Generalized relative entropy, monotone Riemannian metrics, geodesic distance, and trace distance are all known to decrease under the action of quantum channels. We give some new bounds on, and relationships between, the maximal contraction for these quantities.

1.
Ando
,
T.
,
Topics on Operator Inequalities
,
Lecture Notes (Mimeographed)
(
Hokkaido University
,
Sapporo
,
1978
).
2.
Ando
,
T.
, “
Concavity of certain maps on positive definite matrices and applications to Hadamard products
,”
Linear Algebra Appl.
26
,
203
241
(
1979
).
3.
Ando
,
T.
and
Hiai
,
F.
, “
Operator log-convex functions and operator means
,”
Math. Ann.
350
,
611
630
(
2011
).
4.
Bengtsson
,
I.
and
Zyczkowski
,
K.
,
Geometry of Quantum States: An Introduction to Quantum Entanglement
(
Cambridge University Press
,
Cambridge
,
2006
).
5.
Bhatia
,
R.
,
Matrix Analysis
(
Springer
,
New York
,
1996
).
6.
Bhatia
,
R.
,
Positive Definite Matrices
(
Princeton University Press
,
Princeton
,
2007
).
7.
Bures
,
D.
, “
An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite w-algebras
,”
Trans. Am. Math. Soc.
135
,
199
212
(
1969
).
8.
Choi
,
M.-D.
, “
Some assorted inequalities for positive linear maps on C-algebras
,”
J. Oper. Theory
4
,
271
285
(
1980
), http://www.theta.ro/jot/archive/1980-004-002/1980-004-002-006.html.
9.
Choi
,
M.-D.
,
Ruskai
,
M. B.
, and
Seneta
,
E.
, “
Equivalence of certain entropy contraction coefficients
,”
Linear Algebra Appl.
208/209
,
29
36
(
1994
).
10.
Cohen
,
J. E.
,
Iwasa
,
Y.
,
Rautu
,
Gh.
,
Ruskai
,
M. B.
,
Seneta
,
E.
, and
Zbaganu
,
Gh.
, “
Relative entropy under mappings by stochastic matrices
,”
Linear Algebra Appl.
179
,
211
235
(
1993
).
11.
Cohen
,
J. E.
,
Kemperman
,
J. H. B.
, and
Zbăganu
,
Gh.
,
Comparisons of Stochastic Matrices with Applications in Information Theory, Statistics, Economics and Population Sciences
(
Brikhäuser
,
Boston
,
1998
).
12.
Dobrushin
,
R. L.
, “
Central limit theorem for nonstationary Markov chains. I
,”
Theory Probab. Appl.
1
,
65
80
(
1956
);
Dobrushin
,
R. L.
, “
Central limit theorem for nonstationary Markov chains. II
,”
Theory Probab. Appl.
1
,
329
383
(
1956
).
13.
Donoghue
, Jr.,
W. F.
,
Monotone Matrix Functions and Analytic Continuation
(
Springer
,
Berlin-Heidelberg-New York
,
1974
).
14.
Franz
,
U.
,
Hiai
,
F.
, and
Ricard
,
É.
, “
Higher order extension of Löwner’s theory: Operator k-tone functions
,”
Trans. Am. Math. Soc.
366
,
3043
3074
(
2014
).
15.
Friedland
,
S.
and
So
,
W.
, “
On the product of matrix exponentials
,”
Linear Algebra Appl.
196
,
193
205
(
1994
).
16.
Fujiwara
,
A.
and
Algoet
,
P.
, “
One-to-one parametrization of quantum channels
,”
Phys. Rev. A
59
,
3290
3294
(
1990
).
17.
Gibilisco
,
P.
and
Isola
,
T.
, “
Wigner-Yanase information on quantum state space: The geometric approach
,”
J. Math. Phys.
44
,
3752
3762
(
2003
).
18.
Hansen
,
F.
, “
Trace functions as Laplace transforms
,”
J. Math. Phys.
47
,
043504
(
2006
).
19.
Hasegawa
,
H.
, “
α-divergence of the non-commutative information geometry
,”
Rep. Math. Phys.
33
,
87
93
(
1993
).
20.
Hiai
,
F.
, “
Equality cases in matrix norm inequalities of Golden-Thompson type
,”
Linear Multilinear Algebra
36
,
239
249
(
1994
).
21.
Hiai
,
F.
, “
Matrix analysis: Matrix monotone functions, matrix means, and majorization
,”
Interdiscip. Inf. Sci.
16
,
139
248
(
2010
).
22.
Hiai
,
F.
,
Kosaki
,
H.
,
Petz
,
D.
, and
Ruskai
,
M. B.
, “
Families of completely positive maps associated with monotone metrics
,”
Linear Algebra Appl.
439
,
1749
1791
(
2013
).
23.
Hiai
,
F.
,
Mosonyi
,
M.
,
Petz
,
D.
, and
Bény
,
C.
, “
Quantum f-divergences and error correction
,”
Rev. Math. Phys.
23
,
691
747
(
2011
).
24.
Hiai
,
F.
and
Petz
,
D.
, “
Riemannian metrics on positive definite matrices related to means
,”
Linear Algebra Appl.
430
,
3105
3130
(
2009
).
25.
Hiai
,
F.
and
Petz
,
D.
