Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators . By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra , one can predict the asymptotic dynamics for a class of operations.
REFERENCES
1.
A. S.
Holevo
and V.
Giovanetti
, “Quantum channels and their entropic characteristics
,” Rep. Prog. Phys.
75
, 046001
(2012
).2.
M.
Wolf
, Quantum channels and operations, Lecture notes available at: http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf, 2012
.3.
R.
Alicki
and K.
Lendi
, Quantum Dynamical Semigroups and Applications
(Springer Verlag
, Berlin
, 1987
), ISBN: 978-0-3871-8276-6.4.
V.
Gorini
, A.
Kossakowski
, and E. C. G.
Sudarshan
, “Completely positive semigroups of N-level systems
,” J. Math. Phys.
17
(5
), 821
(1976
).5.
G.
Lindblad
, “On the generators of quantum dynamical semigroups
,” Commun. Math. Phys.
48
(2
), 119
(1976
).6.
D.
Chruściński
and S.
Pascazio
, “A brief history of the GKLS equation
,” Open Syst. Inf. Dyn.
24
, 1740001
(2017
).7.
L.
Accardi
, “Nonrelativistic quantum mechanics as a noncommutative Markov process
,” Adv. Math.
20
, 329
(1976
).8.
O.
Szehr
, D.
Reeb
, and M. M.
Wolf
, “Spectral convergence bounds for classical and quantum Markov processes
,” Commun. Math. Phys.
333
, 565
(2015
).9.
I.
Nechita
and C.
Pellegrini
, “Random repeated quantum interactions and random invariant states
,” Probab. Theory Relat. Fields
152
, 299
(2012
).10.
D.
Burgarth
, V.
Giovannetti
, A. N.
Kato
, and K.
Yuasa
, “Quantum estimation via sequential measurements
,” New J. Phys.
17
, 113055
(2015
).11.
J.
Novotny
, G.
Alber
, and I.
Jex
, “Asymptotic properties of quantum Markov chains
,” J. Phys. A: Math. Theor.
45
, 485301
(2012
).12.
D.
Burghart
, G.
Chiribella
, V.
Giovanetti
, P.
Perinotti
, and K.
Yuasa
, “Ergodic and mixing quantum channels in finite dimensions
,” New J. Phys.
15
, 073045
(2013
).13.
M. D.
Choi
, “Completely positive linear maps on complex matrices
,” Linear Algorithm Appl.
10
, 285
(1975
).14.
D. R.
Farenick
, “Irreducible positive linear maps on operator algebras
,” Proc. AMS
124
, 3381
(1996
).15.
D.
Evans
and R.
Hoegh-Krohn
, “Spectral properties of positive maps on C⋆-algebras
,” J. London Math. Soc.
s2-17
, 345
(1978
).16.
U.
Groh
, “The peripheral point spectrum of Schwarz operators on C⋆-algebras
,” Math. Z.
176
, 311
(1981
).17.
V.
Lomonosov
and P.
Rosenthal
, “The simplest proof of Burnside’s theorem on matrix algebras
,” Linear Algebra Appl.
383
, 45
(2004
).18.
M.
Sanz
, D.
Perez-Garcia
, M. M.
Wolf
, and J. I.
Cirac
, “A quantum version of Wieleandt’s inequality
,” IEEE Trans. Inf. Theory
56
, 4668
(2010
).19.
G. P.
Barker
, L. Q.
Eifler
, and T. P.
Kezlan
, “A non-commutative spectral theorem
,” Linear Algebra Appl.
20
, 95
(1978
).20.
D. W.
Kribs
, “Quantum channels, wavelets, dilations and representations of On
,” Proc. Edinburgh Math. Soc.
46
, 421
(2003
).21.
M. M.
Wolf
and J. I.
Cirac
, “Dividing quantum channels
,” Commun. Math. Phys.
279
, 147
(2008
).22.
M. M.
Wolf
and D.
Perez-Garcia
, “The inverse eigenvalue problems for quantum channels
,” preprint arXiv:1005.4545.23.
J.
Glück
, “On the peripheral spectrum of positive operators
,” Positivity
20
, 307
(2016
).24.
M.
Rahaman
, “Multiplicative properties of quantum channels
,” J. Phys. A: Math. Theor.
50
, 345302
(2018
).25.
H. P.
Lotz
, “Über das Spektrum positiver Operatoren
,” Math. Z.
108
, 15
(1968
).26.
V.
Paulsen
, Completely Bounded Maps and Operator Algebras
(Cambridge University Press
, 2002
).27.
D.
Shemesh
, “Common eigenvectors of two matrices
,” Linear Algebra Appl.
62
, 11
(1984
).28.
S. A.
Amitsur
and J.
Levitzki
, “Minimal identities for algebras
,” Proc. AMS
1
, 449
(1950
).29.
G.
Pastuszak
and A.
Jamiołkowski
, “Common reducing unitary subspaces and decoherence in quantum systems
,” Electron. J. Linear Algebra
30
, 253
(2015
).30.
C. J.
Pappacena
, “An upper bound for the length of a finite-dimensional algebra
,” J. Algebra
197
, 535
(1997
).31.
A.
Paz
, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables
,” Linear Multilinear Algebra
15
, 161
(1984
).32.
M.
Fannes
, B.
Nachtergaele
, and R. F.
Werner
, “Finitely correlated states on quantum spin chains
,” Commun. Math. Phys.
144
, 490
(1992
).33.
A.
Jamiołkowski
, “On applications of PI-algebras in the analysis of quantum channels
,” Int. J. Quantum Inf.
10
, 1241007-1
–1241007-13
(2012
).34.
A.
Bluhm
, L.
Rauber
, and M. M.
Wolf
, “Quantum compression relative to a set of measurements
,” Annales Henri Poincare
19
, 1891
(2018
).© 2018 Author(s).
2018
Author(s)
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