# Relating Entropies of Quantum Channels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Lemma**

**1.**

**Proof.**

## 3. Quantum Unital Qubit Channels

**Theorem**

**1.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 4. Asymptotic Case

**Conjecture**

**1**.

**Theorem**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Devetak, I.; Winter, A. Distillation of secret key and entanglement from quantum states. Proc. R. Soc. Lond. A
**2005**, 461, 207–235. [Google Scholar] [CrossRef] [Green Version] - Berta, M.; Christandl, M.; Colbeck, R.; Renes, J.M.; Renner, R. The uncertainty principle in the presence of quantum memory. Nat. Phys.
**2010**, 6, 659. [Google Scholar] [CrossRef] - Chitambar, E.; Gour, G. Quantum resource theories. Rev. Mod. Phys.
**2019**, 91, 025001. [Google Scholar] [CrossRef] [Green Version] - Gühne, O. Characterizing entanglement via uncertainty relations. Phys. Rev. Lett.
**2004**, 92, 117903. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Oppenheim, J.; Wehner, S. The uncertainty principle determines the nonlocality of quantum mechanics. Science
**2010**, 330, 1072–1074. [Google Scholar] [CrossRef] [Green Version] - Rastegin, A.E. Separability conditions based on local fine-grained uncertainty relations. Quantum Inf. Process.
**2016**, 15, 2621–2638. [Google Scholar] [CrossRef] [Green Version] - Enríquez, M.; Puchała, Z.; Życzkowski, K. Minimal rényi–ingarden–urbanik entropy of multipartite quantum states. Entropy
**2015**, 17, 5063–5084. [Google Scholar] [CrossRef] [Green Version] - Roga, W.; Życzkowski, K.; Fannes, M. Entropic characterization of quantum operations. Int. J. Quantum Inf.
**2011**, 9, 1031–1045. [Google Scholar] [CrossRef] [Green Version] - Roga, W.; Puchała, Z.; Rudnicki, Ł.; Życzkowski, K. Entropic trade-off relations for quantum operations. Phys. Rev. A
**2013**, 87, 032308. [Google Scholar] [CrossRef] [Green Version] - Shaari, J.S.; Mancini, S. Entropic bounds for unitary testers and mutually unbiased unitary bases. Ann. Phys.
**2020**, 412, 168043. [Google Scholar] [CrossRef] - Rudnicki, Ł.; Puchała, Z.; Życzkowski, K. Strong majorization entropic uncertainty relations. Phys. Rev. A
**2014**, 89, 052115. [Google Scholar] [CrossRef] [Green Version] - Coles, P.J.; Piani, M. Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A
**2014**, 89, 022112. [Google Scholar] [CrossRef] [Green Version] - Rastegin, A.E.; Życzkowski, K. Majorization entropic uncertainty relations for quantum operations. J. Phys. A Math. Theor.
**2016**, 49, 355301. [Google Scholar] [CrossRef] [Green Version] - Kurzyk, D.; Pawela, Ł.; Puchała, Z. Conditional entropic uncertainty relations for tsallis entropies. Quantum Inf. Process.
**2018**, 17, 1–12. [Google Scholar] [CrossRef] [Green Version] - Puchała, Z.; Rudnicki, Ł.; Krawiec, A.; Życzkowski, K. Majorization uncertainty relations for mixed quantum states. J. Phys. A Math. Theor.
**2018**, 51, 175306. [Google Scholar] [CrossRef] [Green Version] - Vedral, V. The role of relative entropy in quantum information theory. Rev. Mod. Phys.
