Concepts and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, are identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight, we provide a rigorous approach to the generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between the Berezin quantization and Weyl quantization.

1.
Bayer
,
D.
and
Gröchenig
,
K.
, “
Time-frequency localization operators and a Berezin transform
,”
Integr. Equations Oper. Theory
82
(
1
),
95
117
(
2015
).
2.
Busch
,
P.
,
Grabowski
,
M.
, and
Lahti
,
P. J.
,
Operational Quantum Physics
, Volume 31 of Lecture Notes in Physics. New Series M: Monographs (
Springer-Verlag
,
Berlin
,
1995
).
3.
Busch
,
P.
,
Lahti
,
P.
,
Pellonpää
,
J.-P.
, and
Ylinen
,
K.
,
Quantum Measurement. Theoretical and Mathematical Physics
(
Springer
,
Cham
,
2016
).
4.
Cassinelli
,
G.
,
De Vito
,
E.
, and
Toigo
,
A.
, “
Positive operator valued measures covariant with respect to an irreducible representation
,”
J. Math. Phys.
44
(
10
),
4768
4775
(
2003
).
5.
Folland
,
G. B.
,
Harmonic Analysis in Phase Space
(
Princeton University Press
,
1989
).
6.
Graven
,
M. W. A.
, “
Banach modules over Banach algebras
,” Ph.D. thesis,
Katholieke Universiteit Nijmegen
,
The Netherlands
,
1974
.
7.
Gröchenig
,
K.
, “
Foundations of time-frequency analysis
,” in
Applied and Numerical Harmonic Analysis
(
Birkhäuser
,
2001
).
8.
Hall
,
B.
,
Quantum Theory for Mathematicians
, Volume 267 of Graduate Texts in Mathematics (
Springer
,
New York
,
2013
).
9.
Holevo
,
A. S.
, “
Covariant measurements and uncertainty relations
,”
Rep. Math. Phys.
16
(
3
),
385
400
(
1979
).
10.
Keller
,
J.
, “
The spectrogram expansion of Wigner functions
,”
Appl. Comput. Harmon. Anal.
(in press).
11.
Keyl
,
M.
,
Kiukas
,
J.
, and
Werner
,
R. F.
, “
Schwartz operators
,”
Rev. Math. Phys.
28
(
03
),
1630001
(
2016
).
12.
Kiukas
,
J.
, “
Covariant observables on a nonunimodular group
,”
J. Math. Anal. Appl.
324
(
1
),
491
503
(
2006
).
13.
Kiukas
,
J.
,
Lahti
,
P.
, and
Ylinen
,
K.
, “
Normal covariant quantization maps
,”
J. Math. Anal. Appl.
319
(
2
),
783
801
(
2006
).
14.
Kiukas
,
J.
,
Lahti
,
P.
,
Schultz
,
J.
, and
Werner
,
R.
, “
Characterization of informational completeness for covariant phase space observables
,”
J. Math. Phys.
53
(
10
),
102103
(
2012
).
15.
Klauder
,
J. R.
and
Skagerstam
,
B.-S.
, “
Generalized phase-space representation of operators
,”
J. Phys. A: Math. Theor.
40
(
9
),
2093
2105
(
2007
).
16.
Klauder
,
J. R.
and
Skagerstam
,
B.-S.
, “
Extension of Berezin-Lieb inequalities
,” in
Excursions in Harmonic Analysis, Volume 2
, Applied and Numerical Harmonic Analysis (
Birkhäuser/Springer
,
New York
,
2013
), pp.
251
266
.
17.
Landsman
,
N. P.
,
Mathematical Topics Between Classical and Quantum Mechanics
, Springer Monographs in Mathematics (
Springer
,
1999
).
18.
Luef
,
F.
and
Skrettingland
,
E.
, “
Convolutions for localization operators
,”
J. Math. Pures Appl.
(in press); e-print arXiv:1705.03253.
19.
Niculescu
,
C.
and
Persson
,
L.-E.
,
Convex Functions and Their Applications: A Contemporary Approach
, CMS Books in Mathematics (
Springer Science & Business Media
,
New York
,
2006
).
20.
Simon
,
B.
, “
The classical limit of quantum partition functions
,”
Commun. Math. Phys.
71
(
3
),
247
276
(
1980
).
21.
Werner
,
R.
, “
Quantum harmonic analysis on phase space
,”
J. Math. Phys.
25
(
5
),
1404
1411
(
1984
).
22.
Wódkiewicz
,
K.
, “
Operational approach to phase-space measurements in quantum mechanics
,”
Phys. Rev. Lett.
52
(
13
),
1064
1067
(
1984
).
23.
Wünsche
,
A.
and
Bužek
,
V.
, “
Reconstruction of quantum states from propensities
,”
Quantum Semiclassical Opt.: J. Eur. Opt. Soc., Part B
9
(
4
),
631
653
(
1997
).
You do not currently have access to this content.