We analyze the set of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an N-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set is shown to be log-convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels, the super-decoherence commutes with the dynamics so that decohering a quantum accessible channel, we obtain a bistochastic matrix from the set of classical maps accessible by a semigroup. Focusing on three-level systems, we investigate the geometry of the sets of quantum accessible maps, its classical counterpart, and the support of their spectra. We demonstrate that the set is not included in the set of quantum unistochastic channels, although an analogous relation holds for N = 2. The set of transition matrices obtained by super-decoherence of unistochastic channels of order N ≥ 3 is shown to be larger than the set of unistochastic matrices of this order and yields a motivation to introduce the larger sets of k-unistochastic matrices.
REFERENCES
1.
H.-P.
Breuer
and F.
Petruccione
, The Theory of Open Quantum Systems
(Oxford University Press
, 2002
).2.
3.
V.
Gorini
, A.
Kossakowski
, and E. C. G.
Sudarshan
, “Completely positive dynamical semigroups of N-level systems
,” J. Math. Phys.
17
, 821
(1976
).4.
G.
Lindblad
, “On the generators of quantum dynamical semigroups
,” Commun. Math. Phys.
48
, 119
(1976
).5.
D.
Chruściński
and S.
Pascazio
, “A brief history of the GKLS equation
,” Open Syst. Inf. Dyn.
24
, 1740001
(2017
).6.
E. C. G.
Sudarshan
, P. M.
Mathews
, and J.
Rau
, “Stochastic dynamics of quantum-mechanical systems
,” Phys. Rev.
121
, 920
(1961
).7.
M.-D.
Choi
, “Completely positive linear maps on complex matrices
,” Linear Algebra Appl.
10
, 285
(1975
).8.
K.
Życzkowski
and I.
Bengtsson
, “On duality between quantum maps and quantum states
,” Open Syst. Inf. Dyn.
11
, 3
(2004
).9.
I.
Bengtsson
and K.
Życzkowski
, Geometry of Quantum States
, 2nd ed. (Cambridge University Press
, 2017
).10.
D.
Chruściński
and F. A.
Wudarski
, “Non-Markovian random unitary qubit dynamics
,” Phys. Lett. A
377
, 1425
(2013
).11.
T. S.
Cubitt
, J.
Eisert
, and M. M.
Wolf
, “The complexity of relating quantum channels to master equations
,” Commun. Math. Phys.
310
, 383
(2012
).12.
S. N.
Filippov
, J.
Piilo
, S.
Maniscalco
, and M.
Ziman
, “Divisibility of quantum dynamical maps and collision models
,” Phys. Rev. A
96
, 032111
(2017
).13.
M. M.
Wolf
and J. I.
Cirac
, “Dividing quantum channels
,” Commun. Math. Phys.
279
, 147
(2008
).14.
M. M.
Wolf
, J.
Eisert
, T. S.
Cubitt
, and J. I.
Cirac
, “Assessing non-Markovian dynamics
,” Phys. Rev. Lett.
101
, 150402
(2008
).15.
Á.
Rivas
, S. F.
Huelga
, and M. B.
Plenio
, “Quantum non-Markovianity: Characterization, quantification and detection
,” Rep. Prog. Phys.
77
, 094001
(2014
).16.
D.
Chruściński
and F. A.
Wudarski
, “Non-Markovianity degree for random unitary evolution
,” Phys. Rev. A
91
, 012104
(2015
).17.
D.
Davalos
, M.
Ziman
, and C.
Pineda
, “Divisibility of qubit channels and dynamical maps
,” Quantum
3
, 144
(2019
).18.
Z.
Puchała
, Ł.
Rudnicki
, and K.
Życzkowski
, “Pauli semigroups and unistochastic quantum channels
,” Phys. Lett. A
383
, 2376
(2019
).19.
M.
Marcin
, M.
Kuś
, and K.
Życzkowski
, “Unitary quantum gates, perfect entanglers, and unistochastic maps
,” Phys. Rev. A
87
, 022111
(2013
).20.
K.
Siudzińska
, “Geometry of Pauli maps and Pauli channels
,” Phys. Rev. A
100
, 062331
(2019
).21.
R.
Alicki
and K.
Lendi
, Quantum Dynamical Semigroups and Applications
(Springer
, 2007
).22.
S. J.
Szarek
, E.
Werner
, and K.
Życzkowski
, “Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive
,” J. Math. Phys.
49
, 032113
(2008
).23.
V.
Jagadish
, R.
Srikanth
, and F.
Petruccione
, “Measure of positive and not completely positive single-qubit Pauli maps
,” Phys. Rev. A
99
, 022321
(2019
).24.
J. R.
Bolaños-Servin
and R.
Carbone
, “Spectral properties of circulant quantum Markov semigroups
,” Open Syst. Inf. Dyn.
21
, 1450007
(2014
).25.
J. R.
Bolaños-Servin
, R.
Carbone
, and R.
Quezada
, “Structure and block representation for circulant quantum processes
,” Infinite Dimens. Anal., Quantum Probab. Relat. Top.
22
, 1950017
(2019
).26.
M.
Schlosshauer
, “Quantum decoherence
,” Phys. Rep.
831
, 1
(2019
).27.
J. T. H.
Runnenburg
, “On Elfving’s problem of imbedding a time-discrete Markov chain in a time-continuous one for finitely many states
,” Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci.
65
, 536
(1962
).28.
E.
Davies
, “Embeddable Markov matrices
,” Electron. J. Probab.
15
, 1474
(2010
).29.
J.
Bausch
and T.
