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On quantum Renyi entropies: a new generalization and some properties
Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, Marco Tomamichel
TL;DR
Existing quantum Rényi extensions are partly incompatible and do not uniformly recover the quantum entropy measures used in information theory. The paper proposes a new quantum Rényi divergence and conditional entropy, showing data processing, monotonicity in α, and duality while recovering isolated measures as special or limiting cases.
Problem
Existing quantum generalizations, including Petz-type relative entropies and conditional min- and max-entropies, are partly incompatible and do not uniformly derive the corresponding classical measures.
Method
The paper defines a new quantum Rényi divergence and derives a conditional Rényi entropy from it.
Results
The proposed entropies satisfy data-processing inequalities, monotonicity in α, and a duality relation for pure tripartite states.
Takeaways & Limitations
One parameterized quantum entropy measure can naturally generate most entropies in current use as special or limiting cases.
Abstract
from arXiv · showhide
The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
I. INTRODUCTION
The paper introduces a quantum Rényi divergence and derives conditional entropies intended to unify established quantum entropy measures. It establishes natural properties including data processing, monotonicity in α, and duality, while addressing incompatibilities among earlier extensions.
- Earlier quantum generalizations are partly incompatible: classical conditional min-entropy derives naturally from Rényi divergence, unlike its quantum counterpart.
- The proposed quantum Rényi divergence generates a conditional Rényi entropy that contains several established entropy measures as special or limiting cases.
- The divergence decreases under completely positive trace-preserving maps, while conditional entropy increases when acting on the conditioning system.
- The divergence is monotonically increasing in α, and the corresponding conditional entropy is monotonically decreasing in α.
- For pure tripartite states, the conditional entropies satisfy a duality relation between complementary subsystems.
- The paper proves these properties, although its data-processing proof applies only to α ∈ (1, 2], despite the property holding for arbitrary α ≥ 1.
III. OVERVIEW OF RESULTS
The paper proposes a different non-commutative generalization of Rényi divergence, defines it for α∈(0,1)∪(1,∞), and derives conditional Rényi entropies from it. The definition is examined against the paper’s desired properties.
- The paper proposes a different non-commutative generalization of the Rényi divergence.
- The quantum Rényi divergence is defined for orders α∈(0,1)∪(1,∞).
- Non-orthogonality of ρ and σ suffices for finiteness when α<1, whereas σ≫ρ implies non-orthogonality.
- The paper verifies that the proposed definition satisfies properties (I)–(VI).
1. First properties
The paper establishes foundational properties and limiting cases for the new quantum Rényi divergence, including positivity, special-case connections, data processing, and monotonicity in α. It also defines related max-entropy and conditional entropies within the unified framework.
- First properties: The new divergence is nonnegative when Tr[ρ]≥Tr[σ], and equals zero when ρ=σ.
- Limits and Important Special Cases: The relative max-entropy is the α→∞ limit, while quantum relative entropy is the α→1 limit of the α-order Rényi divergence.
- Limits and Important Special Cases: At α=2, the construction recovers collision relative entropy, which has applications in randomness extraction and min-entropy sampling.
- Joint Convexity/Concavity and Data-Processing: The new divergence satisfies data processing for α∈(1,2], and the property holds for all α∈[1/2,1)∪(1,∞).
- Conditional entropies: For conditional entropies, eHα(A|B)ρ is defined by optimizing −eDα(ρAB∥idA⊗σB) over σB.
- Monotonicity: The map α↦eDα(ρ∥σ) is monotonically increasing.
B. From Divergence to Conditional Entropy
The paper defines conditional Rényi entropy by optimizing the new Rényi divergence, thereby deriving a unified family that includes established conditional entropies and obeys natural processing and chain-rule properties.
- Interpretation: The conditional entropy can be interpreted as maximal entropy on A minus the Rényi divergence from a maximally mixed state on A tensored with a state on B.The maximal entropy is log d_A, where d_A is the rank of ρ_A.
- Properties: Data processing makes conditional Rényi entropy increase when the conditioning system is processed, implying that conditioning on more systems can only reduce the entropy.For a tripartite state, eH_α(A|B) is at least eH_α(A|BC).
- Properties: The chain rule bounds how much conditional Rényi entropy can drop when additional systems are included.The bound depends on d_C, the rank of ρ_C.
- Special cases: This construction recovers isolated conditional entropy measures as special or limiting cases, including the classical conditional Rényi entropy in the classical-register setting.For a classical register, the divergence and conditional entropy decompose through the conditional states; the empty-conditioning case yields Arimoto’s classical conditional Rényi entropy.
3. Duality Relation
For pure tripartite states, the paper establishes a duality relation for conditional Rényi entropies, extending known von Neumann, min-, and max-entropy relations and supporting a generalized uncertainty relation.
- Duality Relation: Conditional Rényi entropies satisfy a duality relation for every pure tripartite quantum state.The von Neumann and min/max dualities are identified as limiting cases of the generalized relation.
- Duality Relation: The generalized relation connects the conditional entropies of complementary subsystems in a pure tripartite state.The paper notes that the von Neumann case follows from the Schmidt decomposition, while the min/max case was previously established.
- Uncertainty Relation: Because the generalized entropies satisfy the properties required by prior uncertainty-relation arguments, they yield an uncertainty relation for conditional Rényi entropies.The theorem applies to any two positive operator-valued measures and parameters α, β in the stated range with 1/α + 1/β = 2.
