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

Cheng, Hao-Chung; Hsieh, Min-Hsiu

doi:10.1098/rspa.2015.0563

Cited by 10 publications

(6 citation statements)

References 42 publications

(89 reference statements)

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“…With the exception of few, this limited matrix concentration inequalities to dealing with independent random variables and to sums of independent random matrices. Several papers were devoted to properly defining the matrix entropy and establishing its basic properties such as the subadditivity [5,6,7]. However, as was noted in [6], it is not clear in general how to derive concentration inequalities from matrix functional inequalities, such as log-Sobolev inequality.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Matrix Poincaré inequalities and concentration

Aoun¹,

Banna²,

Youssef³

2019

Preprint

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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 associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Matrix Poincaré inequalities and concentration

Aoun¹,

Banna²,

Youssef³

2019

Preprint

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“…We then extend the matrix Poincaré inequality to Gaussian distribution (called matrix Gaussian Poincaré inequality, Theorem 4). These results rely on the matrix Efron-Stein inequality, which was first proven in our previous work [63,Theorem 5.1].…”

Section: Matrix Functional Inequalitiesmentioning

confidence: 98%

“…In the following, we recall the result from our previous work [63] and Eq. ( 7) to present a matrix Efron-Stein inequality, which plays a major role in the matrix Poincaré inequality.…”

Section: Matrix Functional Inequalitiesmentioning

confidence: 99%

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Cheng

Hsieh

2019

Journal of Mathematical Physics

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Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative Lp spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among many other applications. In this paper, we consider yet another fundamental setting -matrix-valued functions -and prove new Sobolev-type inequalities for them. Our technical contributions are two-fold: (i) we establish a series of matrix Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and (ii) we derive Φ-Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and for those with Gaussian distributions. Our results recover the corresponding classical inequalities (i.e. real-valued functions) when the matrix has one dimension. Finally, as an application of our technical outcomes, we derive the upper bounds for a fundamental entropic quantity -the Holevo quantity -in quantum information science since classical-quantum channels are a special instance of matrix-valued functions. This is obtained through the equivalence between the constants in the strong data processing inequality and the Φ-Sobolev inequality.16

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“…(66) 2 We note that the Fréchet derivative of functions involving matrices has other applications in quantum information theory; see e.g. [35,36,37].…”

Section: Appendix a A Tight Concentration Inequalitymentioning

confidence: 99%

Sphere-packing bound for symmetric classical-quantum channels

Cheng

Hsieh

Tomamichel

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial. This established pre-factor is essentially optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.

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

Cited by 10 publications

References 42 publications

Matrix Poincaré inequalities and concentration

Matrix Poincaré inequalities and concentration

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Sphere-packing bound for symmetric classical-quantum channels

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