# A new quantum version of f-divergence

@article{Matsumoto2013ANQ,
title={A new quantum version of f-divergence},
author={Keiji Matsumoto},
journal={arXiv: Quantum Physics},
year={2013}
}
This paper proposes and studies new quantum version of $f$-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version ($\mathrm{D}_{f}^{\max}$, below), defined as the solution to an optimization problem, or the minimum classical $f$- divergence necessary to generate a given pair of quantum… Expand
Different quantum f-divergences and the reversibility of quantum operations
• Mathematics, Physics
• 2016
This paper compares the standard and the maximal $f-divergences regarding their ability to detect the reversibility of quantum operations, and studies the monotonicity of the Renyi divergences under the special class of bistochastic maps that leave one of the arguments of theRenyi divergence invariant. Expand On variational expressions for quantum relative entropies • Mathematics, Physics • ArXiv • 2015 A new variational expression is created for the measured Rényi relative entropy, which is exploited to show that certain lower bounds on quantum conditional mutual information are superadditive. Expand Reversibility of distance mesures of states with some focus on total variation distance. Consider a classical system, which is in the state described by probability distribution$p$or$q$, and embed these classical informations into quantum system by a physical map$\Gamma$,Expand Relations between different quantum R\'enyi divergences. Quantum generalizations of Renyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to beExpand Optimized Quantum F-Divergences • M. Wilde • Computer Science, Physics • 2018 IEEE International Symposium on Information Theory (ISIT) • 2018 The optimized quantum$f-divergence is introduced as a related generalization of quantum relative entropy and it is proved that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Expand
Quantum Hellinger distances revisited
• Mathematics, Physics
• 2019
This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce theExpand
Geometric Rényi Divergence and its Applications in Quantum Channel Capacities
• Computer Science, Mathematics
• ArXiv
• 2019
A chain rule inequality that immediately implies the "amortization collapse" for the geometric R\'enyi divergence is proved, addressing an open question by Berta et al. in the area of quantum channel discrimination. Expand
Optimized quantum f-divergences and data processing
The optimized quantum f-divergence is introduced as a related generalization of quantum relative entropy and it is proved that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Expand
Concentration of quantum states from quantum functional and Talagrand inequalities
• Mathematics
• 2017
Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincar\'{e} inequalities) have found widespread application in the study of the behavior of ergodic quantum Markov semigroups. TheExpand
The α → 1 limit of the Sharp Quantum Rényi Divergence
Fawzi and Fawzi [1] recently defined the sharp Rényi divergence, D α , for α ∈ (1,∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channelExpand

#### References

SHOWING 1-10 OF 26 REFERENCES
On maximization of measured $f$-divergence between a given pair of quantum states
This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probabilityExpand
Quantum f-divergences and error correction
• Mathematics, Physics
• 2010
It is shown that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex, and an integral representation for operator conveX functions on the positive half-line is provided, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. Expand
Reverse test and quantum analogue of classical fidelity and generalized fidelity
The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic.Expand
REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONS
We give a list of equivalent conditions for reversibility of the adjoint of a unital Schwarz map, with respect to a set of quantum states. A large class of such conditions is given by preservation ofExpand
Reverse estimation theory, Complementality between RLD and SLD, and monotone distances
Many problems in quantum information theory can be vied as interconversion between resources. In this talk, we apply this view point to state estimation theory, motivated by the followingExpand
The proper formula for relative entropy and its asymptotics in quantum probability
• Mathematics
• 1991
Umegaki's relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andDϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesisExpand
Affine connections, duality and divergences for a von Neumann algebra
On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convexExpand
Characterization of several kinds of quantum analogues of relative entropy
• M. Hayashi
• Mathematics, Computer Science
• Quantum Inf. Comput.
• 2006
This paper characterize these quantum analogues of relative entropy from information geometrical viewpoint and considers the naturalness of quantum relative entropy among these analogues. Expand
Deterministic quantum state transformations
Abstract We derive necessary conditions for the existence of a completely positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another.Expand
Introduction to Tensor Products of Banach Spaces
This volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish toExpand