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Regularized barycentric Rényi divergences
Milán Mosonyi
TL;DR
Multivariate quantum Rényi divergences need constructions that handle non-commuting states while retaining properties such as data processing and additivity, but barycentric divergences previously lacked a general additivity result. The paper characterizes their regularization for additive and monotone quantum relative entropies, showing that on strictly positive inputs it equals the minimal barycentric divergence and thereby identifying additive barycentric divergences with the minimal one.
Problem
General additivity was unknown for barycentric Rényi divergences beyond the Umegaki case, which was also the only case with an explicit expression.
Method
The paper derives an explicit expression for the regularized barycentric Rényi divergence using monotone and additive quantum relative entropies.
Results
For monotone and additive relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric divergence on strictly positive inputs.
Takeaways & Limitations
Any additive barycentric Rényi divergence therefore coincides with the minimal one on strictly positive inputs.
Abstract
from arXiv · showhide
Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) admits an explicit expression. Here we settle the problem of additivity by showing that for any choice of additive and monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on strictly positive inputs. This in turn implies that any additive barycentric Rényi divergence coincides with the minimal one on strictly positive inputs.
I. INTRODUCTION
The paper motivates barycentric multivariate quantum Rényi divergences as a variational alternative to explicit matrix-geometric constructions, whose additivity and operational behavior are difficult in the multivariate setting. It resolves the additivity problem by giving an explicit regularized expression under monotonicity, additivity, and support conditions.
- I. INTRODUCTION: Multivariate quantum Rényi divergences are needed to extend classical problems such as state exclusion and distribution convertibility to non-commuting quantum states.
- I. INTRODUCTION: Kubo–Ando-based multivariate constructions can satisfy additivity and data processing but are often non-explicit and may fail to recover operationally relevant bivariate quantities.
- I. INTRODUCTION: Barycentric Rényi divergences instead use a variational optimization over quantum states and a finite collection of quantum relative entropies.
- I. INTRODUCTION: For Umegaki relative entropies, the barycentric optimization has an explicit solution involving the P-weighted log-Euclidean geometric mean, yielding the minimal barycentric divergence.
- I. INTRODUCTION: The paper settles additivity by explicitly characterizing the regularized barycentric divergence for monotone and additive relative entropies, including common-support and maximal-relative-entropy cases.
A. Quantum relative entropies
The paper defines quantum relative entropies through structural properties including isometric invariance, classical reduction, and positivity, then distinguishes monotonicity and additivity under tensor products. Umegaki and Belavkin–Staszewski relative entropies provide the smallest and largest members of the relevant monotone families.
- A. Quantum relative entropies: A quantum relative entropy is required to satisfy isometric invariance, classical reduction, and positivity.
- A. Quantum relative entropies: Isometric invariance extends a quantum relative entropy consistently to non-zero positive semidefinite operators on arbitrary finite-dimensional Hilbert spaces.
- A. Quantum relative entropies: Additivity specifies the tensor-product decomposition of the relative entropy, while weak additivity concerns tensor powers.
- A. Quantum relative entropies: Monotonicity means non-increase under completely positive trace-preserving maps.
- A. Quantum relative entropies: The Umegaki relative entropy is the smallest, and the Belavkin–Staszewski relative entropy the largest, among weakly additive and monotone quantum relative entropies.
B. Barycentric R´enyi divergences
The barycentric Rényi divergence is defined variationally from a finite collection of quantum relative entropies, with monotonicity inherited when those entropies are monotone. General explicit formulas remain unavailable, and normalization matters for strong cq-additivity.
- For finite X, probability distribution P, and non-zero PSD tuple W, the barycentric Rényi divergence is defined from a collection of quantum relative entropies D^{q_x}.
- The optimization can be restricted to states whose support is contained in the joint support of the W_x with x in supp P.
- When all D^{q_x} are monotone, the barycentric divergence is a quantum extension of the multivariate classical Rényi divergence.
- No general explicit formula is known, and normalization is irrelevant for weak additivity but relevant to strong cq-additivity.
P (W) (III.5)
The Umegaki choice yields the minimal barycentric Rényi divergence and is the only previously known case with an explicit optimizer and formula, subject to support conditions.
- The Umegaki-based divergence is called the minimal barycentric Rényi divergence, while the maximal choice is defined analogously through the stated ordering.
