliation and lets key-distribution parties bound smooth min-entropy using smooth max-entropy without the eavesdropper’s system.

Abstract

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In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this work, a recently discovered duality relation between (non-smooth) min- and max-entropies is extended to the smooth case. More precisely, it is shown that the smooth min-entropy of a system A conditioned on a system B equals the negative of the smooth max-entropy of A conditioned on a purifying system C. This result immediately implies that certain operational quantities (such as the amount of compression and the amount of randomness that can be extracted from given data) are related. Such relations may, for example, have applications in cryptographic security proofs.

I. INTRODUCTION

Smooth min- and max-entropies extend entropy-based characterization beyond the i.i.d. setting, supporting operational tasks with small error tolerances. The paper extends min–max duality to the smooth case using purified distance, linking randomness extraction, information reconciliation, and cryptographic applications.

  • I. INTRODUCTION: Smooth min-entropy characterizes randomness extraction, while smooth max-entropy characterizes the communication required for information reconciliation.The former measures extractable uniform-string length; the latter measures communication needed to create a perfectly correlated string.
  • I. INTRODUCTION: Smooth entropies optimize non-smooth min- and max-entropies over states within a smoothing-parameter-dependent distance from the original state.They are often appropriate when a small error tolerance is allowed, whereas non-smooth entropies describe the zero-error case.
  • I. INTRODUCTION: For pure ρABC, conditional min-entropy of A given B equals the negative conditional max-entropy of A given C.The paper extends this established non-smooth duality to smooth min- and max-entropies using purified distance.
  • I. INTRODUCTION: Purified distance forms a metric on sub-normalized states and yields smooth entropies with properties including local-isometry invariance and data processing.The data processing inequality states that quantum operations cannot decrease entropy.
  • I. INTRODUCTION: The duality connects randomness extraction and information reconciliation through a common entropy characterization.In quantum key distribution, Alice and Bob can bound smooth min-entropy using smooth max-entropy conditioned on their own system, without access to the eavesdropper’s system.
  • I. INTRODUCTION: The paper’s results also imply a corresponding duality between inf-spectral and sup-spectral entropy rates beyond the i.i.d. regime.These spectral quantities are related to smooth min-entropy and smooth max-entropy, respectively.

II. METRICS ON THE SET OF SUB-NORMALIZED STATES

This section develops fidelity- and trace-distance tools for sub-normalized quantum states. It introduces purified distance as a metric with monotonicity and purification-preserving properties needed for later smooth-entropy constructions.

  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: The framework distinguishes linear operators, positive semidefinite operators, normalized states, and sub-normalized states on finite-dimensional Hilbert spaces.Sub-normalized states are positive semidefinite operators with trace strictly between zero and one inclusive.
  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: Generalized trace distance extends trace-distance constructions to positive semidefinite operators, while fidelity is extended to sub-normalized states through larger-space extensions.For normalized states, the generalized trace distance recovers the usual trace distance and retains its state-distinguishing interpretation.
  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: Purified distance is defined from the generalized fidelity and reduces to the minimum trace distance between purifications for normalized states.The fidelity-based construction supports comparing sub-normalized states through normalized extensions.
  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: The purified distance is a metric on the set of sub-normalized states.Its metric properties include identity of indiscernibles, symmetry, and the triangle inequality.
  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: Purified distance does not increase under trace non-increasing quantum operations, including projections.This follows from fidelity monotonicity for trace-preserving maps together with the treatment of projections and extensions.
  • II. METRICS ON THE SET OF SUB-NORMALIZED STATES: Any purification of one state can be matched by a purification of another with the same purified distance, and the same preservation holds for suitable extensions.These properties provide distance-preserving purifications and extensions for later arguments.

III. THE ε-NEIGHBORHOOD INDUCED BY P

The paper defines ε-neighborhoods using purified distance to support smooth-entropy constructions on sub-normalized states. These neighborhoods preserve representation independence and provide compatible pure-state purifications, enabling the later smooth min–max duality.