Cubitt
, “The complexity of divisibility
,” Linear Algebra Appl.
504
, 64
(2016
).30.
M.
Lostaglio
, K.
Korzekwa
, and A.
Milne
, “Markovian evolution of quantum coherence under symmetric dynamics
,” Phys. Rev. A
96
, 032109
(2017
).31.
K.
Korzekwa
and M.
Lostaglio
, “Quantum advantage in simulating stochastic processes
,” Phys. Rev. X
11
, 021019
(2021
).32.
H.
Weyl
, “Quantenmechanik und Gruppentheorie
,” Z. Phys.
46
, 1
(1927
).33.
J.
Schwinger
, “Unitary operator bases
,” Proc. Natl. Acad. Sci. U. S. A.
46
, 570
(1960
).34.
J. J.
Sylvester
, The Collected Mathematical Papers of James Joseph Sylvester
(Cambridge University Press
, 1909
).35.
L. J.
Landau
and R. F.
Streater
, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras
,” Linear Algebra Appl.
193
, 107
(1993
).36.
37.
W. J.
Culver
, “On the existence and uniqueness of the real logarithm of a matrix
,” Proc. Am. Math. Soc.
17
, 1146
(1966
).38.
K.
Życzkowski
, “Quartic quantum theory: An extension of the standard quantum mechanics
,” J. Phys. A: Math. Theor.
41
, 355302
(2008
).39.
G.
Chiribella
, G. M.
D’Ariano
, and P.
Perinotti
, “Transforming quantum operations: Quantum supermaps
,” Europhys. Lett.
83
, 30004
(2008
).40.
G.
Gour
, “Comparison of quantum channels by superchannels
,” IEEE Trans. Inf. Theory
65
, 5880
(2019
).41.
C.
Jarlskog
, “Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation
,” Phys. Rev. Lett.
55
, 1039
(1985
).42.
C.
Dunkl
and K.
Życzkowski
, “Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant
,” J. Math. Phys.
50
, 123521
(2009
).43.
I.
Bengtsson
, Å.
Ericsson
, M.
Kuś
, W.
Tadej
, and K.
Życzkowski
, “Birkhoff’s polytope and unistochastic matrices, N = 3 and N = 4
,” Commun. Math. Phys.
259
, 307
(2005
).44.
G.
Rajchel
, A.
Gąsiorowski
, and K.
Życzkowski
, “Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces
,” Math. Comput. Sci.
12
, 473
(2018
).45.
G.
Narang
and Arvind
, “Simulating a single-qubit channel using a mixed-state environment
,” Phys. Rev. A
75
, 032305
(2007
).46.
47.
V.
Jagadish
, R.
Srikanth
, and F.
Petruccione
, “Mixing Markovian Pauli channels: The general case
” Phys. Lett. A
384
(35
), 126907
(2020
).48.
S. N.
Filippov
, A. N.
Glinov
, and L.
Leppäjärvi
, “Phase covariant qubit dynamics and divisibility
,” Lobachevskii J. Math.
41
, 617
–630
(2020
).49.
K.
Korzekwa
, S.
Czachórski
, Z.
Puchała
, and K.
Życzkowski
, “Coherifying quantum channels
,” New J. Phys.
20
, 043028
(2018
).50.
D. M.
de Lachapelle
, D.
Gfeller
, and P. D. L.
Rios
, “Spectral coarse graining of matrices with application to graph theory
,” SIAM J. Matrix Anal. Appl.
1
, 53
–78
(2008
).51.
Z.
Puchała
, K.
Korzekwa
, R.
Salazar
, P.
Horodecki
, and K.
Życzkowski
, “Schur products and Schur superchannels
” (to be published).52.
D.
Chruściński
, G.
Kimura
, A.
Kossakowski
, and Y.
Shishido
, “On the universal constraints for relaxation rates for quantum dynamical semigroup
,” Phys. Rev. Lett.
(in press); arXiv:2011.10159.53.
W.
Tadej
and K.
Życzkowski
, “A concise guide to complex Hadamard matrices
,” Open Syst. Inf. Dyn.
13
, 133
(2006
).54.
A.
Winter
and D.
Yang
, “Operational resource theory of coherence
,” Phys. Rev. Lett.
116
, 120404
(2016
).55.
B.
Yadin
, J.
Ma
, D.
Girolami
, M.
Gu
, and V.
Vedral
, “Quantum processes which do not use coherence
,” Phys. Rev. X
6
, 041028
(2016
).56.
E.
Chitambar
and G.
Gour
, “Comparison of incoherent operations and measures of coherence
,” Phys. Rev. A
94
, 052336
(2016
).57.
D.
Chruściński
, V. I.
Man’ko
, G.
Marmo
, and F.
Ventriglia
, “Stochastic evolution of finite level systems: Classical vs. quantum
,” Phys. Scr.
87
, 045015
(2013
).58.
A.
Kossakowski
, “On necessary and sufficient conditions for a generator of a quantum dynamical semi-group
,” Bull. Acad. Pol. Sci., Ser. Sci., Math., Astron. Phys.
20
, 1021
(1972
).59.
K.
Siudzińska
, “Geometry of generalized Pauli channels
,” Phys. Rev. A
101
, 062323
(2020
).60.
M.
Snamina
and E. J.
Żak
, “Dynamical semigroups in the Birkhoff polytope of order 3 as a tool for analysis of quantum channels
,” Open Syst. Info. Dyn.
27
(1
), 2050001
(2020
).61.
J.
Tolar
, “On Clifford groups in quantum computing
,” J. Phys.: Conf. Ser.
1071
, 012022
(2018
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
