Review PaperNo Access

SPIN SQUEEZING AND ENTANGLEMENT VIA FINITE-DIMENSIONAL DISCRETE PHASE-SPACE DESCRIPTION

Avenida General Osório 414, centro, Jaboticabal, SP 14870-100, Brazil

Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, São Paulo, SP 01140-070, Brazil

Search for more papers by this author

, and

TIAGO DEBARBA

Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Departamento de Física, Avenida Antônio Carlos 6627, Belo Horizonte, MG 31270-901, Brazil

Search for more papers by this author

https://doi.org/10.1142/S0219749913300015Cited by:17 (Source: Crossref)

Abstract

We show how mapping techniques inherent to N²-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin–Meshkov–Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson–Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations ↦ entanglement ↦ squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems.

Keywords:

References

E. Schrödinger , Proceedings of the Cambridge Philosophical Society 31 , 555 ( 1935 ) . Crossref, Google Scholar
A. Einstein , B. Podolsky and N. Rosen , Phys. Rev. 47 , 777 ( 1935 ) . Crossref, Google Scholar
S. J. Freedman and J. F. Clauser , Phys. Rev. Lett. 28 , 938 ( 1972 ) . Crossref, Web of Science, Google Scholar
R. Omnès , The Interpretation of Quantum Mechanics ( Princeton University Press , New Jersey , 1994 ) . Crossref, Google Scholar
A. Peres , Quantum Theory: Concepts and Methods ( Kluwer Academic Publishers , Netherlands , 1995 ) . Google Scholar
M. Horodecki , P. Horodecki and R. Horodecki , Phys. Lett. A 223 , 1 ( 1996 ) . Crossref, Web of Science, Google Scholar
F. G. S. L. Brandão , Phys. Rev. A 72 , 022310 ( 2005 ) . Crossref, Web of Science, Google Scholar
A. K. Ekert , Phys. Rev. Lett. 67 , 661 ( 1991 ) . Crossref, Web of Science, Google Scholar
A. Abeyesinghe et al. , Proc. R. Soc. Lond. A 465 , 2537 ( 2009 ) . Crossref, Web of Science, Google Scholar
K. Horodecki et al. , IEEE Transactions on Information Theory 55 , 1898 ( 2009 ) . Crossref, Web of Science, Google Scholar
P. W. Lamberti , M. Portesi and J. Sparacino , Int. J. Quantum Inform 7 , 1009 ( 2009 ) . Link, Web of Science, Google Scholar
L. Henderson and V. Vedral , J. Phys. A: Math. Gen. 34 , 6899 ( 2001 ) . Crossref, Web of Science, Google Scholar
H. Ollivier and W. H. Zurek , Phys. Rev. Lett. 88 , 017901 ( 2001 ) . Crossref, Web of Science, Google Scholar
R. Jozsa and N. Linden , Proc. R. Soc. Lond. A 459 , 2011 ( 2003 ) . Crossref, Web of Science, Google Scholar
D. Kaszlikowski et al. , Phys. Rev. Lett. 101 , 070502 ( 2008 ) . Crossref, Web of Science, Google Scholar
S. Luo , Phys. Rev. A 77 , 022301 ( 2008 ) . Crossref, Web of Science, Google Scholar
C. Wang , Y.-Y. Zhang and Q.-H. Chen , Phys. Rev. A 85 , 052112 ( 2012 ) . Crossref, Web of Science, Google Scholar
S. Luo , Phys. Rev. A 77 , 042303 ( 2008 ) . Crossref, Web of Science, Google Scholar
K. Modi et al. , Phys. Rev. Lett. 104 , 080501 ( 2010 ) . Crossref, Web of Science, Google Scholar
M. F. Cornelio , M. C. de Oliveira and F. F. Fanchini , Phys. Rev. Lett. 107 , 020502 ( 2011 ) . Crossref, Web of Science, Google Scholar
T. Debarba , T. O. Maciel and R. O. Vianna , Phys. Rev. A 86 , 024302 ( 2012 ) , arXiv:1207.1298 . Crossref, Web of Science, Google Scholar
