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.

1.
E.
Nelson
, “
A quartic interaction in two dimensions
,” in
Mathematical Theory of Elementary Particles
(
MIT Press
,
1964
), pp.
69
73
.
2.
B.
Simon
and
R.
Høegh-Krohn
, “
Hypercontractive semigroups and two dimensional self-coupled Bose fields
,”
J. Funct. Anal.
9
(
2
),
121
180
(
1972
).
3.
L.
Gross
, “
Existence and uniqueness of physical ground states
,”
J. Funct. Anal.
10
(
1
),
52
109
(
1972
).
4.
A.
Bonami
, “
Étude des coefficients de Fourier des fonctions de Lp(G)
,”
Ann. l’Institut Fourier
20
(
2
),
335
402
(
1970
).
5.
W.
Beckner
, “
Inequalities in Fourier analysis
,”
Ann. Math.
102
(
1
),
159
182
(
1975
).
6.
E. B.
Davies
,
L.
Gross
, and
B.
Simon
,
Hypercontractivity: A Bibliographic Review
(
Cambridge University Press
,
1992
), pp.
370
389
.
7.
L.
Gross
, “
Logarithmic sobolev inequalities—A survey
,” in
Lecture Notes in Mathematics
(
Springer Nature
,
1978
), pp.
196
203
.
8.
L.
Gross
, “
Logarithmic sobolev inequalities and contractivity properties of semigroups
,” in
Dirichlet Forms
(
Springer
,
1993
), pp.
54
88
.
9.
D.
Bakry
, “
L’hypercontractivité et son utilisation en théorie des semigroupes
,” in
Lectures on Probability Theory
(
Springer
,
1994
), pp.
1
114
.
10.
A.
Guionnet
and
B.
Zegarlińksi
, “
Lectures on logarithmic Sobolev inequalities
,” in
Lecture Notes in Mathematics
(
Springer
,
2003
), pp.
1
134
.
11.
L.
Gross
,
Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys
(
Princeton University Press
,
2014
), Chap. 2, pp.
45
73
.
12.
D.
Bakry
,
I.
Gentil
, and
M.
Ledoux
,
Analysis and Geometry of Markov Diffusion Operators
(
Springer International Publishing
,
2013
).
13.
P.
Federbush
, “
Partially alternate derivation of a result of Nelson
,”
J. Math. Phys.
10
(
1
),
50
(
1969
).
14.
L.
Gross
, “
Logarithmic Sobolev inequalities
,”
Am. J. Math.
97
(
4
),
1061
(
1975
).
15.
C.
Ané
,
S.
Blachère
,
P.
Fougères
,
I.
Gentil
,
F.
Malrieu
,
C.
Roberto
, and
G.
Scheffer
,
Sur les inégalités de Sobolev logarithmiques, ser. Panoramas et Synthéses
(
Société Mathématique de France
,
Paris
,
2010
), Vol. 10 (in French), Available: http://www.ams.org/bookstore-getitem/item=PASY-10.
16.
M.
Raginsky
, “
Concentration of measure inequalities in information theory, communications, and coding
,”
Found. Trends® Commun. Inf. Theory
10
(
1-2
),
1
247
(
2013
).
17.
M.
Raginsky
, “
Strong data processing inequalities and Φ-Sobolev inequalities for discrete channels
,”
IEEE Trans. Inf. Theory
62
(
6
),
3355
3389
(
2016
).
18.
J.
Kahn
,
G.
Kalai
, and
N.
Linial
, “
The influence of variables on Boolean functions
,” in
29th Annual Symposium on Foundations of Computer Science (FOCS)
(
IEEE
,
1988
), pp.
68
80
.
19.
A.
Ben-Aroya
,
O.
Regev
, and
R.
de Wolf
, “
A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs
,” in
2008 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS)
(
IEEE
,
2008
), Vol. 10, pp.
477
486
.
20.
A.
Montanaro
and
T. J.
Osborne
, “
Quantum Boolean functions
,”
Chicago J. Theor. Comput. Sci.
2010
,
1
45
.
21.
B.
Zegarlinski
, “
Log-Sobolev inequalities for infinite one dimensional lattice systems
,”
Commun. Math. Phys.
133
(
1
),
147
162
(
1990
).
22.
P.
Diaconis
and
L.
Saloff-Coste
, “
Logarithmic Sobolev inequalities for finite Markov chains
,”
Ann. Appl. Probab.
6
(
3
),
695
750
(
1996
).
23.
E. A.
Carlen
and
E. H.
Lieb
, “
Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities
,”
Commun. Math. Phys.
155
(
1
),
27
46
(
1993
).
24.
P.
Biane
, “
Free hypercontractivity
,”
Commun. Math. Phys.
184
(
2
),
457
474
(
1997
).
25.
R.
Olkiewicz
and
B.
Zegarlinski
, “
Hypercontractivity in noncommutative Lp spaces
,”
J. Funct. Anal.
161
(
1
),
246
285
(
1999
).
26.
T.
Bodineau
and
B.
Zegarlinski
, “
Hypercontractivity via spectral theory
,”