  • III. THE ε-NEIGHBORHOOD INDUCED BY P: Smooth min-entropy is defined by maximizing min-entropy over an ε-ball of states close to the original state.The ε-ball is determined by a smoothing parameter and a fidelity-based distance.
  • III. THE ε-NEIGHBORHOOD INDUCED BY P: The ε-ball allows sub-normalized states so embedding a density operator into a larger Hilbert space leaves smooth entropies unchanged.This addresses representation dependence that can arise when neighborhoods contain only normalized states.
  • III. THE ε-NEIGHBORHOOD INDUCED BY P: A fidelity-based metric is required to define an ε-ball of pure states containing purifications of all states in the ε-ball.This purification property is used to establish duality between smooth min- and max-entropies.
  • III. THE ε-NEIGHBORHOOD INDUCED BY P: The construction includes ε-balls of pure states and establishes compactness, convexity, monotonic growth with ε, and invariance under isometries.Additional properties describe behavior under partial trace and purification.
  • III. THE ε-NEIGHBORHOOD INDUCED BY P: The ε-balls are invariant under isometries and behave compatibly with partial trace and purification because purified distance is monotone under quantum operations.These properties are later used for smooth-entropy invariance and duality arguments.
  • III. THE ε-NEIGHBORHOOD INDUCED BY P: Trace-distance-based ε-balls do not provide the purification property needed for the smooth-entropy duality proof.The paper specifically identifies this as a reason for using a fidelity-based metric.

IV. SMOOTH CONDITIONAL MIN- AND MAX-ENTROPIES

This section defines smooth conditional min- and max-entropies by optimizing non-smooth quantities over ε-balls, establishes representation invariance, and proves their duality. The smooth max-entropy is defined through purification and is equivalently an ε-ball optimization of the non-smooth max-entropy.

  • Properties: The resulting smooth min- and max-entropies are invariant under local isometries and satisfy a duality relation.These properties are consequences of the ε-ball definition and the purification-based construction.
  • Smooth min-entropy: The ε-smooth min-entropy maximizes the non-smooth conditional min-entropy over states in an ε-ball around the original state.It is monotonically increasing in ε and reduces to the non-smooth min-entropy at ε = 0.
  • Properties: The smooth min-entropy is independent of the local Hilbert-space representations of the density operator under local isometries.The proof uses invariance of the ε-ball and maps optimizing candidates between isometrically embedded spaces.
  • Smooth max-entropy: The ε-smooth max-entropy is defined as the dual of smooth min-entropy using an arbitrary purification of the conditioned state.All purifications are equivalent up to an isometry on the purifying system, so the definition is well-defined.
  • Smooth max-entropy: The smooth max-entropy is equivalently the minimum non-smooth max-entropy over states in an ε-ball around the original state.The equivalence follows by characterizing ε-balls through purifications and proving both optimization directions.

V. DATA-PROCESSING INEQUALITIES

This section derives data-processing inequalities for smooth conditional min- and max-entropies under physical operations. It also shows that projective measurements on the conditioned system do not decrease the relevant uncertainty measures.

  • General operations: Smooth conditional min- and max-entropies are non-decreasing under local trace-preserving completely positive maps applied to the conditioning system.The proof decomposes the map into an isometry followed by a partial trace and uses invariance under local isometries.
  • General operations: This data-processing property is also described as strong sub-additivity for the smooth min- and max-entropies.For the von Neumann entropy, the analogous inequality is equivalent to strong sub-additivity.
  • Projective measurements: Projective measurements of system A are modeled by an isometric map followed by partial trace over an auxiliary system.The measurement maps basis states |i⟩A to |i⟩X ⊗ |i⟩X′ before tracing out X′.
  • Projective measurements: Theorem 19 establishes the corresponding data-processing inequalities for smooth entropies after a projective measurement of A.The min-entropy proof uses ε-ball preservation under the measurement, while the max-entropy proof uses purification extension and local-isometry invariance.
  • Consequences: Together with the fully quantum asymptotic equipartition property, these inequalities imply analogous inequalities for the von Neumann entropy.This connection is stated as a consequence of Theorems 18 and 19.

