We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi [Formula: see text]-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.
We introduce the geodesic quantum
f-divergences
Dft,
0≤t≤1, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical
f-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield
(t,α)-Rényi divergences joining the Petz and geometric families.
Our first main result is data processing of
Dft for every finite operator-convex generator
f and every
t∈[0,1]. In particular, this gives DPI for the geodesic relative entropies and for the
(t,α)-Rényi divergences when
0<α<1 or
1<α≤2. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter
0≤t<1, exactly the Petz sufficiency class; at
t=1, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every
t. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the
t-quantum Markov chains are precisely the quantum Markov chains for
0≤t<1, whereas at
t=1 they are the invertible BS quantum Markov chains, a class that can be strictly larger.
We also determine parameter-monotonicity regimes and the intersections with the
(α,z) family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.
There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold P of positive definite matrices. The geodesic connecting two points on P is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on P. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮α-Rényi divergence of real order α∈R, including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α–z-Rényi divergence and ♮α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.
We study the differential, statistical, and geometrical consequences generated by the geodesic quantum
f-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter
t∈[0,1]. For an invertible state
ρ and an operator convex function
f, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric
gρ,t(f). Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter
t in this new geometry. We next show that the interpolation of relative modular operators
Γt in the reference purification of a state
σ defines a canonical finite binary experiment
(pt,qt), and introduce a log-likelihood cumulant function
Ψρ,σ(t,s), recovering the Nussbaum-Szkoła distributions at
t=0 and the Matsumoto construction at
t=1. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.
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
Dqx. 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
Dqx are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression.
Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the
Dqx are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.
The same classical f-divergence admits inequivalent extensions to quantum states. The question is whether these extensions can be obtained from a common classical variational principle with an operational interpretation. Here we generalize Matsumoto's reverse-test optimization by introducing retrodictive tests relative to a reference state. A retrodictive test compares a forward and a reverse stochastic process, realized by quantum preparation and measurement, related by Petz transposition relative to the chosen reference, and subject to prescribed quantum averages. Equivalently, it compares two reverse tests sharing a reference output, whose labels are coupled by retrodiction. For faithful states in finite dimension, we prove that the same spectral implementation attains the minimum classical f-divergence over all admissible implementations simultaneously for every operator-convex f. A flat reference yields the standard quantum f-divergences, thereby giving the Nussbaum-Szkoła distributions a variational characterization. Choosing either prescribed state as reference recovers maximal quantum f-divergences and Matsumoto's reverse-test optimization, while power references interpolate between the standard and maximal divergences along the geodesic family. In the relative-entropy case, this gives the passage from Umegaki to Belavkin-Staszewski relative entropy by changing the reference state. The same optimization extends to arbitrary quantum channels and, for channels with faithful Choi operator, gives a variational characterization of quantum entropy production.