A. Preliminaries
The preliminaries establish the operator-analytic framework for the quantum Rényi divergence, including generalized Schatten quantities, continuity across rank changes, and proofs of its structural properties.
- A. Preliminaries: The paper works with finite-dimensional quantum systems and defines a generalized Schatten p-norm for positive semidefinite operators.The definition covers p ∈ (0, ∞), including values below one where the quantity is not a norm.
- A. Preliminaries: Auxiliary variational and operator inequalities support the analysis of the Rényi divergence across α-regimes.The preliminaries use norm duality, Lagrange multipliers, pinching, and operator monotonicity or concavity arguments.
- Continuity: The generalized inverse provides a continuous extension of the divergence when operators are noninvertible or their ranks change.Lemma 13 treats the limiting behavior as a regularization parameter tends to zero, including divergence to +∞ when support conditions fail.
- Structural properties: The proofs establish continuity, unitary invariance, order behavior, additivity, and direct-sum additivity for the divergence.The order proof uses operator monotonicity, while tensor-product and direct-sum identities yield the additivity properties.
- Structural properties: The divergence is nonnegative and equals zero when the two arguments are equal.Nonnegativity follows from the Schatten-norm argument, and equality at identical arguments follows from the order property.
C. Limits for α →1 and α →∞
The paper analyzes limiting behavior of the quantum Rényi divergence, proving convergence to the von Neumann and max-relative entropies under the relevant parameter limits.
- Limits for α → 1: The α → 1 analysis uses a derivative evaluation and L’Hôpital’s rule to establish convergence toward the von Neumann entropy.The argument first treats positive definite operators and then extends to suitable noninvertible cases by restricting to the support.
- Proof strategy: The limiting proofs rely on integral representations of the logarithm and cyclicity of the trace to evaluate derivative expressions.These identities are applied successively in the displayed-equation calculations.
- Limits for α → 1: When the support condition fails, the divergence is infinite at α = 1 and diverges correspondingly for α > 1 as α approaches one.The limiting trace expression is strictly below one, producing divergence of the prefactor-based expression.
- Limits for α → ∞: The α → ∞ analysis is organized around norm inequalities to prove convergence to the max-relative entropy.The proof invokes the reverse triangle inequality for the α-norm and subsequent trace manipulations.
D. Joint Convexity and Data-Processing
The section establishes joint convexity of the relevant exponential divergence functional through operator-concavity and convexity lemmas, providing the basis for Theorem 6.
- Joint Convexity: Joint convexity of exp((α −1) eDα(·∥·)) is sufficient to prove Theorem 6.
- Proof Strategy: The proof uses joint operator concavity of F(L, R)=Lβ ⊗(R^T)1−β for β ∈[0, 1].
- Proof Strategy: The argument extends from strictly positive operators to positive semidefinite ones by approximating with invertible matrices and using continuity.
- Proof Strategy: Applying an operator-convexity theorem to x^α and the positive operator-concave map R^T1−β establishes the required operator-convexity claim.
- Joint Convexity: For α ∈[1, 2], the functional exp((α −1) eDα(·∥·)) is jointly convex.
E. Monotonicity in α
The section proves that the auxiliary divergence is monotonically increasing in α and transfers this property to the divergence itself by optimizing over the auxiliary operator.
- Monotonicity: The auxiliary quantity eDα(ρ∥σ; τ) is introduced to prove monotonicity of eDα(ρ∥σ).
- Monotonicity: For normalized ρ and arbitrary σ, τ ≥0, α 7→eDα(ρ∥σ; τ) is monotonically increasing.
- Proof Strategy: The proof uses a purification of ρ, an auxiliary operator X, and Jensen’s inequality applied to the convex function x log x.
- Boundary Cases: When σ or τ does not dominate ρ, the auxiliary divergence takes infinite values in the relevant α ranges, so monotonicity holds trivially.
- Conclusion: For α′ ≥α, optimizing the auxiliary expression over τ yields Dα′(ρ∥σ; τ) ≥eDα(ρ∥σ; τ) and proves Theorem 7.
F. Duality of the Conditional R´enyi Entropy
The section derives a minimax representation of the conditional Rényi entropy using a transpose identity and establishes the convexity structure needed to interchange optimization order.
- Duality: A transpose identity relates the action of DAB on a maximally entangled state to the corresponding operator DC on the auxiliary system.
- Minimax Representation: The conditional Rényi entropy is represented as a minimax problem over positive operators σB and τC with trace at most one.
- Minimax Representation: The objective is concave in σB and convex in τC for α < 1, with the roles reversed for α > 1.
- Minimax Representation: This convexity structure permits interchanging the infimum and supremum in both α regimes.
G. Conditioning on Classical Information
The section analyzes conditioning on a classical register by restricting the optimization to classical conditional states and decomposing the divergence across conditional components.
- Classical Conditioning: For a tripartite state ρABY with classical Y, the proof starts from its classical-register decomposition.
- Classical Conditioning: Data processing allows the infimum over σBY to be restricted to states with classical Y.
- Classical Conditioning: The exponential divergence is decomposed into divergences of the conditional states indexed by the classical register.
- Optimization: A Lagrange multiplier calculation shows that the infimum is attained by a distribution satisfying ˆqy = ry/ P.