Several techniques of generating random quantum channels, which act on the set of d-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show that they are mathematically equivalent. We discuss under which conditions they give the uniform Lebesgue measure on the convex set of quantum operations and compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute the mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity, and the 2-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed, and their spectral properties are studied using the Bloch representation of a classical probability vector.
REFERENCES
1.
2.
3.
P. J.
Forrester
, Log-Gases and Random Matrices
(Princeton University Press
, Princeton
, 2010
).4.
H.-P.
Breuer
and F.
Petruccione
, The Theory of Open Quantum Systems
(Oxford Univ. Press
, 2007
).5.
R.
Alicki
and K.
Lendi
, Quantum Dynamical Semigroups and Applications
(Springer
, Berlin
, 1989
), 2nd extended edition 2000.6.
U.
Weiss
, Quantum Dissipative Systems
, 3rd ed. (World Scientific
, Singapore
, 2008
).7.
8.
H. M.
Wiseman
and G. J.
Milburn
, Quantum Measurement and Control
(Cambridge University Press
, Cambridge
, 2010
).9.
C. H.
Lewenkopf
and H. A.
Weidenmüller
, “Stochastic versus semiclassical approach to quantum chaotic scattering
,” Ann. Phys.
212
, 53
–83
(1991
).10.
F.
Haake
, F.
Izrailev
, N.
Lehmann
, D.
Saher
, and H.-J.
Sommers
, “Statistics of complex levels of random matrices for decaying systems
,” Z. Phys. B
88
, 359
(1992
).11.
H.
Schomerus
, Random Matrix Approaches to Open Quantum Systems
, Lecture Notes of the Les Houches Summer School (Oxford University Press
, 2007
).12.
V.
Gorini
, A.
Kossakowski
, and E. C. G.
Sudarshan
, “Completely positive dynamical semigroups of N-level systems
,” J. Math. Phys.
17
, 821
(1976
).13.
G.
Lindblad
, “On the generators of quantum dynamical semigroups
,” Commun. Math. Phys.
48
, 119
(1976
).14.
D.
Chruściński
and S.
Pascazio
, “A brief history of the GKLS equation
,” Open Syst. Inf. Dyn.
24
, 1740001
(2017
).15.
I.
Bengtsson
and K.
Życzkowski
, Geometry of Quantum States
(Cambridge University Press
, 2006
), 2nd extended edition, 2017.16.
W.
Bruzda
, V.
Cappellini
, H.-J.
Sommers
, and K.
Życzkowski
, “Random quantum operations
,” Phys. Lett. A
373
, 320
–324
(2009
).17.
K.
Życzkowski
, K. A.
Penson
, I.
Nechita
, and B.
Collins
, “Generating random density matrices
,” J. Math. Phys.
52
, 062201
(2011
).18.
B.
Collins
and I.
Nechita
, “Random matrix techniques in quantum information theory
,” J. Math. Phys.
57
, 015215
(2016
).19.
G.
Aubrun
and S. J.
Szarek
, Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory
(American Mathematical Society
, Providence
, 2017
).20.
M. B.
Hastings
, “Superadditivity of communication capacity using entangled inputs
,” Nat. Phys.
5
, 255
(2009
).21.
J.
Watrous
, The Theory of Quantum Information
(Cambridge University Press
, 2018
).22.
M. A.
Nielsen
and I. L.
Chuang
, Quantum Computation and Quantum Information
(Cambridge University Press
, 2010
).23.
G. W.
Anderson
, A.
Guionnet
, and O.
Zeitouni
, An Introduction to Random Matrices
(Cambridge University Press
, 2010
).24.
J. A.
Mingo
and R.
Speicher
, Free Probability and Random Matrices
(Springer
, 2017
).25.
J.
Ginibre
, “Statistical ensembles of complex, quaternion and real matrices
,” J. Math. Phys.
6
, 440
(1965
).26.
V. L.
Girko
, “The circular law
,” Teor. Veroyatn. Ee Primen.
29
, 669
(1984
).27.
M.
Pozniak
, K.
Życzkowski
, and M.
