# New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity

@article{Cheng2015NewCO, title={New Characterizations of Matrix \$$\Phi$\$-Entropies, Poincar{\'e} and Sobolev Inequalities and an Upper Bound to Holevo Quantity}, author={Hao-Chung Cheng and Min-Hsiu Hsieh}, journal={ArXiv}, year={2015}, volume={abs/1506.06801} }

We derive new characterizations of the matrix $\Phi$-entropies introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014}]. These characterizations help to better understand the properties of matrix $\Phi$-entropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove a Poincar\'e inequality for the matrix $\Phi$-entropies. We also provide a new proof for…

## Tables from this paper

## 8 Citations

### Characterizations of matrix and operator-valued Φ-entropies, and operator Efron–Stein inequalities

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2016

The results demonstrate that the subadditivity of operator-valued Φ-entropies is equivalent to the convexity, and the operator Efron–Stein inequality is derived.

### Exponential Decay of Matrix $Φ$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles

- MathematicsArXiv
- 2015

The results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Τ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium.

### M ay 2 01 9 Quantum Sphere-Packing Bounds with Polynomial Prefactors

- Computer Science
- 2019

A finite blocklength sphere-packing bound for classical-quantum channels is established, which significantly improves Dalai’s prefactor from the order of subexponential to polynomial and the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes, indicating the sphere- packing bound is almost exact in the high rate regime.

### Quantum Sphere-Packing Bounds With Polynomial Prefactors

- Computer ScienceIEEE Transactions on Information Theory
- 2019

A finite blocklength sphere-packing bound for classical-quantum channels is established, which significantly improves Dalai’s prefactor from the order of subexponential to polynomial and the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes, indicating the sphere- packing bound is almost exact in the high rate regime.

### PR ] 3 0 O ct 2 01 9 MATRIX POINCARÉ INEQUALITIES AND CONCENTRATION

- Mathematics
- 2019

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the…

### Concentration Inequalities of Random Matrices and Solving Ptychography with a Convex Relaxation

- Computer Science
- 2017

A new concept of matrix phi-entropy is constructed and it is proved that matrix phoentropy also satisfies a subadditivity property similar to the scalar form, which is a powerful approach in developing scalar concentration inequalities.

### Sphere-packing bound for symmetric classical-quantum channels

- Computer Science2017 IEEE International Symposium on Information Theory (ISIT)
- 2017

This work provides a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel and shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial.

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