Möbius gyrovector spaces in quantum information and computation

Abraham A. Ungar

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 341-356
  • ISSN: 0010-2628

Abstract

top
Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball 𝔹 2 of a Euclidean 2-space ℝ 2 is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball 𝔹 n of a Euclidean n -space ℝ n for all n ≥ 2 . Similarly, the decomplexification of the complex 2 × 2 qubit density matrix of QIC, which is parametrized by the real, 3-dimensional Bloch gyrovector, into an equivalent (in a specified sense) real 4 × 4 matrix is presented. As in the case of Möbius addition, this decomplexification proves useful, enabling the resulting real matrix to be generalized into a corresponding matrix parametrized by a real, n -dimensional Bloch gyrovector, for all n ≥ 2 . The applicability of the n -dimensional Bloch gyrovector with n = 3 to QIC is well known. The problem as to whether the n -dimensional Bloch gyrovector with n > 3 is applicable to QIC as well remains to be explored.

How to cite

top

Ungar, Abraham A.. "Möbius gyrovector spaces in quantum information and computation." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 341-356. <http://eudml.org/doc/250484>.

@article{Ungar2008,
abstract = {Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball $\mathbb \{B\}^2$ of a Euclidean 2-space $\mathbb \{R\}^2$ is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball $\mathbb \{B\}^n$ of a Euclidean $n$-space $\mathbb \{R\}^n$ for all $n\ge 2$. Similarly, the decomplexification of the complex $2\times 2$ qubit density matrix of QIC, which is parametrized by the real, 3-dimensional Bloch gyrovector, into an equivalent (in a specified sense) real $4\times 4$ matrix is presented. As in the case of Möbius addition, this decomplexification proves useful, enabling the resulting real matrix to be generalized into a corresponding matrix parametrized by a real, $n$-dimensional Bloch gyrovector, for all $n\ge 2$. The applicability of the $n$-dimensional Bloch gyrovector with $n=3$ to QIC is well known. The problem as to whether the $n$-dimensional Bloch gyrovector with $n>3$ is applicable to QIC as well remains to be explored.},
author = {Ungar, Abraham A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces; quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces},
language = {eng},
number = {2},
pages = {341-356},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Möbius gyrovector spaces in quantum information and computation},
url = {http://eudml.org/doc/250484},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Ungar, Abraham A.
TI - Möbius gyrovector spaces in quantum information and computation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 341
EP - 356
AB - Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball $\mathbb {B}^2$ of a Euclidean 2-space $\mathbb {R}^2$ is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball $\mathbb {B}^n$ of a Euclidean $n$-space $\mathbb {R}^n$ for all $n\ge 2$. Similarly, the decomplexification of the complex $2\times 2$ qubit density matrix of QIC, which is parametrized by the real, 3-dimensional Bloch gyrovector, into an equivalent (in a specified sense) real $4\times 4$ matrix is presented. As in the case of Möbius addition, this decomplexification proves useful, enabling the resulting real matrix to be generalized into a corresponding matrix parametrized by a real, $n$-dimensional Bloch gyrovector, for all $n\ge 2$. The applicability of the $n$-dimensional Bloch gyrovector with $n=3$ to QIC is well known. The problem as to whether the $n$-dimensional Bloch gyrovector with $n>3$ is applicable to QIC as well remains to be explored.
LA - eng
KW - quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces; quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces
UR - http://eudml.org/doc/250484
ER -

