Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincaré inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.

1.
R.
Alicki
, “
On the detailed balance condition for non-Hamiltonian systems
,”
Rep. Math. Phys.
10
(
2
),
249
258
(
1976
).
2.
L.
Ambrosio
,
N.
Gigli
, and
G.
Savaré
,
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
(
Springer Science & Business Media
,
2008
).
3.
S.
Beigi
and
C.
King
, “
Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
,”
J. Math. Phys.
57
(
1
),
015206
(
2016
).
4.
J.-D.
Benamou
and
Y.
Brenier
, “
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
,”
Numerisch. Math.
84
(
3
),
375
393
(
2000
).
5.
S.
Bobkov
and
F.
Götze
, “
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
,”
J. Funct. Anal.
163
(
1
),
1
28
(
1999
).
6.
S. G.
Bobkov
,
I.
Gentil
, and
M.
Ledoux
, “
Hypercontractivity of Hamilton-Jacobi equations
,”
J. Math. Pures Appl.
80
(
7
),
669
696
(
2001
).
7.
R.
Carbone
and
E.
Sasso
, “
Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup
,”
Probab. Theory Relat. Fields
140
(
3-4
),
505
522
(
2008
).
8.
E. A.
Carlen
and
J.
Maas
, “
An analog of the 2-Wasserstein metric in non-commutative probability under which the Fermionic Fokker–Planck equation is gradient flow for the entropy
,”
Commun. Math. Phys.
331
(
3
),
887
926
(
2014
).
9.
E. A.
Carlen
and
J.
Maas
, “
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
,”
J. Funct. Anal.
273
(
5
),
1810
1869
(
2017
).
10.
Y.
Chen
,
T. T.
Georgiou
,
L.
Ning
, and
A.
Tannenbaum
, “
Matricial wasserstein-1 distance
,”
IEEE Control Syst. Lett.
1
(
1
),
14
19
(
2017
).
11.
Y.
Chen
,
T. T.
Georgiou
, and
A.
Tannenbaum
, “
Matrix optimal mass transport: A quantum mechanical approach
,”
IEEE Trans. Auto. Control
63
(
8
) (
2018
).
12.
E.
Christensen
and
D. E.
Evans
, “
Cohomology of operator algebras and quantum dynamical semigroups
,”
J. London Math. Soc.
s2-20
(
2
),
358
368
(
1979
).
13.
T. M.
Cover
and
J. A.
Thomas
,
Elements of Information Theory
(
John Wiley & Sons
,
2012
).
14.
T.
Cubitt
,
M.
Kastoryano
,
A.
Montanaro
, and
K.
Temme
, “
Quantum reverse hypercontractivity
,”
J. Math. Phys.
56
(
10
),
102204
(
2015
).
15.
E. B.
Davies
and
J. M.
Lindsay
, “
Non-commutative symmetric Markov semigroups
,”
Math. Z.
210
(
1
),
379
411
(
1992
).
16.
P.
Delgosha
and
S.
Beigi
, “
Impossibility of local state transformation via hypercontractivity
,”
Commun. Math. Phys.
332
(
1
),
449
476
(
2014
).
17.
P.
Diaconis
and
L.
Saloff-Coste
, “
Logarithmic Sobolev inequalities for finite Markov chains
,”
Ann. Appl. Probab.
6
(
3
),
695
750
(
1996
).
18.
M.
Erbar
and
J.
Maas
, “
Ricci curvature of finite Markov chains via convexity of the entropy
,”
Arch. Ration. Mech. Anal.
206
(
3
),
997
1038
(
2012
).
19.
V.
Gorini
,
A.
Kossakowski
, and
E. C. G.
Sudarshan
, “
Complete positive dynamical semigroups of N-level systems
,”
J. Math. Phys.
17
,
821
(
1976
).
20.
N.
Gozlan
, “
A characterization of dimension free concentration in terms of transportation inequalities
,”
Ann. Probab.
37
(
6
),
2480
2498
(
2009
).
21.
M.
Gromov
and
V. D.
Milman
, “
A topological application of the isoperimetric inequality
,”
Am. J. Math.
105
(
4
),
843
854
(
1983
).
22.
M.
Hayashi
, “
Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation
,”
J. Phys. A: Math. Gen.
35
,
7689
7727
(
2002
).
23.
M.
Hayashi
,
Asymptotic Theory of Quantum Statistical Inference
(
World Scientific
,
2005
).
24.
F.
Hiai
and
H.
Kosaki
, “
Means for matrices and comparison of their norms
,”
Indiana Univ. Math. J.
48
(
3
),
899
936
(
1999
).
25.
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
).
26.
F.
Hiai
and
M.
Mosonyi
, “
Different quantum f-divergences and the reversibility of quantum operations
,”
Rev. Math. Phys.
29
(
7
),
1750023
(
2017
).
27.
M.
Junge
and
Q.
Zeng
, “
Noncommutative martingale deviation and Poincaré type inequalities with applications
,”
Probab. Theory Relat. Fields
161
(
3
),
449
507
(
2015
).
28.
M.
Kastoryano
and
K.
Temme
, “
Non-commutative Nash inequalities
,”
J. Math. Phys.
57
(
1
),
015217
(
2016
).
29.
M. J.
Kastoryano
and
K.
Temme
, “
Quantum logarithmic Sobolev inequalities and rapid mixing
,”
J. Math. Phys.
54
(
5
),
052202
(
2013
).
30.
A.
