A Collection of Pinsker-type Inequalities for Quantum Divergences
- URL: http://arxiv.org/abs/2601.10395v1
- Date: Thu, 15 Jan 2026 13:45:51 GMT
- Title: A Collection of Pinsker-type Inequalities for Quantum Divergences
- Authors: Kläre Wienecke, Gereon Koßmann, René Schwonnek,
- Abstract summary: Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states.<n>We formulate corresponding estimates for a variety of quantum and classical divergences.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including $f$-divergences like Hellinger and $χ^2$-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.
Related papers
- Error exponents of quantum state discrimination with composite correlated hypotheses [40.82628972269358]
We study the error exponents in quantum hypothesis testing between two sets of quantum states.<n>We introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states.
arXiv Detail & Related papers (2025-08-18T13:04:06Z) - Generalization Bounds for Quantum Learning via Rényi Divergences [45.45698347373077]
This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms.<n>Our primary contribution is the derivation of these bounds in terms of quantum and classical R'enyi divergences.
arXiv Detail & Related papers (2025-05-16T09:21:31Z) - Weak coupling limit for quantum systems with unbounded weakly commuting system operators [50.24983453990065]
This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles.<n>We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir are non-zero in the WCL.<n>We prove that the resulting reduced system dynamics converges to unitary dynamics with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian.
arXiv Detail & Related papers (2025-05-13T05:32:34Z) - Normal quantum channels and Markovian correlated two-qubit quantum
errors [77.34726150561087]
We study general normally'' distributed random unitary transformations.
On the one hand, a normal distribution induces a unital quantum channel.
On the other hand, the diffusive random walk defines a unital quantum process.
arXiv Detail & Related papers (2023-07-25T15:33:28Z) - Quantum Rényi and $f$-divergences from integral representations [11.74020933567308]
Smooth Csisz'ar $f$-divergences can be expressed as integrals over so-called hockey stick divergences.
We find that the R'enyi divergences defined via our new quantum $f$-divergences are not additive in general.
We derive various inequalities, including new reverse Pinsker inequalities with applications in differential privacy.
arXiv Detail & Related papers (2023-06-21T15:39:38Z) - Experimental violations of Leggett-Garg's inequalities on a quantum
computer [77.34726150561087]
We experimentally observe the violations of Leggett-Garg-Bell's inequalities on single and multi-qubit systems.
Our analysis highlights the limits of nowadays quantum platforms, showing that the above-mentioned correlation functions deviate from theoretical prediction as the number of qubits and the depth of the circuit grow.
arXiv Detail & Related papers (2021-09-06T14:35:15Z) - Dissipative evolution of quantum Gaussian states [68.8204255655161]
We derive a new model of dissipative time evolution based on unitary Lindblad operators.
As we demonstrate, the considered evolution proves useful both as a description for random scattering and as a tool in dissipator engineering.
arXiv Detail & Related papers (2021-05-26T16:03:34Z) - Monotonic multi-state quantum $f$-divergences [0.0]
We write down a class of multi-state quantum $f$-divergences and prove that they satisfy the data processing inequality.
For two states, this class includes the $(alpha,z)$-R'enyi divergences, the $f$-divergences of Petz, and the measures in citematsumoto2015new as special cases.
We conjecture that these multi-state R'enyi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.
arXiv Detail & Related papers (2021-03-17T20:10:04Z) - Sequential Estimation of Convex Divergences using Reverse Submartingales
and Exchangeable Filtrations [31.088836418378534]
We present a unified technique for sequential estimation of convex divergences between distributions.
The technical underpinnings of our approach lie in the observation that empirical convex divergences are (partially ordered) reverse submartingales.
These techniques appear to be powerful additions to the existing literature on both confidence sequences and convex divergences.
arXiv Detail & Related papers (2021-03-16T18:22:14Z) - The $\alpha \ o 1$ Limit of the Sharp Quantum R\'enyi Divergence [6.553031877558699]
Fawzi and Fawzi recently defined the sharp R'enyi divergence, $D_alpha#$, for $alpha in (1, infty)$.
By finding a new expression of the sharp divergence in terms of a minimization of the geometric R'enyi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy.
arXiv Detail & Related papers (2021-02-12T15:35:14Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.