**2002**, 74, 197. [Google Scholar] [CrossRef] [Green Version] - Yuan, X. Hypothesis testing and entropies of quantum channels. Phys. Rev. A
**2019**, 99, 032317. [Google Scholar] [CrossRef] [Green Version] - Gour, G.; Wilde, M.M. Entropy of a quantum channel. Phys. Rev. Res.
**2021**, 3, 023096. [Google Scholar] [CrossRef] - Liu, Z.-W.; Winter, A. Resource theories of quantum channels and the universal role of resource erasure. arXiv
**2019**, arXiv:1904.04201. [Google Scholar] - Katariya, V.; Wilde, M.M. Geometric distinguishability measures limit quantum channel estimation and discrimination. Quantum Inf. Process.
**2021**, 20, 1–170. [Google Scholar] [CrossRef] - Fang, K.; Fawzi, O.; Renner, R.; Sutter, D. Chain rule for the quantum relative entropy. Phys. Rev. Lett.
**2020**, 124, 100501. [Google Scholar] [CrossRef] [Green Version] - Leditzky, F.; Kaur, E.; Datta, N.; Wilde, M.M. Approaches for approximate additivity of the holevo information of quantum channels. Phys. Rev. A
**2018**, 97, 012332. [Google Scholar] [CrossRef] [Green Version] - Fang, K.; Fawzi, H. Geometric rényi divergence and its applications in quantum channel capacities. Commun. Math. Phys.
**2021**, 384, 1615–1677. [Google Scholar] [CrossRef] - Umegaki, H. Conditional Expectation in an Operator Algebra, IV (Entropy and Information). In Kodai Mathematical Seminar Reports; Department of Mathematics, Tokyo Institute of Technology: Tokyo, Japan, 1962; Volume 14, pp. 59–85. [Google Scholar]
- Haber, H.E. Notes on the Matrix Exponential and Logarithm. Available online: http://scipp.ucsc.edu/~haber/webpage/MatrixExpLog.pdf (accessed on 14 June 2021).
- Nechita, I.; Puchała, Z.; Pawela, Ł.; Życzkowski, K. Almost all quantum channels are equidistant. J. Math. Phys.
**2018**, 59, 052201. [Google Scholar] [CrossRef] [Green Version] - Voiculescu, D. Multiplication of certain non-commuting random variables. J. Oper. Theory
**1987**, 18, 223–235. [Google Scholar] - Życzkowski, K.; Penson, K.A.; Nechita, I.; Collins, B. Generating random density matrices. J. Math. Phys.
**2011**, 52, 062201. [Google Scholar] [CrossRef] - Puchała, Z.; Pawela, Ł.; Życzkowski, K. Distinguishability of generic quantum states. Phys. Rev. A
**2016**, 93, 062112. [Google Scholar] [CrossRef] [Green Version] - Kukulski, R.; Nechita, I.; Pawela, Ł.; Puchała, Z.; Życzkowski, K. Generating random quantum channels. J. Math. Phys.
**2021**, 62, 062201. [Google Scholar] [CrossRef]

**Figure 1.**The quantity $D(\sigma \parallel \gamma )$ where $\sigma $ and $\gamma $ are as in Equations (37) and (38) respectively. The plots are for $\nu =\mathrm{Dir}(d,1)$ (red), $\nu =\mathrm{Dir}(2,1)$ (blue), $\nu =\mathrm{Dir}(d,2)$ (yellow) and $\nu =\delta (1/d)$ (green). The dashed line is the quantity $log\left(d\right)-\frac{1}{2}$.

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Kurzyk, D.; Pawela, Ł.; Puchała, Z.
Relating Entropies of Quantum Channels. *Entropy* **2021**, *23*, 1028.
https://doi.org/10.3390/e23081028

**AMA Style**

Kurzyk D, Pawela Ł, Puchała Z.
Relating Entropies of Quantum Channels. *Entropy*. 2021; 23(8):1028.
https://doi.org/10.3390/e23081028

**Chicago/Turabian Style**

Kurzyk, Dariusz, Łukasz Pawela, and Zbigniew Puchała.
2021. "Relating Entropies of Quantum Channels" *Entropy* 23, no. 8: 1028.
https://doi.org/10.3390/e23081028