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Sandwiched Rényi Divergence Satisfies Data Processing Inequality
Salman Beigi
TL;DR
The paper examines whether sandwiched Rényi divergence has expected structural properties, including data processing. It develops interpolation- and minimax-based proofs establishing data processing for α > 1 alongside monotonicity, positivity, convexity, and duality results.
Problem
Sandwiched Rényi divergence is expected to satisfy structural properties such as the data processing inequality, but earlier results covered only selected α ranges.
Method
The proofs use complex interpolation, including a specialized Riesz-Thorin theorem, and Sion’s minimax theorem for an equality involving conditional entropies.
Results
The paper proves data processing for all α > 1 and establishes monotonicity in α, positivity with equality only when ρ = σ, convexity, and a duality relation.
Takeaways & Limitations
The results establish several fundamental properties of sandwiched Rényi divergence beyond the previously identified special α regimes.
Abstract
from arXiv · showhide
Sandwiched (quantum) $α$-Rényi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and Müller-Lennert et al (arXiv:1306.3142v1). This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular we show that sandwiched $α$-Rényi divergence satisfies the data processing inequality for all values of $α> 1$. Moreover we prove that $α$-Holevo information, a variant of Holevo information defined in terms of sandwiched $α$-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.
I. A NON-COMMUTATIVE R´ENYI DIVERGENCE
The paper studies sandwiched Rényi divergence, a non-commutative quantum generalization, and establishes several conjectured properties beyond previously known parameter ranges.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: Sandwiched Rényi divergence extends Rényi divergence to quantum density matrices and reduces to the earlier divergence when ρ and σ commute.The paper adopts this name to distinguish it from the previous quantum Rényi divergence.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: Its data processing inequality was previously proved only for 1 < α ≤ 2, while related properties were conjectured elsewhere for other α values.These properties include positivity, equality conditions, and data processing.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: For all positive α ≠ 1, the paper proves positivity and equality if and only if ρ = σ.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: For every α > 1, it proves the data processing inequality for sandwiched Rényi divergence.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: The paper also proves monotonicity in α for general ρ when α > 1, extending an earlier rank-one result.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: Additional results include general duality for conditional Rényi entropy and super-additivity of α-Holevo information.
- I. A NON-COMMUTATIVE R´ENYI DIVERGENCE: The proofs use Hölder inequalities, Riesz-Thorin interpolation, complex interpolation ideas, and Sion’s minimax theorem.
II. H¨OLDER’S INEQUALITIES
This section develops the Schatten-norm and weighted-norm inequalities used in the paper’s proofs, including Hölder, reverse Hölder, and duality relations.
- II. H¨OLDER’S INEQUALITIES: For 1 ≤ p ≤ ∞, Schatten p-norms satisfy the triangle inequality and form the spaces Lp(H).
- II. H¨OLDER’S INEQUALITIES: Hölder’s inequality bounds |tr(XY)| by the product of the Schatten p-norms of X and Y.It also identifies Lp′(H) as the dual space of Lp(H).
- II. H¨OLDER’S INEQUALITIES: The section extends Hölder’s framework to reverse Hölder inequalities for combinations of positive and negative exponents.
- II. H¨OLDER’S INEQUALITIES: For 0 < p < 1, the Schatten p-quasi-norm of a positive operator admits an infimum characterization over positive operators with dual exponent p′.
- II. H¨OLDER’S INEQUALITIES: Weighted norms are defined using the super-operator Γσ(X) = σ^1/2Xσ^1/2 and, for full-rank σ, satisfy norm and duality properties.
III. RIESZ-THORIN THEOREM
The paper develops a Riesz-Thorin theorem for weighted Schatten spaces using complex interpolation, providing the interpolation machinery required for its quantum-information results.
- III. RIESZ-THORIN THEOREM: The proofs rely primarily on complex interpolation, especially a Riesz-Thorin theorem tailored to quantum-information applications.
- III. RIESZ-THORIN THEOREM: The section derives the weighted-space interpolation result from Hadamard’s three-line theorem for bounded holomorphic operator-valued maps.
- III. RIESZ-THORIN THEOREM: A corollary identifies Lpθ,σ(H) as the complex interpolation space between the endpoint weighted Schatten spaces.
- III. RIESZ-THORIN THEOREM: The generalized theorem interpolates super-operator bounds between Lp0,σ(H) to Lq0,σ′(H′) and Lp1,σ(H) to Lq1,σ′(H′).
- III. RIESZ-THORIN THEOREM: Complete positivity ensures that the relevant weighted super-operator norm supremum is attained at a positive semi-definite operator.
- III. RIESZ-THORIN THEOREM: The proof constructs a holomorphic operator-valued map whose intermediate value is the target operator and applies boundary estimates through the three-line theorem.
IV. STATEMENTS AND PROOFS OF THE MAIN RESULTS
The paper establishes positivity, equality, data processing, monotonicity, convexity, and duality properties for sandwiched Rényi divergence and related conditional entropy. The proofs combine Schatten-norm inequalities, interpolation, Hölder duality, and minimax arguments.
- Main results: Dα(ρ||σ) ≥ 0 for all positive α ≠ 1, with equality if and only if ρ = σ.The equality argument uses the equality condition in Hölder’s inequality and concludes that the density matrices must coincide.
- Main results: For every α > 1, sandwiched Rényi divergence satisfies the data processing inequality under any completely positive trace-preserving map.Riesz-Thorin interpolation reduces the proof to endpoint bounds at α = 1 and α = ∞; trace preservation supplies the α = 1 bound.
- Main results: For fixed density matrices ρ and σ, α ↦ Dα(ρ||σ) is increasing for α > 1, and the proof yields monotonicity of α ↦ exp(Dα(ρ||σ)).The argument applies an interpolation inequality with p0 = 1, p1 = β, and pθ = α.
- Main results: The authors obtain a convexity property equivalent to convexity of 1/α ↦ Dα(ρ∥σ)/α′.This follows by allowing p0 > 1 in the same interpolation proof used for monotonicity.
- Main results: For 1/2 ≤ α, β ≤ ∞ with α, β ≠ 1 and satisfying the stated conjugacy relation, tripartite pure states obey Hα(A|B) = −Hβ(A|C).The duality proof uses Hölder’s duality and Sion’s minimax theorem, with compact convex optimization domains.
V. α-HOLEVO INFORMATION IS SUPER-ADDITIVE
The paper defines α-Rényi mutual information using sandwiched divergence and proves it is additive, then uses this result to establish super-additivity of α-Holevo information for quantum channels.
- α-Rényi mutual information: α-Rényi mutual information is defined through an infimum over density matrices σB, reducing to ordinary mutual information when α = 1.The definition uses ρA = trB(ρAB).
- α-Holevo information: Theorem 10 supplies the intermediate relation used to connect α-Rényi mutual information with the channel quantities considered later.It applies for α > 1 and 1/2 ≤ β < 1 under the stated parameter relation, using a purification of ρAB.
- α-Rényi mutual information: α-Rényi mutual information is additive for α ≥ 1 on tensor-product bipartite states.The proof uses product choices in the minimization and a product purification for the reverse direction.
- α-Holevo information: α-Holevo information is defined for a noisy quantum channel by taking a supremum over classical-quantum input states.At α = 1, it reduces to the usual Holevo information.
- α-Holevo information: α-Holevo information is super-additive for α ≥ 1 across two quantum channels.The proof restricts the supremum to tensor-product input states and invokes additivity of α-Rényi mutual information.