Min- and Max- Relative Entropies and a New Entanglement Monotone

Nilanjana Datta

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Min- and Max- Relative Entropies and a New Entanglement Monotone

Nilanjana Datta

arXiv:0803.2770v3Mar 19, 2008quant-ph

TL;DR

The paper addresses entropy and rate constructions beyond memoryless assumptions by introducing min- and max-relative entropies and their smooth versions. These quantities recover Renner’s entropies, support mutual-information and entanglement constructions, and connect asymptotically to spectral divergence rates.

Problem
Existing information-spectrum and smooth-entropy approaches address non-memoryless settings through different quantities and formalisms.
Method
The paper defines min- and max-relative entropies, smooth extensions, induced mutual informations, and the max-relative entropy of entanglement.
Results
The new relative entropies recover Renner’s min- and max-entropies and their smooth versions are asymptotically related to spectral divergence rates.
Takeaways & Limitations
The quantities provide parent constructions linking smooth Rényi entropies, mutual informations, entanglement monotones, and information-spectrum rates.

Abstract

from arXiv · show

Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies, and allow us to define the analogues of the mutual information, in the Smooth Renyi Entropy framework. Further, the spectral divergence rates of the Information Spectrum approach are shown to be obtained from the smooth min- and max-relative entropies in the asymptotic limit.

I. INTRODUCTION

The paper develops min- and max-relative entropies as parent quantities for Renner’s smooth entropy framework, a new entanglement monotone, and asymptotic links to spectral divergence rates.

Relative entropy is a foundational quantum-information quantity that measures distinguishability and generates several other entropic quantities.
The Information Spectrum approach evaluates optimal information-theoretic rates without assuming memoryless sources, channels, or entanglement resources.
The Smooth Entropy framework defines non-asymptotic smooth min- and max-entropies for individual states using a smoothness parameter ε.
The paper introduces min- and max-relative entropies, whose suitable substitutions recover Renner’s unconditional and conditional min- and max-entropies.
The results include a max-relative entropy of entanglement and asymptotic relations between smooth relative entropies and spectral divergence rates.
The min-relative entropy characterizes zero-first-kind-error state discrimination, while the max-relative entropy relates to optimal Bayesian discrimination among finitely many known states.

II. MATHEMATICAL PRELIMINARIES

The preliminaries establish finite-dimensional operator notation, spectral projections, and basic inequalities used throughout the paper’s proofs.

The paper works with positive unit-trace states on finite-dimensional Hilbert spaces and uses logarithms to base 2.
Self-adjoint operators are decomposed spectrally, with projections such as {A ≥ B} defined through the positive spectrum of A − B.
A key lemma establishes monotonicity-related inequalities for self-adjoint operators under completely positive trace-preserving maps.
The preliminaries collect operator inequalities, trace-distance and fidelity relations, and the gentle measurement lemma.

III. DEFINITIONS OF MIN- AND MAX-RELATIVE ENTROPIES

The paper defines min- and max-relative entropies and derives Renner-style entropies and mutual informations by choosing appropriate reference operators.

Dmax(ρ||σ) is defined for positive operators and is well-defined when supp ρ ⊆ supp σ.
Dmax(ρ||σ) also has an eigenvalue-based equivalent form and a spectral-projection characterization using {ρ ≥ λσ}.
Dmin(ρ||σ) is defined for positive operators and requires a non-zero intersection between the supports of ρ and σ.
Setting σ = I yields the state min- and max-entropies, while σ = I_A ⊗ σ_B yields bipartite conditional versions.
Setting σ = ρ_A ⊗ ρ_B defines min- and max-mutual informations for bipartite states.
Smooth relative entropies generate smooth entropies and mutual informations, and their asymptotic limits yield spectral divergence rates.

IV. PROPERTIES OF MIN- AND MAX-RELATIVE ENTROPIES

The min- and max-relative entropies obey structural properties including monotonicity, convexity, non-negativity, and bounds relative to quantum relative entropy and discrimination quantities.

For states, both quantities are non-negative and vanish when the states are identical; Dmin also vanishes for identical supports.
The min- and max-relative entropies are monotonic under completely positive trace-preserving maps.
The min-relative entropy is jointly convex, and the max-relative entropy satisfies a corresponding bound for mixtures of states.
The quantum relative entropy is bounded between the min- and max-relative entropies: Dmin(ρ||σ) ≤ S(ρ||σ) ≤ Dmax(ρ||σ).
The min- and max-relative entropies are invariant under joint unitary transformations.
For supported states, Dmax(ρ||σ) is bounded above by −log μmin(σ), where μmin(σ) is σ’s smallest non-zero eigenvalue.
Dmin is connected to binary state discrimination with vanishing Type I error and provides a lower bound to the quantum Chernoff bound.

V. A NEW ENTANGLEMENT MONOTONE

The paper introduces the max-relative entropy of entanglement as a new entanglement monotone, establishing it through sufficient conditions and showing it upper-bounds the relative entropy of entanglement.