Infinite Dimens. Anal. Quantum Probab. Relat. Top.
03
(
01
),
15
31
(
2000
).
27.
R.
Carbone
, “
Optimal Log-Sobolev inequality and hypercontractivity for positive semigroups on M2(C)
,”
Infinite Dimensional Analysis Quantum Probab. Relat. Top.
07
(
03
),
317
335
(
2004
).
28.
R.
Carbone
and
E.
Sasso
, “
Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup
,”
Probab. Theory Relat Fields
140
(
3-4
),
505
522
(
2007
).
29.
M. H. A.
Al-Rashed
and
B.
Zegarliński
, “
Monotone norms and finsler structures in noncommutative spaces
,”
Infinite Dimensional Analysis Quantum Probab. Relat. Top.
17
(
04
),
1450029
(
2014
).
30.
R.
Carbone
and
A.
Martinelli
, “
Logarithmic Sobolev inequalities in non-commutative algebras
,”
Infinite Dimensional Analysis Quantum Probab. Relat. Top.
18
(
02
),
1550011
(
2015
).
31.
B.
Zegarliński
, “
Linear and nonlinear dissipative dynamics
,”
Rep. Math. Phys.
77
(
3
),
377
397
(
2016
).
32.
G.
Lindblad
, “
On the generators of quantum dynamical semigroups
,”
Commun. Math. Phys.
48
(
2
),
119
130
(
1976
).
33.
M. J.
Kastoryano
and
K.
Temme
, “
Quantum logarithmic Sobolev inequalities and rapid mixing
,”
J. Math. Phys.
54
(
5
),
052202
(
2013
).
34.
K.
Temme
,
M. J.
Kastoryano
,
M. B.
Ruskai
,
M. M.
Wolf
, and
F.
Verstraete
, “
The χ2-divergence and mixing times of quantum Markov processes
,”
J. Math. Phys.
51
(
12
),
122201
(
2010
).
35.
A.
Montanaro
, “
Some applications of hypercontractive inequalities in quantum information theory
,”
J. Math. Phys.
53
(
12
),
122206
(
2012
).
36.
T.
Cubitt
,
M.
Kastoryano
,
A.
Montanaro
, and
K.
Temme
, “
Quantum reverse hypercontractivity
,”
J. Math. Phys.
56
(
10
),
102204
(
2015
).
37.
A.
Müller-Hermes
,
D. S.
França
, and
M. M.
Wolf
, “
Entropy production of doubly stochastic quantum channels
,”
J. Math. Phys.
57
(
2
),
022203
(
2016
).
38.
A.
Müller-Hermes
,
D. S.
França
, and
M. M.
Wolf
, “
Relative entropy convergence for depolarizing channels
,”
J. Math. Phys.
57
(
2
),
022202
(
2016
).
39.
A.
Müller
and
D. S.
Franca
, “
Sandwiched rényi convergence for quantum evolutions
,”
Quantum
2
,
55
(
2018
).
40.
C.
King
, “
Hypercontractivity for semigroups of unital qubit channels
,”
Commun. Math. Phys.
328
(
1
),
285
301
(
2014
).
41.
S.
Beigi
and
C.
King
, “
Hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm
,”
J. Math. Phys.
57
(
1
),
015206
(
2016
).
42.
H.-C.
Cheng
,
M.-H.
Hsieh
, and
M.
Tomamichel
, “
Exponential decay of matrix Φ-entropies on Markov semigroups with applications to dynamical evolutions of quantum ensembles
,”
J. Math. Phys.
58
(
9
),
092202
(
sep 2017
).
43.
F. J.
Dyson
, “
The threefold way: Algebraic structure of symmetry groups and ensembles in quantum mechanics
,”
J. Math. Phys.
3
(
6
),
1199
(
1962
).
44.
J. D. M.
Rennie
and
N.
Srebro
, “
Fast maximum margin matrix factorization for collaborative prediction
,” in
Proceedings of the 22nd international conference on Machine learning - ICML ’05
(
ACM Press
,
2005
).
45.
M.
Yuan
and
Y.
Lin
, “
Model selection and estimation in regression with grouped variables
,”
J. R. Stat. Soc.: Ser. B
68
(
1
),
49
67
(
2006
).
46.
A.
Argyriou
,
T.
Evgeniou
, and
M.
Pontil
, “
Convex multi-task feature learning
,”
Mach. Learn.
73
(
3
),
243
272
(
2008
).
47.
J. D.
Fonseca
,
M.
Grasselli
, and
C.
Tebaldi
, “
Option pricing when correlations are stochastic: An analytical framework
,”
Rev. Deriv. Res.
10
(
2
),
151
180
(
2007
).
48.
S.
Boucheron
,
G.
Lugosi
, and
P.
Massart
,
Concentration Inequalities: A Nonasymptotic Theory of Independence
(
Oxford University Press
,
2013
).
49.
J. A.
Tropp
, “
An introduction to matrix concentration inequalities
,”
Found. Trends Mach. Learn.
8
(
1-2
),
1
230
(
2015
).
50.
H.-C.
Cheng
,
M.-H.
Hsieh
, and
P.-C.
Yeh
, “
The learnability of unknown quantum measurements
,”
Quantum Inf. Comput.
16
(
7-8
),
0615
0656
(
2016