R. Dorner and V. Vedral , arXiv:1208.4961 . Google Scholar
W. H. Zurek , Rev. Mod. Phys. 75 , 715 ( 2003 ) . Crossref, Web of Science, Google Scholar
T. Tufarelli, D. Girolami, R. Vasile, S. Bose and G. Adesso , arXiv:1205.0251 . Google Scholar
A. R. U. Devi , X. Wang and B. C. Sanders , Quantum Inf. Process. 2 , 207 ( 2003 ) . Crossref, Web of Science, Google Scholar
J. K. Korbicz et al. , Phys. Rev. A 74 , 052319 ( 2006 ) . Crossref, Web of Science, Google Scholar
G. Tóth et al. , Phys. Rev. Lett. 99 , 250405 ( 2007 ) . Crossref, Web of Science, Google Scholar
P. K. Pathak et al. , J. Phys. A: Math. Theor. 41 , 145302 ( 2008 ) . Crossref, Web of Science, Google Scholar
G. Tóth et al. , Phys. Rev. A 79 , 042334 ( 2009 ) . Crossref, Web of Science, Google Scholar
G. Vitagliano et al. , Phys. Rev. Lett. 107 , 240502 ( 2011 ) . Crossref, Web of Science, Google Scholar
D. J. Wineland et al. , Phys. Rev. A 46 , R6797 ( 1992 ) . Crossref, Web of Science, Google Scholar
D. J. Wineland et al. , Phys. Rev. A 50 , 67 ( 1994 ) . Crossref, Web of Science, Google Scholar
J. Hald et al. , Phys. Rev. Lett. 83 , 1319 ( 1999 ) . Crossref, Web of Science, Google Scholar
A. Sørensen and K. Mølmer , Phys. Rev. Lett. 83 , 2274 ( 1999 ) . Crossref, Web of Science, Google Scholar
A. S. Sørensen and K. Mølmer , Phys. Rev. Lett. 86 , 4431 ( 2001 ) . Crossref, Web of Science, Google Scholar
A. Sørensen et al. , Nature 409 , 63 ( 2001 ) . Crossref, Web of Science, Google Scholar
V. Meyer et al. , Phys. Rev. Lett. 86 , 5870 ( 2001 ) . Crossref, Web of Science, Google Scholar
M. Kitagawa and M. Ueda , Phys. Rev. A 47 , 5138 ( 1993 ) . Crossref, Web of Science, Google Scholar
A. Vourdas , Rep. Prog. Phys. 67 , 267 ( 2004 ) . Crossref, Web of Science, Google Scholar
C. Ferrie , Rep. Prog. Phys. 74 , 116001 ( 2011 ) . Crossref, Web of Science, Google Scholar
D. Galetti and A. F. R. Toledo Piza , Physica A 149 , 267 ( 1988 ) . Crossref, Web of Science, Google Scholar
R. Aldrovandi and D. Galetti , J. Math. Phys. 31 , 2987 ( 1990 ) . Crossref, Web of Science, Google Scholar
D. Galetti and A. F. R. Toledo Piza , Physica A 186 , 513 ( 1992 ) . Crossref, Web of Science, Google Scholar
D. Galetti and A. F. R. Toledo Piza , Physica A 214 , 207 ( 1995 ) . Crossref, Web of Science, Google Scholar
D. Galetti and M. A. Marchiolli , Ann. Phys. 249 , 454 ( 1996 ) . Crossref, Web of Science, Google Scholar
D. Galetti and M. Ruzzi , Physica A 264 , 473 ( 1999 ) . Crossref, Web of Science, Google Scholar
D. Galetti and M. Ruzzi , J. Phys. A: Math. Gen. 33 , 2799 ( 2000 ) . Crossref, Google Scholar
M. Ruzzi , D. Galetti and M. A. Marchiolli , J. Phys. A: Math. Gen. 38 , 6239 ( 2005 ) . Crossref, Google Scholar
M. A. Marchiolli , M. Ruzzi and D. Galetti , Phys. Rev. A 72 , 042308 ( 2005 ) . Crossref, Web of Science, Google Scholar
M. A. Marchiolli , M. Ruzzi and D. Galetti , Phys. Rev. A 76 , 032102 ( 2007 ) . Crossref, Web of Science, Google Scholar
M. A. Marchiolli , E. C. Silva and D. Galetti , Phys. Rev. A 79 , 022114 ( 2009 ) . Crossref, Web of Science, Google Scholar
M. A. Marchiolli and M. Ruzzi , Ann. Phys. (N.Y.) 327 , 1538 ( 2012 ) . Crossref, Web of Science, Google Scholar
J. Schwinger , Quantum Mechanics: Symbolism of Atomic Measurements ( Springer-Verlag , Berlin , 2001 ) . Crossref, Google Scholar
