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The Smooth Entropy Formalism for von Neumann Algebras

Mario Berta, Fabian Furrer, Volkher B. Scholz

arXiv:1107.5460v3quant-phmath-phmath.MP

TL;DR

Information-theoretic methods for infinite-dimensional quantum systems call for an extension beyond the finite-dimensional smooth entropy formalism. This paper extends that formalism to von Neumann algebras and reports that conditional min- and max-entropies retain task-characterizing and operational roles in the algebraic setting.

  • Problem

    The smooth entropy formalism had been developed for finite-dimensional systems, while information-theoretic tasks for more general quantum systems such as quantum fields require a von Neumann algebra framework.

    Source: §I · pp. 1–2

  • Method

    The paper extends the smooth entropy formalism to von Neumann algebras and extends the purified distance to that setting.

    Source: §I · p. 2, B. Purified Distance · pp. 8–9

  • Results

    For quantum key distribution modeled with von Neumann algebras, the paper states that the basic information-theoretic tasks remain characterized by smooth conditional min- and max-entropies.

    Source: D. Quantum Key Distribution · p. 22

  • Takeaways & Limitations

    The framework is intended to support information-theoretic problems for infinite-dimensional systems, including bosonic and fermionic quantum fields.

    Source: §I · p. 2

Abstract

from arXiv · show

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

I. INTRODUCTION

The paper extends smooth entropy methods beyond finite-dimensional Hilbert spaces to quantum systems modeled by von Neumann algebras. Its motivation is a framework for information-theoretic tasks involving systems such as quantum fields.

I. INTRODUCTION

  • Smooth conditional min- and max-entropies characterize optimal performance for arbitrary resources and recover conditional von Neumann entropy asymptotically for independent, identically distributed resources.The paper presents these entropies as operational measures of information-theoretic tasks.Source: §I · pp. 1–2
  • The smooth entropy formalism is extended to von Neumann algebras, including settings beyond full operator algebras used in earlier infinite-dimensional work.The paper notes that a finite-temperature free-boson field algebra is not of the full-algebra type.Source: §I · pp. 1–2

A. Mathematical Background

  • A von Neumann algebra is a *-subalgebra of bounded operators equal to its bicommutant; it is σ-weakly closed, unlike every norm-closed C*-subalgebra.The text also describes von Neumann algebras as C*-algebras that are dual spaces of Banach spaces.Source: A. Mathematical Background · p. 3
  • Normal positive functionals provide the state framework, while weights extend the treatment to positive elements and may take infinite values.Normal functionals are σ-weakly continuous; a semifinite normal trace on an infinite-dimensional B(H) is given as an example of a weight.Source: A. Mathematical Background · pp. 3–4

B. Algebraic Quantum Theory

The algebraic framework represents systems through their observable algebras and accommodates composites whose subsystems commute. It also sets up the general system model used for the paper’s entropy definitions.

B. Algebraic Quantum Theory

  • A physical system is represented by a von Neumann algebra generated by its observables, with measurement probabilities given by applying the state to the outcome operators.The paper models discrete-outcome measurements by positive operators summing to the identity.Source: B. Algebraic Quantum Theory · p. 4
  • Quantum channels are normal completely positive unital maps; their pre-dual maps preserve the norm of positive functionals, including where no trace is available.For full operator algebras, this norm conservation corresponds to trace preservation.Source: B. Algebraic Quantum Theory · pp. 4–5
  • Composite subsystems are represented by mutually commuting von Neumann algebras, a description the paper says covers both bosonic and fermionic theories.Their joint algebra is generated by the subsystem algebras.Source: B. Algebraic Quantum Theory · p. 5
  • Purification represents a state as a vector state in a representation, with the represented algebra and its commutant serving as relevant and complementary systems.The paper notes that its purification notion need not yield a pure state on the combined relevant and complementary algebra.Source: B. Algebraic Quantum Theory · p. 5

III. MIN- AND MAX-ENTROPY

  • The conditional entropy definitions that follow require a trace on system A, so the paper takes A to be a full algebra while allowing B to remain general.For finite-dimensional systems, the paper uses the full-algebra representation B(C^n).Source: §III · p. 6, B. Algebraic Quantum Theory · p. 6

A. Conditional Min- and Max-Entropy

The conditional min-entropy is defined in a setting with a full-algebra A system and general side information B. The conditional max-entropy is obtained by applying the min-entropy to a purification, with the definition independent of the purification chosen.

