Quantum entropies, relative entropies, and related preserver problems

Abstract A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of $L_1$ L 1 -distance.Abstract A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of $$L_1$$ L 1 -distance.

arXiv:2303.11707Letters in Mathematical PhysicsFeb 16, 202412 citations

Quantum entropies

Sarah Chehade, Anna Vershynina

ScholarpediaJan 1, 201919 citations

Reversibility of stochastic maps via quantum divergences

Fumio Hiai, Milán Mosonyi

We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum $f$ -divergences, sandwiched R\'enyi divergences, and $\alpha$ - $z$ -R\'enyi relative entropies.We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum $f$-divergences, sandwiched R\'enyi divergences, and $\alpha$-$z$-R\'enyi relative entropies.

Apr 11, 20165 citations

Maximal Rényi Relative Entropy for $α>2$ Maximal Rényi Relative Entropy for $α>2$

Roberto Rubboli

Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for $α\in[0,1/2)$ and the sandwiched Rényi relative entropy for $α\geq 1/2$ . In contrast, the maximal extension had previously been identified only for $α\in[0,2]$ , where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for $α>2$ , the maximal extension is given by the $α$ - $z$ Rényi relative entropy with $z=α-1$ . As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for $α\in[0,1/2)$ and the sandwiched Rényi relative entropy for $α\geq 1/2$. In contrast, the maximal extension had previously been identified only for $α\in[0,2]$, where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for $α>2$, the maximal extension is given by the $α$-$z$ Rényi relative entropy with $z=α-1$. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.

quant-pharXiv:2608.01124Aug 4, 20260 citations

Classical and Quantum Entropies:

Fabio Benatti

Princeton University Press eBooksJul 18, 20180 citations

L_1

L 1 -distance.

We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum

f

-divergences, sandwiched R\'enyi divergences, and

\alpha

z

-R\'enyi relative entropies.

Maximal Rényi Relative Entropy for

α>2

α\in[0,1/2)

and the sandwiched Rényi relative entropy for

α\geq 1/2

. In contrast, the maximal extension had previously been identified only for

α\in[0,2]

, where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for

α>2

, the maximal extension is given by the

α

z

Rényi relative entropy with

z=α-1

. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.