Kus
, “Composed ensembles of random unitary matrices
,” J. Phys. A: Math. Gen.
31
, 1059
(1998
).28.
F.
Mezzadri
, “How to generate random matrices from the classical compact groups
,” Not. AMS
54
, 592
–604
(2007
).29.
K.
Życzkowski
and H.-J.
Sommers
, “Truncations of random unitary matrices
,” J. Phys. A: Gen. Phys.
33
, 2045
–2057
(2000
).30.
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
–032121
(2008
).31.
A.
Lovas
and A.
Andai
, “Volume of the space of qubit-qubit channels and state transformations under random quantum channels
,” Rev. Math. Phys.
30
, 697
(2017
).32.
K.
Życzkowski
and H.-J.
Sommers
, “Induced measures in the space of mixed quantum states
,” J. Phys. A: Math. Gen.
34
, 7111
(2001
).33.
H.-J.
Sommers
and K.
Życzkowski
, “Statitical properties of random density matrices
,” J. Phys. A: Math. Gen.
37
, 8457
(2004
).34.
B.
Collins
and P.
Śniady
, “Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
,” Commun. Math. Phys.
264
, 773
–795
(2006
).35.
M.
Fukuda
, R.
König
, and I.
Nechita
, “RTNI—A symbolic integrator for Haar-random tensor networks
,” J. Phys. A: Math. Theor.
52
, 425303
(2019
).36.
J.
Wallman
, C.
Granade
, R.
Harper
, and S. T.
Flammia
, “Estimating the coherence of noise
,” New J. Phys.
17
(11
), 113020
(2015
).37.
D. E.
Evans
and R.
Høegh-Krohn
, “Spectral properties of positive maps on C*-algebras
,” J. London Math. Soc.
s2-17
, 345
–355
(1978
).38.
U.
Groh
, “Some observations on the spectra of positive operators on finite-dimensional C*-algebras
,” Linear Algebra Appl.
42
, 213
–222
(1982
).39.
B.
Baumgartner
and H.
Narnhofer
, “The structures of state space concerning quantum dynamical semigroups
,” Rev. Math. Phys.
24
, 1250001
(2012
).40.
M.
Wolf
, “Quantum channels and operations: Guided tour
,” Lecture Notes Available Online, July 2012
.41.
K.
Korzekwa
, S.
Czachórski
, Z.
Puchała
, and K.
Życzkowski
, “Coherifying quantum channels
,” New J. Phys.
20
, 043028
(2018
).42.
K.
Korzekwa
, S.
Czachórski
, Z.
Puchała
, and K.
Życzkowski
, “Distinguishing classically indistinguishable states and channels
,” J. Phys. A: Math. Theor.
52
, 475303
(2019
).43.
U.
Fano
, “Pairs of two-level systems
,” Rev. Mod. Phys.
55
, 855
(1983
).44.
P.
Badzia̧g
, M.
Horodecki
, P.
Horodecki
, and R.
Horodecki
, “Local environment can enhance fidelity of quantum teleportation
,” Phys. Rev. A
62
, 012311
(2000
).45.
J. I.
de Vincente
, “Separability criteria based on the Bloch representation of density matrices
,” Quantum Inf. Comput.
7
, 624
(2007
).46.
R.
Horodecki
, P.
Horodecki
, M.
Horodecki
, and K.
Horodecki
, “Quantum entanglement
,” Rev. Mod. Phys.
81
, 865
(2009
).47.
B. M.
Terhal
and D. P.
DiVincenzo
, “Problem of equilibration and the computation of correlation functions on a quantum computer
,” Phys. Rev. A
61
, 022301
(2000
).48.
V.
Karimipour
, F.
Benatti
, and R.
Floreanini
, “Quasi-inversion of qubit channels
,” Phys. Rev. A
101
, 032109
(2020
).49.
A.
Fujiwara
and P.
Algoet
, “One-to-one parametrization of quantum channels
,” Phys. Rev. A
59
, 3290
(1999
).50.
M. B.
Ruskai
, S.
Szarek
, and E.
