We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
REFERENCES
1.
F.
Haake
, Quantum Signatures of Chaos
, III ed. (Springer
, Berlin
, 2010
).2.
M. B.
Hastings
, Nat. Phys.
5
, 255
(2009
).3.
S. L.
Braunstein
, Phys. Lett. A
219
, 169
(1996
).4.
K.
Życzkowski
and H.-J.
Sommers
, J. Phys. A
34
, 7111
(2001
).5.
B.
Collins
, I.
Nechita
, and K.
Życzkowski
, J. Phys. A
43
, 275303
(2010
).6.
7.
J. P.
Forrester
, Log-Gases and Random matrices
(Princeton University
, Princeton, NJ
, 2010
).8.
D.
Page
, Phys. Rev. Lett.
71
, 1291
(1993
).9.
I.
Bengtsson
and K.
Życzkowski
, Geometry of Quantum States: An Introduction to Quantum Entanglement
(Cambridge University Press
, Cambridge, England
, 2006
).10.
D.
Armstrong
, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
MWMO 949 (Providence RI
: AMS Bookstore
, 2009
).11.
T.
Banica
, S.
Belinschi
, M.
Capitaine
, and B.
Collins
, Can. J. Math.
63
, 3
(2011
).12.
13.
M. J.
Hall
, Phys. Lett. A
242
, 123
(1998
).14.
P. B.
Slater
, J. Phys. A
32
, 8231
(1999
).15.
V. A.
Osipov
, H.-J.
Sommers
, and K.
Życzkowski
, J. Phys. A
43
, 055302
(2010
).16.
J. A.
Miszczak
, Z.
Puchała
, and P.
Gawron
, Quantum Information package for Mathematica
, also see http://zksi.iitis.pl/wiki/projects:mathematica-qi (2010
).17.
J. A.
Miszczak
, “Generating and using truly random quantum states in MATHEMATICA
,” preprint; e-print arXiv:1102.4598.18.
J.
Ginibre
, J. Math. Phys.
6
, 440
(1965
).19.
20.
H.-J.
Sommers
and K.
Życzkowski
, J. Phys. A
37
, 8457
(2004
).21.
F.
Hiai
and D.
Petz
, The Semicircle Law, Free Random Variables, and Entropy
(American Mathematical Society
, Providence
, 2000
).22.
23.
M.
Bożejko
and W.
Bryc
, J. Funct. Anal.
236
, 59
(2006
).24.
E.
Lubkin
, J. Math. Phys.
19
, 1028
(1978
).25.
V.
Cappellini
, H.-J.
Sommers
, and K.
Życzkowski
, J. Math. Phys.
48
, 052110
(2007
).26.
27.
A.
Uhlmann
, “The metric of Bures and the geometric phase
,” in Groups and Related Topics
, edited by Gierelak
, R.
, Lukierski
, J.
, and Popowicz
, Z.
(Kluver
, Dodrecht
, 1992
).28.
D.
Petz
and C.
Sudár
, J. Math. Phys.
37
, 2662
(1996
).29.
H.-J.
Sommers
and K.
Życzkowski
, J. Phys. A
36
, 10083
(2003
).30.
M.
Poźniak
, K.
Życzkowski
, and M.
Kuś
, J. Phys. A
31
, 1059
(1998
).31.
M.
Fannes
, B.
Nachtergaele
, and R. F.
Werner
, Commun. Math. Phys.
144
, 443
(1992
).32.
F.
Verstraete
and J. I.
Cirac
, “Renormalization algorithms for quantum - many body systems in two and higher dimensions
,” preprint; e-print arXiv:cond-mat/0407066.33.
N.
Schuch
, D.
Pérez-Garcıa
, and J. I.
Cirac
, “Classifying quantum phases using Matrix Product States and PEPS
,” preprint; e-print arXiv:1010.3732.34.
M.
Żukowski
, A.
Zeilinger
, M. A.
Horne
, and A. K.
Ekert
, Phys. Rev. Lett.
71
, 4287
(1993
).35.
R. F.
Werner
, J. Phys. A
34
, 7081
(2001
).36.
N.
Alexeev
, F.
Götze
, and A.
Tikhomirov
, Lith. Math. J.
50
, 121
(2010
).37.
F.
Benaych-Georges
, Ann. I.H.P. Probab. Stat.
46
, 644
(2010
).38.
K. A.
Penson
and A. I
Solomon
, “Coherent states from combinatorial sequences
,” in Quantum Theory and Symmetries
, Kraków 2001, (World Scientific
, River Edge, NJ
, 2002
); preprint; e-print arXiv:quant-ph/0111151.39.
Z.
Burda
, A.
Jarosz
, G.
Livan
, M. A.
Nowak
, and A.
Swiech
, Phys. Rev. E
82
, 061114
(2010
).40.
D.-Z.
Liu
, C.
Song
, and Z.-D.
Wang
, “On Explicit Probability Densities Associated with Fuss-Catalan Numbers
,” preprint; e-print arXiv:1008.0271.41.
K. A.
Penson
and K.
Życzkowski
, “Products of Ginibre matrices: Fuss-Catalan and Raney distributions
,” Phys. Rev. E
(in press);e-print arXiv:1103.3453.
42.
A.
Erdélyi
, W.
Magnus
, F.
Oberhettinger
, and F. G.
Tricomi
, Higher Transcendental Functions
(McGraw–Hill
, New York
, 1953
).43.
T.-C.
Wei
and P. M.
Goldbart
, Phys. Rev. A
68
, 042307
(2003
).44.
W.
Bruzda
, V.
Cappellini
, H.-J.
Sommers
, and K.
Życzkowski
, Phys. Lett. A
373
, 320
(2009
).45.
W.
Bruzda
, M.
Smaczyński
, V.
Cappellini
, H.-J.
Sommers
, and K.
Życzkowski
, Phys. Rev. E
81
, 066209
(2010
).46.
A.
De Pasquale
, P.
Facchi
, G.
Parisi
, S.
Pascazio
, and A.
Scardicchio
, Phys. Rev. A
81
, 052324
(2010
).47.
C.
Nadal
, S. N.
Majumdar
, and M.
Vergassola
, Phys. Rev. Lett.
104
, 110501
(2010
).48.
S.
Garnerone
, T. R.
de Oliveira
, S.
Haas
, and P.
Zanardi
, Phys. Rev. A
82
, 052312
(2010
).49.
D. V.
Voiculescu
, K. J.
Dykema
, and A.
Nica
, Free random variables
(American Mathematical Society
, Providence
, 1992
).50.
A.
Nica
and R.
Speicher
, Lectures on the combinatorics of free probability
, London Mathematical Society Lecture Note Series
, 335
(Cambridge University Press
, Cambridge, England
, 2006
).51.
H.
Kesten
, Trans. Am. Math. Soc.
92
, 336
(1959
).52.
U.
Haagerup
and F.
Larsen
, J. Funct. Anal.
176
, 331
(2000
).53.
54.
K.
Życzkowski
and H.-J.
Sommers
, J. Phys. A
36
, 10115
(2003
).55.
B.
Collins
and P.
Śniady
, Commun. Math. Phys.
264
, 773
(2006
).56.
A.
Zvonkin
, Math. Comput. Modell.
26
, 281
(1997
).57.
A.
Okounkov
, Int. Math. Res. Notices
20
, 1043
(2000
).© 2011 American Institute of Physics.
2011
American Institute of Physics
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