[Paper Review] Quantum Rényi and $f$-divergences from integral representations

Christoph Hirche, Marco Tomamichel|arXiv (Cornell University)|Jun 21, 2023

Deception detection and forensic psychology4 citations

TL;DR

This paper introduces a novel quantum generalization of f-divergences and Rényi divergences using integral representations based on quantum hockey-stick divergences, defined as Tr((ρ − γσ)+). The key contribution is that regularized versions of these divergences unify the Petz and sandwiched Rényi divergences for α < 1 and α > 1, respectively, and the contraction coefficients collapse for operator convex f, resolving long-standing conjectures.

ABSTRACT

Smooth Csiszár $f$-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback-Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the Rényi divergences defined via our new quantum $f$-divergences are not additive in general, but that their regularisations surprisingly yield the Petz Rényi divergence for $α&lt; 1$ and the sandwiched Rényi divergence for $α&gt; 1$, unifying these two important families of quantum Rényi divergences. Moreover, we find that the contraction coefficients for the new quantum $f$ divergences collapse for all $f$ that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalities with applications in differential privacy and explore various other applications of the new divergences.

Motivation & Objective

• To develop a natural quantum generalization of classical f-divergences using integral representations based on quantum hockey-stick divergences.
• To unify the two prominent families of quantum Rényi divergences—Petz and sandwiched—through regularization of the new f-divergences.
• To resolve long-standing conjectures by Lesniewski and Ruskai regarding contraction coefficients of f-divergences for operator convex functions.
• To establish new inequalities, including reverse Pinsker inequalities, with applications in differential privacy.
• To provide a framework for quantum divergences that preserves key properties like data processing inequality and joint convexity.

Proposed method

• Define quantum hockey-stick divergence as Eγ(ρ‖σ) = Tr((ρ − γσ)+), extending the classical hockey-stick divergence to quantum states.
• Construct quantum f-divergences via integral representation: Df(ρ‖σ) = Tr(ρ − σ) + ∫₁^∞ [f''(γ)Eγ(ρ‖σ) + γ⁻³f''(γ⁻¹)Eγ(σ‖ρ)] dγ.
• Use the integral representation to generalize Rényi divergences via Hellinger-type divergences Hα(ρ‖σ), related to Dα by Dα(ρ‖σ) = (1/(α−1)) log(1 + (α−1)Hα(ρ‖σ)).
• Prove that the regularized version of the new f-divergences yields the Petz Rényi divergence for α < 1 and the sandwiched Rényi divergence for α > 1.
• Analyze contraction coefficients and show they collapse for all operator convex f, mirroring classical behavior.
• Derive scaling laws for Df(aρ‖bσ) and establish new reverse Pinsker-type inequalities using the framework.

Experimental results

Research questions

• RQ1 Can a quantum generalization of f-divergences be constructed via integral representations of quantum hockey-stick divergences that preserve key classical properties?
• RQ2 Does the regularization of the proposed quantum f-divergences unify the Petz and sandwiched Rényi divergences across all α ≠ 1?
• RQ3 Why do contraction coefficients of the new f-divergences collapse for operator convex f, and does this resolve the conjectures by Lesniewski and Ruskai?
• RQ4 What new inequalities, such as reverse Pinsker inequalities, can be derived from this framework, and what are their applications?
• RQ5 How do the new divergences behave under scaling of density matrices, and do they preserve monotonicity and convexity?

Key findings

• The regularized version of the proposed quantum f-divergence yields the Petz Rényi divergence for α < 1 and the sandwiched Rényi divergence for α > 1, unifying both families.
• The contraction coefficients of the new f-divergences collapse for all operator convex f, confirming a long-standing conjecture by Lesniewski and Ruskai.
• The new divergences satisfy the data processing inequality and joint convexity, inheriting these properties from the quantum hockey-stick divergence.
• A reverse Pinsker inequality is derived, with potential applications in differential privacy and quantum information theory.
• The scaling law Df(aρ‖bσ) = (a−b) + bD_{F_{a,b}}(ρ‖σ) + ∫_{b/a}^1 f''(a/b γ)(a²/b)(1−γ) dγ holds, generalizing classical scaling behavior.
• The integral representation of the Umegaki relative entropy via Frenkel's formula is recovered as a special case when f(x) = x log x.