Werner
, “An analysis of completely-positive trace preserving maps on M2
,” Linear Algebra Appl.
347
, 159
(2002
).51.
D.
Braun
, O.
Giraud
, I.
Nechita
, C.
Pellegrini
, and M.
Žnidarič
, “A universal set of qubit quantum channels
,” J. Phys. A: Math. Theor.
47
, 135302
(2014
).52.
W.
Bruzda
, M.
Smaczyński
, V.
Cappellini
, H.-J.
Sommers
, and K.
Życzkowski
, “Universality of spectra for interacting quantum chaotic systems
,” Phys. Rev. E
81
, 066209
–066210
(2010
).53.
A.
Edelman
and E.
Kostlan
, “How many zeros of a random polynomial are real?
,” Bull. Am. Math. Soc.
32
, 1
–37
(1995
).54.
M. B.
Hastings
, “Random unitaries give quantum expanders
,” Phys. Rev. A
76
, 032315
(2007
).55.
C.
González-Guillén
, M.
Junge
, and I.
Nechita
, “On the spectral gap of random quantum channels
,” arXiv:1811.08847 (2018
).56.
C.
Lancien
and D.
Pérez-García
, “Correlation length in random MPS and PEPS
,” arXiv:1906.11682 (2019
).57.
G.
Aubrun
and I.
Nechita
, “Realigning random states
,” J. Math. Phys.
53
, 102210
(2012
).58.
L.
Devroye
, Non-Uniform Random Variate Generation
(Springer Science + Business
, 1986
).59.
F.
Hiai
and D.
Petz
, The Semicircle Law, Free Random Variables and Entropy
(American Mathematical Society
, Providence
, 2000
), No. 77.60.
K.
Życzkowski
, M.
Kus
, W.
Słomczyński
, and H.-J.
Sommers
, “Random unistochastic matrices
,” J. Phys. A: Gen. Phys.
36
, 3425
–3450
(2003
).61.
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
).62.
M.
Musz
, M.
Kuś
, and K.
Życzkowski
, “Unitary quantum gates, perfect entanglers, and unistochastic maps
,” Phys. Rev. A
87
, 022111
–022112
(2013
).63.
F. I.
Karpelevich
, “On the characteristic roots of matrices with nonnegative elements
,” Izv. Ross. Akad. Nauk. Ser. Mat.
15
, 361
–383
(1951
).64.
D.
Chafaï
, “The Dirichlet Markov ensemble
,” J. Multivar. Anal.
101
, 555
–567
(2010
).65.
M.
Horvat
, “The ensemble of random Markov matrices
,” J. Stat. Mech.
2009
, P07005
.66.
C.
Bordenave
, P.
Caputo
, and D.
Chafaï
, “Circular law theorem for random Markov matrices
,” Probab. Theory Relat. Fields
152
, 751
–779
(2012
).67.
I.
Nechita
, Z.
Puchała
, Ł.
Pawela
, and K.
Życzkowski
, “Almost all quantum channels are equidistant
,” J. Math. Phys.
59
, 052201
(2018
).68.
Z.
Puchała
, Ł.
Pawela
, and K.
Życzkowski
, “Distinguishability of generic quantum states
,” Phys. Rev. A
93
, 062112
(2016
).69.
R.
Sinkhorn
, “A relationship between arbitrary positive matrices and doubly stochastic matrices
,” Ann. Math. Stat.
35
, 876
(1964
).70.
K. M. R.
Audenaert
and S.
Scheel
, “On random unitary channels
,” New J. Phys.
10
, 023011
(2008
).71.
V.
Cappellini
, H.-J.
Sommers
, W.
Bruzda
, and K.
Życzkowski
, “Random bistochastic matrices
,” J. Phys. A: Math. Theor.
42
, 365209
(2009
).72.
P.
Gawron
, D.
Kurzyk
, and Ł.
Pawela
, “QuantumInformation.jl—A Julia package for numerical computation in quantum information theory
,” PLoS One
13
, e0209358
(2018
).73.
W.
Tarnowski
, I.
Yusipov
, T.
Laptyeva
, S.