).
51.
S. G.
Bobkov
, “
Some extremal properties of the Bernoulli distribution
,”
Theory Probab. Appl.
41
(
4
),
748
755
(
1997
).
52.
M.
Ledoux
, “
On Talagrand’s deviation inequalities for product measures
,”
ESAIM: Probab. Stat.
1
,
63
87
(
1997
).
53.
M.
Raginsky
, “
Logarithmic Sobolev inequalities and strong data processing theorems for discrete channels
,” in
2013 IEEE International Symposium on Information Theory (ISIT)
(
IEEE
,
2013
), Vol. 07, pp.
419
423
.
54.
D.
Paulin
,
L.
Mackey
, and
J. A.
Tropp
, “
Efron–Stein inequalities for random matrices
,”
Ann. Probab.
44
(
5
),
3431
3473
(
2016
).
55.
M. S.
Berger
,
Nonlinearity and Functional Analysis
(
Academic Press
,
1977
).
56.
L. V.
Kantorovich
and
G. P.
Akilov
,
Functional Analysis in Normed Spaces
(
Pergamon Press
,
New York
,
1982
).
57.
R.
Bhatia
,
Matrix Analysis
(
Springer
,
New York
,
1997
).
58.
K.
Atkinson
and
W.
Han
,
Theoretical Numerical Analysis: A Functional Analysis Framework
(
Springer International Publishing
,
2009
).
59.
N. J.
Higham
,
Functions of Matrices: Theory and Computation
(
Society for Industrial & Applied Mathematics (SIAM)
,
2008
).
60.
V. V.
Peller
, “
Hankel operators in the perturbation theory of unitary and self-adjoint operators
,”
Funct. Anal. Appl.
19
(
2
),
111
123
(
1985
).
61.
K.
Bickel
, “
Differentiating matrix functions
,”
Oper. Matrices
7
(
1
),
71
90
(
2007
).
62.
R. Y.
Chen
and
J. A.
Tropp
, “
Subadditivity of matrix φ-entropy and concentration of random matrices
,”
Electron. J. Probab.
19
(
27
),
01
(
2014
).
63.
H.-C.
Cheng
and
M.-H.
Hsieh
, “
Characterizations of matrix and operator-valued Φ-entropies, and operator Efron–Stein inequalities
,”
Proc. R. Soc. London, Ser. A
472
,
20150563
(
2016
).
64.
T. M.
Flett
,
Differential Analysis
(
Cambridge University Press
,
1980
).
65.
J.
Schur
, “
Bemerkungen zur theorie der beschränkten bilinearformen mit unendlich vielen veränderlichen
,”
J. Reine Angew. Math.
1911
(
140
),
1
28
.
66.
T.
Tao
,
Topics in Random Matrix Theory
(
American Mathematical Society
,
2012
), Vol. 132, Available: https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf.
67.
I.
Bardet
and
C.
Rouzé
, “
Hypercontractivity and logarithmic Sobolev inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates
,” e-print arXiv:1803.05379 [quant-ph].
68.
R.
Latała
and
K.
Oleszkiewicz
, “
Between Sobolev and Poincaré
,” in
Geometric Aspects of Functional Analysis
(
Springer Berlin Heidelberg
,
2000
), pp.
147
168
.
69.
C.
Cuchiero
,
D.
Filipović
,
E.
Mayerhofer
, and
J.
Teichmann
, “
Affine processes on positive semidefinite matrices
,”
Ann. Appl. Probab.
21
(
2
),
397
463
(
2011
).
70.
F.
Hiai
,
H.
Kosaki
,
D.
Petz
, and
M. B.
Ruskai
, “
Families of completely positive maps associated with monotone metrics
,”
Linear Algebra Appl.
439
(
7
),
1749
1791
(
2013
).
71.
D. R.
Farenick
and
F.
Zhou
, “
Jensen’s inequality relative to matrix-valued measures
,”
J. Math. Anal. Appl.
327
(
2
),
919
929
(
2007
).
72.

We assume that the functions considered in the paper are Fréchet differentiable. We refer readers to studies such as Refs. 60 and 61 for the conditions under which a function is Fréchet differentiable.

73.

Note that f here is a multivariate super-operator. The separate convexity means that for 0 ≤ t ≤ 1,for Y=(Y1,,Yn)Mdsan, and Ỹ(i)=(Y1,,Yi1,Yi,Yi+1,,Yn)Mdsan. The separate monotonicity is defined similarly.

tfY+(1t)fỸ(i)ftY+(1t)Ỹ(i) 
tfY+(1t)fỸ(i)ftY+(1t)Ỹ(i) 

74.

The Gaussian unitary ensembles are a family of random Hermitian matrices whose upper-triangular entries are independently and identically distributed (i.i.d.) complex standard Gaussian random variables, while the diagonal entries are i.i.d. real standard Gaussian random variables; see, e.g., Ref. 66, Sec. 2.6.

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