Kossakowski
,
A.
Frigerio
,
V.
Gorini
, and
M.
Verri
, “
Quantum detailed balance and KMS condition
,”
Commun. Math. Phys.
57
(
2
),
97
110
(
1977
).
31.
M.
Ledoux
, “
On Talagrand’s deviation inequalities for product measures
,”
ESAIM: Probab. Stat.
1
,
63
87
(
1997
).
32.
M.
Ledoux
and
M.
Talagrand
,
Probability in Banach Spaces: Isoperimetry and Processes
(
Springer Science & Business Media
,
2013
).
33.
A.
Lesniewski
and
M. B.
Ruskai
, “
Monotone riemannian metrics and relative entropy on noncommutative probability spaces
,”
J. Math. Phys.
40
(
11
),
5702
5724
(
1999
).
34.
G.
Lindblad
, “
Brownian motion of a quantum harmonic oscillator
,”
Rep. Math. Phys.
10
(
3
),
393
406
(
1976
).
35.
G.
Lindblad
, “
On the generators of quantum dynamical semigroups
,”
Commun. Math. Phys.
48
(
2
),
119
130
(
1976
).
36.
K.
Marton
, “
Bounding d¯-distance by informational divergence: A method to prove measure concentration
,”
Ann. Probab.
24
(
2
),
857
866
(
1996
).
37.
K.
Matsumoto
, “
A new quantum version of f-divergence
,” e-print arXiv:1311.4722 (
2013
).
38.
E.
Milman
, “
On the role of convexity in isoperimetry, spectral gap and concentration
,”
Invent. Math.
177
(
1
),
1
43
(
2009
).
39.
M.
Mittnenzweig
and
A.
Mielke
, “
An entropic gradient structure for lindblad equations and couplings of quantum systems to macroscopic models
,”
J. Stat. Phys.
167
(
2
),
205
233
(
2017
).
40.
A.
Montanaro
, “
Some applications of hypercontractive inequalities in quantum information theory
,”
J. Math. Phys.
53
(
12
),
122206
(
2012
).
41.
A.
Müller-Hermes
and
D. S.
Franca
, “
Sandwiched Rényi convergence for quantum evolutions
,”
Quantum
2
,
55
(
2018
).
42.
A.
Müller-Hermes
,
D. S.
França
, and
M. M.
Wolf
, “
Relative entropy convergence for depolarizing channels
,”
J. Math. Phys.
57
(
2
),
022202
(
2016
).
43.
H.
Nagaoka
,
On the Relation Between Kullback Divergence and Fisher Information: From Classical Systems to Quantum Systems
(
World Scientific Publishing Co
,
2005
), pp.
399
419
.
44.
J. R.
Norris
,
Markov Chains
(
Cambridge University Press
,
1998
).
45.
R.
Olkiewicz
and
B.
Zegarlinski
, “
Hypercontractivity in noncommutative Lp Spaces
,”
J. Funct. Anal.
161
(
1
),
246
285
(
1999
).
46.
T.
Osborne
and
A.
Winter
, A quantum generalisation of Talagrand’s inequality, https://tjoresearchnotes.wordpress.com/2009/02/13/.
47.
F.
Otto
and
C.
Villani
, “
Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
,”
J. Funct. Anal.
173
(
2
),
361
400
(
2000
).
48.
A.
Pazy
,
Semigroups of Linear Operators and Applications to Partial Differential Equations
(
Springer Science & Business Media
,
2012
), Vol. 44.
49.
D.
Perez-Garcia
,
M. M.
Wolf
,
D.
Petz
, and
M. B.
Ruskai
, “
Contractivity of positive and trace-preserving maps under Lp norms
,”
J. Math. Phys.
47
(
8
),
083506
(
2006
).
50.
D.
Petz
, “
Monotone metrics on matrix spaces
,”
Linear Algebra Appl.
244
,
81
96
(
1996
).
51.
D.
Petz
and
M.
Ruskai
, “
Contraction of generalized relative entropy under stochastic mappings on matrices
,”
Infinite Dimens. Anal. Quantum Probab. Relat. Top.
1
(
1
),
83
89
(
1998
).
52.
M.
Raginsky
and
I.
Sason
, “
Concentration of measure inequalities in information theory, communications, and coding
,”
Found. Trends Commun. Inf. Theory
10
(
1-2
),
1
247
(
2014
).
53.
E. K.
Ryu
,
Y.
Chen
,
W.
Li
, and
S.
Osher
, “
Vector and matrix optimal mass transport: Theory, algorithm, and applications
,”
SIAM J. Sci. Comp.
40
(
5
),
A3675
A3698
(
2018
).
54.
H.
Spohn
, “
Entropy production for quantum dynamical semigroups
,”
J. Math. Phys.
19
(
5
),
1227
1230
(
1978
).
55.
K.
Temme
,
M. J.
Kastoryano
,
M.
Ruskai
,
M. M.
Wolf
, and
F.
Verstraete
, “
The χ2-divergence and mixing times of quantum Markov processes
,”
J. Math. Phys.
51
(
12
),
122201
(
2010
).
56.
K.
Temme
,
F.
Pastawski
, and
M. J.
Kastoryano
, “
Hypercontractivity of quasi-free quantum semigroups
,”
J. Phys. A: Math. Theor.
47
(
40
),
405303
(
2014
).
57.
C.
Villani
,
Optimal Transport, Old and New
(
Springer Science & Business Media
,
2008
), Vol. 338.
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