ML p(r)ior | Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities

### Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities

2016-01-31
1602.00233 | math-ph
We derive new characterisations of the matrix $\mathrm{\Phi}$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $\Phi$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $\Phi$-entropy functionals, and prove their subadditivity under L\"owner partial ordering. Our results demonstrate that the subadditivity of operator-valued $\Phi$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an interesting result in quantum information theory: the matrix $\Phi$-entropy of a quantum ensemble is monotone under unital quantum channels. Finally, we derive the operator Efron-Stein inequality to bound the operator-valued variance of a random matrix.

Highlights - Most important sentences from the article

# Related Articles

2019-04-05
1904.03213 | cs.DS

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input m… show more

Highlights - Most important sentences from the article

2019-02-22
1902.08428 | math.PR

It is known that the empirical spectral distribution of random matrices obtained from linear codes o… show more

Highlights - Most important sentences from the article

2019-04-18
1904.08832 | quant-ph

This paper initiates the study of a class of entangled-games, mono-state games, denoted by $(G,\psi)… show more Highlights - Most important sentences from the article 2019-02-17 1902.06305 | cs.IT In this paper I investigate the construction and the properties of the so-called marginal perspectiv… show more Highlights - Most important sentences from the article 2015-06-22 1506.06801 | quant-ph Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions a… show more Highlights - Most important sentences from the article 2018-11-01 1811.00345 | cs.IT We show that the uniform distribution minimises entropy among all symmetric log-concave distribution… show more Highlights - Most important sentences from the article 2018-10-22 1810.09503 | cs.IT This paper proposes a generalization of Tsallis entropy and Tsallis relative entropy and discusses s… show more Highlights - Most important sentences from the article 2019-04-11 1904.05510 | cs.LG Matrices satisfying the Restricted Isometry Property (RIP) play an important role in the areas of co… show more Highlights - Most important sentences from the article 2017-12-04 1712.00913 | math.CO A natural link between the notions of majorization and strongly Sperner posets is elucidated. It is … show more Highlights - Most important sentences from the article 2018-09-04 1809.00781 | cs.IT In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible… show more Highlights - Most important sentences from the article 2018-08-10 1808.05550 | math.ST In this paper we prove the concavity of the$k$-trace functions,$A\mapsto (\text{Tr}_k[\exp(H+\ln A… show more

Highlights - Most important sentences from the article

2019-04-04
1904.02314 | cs.IT

This note contributes to the understanding of generalized entropy power inequalities. Our main goal … show more

Highlights - Most important sentences from the article

2019-03-16
1903.06961 | math.NT

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability dis… show more

Highlights - Most important sentences from the article

2018-10-23
1810.09791 | math.PR

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution… show more

Highlights - Most important sentences from the article

2017-03-30
1703.10326 | cs.IT

Randomness extraction against side information is the art of distilling from a given source a key wh… show more

Highlights - Most important sentences from the article

2018-11-14
1811.06122 | cs.IT

We introduce a variant of the R\'enyi entropy definition that aligns it with the well-known H\"older… show more

Highlights - Most important sentences from the article