- Before this work, an explicit formula and optimal ω were known only when every component relative entropy equals the Umegaki relative entropy.
- In the Umegaki case, the optimizer is associated with the P-weighted log-Euclidean geometric mean of W.
- When the relevant support is zero, any state is optimal and the value is +∞; otherwise the stated optimizer characterization applies under the support condition.
P (W)/ Tr GLE
The displayed derivation decomposes trace-log terms involving tensor powers and combines intermediate identities to obtain the target relation.
- The derivation expands trace expressions involving ω_n, tensor powers of ω, and tensor powers of W.
- Support and tensor-product identities are used in an intermediate equality before the remaining P-weighted trace-log terms are handled.
- Combining equations (III.8) and (III.9) yields equation (III.7).
C. Additivity and regularization
The section defines several additivity notions, records prior Umegaki-specific strong cq-additivity, and introduces regularization to establish weak additivity more generally.
- The section distinguishes additive, cq-additive, strongly cq-additive, and weakly additive behavior for tensor-product constructions.
- Strong cq-additivity implies both cq-additivity and ordinary additivity for barycentric Rényi divergences.
- Previously, general additivity was unknown beyond the Umegaki example, including weak additivity for a specific relative entropy and non-Dirac P.
- If the component relative entropies are weakly subadditive, the barycentric divergence is weakly subadditive, and its regularized version is weakly additive by a Fekete-lemma argument.
- The paper states that the regularized divergence has an explicit expression and will coincide with the minimal divergence except in the all-Umegaki case distinction described in the section.
- No explicit formula is known for the unregularized divergence, at least on tuples of invertible PSD operators.
D. Regularized barycentric R´enyi divergences for equally supported arguments
For equally supported arguments, the section develops universal symmetric-state tools to compare regularized divergences with the minimal barycentric Rényi divergence and establishes the resulting equality under monotone, additive relative entropies.
- D. Regularized barycentric Rényi divergences for equally supported arguments: Universal symmetric states are symmetric, commute with every symmetric state, and exist with polynomial dimension-dependent overhead.These states provide the auxiliary sequence used in the regularization argument.
- D. Regularized barycentric Rényi divergences for equally supported arguments: The log-Euclidean interpolation ω_n,t between ω_n and Ω_n satisfies an exact logarithmic trace identity involving the normalization constant c_n,t.This identity is used to relate the interpolated state to the component states in the proof.
- D. Regularized barycentric Rényi divergences for equally supported arguments: The maximal and Umegaki relative entropies differ by a nonnegative quantity bounded by terms that grow at most linearly in n plus logarithmic state-dependent corrections.The comparison is the key estimate used to control the regularized construction.
- D. Regularized barycentric Rényi divergences for equally supported arguments: For monotone and additive relative entropies, the constructed bounds are obtained by taking n to infinity and then t down to zero.The proof combines additivity, monotonicity, and the interpolation estimates to establish the limiting inequalities.
- D. Regularized barycentric Rényi divergences for equally supported arguments: Theorem III.10 concludes the relevant equality for equally supported arguments under the stated positivity and monotonicity assumptions.Its proof is immediate from Lemma III.9 and the preceding equation.
E. Regularized maximal barycentric R´enyi divergences
The maximal barycentric Rényi divergence is extended to absolutely continuous components, with the construction based on the part of each W_x supported by the aggregate operator.
- E. Regularized maximal barycentric R´enyi divergences: Each W_x is decomposed using its absolutely continuous part with respect to the aggregate operator W̄.The associated quantities are used to formulate the maximal barycentric divergence for general supports.
P (f W). (III.27)
Under monotone and additive relative entropies, the regularized expression is identified with the barycentric quantity built from the absolutely continuous components.
- P (f W). (III.27): Theorem III.11 establishes the regularized barycentric divergence in terms of the absolutely continuous components fW_x under monotone and additive relative entropies.The theorem applies to finite index sets and nonzero positive semidefinite operator collections satisfying the stated support condition.
P (W) (III.28)
The section presents the displayed relation “P (c log f Wx)W 0” and states that its proof follows immediately from earlier results.
- The passage identifies the displayed statement as a proved result rather than providing additional derivation.
- The proof is obtained immediately from Lemma III.9, Theorem III.10, and equation (III.27).