Regularized barycentric Rényi divergences

I. INTRODUCTION

• Multivariate quantum Rényi divergences are needed to extend classical problems such as state exclusion and distribution convertibility to non-commuting quantum states.Source: §I · p. 1
• 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.Source: §I · p. 2
• Barycentric Rényi divergences instead use a variational optimization over quantum states and a finite collection of quantum relative entropies.Source: §I · p. 2
• For Umegaki relative entropies, the barycentric optimization has an explicit solution involving the P-weighted log-Euclidean geometric mean, yielding the minimal barycentric divergence.Source: §I · p. 2, P (W) Tr GLE · p. 2
• 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.Source: P (W) Tr GLE · p. 2

A. Quantum relative entropies

• A quantum relative entropy is required to satisfy isometric invariance, classical reduction, and positivity.Source: A. Quantum relative entropies · pp. 3–4
• Isometric invariance extends a quantum relative entropy consistently to non-zero positive semidefinite operators on arbitrary finite-dimensional Hilbert spaces.Source: A. Quantum relative entropies · p. 4
• Additivity specifies the tensor-product decomposition of the relative entropy, while weak additivity concerns tensor powers.Source: A. Quantum relative entropies · p. 4
• Monotonicity means non-increase under completely positive trace-preserving maps.Source: A. Quantum relative entropies · p. 4
• The Umegaki relative entropy is the smallest, and the Belavkin–Staszewski relative entropy the largest, among weakly additive and monotone quantum relative entropies.Source: A. Quantum relative entropies · p. 4

• 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}.Source: B. Barycentric R´enyi divergences · p. 4
• The optimization can be restricted to states whose support is contained in the joint support of the W_x with x in supp P.Source: B. Barycentric R´enyi divergences · p. 4
• When all D^{q_x} are monotone, the barycentric divergence is a quantum extension of the multivariate classical Rényi divergence.Source: B. Barycentric R´enyi divergences · p. 5
• No general explicit formula is known, and normalization is irrelevant for weak additivity but relevant to strong cq-additivity.Source: B. Barycentric R´enyi divergences · p. 5

• The Umegaki-based divergence is called the minimal barycentric Rényi divergence, while the maximal choice is defined analogously through the stated ordering.Source: P (W) (III.5) · p. 5
• Before this work, an explicit formula and optimal ω were known only when every component relative entropy equals the Umegaki relative entropy.Source: P (W) (III.5) · p. 5
• In the Umegaki case, the optimizer is associated with the P-weighted log-Euclidean geometric mean of W.Source: P (W) (III.5) · p. 5, P (W) Tr GLE · p. 5
• When the relevant support is zero, any state is optimal and the value is +∞; otherwise the stated optimizer characterization applies under the support condition.Source: P (W). (III.6) · p. 5

• The derivation expands trace expressions involving ω_n, tensor powers of ω, and tensor powers of W.Source: P (W)/ Tr GLE · p. 5
• Support and tensor-product identities are used in an intermediate equality before the remaining P-weighted trace-log terms are handled.Source: P (W)/ Tr GLE · pp. 5–6
• Combining equations (III.8) and (III.9) yields equation (III.7).Source: P (W)/ Tr GLE · p. 6

• The section distinguishes additive, cq-additive, strongly cq-additive, and weakly additive behavior for tensor-product constructions.Source: C. Additivity and regularization · p. 6
• Strong cq-additivity implies both cq-additivity and ordinary additivity for barycentric Rényi divergences.Source: C. Additivity and regularization · pp. 6–7
• Previously, general additivity was unknown beyond the Umegaki example, including weak additivity for a specific relative entropy and non-Dirac P.Source: C. Additivity and regularization · p. 7
• 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.Source: C. Additivity and regularization · p. 7
• 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.Source: C. Additivity and regularization · p. 7
• No explicit formula is known for the unregularized divergence, at least on tuples of invertible PSD operators.Source: C. Additivity and regularization · p. 7

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.Source: D. Regularized barycentric R´enyi… · p. 7
• 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.Source: D. Regularized barycentric R´enyi… · pp. 7–8
• 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.Source: D. Regularized barycentric R´enyi… · pp. 8–9
• 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.Source: P (W), GLE · pp. 9–10
• 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.Source: P (W), GLE · p. 10

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.Source: E. Regularized maximal barycentric… · p. 10

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.Source: P (f W). (III.27) · p. 10

• The passage identifies the displayed statement as a proved result rather than providing additional derivation.Source: P (W) (III.28) · p. 10
• The proof is obtained immediately from Lemma III.9, Theorem III.10, and equation (III.27).Source: P (W) (III.28) · p. 10