, “
From quasi-entropy to various quantum information quantities
,”
Publ. Res. Inst. Math. Sci.
48
,
525
542
(
2012
).
26.
Hiai
,
F.
and
Petz
,
D.
, “
Convexity of quasi-entropy type functions: Lieb’s and Ando’s convexity theorems revisited
,”
J. Math. Phys.
54
,
062201
(
2013
).
27.
Holevo
,
A. S.
, “
Coding theorems for quantum channels
,”
Russ. Math. Surv.
53
,
1295
1331
(
1999
); e-print arXiv:quant-ph/9809023.
28.
Horodecki
,
M.
,
Shor
,
P. W.
, and
Ruskai
,
M. B.
, “
Entanglement breaking channels
,”
Rev. Math. Phys.
15
,
629
641
(
2003
).
29.
Jenčová
,
A.
and
Ruskai
,
M. B.
, “
A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality
,”
Rev. Math. Phys.
22
,
1099
1121
(
2010
).
30.
Kadison
,
R.
, “
A generalized Schwarz inequality and algebraic invariants for operator algebras
,”
Ann. Math.
56
,
494
503
(
1952
).
31.
Kastoryano
,
M. J.
and
Temme
,
K.
, private communication (2012).
32.
Kastoryano
,
M. J.
and
Temme
,
K.
, “
Quantum logarithmic Sobolev inequalities and rapid mixing
,”
J. Math. Phys.
54
,
052202
(
2013
).
33.
Kiefer
,
J.
, “
Optimum experimental designs
,”
J. R. Stat. Soc. Ser. B (Methodol.)
21
,
272
310
(
1959
), http://www.jstor.org/stable/2983802.
34.
King
,
C.
and
Ruskai
,
M. B.
, “
Minimal entropy of states emerging from noisy quantum channels
,”
IEEE Trans. Inf. Theory
47
,
192
209
(
2001
).
35.
Kobayashi
,
S.
and
Nomizu
,
K.
,
Foundations of Differential Geometry
(
Wiley Interscience
,
New York-London
,
1963
), Vol.
1
.
36.
Kosaki
,
H.
, “
Interpolation theory and the Wigner-Yanase-Dyson-Lieb concavity
,”
Commun. Math. Phys.
87
,
315
329
(
1982
).
37.
Kraus
,
F.
, “
Über konvexe matrix funktionen
,”
Math. Z.
41
,
18
42
(
1936
).
38.
Lesniewski
,
A.
and
Ruskai
,
M. B.
, “
Monotone Riemannian metrics and relative entropy on noncommutative probability spaces
,”
J. Math. Phys.
40
,
5702
5724
(
1999
).
39.
Lieb
,
E. H.
, “
Convex trace functions and the Wigner-Yanase-Dyson conjecture
,”
Adv. Math.
11
,
267
288
(
1973
).
40.
Lieb
,
E. H.
and
Ruskai
,
M. B.
, “
Some operator inequalities of the Schwarz type
,”
Adv. Math.
12
,
269
273
(
1974
).
41.
Löwner
,
K.
, “
Über monotone matrix funktionen
,”
Math. Z.
38
,
177
216
(
1934
).
42.
Morozova
,
E. A.
and
Chentsov
,
N. N.
, “
Markov invariant geometry on state manifolds (Russian)
,”
Itogi Nauki Tekh.
36
,
69
102
(
1989
)
[
Morozova
,
E. A.
and
Chentsov
,
N. N.
,
J. Sov. Math.
56
,
2648
2669
(
1991
)].
43.
Nielsen
,
M. A.
and
Chuang
,
I. L.
,
Quantum Computation and Quantum Information
(
Cambridge University Press
,
Cambridge
,
2000
).
44.
Petz
,
D.
, “
Quasi-entropies for states of a von Neumann algebra
,”
Publ. Res. Inst. Math. Sci.
21
,
781
800
(
1985
).
45.
Petz
,
D.
, “
Quasi-entropies for finite quantum systems
,”
Rep. Math. Phys.
23
,
57
65
(
1986
).
46.
Petz
,
D.
, “
Monotone metrics on matrix spaces
,”
Linear Algebra Appl.
244
,
81
96
(
1996
).
47.
Petz
,
D.
,
Quantum Information Theory and Quantum Statistics
(
Springer
,
Berlin-Heidelberg
,
2008
).
48.
Ruskai
,
M. B.
, “
Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy
,”
Rev. Math. Phys.
6
,
1147
1161
(
1994
).
49.
Ruskai
,
M. B.
,
Szarek
,
S.
, and
Werner
,
E.
, “
An analysis of completely positive trace-preserving maps on M 2
,”
Linear Algebra Appl.
347
,
159
187
(
2002
).
50.
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
,
122201
(
2010
).
51.
Tomamichel
,
M.
,
Colbeck
,
R.
, and
Renner
,
R.
, “
A fully quantum asymptotic equipartition property
,”
IEEE Trans. Inf. Theory
55
,
5840
5847
(
2009
).
52.
Uhlmann
,
A.
, “
Density operators an arena for differential geometry
,”
Rep. Math. Phys.
33
,
253
263
(
1993
).
53.
Uhlmann
,
A.
, “
Geometric phases and related structures
,”
Rep. Math. Phys.
36
,
461
481
(
1995
).
You do not currently have access to this content.