APPENDIX A: TECHNICAL RESULTS

The appendix establishes technical properties of min- and max-entropies, including dimension bounds, continuity, and concavity of the max-entropy, paralleling known properties of von Neumann entropy.

  • Technical results: The appendix gives bounds on min- and max-entropies in terms of Hilbert-space dimensions, proves continuity as a function of the state, and shows that max-entropy is concave.It notes that analogous properties hold for the von Neumann entropy.

1. Preliminaries

The preliminaries introduce the functional Φ(ρAB) = 2^-Hmin(A|B)ρ and establish its structural properties. These include scaling, monotonicity, sub-additivity, convexity, and dimension-dependent bounds.

  • Functional properties: The functional Φ is positively homogeneous: multiplying the state by λ ≥ 0 multiplies Φ by λ.This is property i) of the functional.
  • Functional properties: Φ is monotone under the operator order: ρAB ≥ τAB implies Φ(ρAB) ≥ Φ(τAB).This is property ii) of the functional.
  • Functional properties: Φ is sub-additive, satisfying Φ(ρAB + τAB) ≤ Φ(ρAB) + Φ(τAB).This is property iii) of the functional.
  • Bounds: For dA = dim HA and dmin = min{dA, dim HB}, Φ obeys dimension-dependent lower and upper bounds involving tr ρAB and the positive part of its trace.The stated bounds are 1/dA tr ρAB ≤ Φ(ρAB) ≤ dmin tr{ρAB}+.
  • Functional properties: Convexity follows from the listed properties: Φ(λρAB + (1 − λ)τAB) ≤ λΦ(ρAB) + (1 − λ)Φ(τAB).The result is derived from properties i) and iii).

2. Bounds on the Conditional Entropies

The section establishes dimension- and trace-dependent bounds relating conditional min- and max-entropies, deriving the max-entropy bounds through duality.

  • For sub-normalized states, Hmin(A|B)ρ + log tr ρAB is bounded above by Hmax(A|B)ρ − log tr ρAB.
  • −log dmin ≤ Hmin(A|B)ρ + log tr ρAB ≤ log dA − log dmin ≤ Hmax(A|B)ρ − log tr ρAB ≤ log dA.Here dA = dim HA and dmin = min{dA, dim HB}.
  • The min-entropy bounds follow from earlier properties, while the max-entropy bounds follow by duality.

3. Continuity of the Conditional Entropies

The section develops continuity results for conditional min- and max-entropies, including uniform Lipschitz continuity of the conditional min-entropy and continuity of the smooth entropies by duality.

  • The conditional min-entropy has an operational interpretation as a guessing probability, which implies continuity in the state.
  • Lemma 21 establishes a continuity bound for conditional min-entropy under generalized distance between sub-normalized states.The lemma applies to ρAB, τAB ∈ S≤(HAB) with δ := ¯D(ρAB, τAB).
  • The continuity bound is tight for a construction involving a normalized fully entangled state and a perturbation by δ ψAB.
  • Lemma 21 implies that conditional min-entropy is uniformly Lipschitz continuous on normalized states and within any ε-ball.Because ¯D(ρ, τ) ≤ P(ρ, τ), the lemma also applies when δ is the purified distance.
  • The continuity of the smooth min- and max-entropies follows by selecting an optimizer in an ε-ball and using purifications together with duality.

4. Concavity of the Max-Entropy

The section proves that max-entropy is concave and uses data processing, properties of Φ, and logarithm concavity in the argument.

  • The max-entropy is a concave function of the state.
  • The proof uses a construction involving purifications and auxiliary orthonormal bases whose marginals are τAB and τACZ.
  • Data processing for max-entropy, properties of Φ, and concavity of the logarithm yield the stated result.
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