J. M. Jauch , Foundations of Quantum Mechanics ( Addison-Wesley , Massachusetts , 1968 ) . Google Scholar
E. Prugovečki , Quantum Mechanics in Hilbert Space ( Academic Press , New York , 1981 ) . Google Scholar
U. Fano , Rev. Mod. Phys. 29 , 74 ( 1957 ) . Crossref, Web of Science, Google Scholar
H. J. Lipkin , N. Meshkov and A. J. Glick , Nucl. Phys. 62 , 188 ( 1965 ) . Crossref, Web of Science, Google Scholar
N. Meshkov , A. J. Glick and H. J. Lipkin , Nucl. Phys. 62 , 199 ( 1965 ) . Crossref, Web of Science, Google Scholar
A. J. Glick , H. J. Lipkin and N. Meshkov , Nucl. Phys. 62 , 211 ( 1965 ) . Crossref, Web of Science, Google Scholar
S. Dusuel and J. Vidal , Phys. Rev. Lett. 93 , 237204 ( 2004 ) . Crossref, Web of Science, Google Scholar
S. Dusuel and J. Vidal , Phys. Rev. B 71 , 224420 ( 2005 ) . Crossref, Web of Science, Google Scholar
T. Barthel , S. Dusuel and J. Vidal , Phys. Rev. Lett. 97 , 220402 ( 2006 ) . Crossref, Web of Science, Google Scholar
J. Vidal , S. Dusuel and T. Barthel , J. Stat. Mech. P01015 ( 2007 ) . Crossref, Web of Science, Google Scholar
R. Orús , S. Dusuel and J. Vidal , Phys. Rev. Lett. 101 , 025701 ( 2008 ) . Crossref, Web of Science, Google Scholar
J. Wilms et al. , J. Stat. Mech. P01023 ( 2012 ) . Crossref, Web of Science, Google Scholar
V. V. Dodonov and V. I. Man'ko, Proc. Lebedev Phys. Inst. Acad. Sci. USSR Vol. 183: Invariants and the Evolution of Nonstationary Quantum Systems (Nova Science, New York, 1989), pp. 3–101 . Google Scholar
A. Wehrl , Rev. Mod. Phys. 50 , 221 ( 1978 ) . Crossref, Web of Science, Google Scholar
S. S. Mizrahi and M. A. Marchiolli , Physica A 199 , 96 ( 1993 ) . Crossref, Web of Science, Google Scholar
K. Pia̧tek and W. Leoński , J. Phys. A: Math. Gen. 34 , 4951 ( 2001 ) . Crossref, Google Scholar
R. Gilmore , Lie Groups, Lie Algebras, and Some of Their Applications ( Krieger Publishing Company , Florida , 1994 ) . Google Scholar
N. Zettili , Quantum Mechanics: Concepts and Applications ( John Wiley & Sons , Chichester , 2004 ) . Google Scholar
A. Perelomov , Generalized Coherent States and Their Applications ( Springer-Verlag , Berlin , 1986 ) . Crossref, Google Scholar
A. Inomata , H. Kuratsuji and C. C. Gerry , Path Integrals and Coherent States of SU(2) and SU(1, 1) ( World Scientific , Singapore , 1992 ) . Link, Google Scholar
R. R. Puri , Mathematical Methods of Quantum Optics ( Springer , Berlin , 2001 ) . Crossref, Google Scholar
K. Wódkiewicz and J. H. Eberly , J. Opt. Soc. Am. B 2 , 458 ( 1985 ) . Crossref, Web of Science, Google Scholar
M. Ban , J. Opt. Soc. Am. B 10 , 1347 ( 1993 ) . Crossref, Web of Science, Google Scholar
M. A. Marchiolli and D. Galetti , Phys. Scr. 78 , 045007 ( 2008 ) . Crossref, Web of Science, Google Scholar
P. W. Atkins and J. C. Dobson , Proc. R. Soc. Lond. A 321 , 321 ( 1971 ) . Crossref, Web of Science, Google Scholar
P. Ring and P. Schuck , The Nuclear Many-Body Problem ( Springer-Verlag , Berlin , 2004 ) . Google Scholar
F. T. Arecchi et al. , Phys. Rev. A 6 , 2211 ( 1972 ) . Crossref, Web of Science, Google Scholar
L. M. Narducci , C. A. Coulter and C. M. Bowden , Phys. Rev. A 9 , 829 ( 1974 ) . Crossref, Web of Science, Google Scholar
L. M. Narducci et al. , Phys. Rev. A 11 , 973 ( 1975 ) . Crossref, Web of Science, Google Scholar
