We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.
REFERENCES
1.
E. M.
Alfsen
and F. W.
Shultz
, State Spaces of Operator Algebras
(Birkhäuser
, Boston
, 2001
).2.
S.
Amari
, “Information geometry on hierarchy of probability distributions
,” IEEE Trans. Inf. Theory
47
, 1701
–1711
(2001
).3.
S.
Amari
and H.
Nagaoka
, Methods of Information Geometry
, Translations of Mathematical Monographs
Vol. 191
(American Mathematical Society
, Providence
, 2000
).4.
K. M. R.
Audenaert
, M.
Nussbaum
, A.
Szkoła
, and F.
Verstraete
, “Asymptotic error rates in quantum hypothesis testing
,” Commun. Math. Phys.
279
, 251
–283
(2008
).5.
N.
Ay
, “An information-geometric approach to a theory of pragmatic structuring
,” Ann. Probab.
30
, 416
–436
(2002
).6.
7.
N.
Ay
, E.
Olbrich
, N.
Bertschinger
, and J.
Jost
, “A geometric approach to complexity
,” Chaos
21
, 037103
(2011
).8.
Barndorff-Nielsen
, O.
, Information and Exponential Families in Statistical Theory
(Wiley
, New York
, 1978
).9.
Baumgratz
, T.
, Gross
, D.
, Cramer
, M.
, and Plenio
, M. B.
, “Scalable reconstruction of density matrices
,” e-print arXiv:1207.0358 [quant-ph].10.
Bengtsson
, I.
, Weis
, S.
, and Życzkowski
, K.
, “Geometry of the set of mixed quantum states: An apophatic approach
,” in Proceedings of the XXX Workshop on Geometric Methods in Physics, Białowieża, 26.06–02.07.2011 (in press).11.
Bjelaković
, I.
and Szkoła
, A.
, “The data compression theorem for ergodic quantum information sources
,” Quantum. Inf. Process.
4
, 49
–63
(2005
).12.
A.
Caticha
, Entropic Inference and the Foundations of Physics
, Brazilian Chapter of the International Society for Bayesian Analysis-ISBrA
, Sao Paulo, Brazil
(2012
).13.
Csiszár
, I.
and Matúš
, F.
, “Information projections revisited
,” IEEE Trans. Inf. Theory
49
, 1474
–1490
(2003
).14.
Davidson
, K. R.
, C*-Algebras by Example
(American Mathematical Society
, Providence
, 1996
).15.
Ellis
, R.
, Entropy, Large Deviations, and Statistical Mechanics
, Classics in Mathematics
(Springer
, 2006
).16.
Erb
, I.
and Ay
, N.
, “Multi-information in the thermodynamic limit
,” J. Stat. Phys.
115
, 949
–976
(2004
).17.
Grünbaum
, B.
, Convex Polytopes
, 2nd ed. (Springer-Verlag
, New York
, 2003
).18.
Heinosaari
, T.
, Mazzarella
, L.
, and Wolf
, M. M.
, “Quantum tomography under prior information
,” e-print arXiv:1109.5478 [quant-ph].19.
R.
Horodecki
, P.
Horodecki
, M.
Horodecki
, and K.
Horodecki
, “Quantum entanglement
,” Rev. Mod. Phys.
81
, 865
–942
(2009
).20.
Ingarden
, R. S.
, Kossakowski
, A.
, and Ohya
, M.
, Information Dynamics and Open Systems
(Kluwer
, Dordrecht
, 1997
).21.
Jaynes
,E. T.
, “Information theory and statistical mechanics I/II
,” Phys. Rev.
106
, 620
–630
;Jaynes
, E. T.
, Phys. Rev.
108
, 171
–190
(1957
).22.
Kojima
, M.
, Kojima
, S.
, and Hara
, S.
, “Linear algebra for semidefinite programming
,” Sūrikaisekikenkyūsho Kōkyūroku
1004
, 1
–23
(1997
).23.
Kuperberg
, G.
, “The capacity of hybrid quantum memory
,” IEEE Trans. Inf. Theory
49
, 1465
–1473
(2003
).24.
Lieb
, E. H.
, “Convex trace functions and the Wigner-Yanase-Dyson conjecture
,” Adv. Math.
11
, 267
–288
(1973
).25.
Matsuda
, H.
, Kudo
, K.
, Kiyoshi
, N.
, Nakamura
, R.
, Yamakawa
, O.
, and Murata
, T.
, “Mutual information of Ising systems
,” Int. J. Theor. Phys.
35
, 839
–845
(1996
).26.
Matúš
, F.
, “Optimality conditions for maximizers of the information divergence from an exponential family
,” Kybernetika
43
, 731
–746
(2007
).27.
Matúš
, F.
and Rauh
, J.
, “Maximization of the information divergence from an exponential family and criticality
,” in Proceedings of IEEE International Symposium on Information Theory
(July 31–Aug. 5, 2011
).28.
Netzer
, T.
, “Spectrahedra and their shadows
,” Habilitationsschrift, Universität Leipzig (2011
).29.
Nussbaum
, M.
and Szkoła
, A.
, “An assymptotic error bound for testing multiple quantum hypothesis
,” Ann. Stat.
39
, 3211
–3233
(2011
).30.
Petz
, D.
, “Geometry of canonical correlation on the state space of a quantum system
,” J. Math. Phys.
35
, 780
–795
(1994
).31.
Petz
, D.
, “Monotone metrics on matrix spaces
,” Linear Algebr. Appl.
244
, 81
–96
(1996
).32.
Petz
, D.
, Quantum Information Theory and Quantum Statistics
, Theoretical and Mathematical Physics
(Springer-Verlag
, Berlin
, 2008
).33.
Petz
, D.
and Ruppert
, L.
, “Efficient quantum tomography needs complementary and symmetric measurements
,” Rep. Math. Phys.
69
, 161
–177
(2012
).34.
Rau
, J.
, “Inferring the Gibbs state of a small quantum system
,” Phys. Rev. A
84
, 012101
(2011
).35.
Rauh
, J.
, “Finding the maximizers of the information divergence from an exponential family
,” IEEE Trans. Inf. Theory
57
, 3236
–3247
(2011
).36.
Rockafellar
, R. T.
, Convex Analysis
(Princeton University Press
, Princeton
, 1970
).37.
Ruskai
, M. B.
, “Extremal properties of relative entropy in quantum statistical mechanics
,” Rep. Math. Phys.
26
, 143
–150
(1988
).38.
Schumacher
, B.
, “Quantum coding
,” Phys. Rev. A
51
, 2738
–2747
(1995
).39.
Shannon
, C. E.
, “A mathematical theory of communication
,” Bell Syst. Tech. J.
27
, 379
–423
(1948
).40.
Vedral
, V.
, Plenio
, M. B.
, Rippin
, M. A.
, and Knight
, P. L.
, “Quantifying entanglement
,” Phys. Rev. Lett.
78
, 2275
–2279
(1997
).41.
Wehrl
, A.
, “General properties of entropy
,” Rev. Mod. Phys.
50
, 221
–260
(1978
).42.
43.
Weis
, S.
, “Quantum convex support
,” Linear. Algebr. Appl.
435
, 3168
–3188
(2011
).44.
45.
Wichmann
, E. H.
, “Density matrices arising from incomplete measurements
,” J. Math. Phys.
4
, 884
–896
(1963
).46.
Wootters
, W. K.
and Fields
, B. D.
, “Optimal state-discrimination by mutually unbiased measurements
,” Ann. Phys.
191
, 363
–381
(1989
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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