The max-relative entropy of entanglement, Emax(ρ), is introduced for bipartite states.
Dmax satisfies the sufficient conditions needed for an entanglement monotone: faithfulness, unitary invariance, and nonincrease under local operations and classical communication.
Emax(ρ) is defined by minimizing Dmax(ρ||σ) over separable states.
The max-relative entropy of entanglement is an upper bound to the relative entropy of entanglement.
Further properties of Emax(ρ) are deferred to a forthcoming paper.

VI. SMOOTH MIN- AND MAX- RELATIVE ENTROPIES

The paper defines smooth min- and max-relative entropies by introducing a smoothness parameter and optimizing over nearby subnormalized states.

Smooth min- and max-relative entropies generalize the corresponding non-smooth relative entropy measures using ε ≥ 0.
At ε = 0, the smooth quantities reduce to the non-smooth min- and max-relative entropies.
The ε-smooth quantities are defined for a bipartite state ρ relative to a state σ.
The smoothing set contains positive operators within trace-norm distance ε of ρ and with trace no greater than Tr(ρ).

VII. SPECTRAL DIVERGENCE RATES

This section defines quantum spectral divergence rates using operator sequences and establishes equivalence with earlier definitions of the spectral sup- and inf-divergence rates.

Quantum spectral divergence rates generalize quantum relative entropy for sequences of quantum states.
The rates are defined from difference operators Πn(γ) = ρn − 2^nγσn.
The spectral sup-divergence rate has an equivalent formulation matching its original definition.
The spectral inf-divergence rate likewise has an equivalent formulation matching its original definition.
The alternative definitions are useful because they permit applying key lemmas to derive properties of the divergence rates.
Spectral entropy, conditional entropy, and mutual information generalizations can be expressed as spectral divergence rates with appropriate substitutions.

A. Definition of spectral entropy rates

The paper defines sup- and inf-spectral entropy rates for sequences of quantum states and relates them to spectral divergence rates and von Neumann entropies.

For a sequence of density matrices ρn acting on Hilbert spaces Hn, the sup- and inf-spectral entropy rates are defined.
The identity operators In provide the substitutions needed to obtain spectral entropy rates from spectral divergence rates.
Spectral entropy rates of a state sequence are related to the von Neumann entropies of the individual states ρn.
Conditional spectral entropy rates are defined for bipartite state sequences using partial traces over the relevant Hilbert spaces.
Mutual information rates are defined analogously for bipartite state sequences.
These spectral entropy rates have operational significance through their relation to optimal protocol rates.

VIII. RELATION BETWEEN SPECTRAL DIVERGENCE RATES AND SMOOTH MIN- AND MAX- RELATIVE ENTROPIES

This section proves relations between spectral divergence rates and smooth relative entropies using self-contained arguments based on definitions and earlier lemmas. The sup-spectral divergence rate is related to the smooth max-relative entropy, while corresponding bounds are established asymptotically.

The proofs are self-contained and rely only on the definitions of the entropic quantities and lemmas from Section II.
D(bρ∥bσ), defined by (40) or (42), satisfies the section’s stated relation for the sup-spectral divergence rate.
max(ρn||σn) is the smooth max-relative entropy of ρn and σn for the sequences bρ and bσ.
For any ε > 0, sufficiently large n satisfies the bound obtained from the preceding propositions and lemmas.
Choosing parameters α, γ, and δ makes both terms vanish asymptotically, yielding the required bound involving D(bρ||bσ).

B. Relation between D

This subsection derives a relation involving the inf-spectral divergence rate and smooth min-relative entropy through projection-based bounds and an asymptotic contradiction argument. The proof uses states within an ε-ball and concludes the required bound as ε approaches zero.

The inf-spectral divergence rate D is introduced through the relation stated in the subsection.
min(ρn||σn) is the smooth min-relative entropy of ρn and σn for the sequences bρ and bσ.
The proof defines a projection using γ and applies the Gentle measurement lemma to place eρn,γ in Bε(ρn) for sufficiently large n.
The argument assumes γ > D, derives bounds involving eρn,ε and its support projection, and obtains a contradiction.
The second term tends to zero asymptotically, but the first does not tend to 1 when γ0 > D; the resulting contradiction supports the required bound.

IX. APPENDIX

The appendix proves an auxiliary lemma using a purification of ρAB, transformed states, operator inequalities, and fidelity properties. The argument combines operator monotonicity with the overlap formula for pure-state fidelity.

The proof begins with a purification |Ψ⟩ABR of ρAB and defines a transformed state using TAB ⊗ IR.
It analyzes a Hermitian operator constructed from TAB and its adjoint.
The proof applies αAB ≤ βAB and positivity of IAB − ¯TAB to derive trace inequalities.
Operator monotonicity of the square root supplies a further inequality in the argument.
The final step uses the pure-state fidelity overlap formula together with equation (5).