L. M. Narducci et al. , Phys. Rev. A 18 , 1571 ( 1978 ) . Crossref, Web of Science, Google Scholar
G. S. Agarwal et al. , Phys. Rev. A 20 , 2040 ( 1979 ) . Crossref, Web of Science, Google Scholar
G. S. Agarwal , Phys. Rev. A 24 , 2889 ( 1981 ) . Crossref, Web of Science, Google Scholar
J. P. Dowling , G. S. Agarwal and W. P. Schleich , Phys. Rev. A 49 , 4101 ( 1994 ) . Crossref, Web of Science, Google Scholar
R. H. Dicke , Phys. Rev. 93 , 99 ( 1954 ) . Crossref, Web of Science, Google Scholar
R. Lo Franco et al. , Eur. Phys. J. Special Topics 160 , 247 ( 2008 ) . Crossref, Web of Science, Google Scholar
W. -M. Zhang , D. H. Feng and R. Gilmore , Rev. Mod. Phys. 62 , 867 ( 1990 ) . Crossref, Web of Science, Google Scholar
J. M. Radcliffe , J. Phys. A: Gen. Phys. 4 , 313 ( 1971 ) . Crossref, Google Scholar
E. Majorana , Nuovo Cimento 9 , 43 ( 1932 ) . Crossref, Google Scholar
F. Bloch , Phys. Rev. 70 , 460 ( 1946 ) . Crossref, Web of Science, Google Scholar
R. V. Buniy , S. D. H. Hsu and A. Zee , Phys. Lett. B 630 , 68 ( 2005 ) . Crossref, Web of Science, Google Scholar
R. V. Buniy , S. D. H. Hsu and A. Zee , Phys. Lett. B 640 , 219 ( 2006 ) . Crossref, Web of Science, Google Scholar
X. Calmet , M. L. Graesser and S. D. H. Hsu , Phys. Rev. Lett. 93 , 211101 ( 2004 ) . Crossref, Web of Science, Google Scholar
X. Calmet , M. L. Graesser and S. D. H. Hsu , Int. J. Mod. Phys. D 14 , 2195 ( 2005 ) . Link, Web of Science, Google Scholar
A. F. Ali , S. Das and E. C. Vagenas , Phys. Lett. B 678 , 497 ( 2009 ) . Crossref, Web of Science, Google Scholar
J. Vidal , G. Palacios and R. Mosseri , Phys. Rev. A 69 , 022107 ( 2004 ) . Crossref, Web of Science, Google Scholar
J. I. Latorre et al. , Phys. Rev. A 71 , 064101 ( 2005 ) . Crossref, Web of Science, Google Scholar
H. Wichterich , J. Vidal and S. Bose , Phys. Rev. A 81 , 032311 ( 2010 ) . Crossref, Web of Science, Google Scholar
Quantum Mechanics in Phase Space: An Overview with Selected Papers, World Scientific Series in 20th Century Physics 34, C. K. Zachos, D. B. Fairlie and T. L. Curtright (eds.) (World Scientific Publishing, Singapore, 2005) . Google Scholar
P. Hyllus et al. , Phys. Rev. A 86 , 012337 ( 2012 ) . Crossref, Web of Science, Google Scholar
S. Haroche and J. -M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, New York, 2007) pp. 579–582. Google Scholar
H. Araki and E. H. Lieb , Commun. Math. Phys. 18 , 160 ( 1970 ) . Crossref, Web of Science, Google Scholar
E. H. Lieb and M. B. Ruskai , J. Math. Phys. 14 , 1938 ( 1973 ) . Crossref, Web of Science, Google Scholar
W. H. Zurek , Nature 412 , 712 ( 2001 ) . Crossref, Web of Science, Google Scholar
G. Manfredi and M. R. Feix , Phys. Rev. E 62 , 4665 ( 2000 ) . Crossref, Web of Science, Google Scholar
V. Vedral , Rev. Mod. Phys. 74 , 197 ( 2002 ) . Crossref, Web of Science, Google Scholar
E. H. Lieb and J. P. Solovej, e-print (2012) , arXiv:math-ph/1208.3632 . Google Scholar
M. A. Marchiolli , S. S. Mizrahi and V. V. Dodonov , Phys. Rev. A 56 , 4278 ( 1997 ) . Crossref, Web of Science, Google Scholar
M. A. Marchiolli , S. S. Mizrahi and V. V. Dodonov , Phys. Rev. A 57 , 3885 ( 1998 ) . Crossref, Web of Science, Google Scholar
C. Muñoz and A. B. Klimov , J. Phys. A: Math. Theor. 45 , 045301 ( 2012 ) . Crossref, Web of Science, Google Scholar