A. Conditional Min- and Max-Entropy

  • The conditional entropy setup restricts A to a full algebra so its trace can be used, while the side-information system B remains general.The paper identifies this as the reason for choosing M_AB = B(H_A) ⊗ M_B.Source: A. Conditional Min- and Max-Entropy · p. 6
  • The conditional max-entropy is defined as the conditional min-entropy of A given the complementary system in a purification of the original state.The paper defines it as a dual quantity using a purification.Source: A. Conditional Min- and Max-Entropy · pp. 6–7
  • The conditional max-entropy construction requires that the choice of purification not change the resulting quantity.The paper states this requirement before proving equivalence across purifications.Source: A. Conditional Min- and Max-Entropy · p. 7
  • The conditional min-entropy is unchanged across the compared purifications, with the proof using an intertwining partial isometry and a symmetric argument.The partial isometry maps the purification vectors and intertwines the represented algebra actions.Source: A. Conditional Min- and Max-Entropy · p. 7

B. Purified Distance

The section extends purified distance and generalized fidelity to von Neumann algebras, including subnormalized states. The resulting distance is a metric equivalent to norm distance and is monotone under completely positive contractions.

B. Purified Distance

  • The choice of distance measure is crucial to smoothing conditional min- and max-entropies, motivating the extension of purified distance to von Neumann algebras.Source: B. Purified Distance · p. 8
  • Generalized fidelity extends to subnormalized states using normalized extensions in a larger algebra, with subnormalized states interpretable as post-measurement states conditioned on outcomes.A construction uses the direct sum M ⊕ C to extend subnormalized states.Source: B. Purified Distance · p. 8
  • The purified distance is defined through an infimum over representations with vector purifications and is a metric equivalent to norm distance.It is defined in the same way as for finite-dimensional spaces.Source: B. Purified Distance · p. 9
  • The purified distance is monotone under completely positive contractions.Source: B. Purified Distance · p. 9

C. Smooth Conditional Min- and Max-Entropy

Smooth conditional min- and max-entropies are built using purified-distance neighborhoods of states. Their definitions are purification-independent, and the smooth max-entropy can equivalently be expressed through an optimization over conditional max-entropies.

C. Smooth Conditional Min- and Max-Entropy

D. Properties of Smooth Entropies

The section establishes data-processing properties for smooth conditional entropies and notes that positive smoothing keeps them finite even when the unsmoothed quantities can diverge.

D. Properties of Smooth Entropies

E. Min- and Max-Relative Entropy

The section introduces min- and max-relative entropies for von Neumann algebras, which also provide definitions of unconditional min- and max-entropy. It states that both relative entropies are monotone under quantum channels.

E. Min- and Max-Relative Entropy

IV. APPLICATIONS TO QUANTUM INFORMATION THEORY

The applications use a finite-dimensional A-system while allowing the B-system to be fully general, a setup suited to quantum information theory and quantum cryptography.

IV. APPLICATIONS TO QUANTUM INFORMATION THEORY

  • The A-system is finite-dimensional, but the B-system is fully general, avoiding assumptions about the adversarial system.The authors identify this setup as suited to quantum information theory, particularly quantum cryptography.Source: §IV · p. 14

A. Operational Interpretation of Conditional Min- and Max-Entropy

The section gives operational characterizations of conditional min- and max-entropy for a finite-dimensional A-system and a general von Neumann algebra B. These connect the entropies to optimal entanglement fidelity and decoupling fidelity, and establish that conditional min-entropy does not exceed conditional max-entropy.

Optimal entanglement fidelity

  • Conditional min-entropy is characterized through optimal entanglement fidelity, generalizing its operational meaning from the finite-dimensional setting [25, Theorem 2].The paper notes that the square differs from [25, Theorem 2] because the fidelity definitions differ.Source: A. Operational Interpretation of… · p. 14

Operational interpretation of conditional min-entropy

  • The min-entropy operational characterization is related to quantum channels by identifying completely positive maps with positive operators satisfying a normalization condition.The text describes this correspondence in both directions using matrix elements of the operator and channel.Source: A. Operational Interpretation of… · p. 15

Optimal decoupling fidelity

  • The optimal decoupling fidelity proposition generalizes the operational meaning of conditional max-entropy [25, Theorem 3], stated for M_AB = B(C^n) ⊗ M_B and ω_AB ∈ S≤(M_AB).Source: A. Operational Interpretation of… · p. 15

Ordering of entropies

  • The conditional min-entropy is never larger than the conditional max-entropy for the stated systems and states.The result follows from the ordering of min- and max-relative entropy.Source: A. Operational Interpretation of… · p. 16

B. Classical Quantum Systems

The section treats finite-alphabet classical registers as abelian von Neumann algebras and embeds classical-quantum states into a finite-dimensional A-system with quantum side information B. It relates conditional min-entropy to guessing probability and conditional max-entropy to decoupling fidelity.

