In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.
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.
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.
The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.
Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum
f-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative
χ2-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel
f-divergences, establishing new and strengthening previously known results.
Any reasonable measure of distinguishability of quantum states must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. We can ask about the proportion of information lost or preserved and this leads us to study contraction and expansion coefficients respectively, which can be combined into a single \emph{relative expansion coefficient}. We focus on two prominent families: (i) standard quantum
f divergences and (ii) their local (second-order) behaviour, which induces a monotone Riemannian semi-norm (that is linked to the
χ2 divergence). Building on prior work, we identify new families of
f for which the global (
f divergence) and local (Riemannian) relative expansion coefficients coincide for every pair of channels, and we clarify how exceptional such exact coincidences are. Beyond equality, we introduce an \emph{equivalence} framework that transfers qualitative properties such as strict positivity uniformly across different relative expansion coefficients. Leveraging the link between equality in the data processing inequality (DPI) and channel reversibility, we apply our framework of relative expansion coefficients to approximate recoverability of quantum information. Using our relative expansion results for primitive channels, we prove a reverse quantum Markov convergence theorem, converting positive expansion coefficients into quantitative lower bounds on the convergence rate.