A set of functional inequalities—called Nash inequalities—are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative 𝕃p spaces, where their relationship to Poincaré and log-Sobolev inequalities is fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.

1.
Beigi
,
S.
, “
Sandwiched Rényi divergence satisfies data processing inequality
,”
J. Math. Phys.
54
(
12
),
122202
(
2013
).
2.
Carbone
,
R.
and
Sasso
,
E.
, “
Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup
,”
Probab. Theory Relat. Fields
140
(
3-4
),
505
522
(
2008
).
3.
Carlen
,
E. A.
,
Kusuoka
,
S.
, and
Stroock
,
D. W.
, “
Upper bounds for symmetric Markov transition functions
,” in
Annales de l’IHP Probabilités et Statistiques
(
Gauthier-Villars
,
1987
), Vol.
23
, pp.
245
287
.
4.
Cubitt
,
T. S.
,
Lucia
,
A.
,
Michalakis
,
S.
, and
Perez-Garcia
,
D.
, “
Stability of local quantum dissipative systems
,”
Comm. Math. Phys.
337
,
1275
(
2015
); preprint arXiv:1303.4744 (
2013
).
5.
Davies
,
E. B.
,
Heat Kernels and Spectral Theory
(
Cambridge University Press
,
1990
), Vol.
92
.
6.
Davies
,
E. B.
and
Simon
,
B.
, “
Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians
,”
J. Funct. Anal.
59
(
2
),
335
395
(
1984
).
7.
Diaconis
,
P.
and
Saloff-Coste
,
L.
, “
Nash inequalities for finite Markov chains
,”
J. Theor. Probab.
9
(
2
),
459
510
(
1996
).
8.
Eisler
,
V.
, “
Crossover between ballistic and diffusive transport: The quantum exclusion process
,”
J. Stat. Mech.: Theory Exp.
2011
(
06
),
P06007
.
9.
Gross
,
L.
, “
Logarithmic Sobolev inequalities
,”
Am. J. Math.
97
,
1061
1083
(
1975
).
10.
Kastoryano
,
M. J.
and
Brandao
,
F. G. S. L.
, “
Quantum Gibbs samplers: The commuting case
,” preprint arXiv:1409.3435 (
2014
).
11.
Kastoryano
,
M. J.
and
Eisert
,
J.
, “
Rapid mixing implies exponential decay of correlations
,”
J. Math. Phys.
54
(
10
),
102201
(
2013
).
12.
Kastoryano
,
M. J.
,
Reeb
,
D.
, and
Wolf
,
M. M.
, “
A cutoff phenomenon for quantum Markov chains
,”
J. Phys. A: Math. Theor.
45
(
7
),
075307
(
2012
).
13.
Kastoryano
,
M. J.
and
Temme
,
K.
, “
Quantum logarithmic Sobolev inequalities and rapid mixing
,”
J. Math. Phys.
54
,
052202
(
2013
); e-print arXiv:1207.3261.
14.
King
,
C.
, “
Additivity for unital qubit channels
,”
J. Math. Phys.
43
(
10
),
4641
4653
(
2002
).
15.
King
,
C.
, “
Hypercontractivity for semigroups of unital qubit channels
,”
Commun. Math. Phys.
328
(
1
),
285
301
(
2014
).
16.
King
,
C.
and
Ruskai
,
M. B.
, “
Minimal entropy of states emerging from noisy quantum channels
,”
IEEE Trans. Inf. Theory
47
(
1
),
192
209
(
2001
).
17.
Montanaro
,
A.
and
Osborne
,
T. J.
, “
Quantum boolean functions
,” preprint arXiv:0810.2435 (
2008
).
18.
Muller-Hermes
,
A.
,
Reeb
,
D.
, and
Wolf
,
M. M.
, “
Quantum subdivision capacities and continuous-time quantum coding
,”
IEEE Trans. Inf. Theory
61
,
565
(
2014
).
19.
Pisier
,
G.
and
Xu
,
Q.
, “
Non-commutative Lp-spaces
,” in
Handbook of the Geometry of Banach Spaces
(
Elsevier
,
2003
), Vol.
2
, pp.
1459
1517
.
20.
Saloff-Coste
,
L.
, “
Lectures on finite Markov chains
,” in
Lectures on Probability Theory and Statistics
(
Springer
,
1997
), pp.
301
413
.
21.
Temme
,
K.
, “
Thermalization time bounds for Pauli stabilizer Hamiltonians
,” preprint arXiv:1412.2858 (
2014
).
22.
Temme
,
K.
,
Kastoryano
,
M. J.
,
Ruskai
,
M. B.
,
Wolf
,
M. M.
, and
Verstraete
,
F.
, “
The χ2-divergence and mixing times of quantum Markov processes
,”
J. Math. Phys.
51
(
12
),
122201
(
2010
).
23.
Temme
,
K.
,
Pastawski
,
F.
, and
Kastoryano
,
M. J.
, “
Hypercontractivity of quasi-free quantum semigroups
,”
J. Phys. A: Math. Theor.
47
(
40
),
405303
(
2014
).
24.
Varopoulos
,
N. T.
, “
Hardy-Littlewood theory for semigroups
,”
J. Funct. Anal.
63
(
2
),
240
260
(
1985
).
25.
Watrous
,
J.
, “
Notes on super-operator norms induced by Schatten norms
,” preprint arXiv:quant-ph/0411077 (
2004
).
26.
Zegarlinski
,
B.
, “
Hypercontractivity in non-commutative Lp spaces
,” in
Stochastic Processes, Physics, and Geometry, Max Planck Institute for Mathematics in the Sciences, Leipzig, January 18-22, 1999
(
American Mathematical Society
,
2000
), Vol.
28
, p.
323
.
27.

St will act on observables (the Heisenberg picture), and St on states (the Schrödinger picture). The conjugation is with respect to the Hilbert-Schmidt inner product.

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