Classical quantum systems

  • A finite-alphabet classical system is represented by the abelian algebra ℓ∞(X), and its joint system with B is described by a von Neumann algebra.Classical-quantum states are embedded into B(C^|X|) ⊗ MB to define the conditional entropies.Source: B. Classical Quantum Systems · p. 17

Conditional min-entropy

  • For a finite classical register X and quantum side information B, conditional min-entropy has an operational interpretation as the probability of correctly guessing X using B [25, Theorem 1].The characterization is Hmin(X|B)ω = −log pguess(X|B)ω.Source: B. Classical Quantum Systems · p. 17

Conditional max-entropy

  • The smooth conditional max-entropy can alternatively be smoothed over the embedded set, by data processing for smooth conditional min-entropy.Source: B. Classical Quantum Systems · p. 18
  • Conditional max-entropy for a classical register X is characterized by the decoupling fidelity Fdec(X|B)ω.The section states Hmax(X|B)ω = log Fdec(X|B)ω.Source: B. Classical Quantum Systems · p. 18

C. Entropic Uncertainty Relations with Quantum Side Information

The section extends entropic uncertainty relations with quantum side information to von Neumann algebras and develops a more general relation for quantum channels. It notes applications to continuous-variable quantum key distribution and a follow-up extension to measurements with infinitely many outcomes.

Measurements

  • The uncertainty relation applies to measurements with finite outcome ranges on a fully general tripartite von Neumann algebra, with uncertainty quantified by smooth conditional min- and max-entropy.The authors state that approximation techniques cannot simply lift the result from finite dimensions.Source: C. Entropic Uncertainty Relations with… · p. 19

Quantum Channels

  • The paper derives the measurement relation from a more general uncertainty relation for quantum channels with matrix-algebra domains.The channel formulation concerns uncertainties about the output systems given side information B and C.Source: C. Entropic Uncertainty Relations with… · p. 19

Applications

D. Quantum Key Distribution

The paper extends smooth-entropy characterizations of quantum key distribution tasks to systems modeled by von Neumann algebras. It treats privacy amplification and data compression with quantum side information in this setting.

D. Quantum Key Distribution

  • Privacy amplification and data compression remain characterized by smooth conditional min- and max-entropy when quantum systems are modeled by von Neumann algebras.The paper frames these as the two subtasks of quantum key distribution.Source: D. Quantum Key Distribution · pp. 22–23
  • Alice and Bob seek a shared uniform key that is correct between them and secure from Eve, whose system is modeled as a purification.Alice’s finite-outcome measurement produces a classical-quantum state shared with Bob and Eve.Source: D. Quantum Key Distribution · p. 23
  • Privacy amplification hashes Alice’s input alphabet X into a smaller key alphabet K using a family of two-universal hash functions.The hash family is defined for finite alphabets with |K| < |X| in this description.Source: D. Quantum Key Distribution · p. 23
  • The paper states that its main data-compression result quantifies the achievable error probability for a protocol with an alphabet C.The supplied proposition passage does not include the bound itself.Source: D. Quantum Key Distribution · p. 25
  • For data compression, Alice sends an encoded classical message and Bob decodes using a measurement suited to distinguish the candidate states.The paper selects an encoding from a family of two-universal hash functions and describes a corresponding decoding measurement.Source: D. Quantum Key Distribution · p. 26

V. DISCUSSION AND OUTLOOK

The discussion concludes that smooth-entropy characterizations extend to von Neumann algebras, including privacy amplification and data compression. It points to applications in continuous-variable quantum information and thermodynamics, and to a possible extension using operator systems.

V. DISCUSSION AND OUTLOOK

  • The generalized framework preserves smooth conditional min- and max-entropy characterizations of privacy amplification and data compression.The results can be used for one-shot tasks with continuous-variable systems, including fermionic and bosonic quantum fields.Source: §V · p. 26
  • The framework also has potential relevance to thermodynamics, where systems of interest often have infinitely many degrees of freedom.The paper notes that smooth min- and max-entropy have also been used in thermodynamics.Source: §V · p. 26
  • An operator-system generalization could restrict which measurements are allowed, including measurements used by a decoder in data compression.The paper describes this as a possible extension of the formalism.Source: §V · p. 26

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