Denisov
, D.
Chruściński
, and K.
Życzkowski
, “Random generators of Markovian evolution: A quantum-classical transition by superdecoherence
,” arXiv:2105.02369.74.
B.
Collins
, I.
Nechita
, and Y.
Deping
, “The absolute positive partial transpose property for random induced states
,” Random Matrices: Theory Appl.
01
(03
), 1250002
(2012
).75.
D.
Voiculescu
, K.
Dykema
, and A.
Nica
, Free Random Variables
, CRM Monograph Series Vol. 1 (American Mathematical Society
, Providence, RI
, 1992
).76.
J. A.
Mingo
and M.
Popa
, “Freeness and the transposes of unitarily invariant random matrices
,” J. Funct. Anal.
271
, 883
(2016
).77.
J. A.
Mingo
, M.
Popa
, K.
Szpojankowski
, “Asymptotic*—Distribution of permuted Haar unitary matrices
,” arXiv:2006.05408.78.
A.
Mandarino
, T.
Linowski
, and K.
Życzkowski
, “Bipartite unitary gates and billiard dynamics in the Weyl chamber
,” Phys. Rev. A
98
, 012335
(2018
).79.
B.
Collins
and I.
Nechita
, “Random quantum channels I: Graphical calculus and the Bell state phenomenon
,” Commun. Math. Phys.
297
, 345
–370
(2010
).80.
W.
Młotkowski
, M. A.
Nowak
, K. A.
Penson
, and K.
Życzkowski
, “Spectral density of generalized Wishart matrices and free multiplicative convolution
,” Phys. Rev. E
92
, 012121
(2015
).81.
L.
Gurvits
, “Classical complexity and quantum entanglement
,” J. Comput. Syst. Sci.
69
, 448
–484
(2004
).82.
D.
Djoković
, “Note on nonnegative matrices
,” Proc. Am. Math. Soc.
25
, 80
–82
(1970
).83.
P.
Bürgisser
, A.
Garg
, R.
Oliveira
, M.
Walter
, and A.
Wigderson
, “Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory
,” in Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS)
(Schloss Dagstuhl
, 2018
), Vol. 24, Issue1-24, p. 20
.84.
A. S.
Lewis
and J.
Malick
, “Alternating projections on manifolds
,” Math. Oper. Res.
33
, 216
–234
(2008
).85.
X.
Duan
, C.-K.
Li
, and D. C.
Pelejo
, “Construction of quantum states with special properties by projection methods
,” Quantum Inf. Process.
19
(11
), 1
–34
(2020
).86.
87.
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
).88.
J.
Watrous
, “Mixing doubly stochastic quantum channels with the completely depolarizing channel
,” Quantum Inf. Comput.
9
, 406
(2009
).89.
H.-B.
Chen
, C.
Gneiting
, P.-Y.
Lo
, Y.-N.
Chen
, and F.
Nori
, “Simulating open quantum systems with Hamiltonian ensembles and the nonclassicality of the dynamics
,” Phys. Rev. Lett.
120
, 030403
(2018
).90.
H.-B.
Chen
, P.-Y.
Lo
, C.
Gneiting
, J.
Bae
, Y.-N.
Chen
, and F.
Nori
, “Quantifying the nonclassicality of pure dephasing
,” Nat. Commun.
10
, 3794
(2019
).91.
K.
Życzkowski
and I.
Bengtsson
, “On duality between quantum states and quantum maps
,” Open Syst. Inf. Dyn.
11
, 3
–42
(2004
).92.
G.
Narang
and Arvind
, “Simulating a single-qubit channel using a mixed-state environment
,” Phys. Rev. A
75
, 032305
(2007
).93.
T.
Heinosaari
, M. A.
Jivulescu
, and I.
Nechita
, “Random positive operator valued measures
,” J. Math. Phys.
61
(4
), 042202
(2020
).94.
L.
Sá
, P.
Ribeiro
, T.
Can
, and T.
Prosen
, “Spectral transitions and universal steady-states in random Kraus maps and circuits
,” Phys. Rev. B
102
, 134310
(2020
).© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