References

top
  1. Ahlfors L.V., Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. Zbl0272.30012MR0357743
  2. Altepeter J.B., James D.F.V., Kwiat P.G., 10.1007/978-3-540-44481-7_4, in Quantum State Estimation, Lecture Notes in Phys., vol. 649, Springer, Berlin, 2004, pp.113-145. Zbl1092.81511MR2180163DOI10.1007/978-3-540-44481-7_4
  3. Arvind, Mallesh K.S., Mukunda N., 10.1088/0305-4470/30/7/021, J. Phys. A 30 7 2417-2431 (1997). (1997) MR1457387DOI10.1088/0305-4470/30/7/021
  4. Chen J.-L., Ungar A.A., 10.1023/A:1015032332156, Found. Phys. 32 4 531-565 (2002). (2002) MR1903786DOI10.1023/A:1015032332156
  5. Chruściński D., Jamiołkowski A., Geometric Phases in Classical and Quantum Mechanics, Progress in Mathematical Physics, vol. 36, Birkhäuser Boston Inc., Boston, MA, 2004. 
  6. Feder T., 10.1016/S0021-8693(02)00536-7, J. Algebra 259 1 177-190 (2003). (2003) Zbl1019.20031MR1953714DOI10.1016/S0021-8693(02)00536-7
  7. Fisher S.D., Complex Variables, corrected reprint of the second (1990) edition, Dover Publications Inc., Mineola, NY, 1999. Zbl1079.30500MR1702283
  8. Foguel T., Ungar A.A., 10.1515/jgth.2000.003, J. Group Theory 3 1 27-46 (2000). (2000) Zbl0944.20053MR1736515DOI10.1515/jgth.2000.003
  9. Foguel T., Ungar A.A., 10.2140/pjm.2001.197.1, Pacific J. Math 197 1 1-11 (2001). (2001) Zbl1066.20068MR1810204DOI10.2140/pjm.2001.197.1
  10. Greene R.E., Krantz S.G., Function Theory of One Complex Variable, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1997. Zbl1114.30001MR1443431
  11. Hausner M., A Vector Space Approach to Geometry, reprint of the 1965 original, Dover Publications Inc., Mineola, NY, 1998. Zbl0924.51001MR1651732
  12. James D.F.V., Kwiat P.G., Munro W.J., White A.G., Volume of the set of separable states, Phys. Rev. A (3) 58 2 883-892 (1998). (1998) 
  13. Kinyon M.K., Ungar A.A., 10.2307/2690974, Math. Mag. 73 4 273-284 (2000). (2000) MR1822756DOI10.2307/2690974
  14. Korányi A., 10.2307/2695525, Amer. Math. Monthly 108 2 120-125 (2001). (2001) Zbl1012.15007MR1818185DOI10.2307/2695525
  15. Lévay P., 10.1088/0305-4470/37/5/024, J. Phys. A 37 5 1821-1841 (2004). (2004) Zbl1067.81049MR2044194DOI10.1088/0305-4470/37/5/024
  16. Lévay P., 10.1088/0305-4470/37/16/009, J. Phys. A 37 16 4593-4605 (2004). (2004) Zbl1062.81067MR2065947DOI10.1088/0305-4470/37/16/009
  17. Marsden J.E., Elementary Classical Analysis. With the assistance of Michael Buchner, Amy Erickson, Adam Hausknecht, Dennis Heifetz, Janet Macrae and William Wilson, and with contributions by Paul Chernoff, István Fáry and Robert Gulliver; W.H. Freeman and Co., San Francisco, 1974, . MR0357693
  18. Needham T., Visual Complex Analysis, The Clarendon Press, Oxford University Press, New York, 1997. Zbl1095.30500MR1446490
  19. Nielsen M.A., Chuang I.L., Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. Zbl1049.81015MR1796805
  20. Ungar A.A., Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces, Fundamental Theories of Physics, vol. 117, Kluwer Academic Publishers Group, Dordrecht, 2001. Zbl0972.83002MR1978122
  21. Ungar A.A., The density matrix for mixed state qubits and hyperbolic geometry, Quantum Inf. Comput. 2 6 513-514 (2002). (2002) Zbl1152.81823MR1945452
  22. Ungar A.A., 10.1023/A:1021446605657, Found. Phys. 32 11 1671-1699 2002. MR1954912DOI10.1023/A:1021446605657
  23. Ungar A.A., Analytic Hyperbolic Geometry: Mathematical Foundations and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Zbl1089.51003MR2169236
  24. Ungar A.A., Gyrogroups, the grouplike loops in the service of hyperbolic geometry and Einstein's special theory of relativity, Quasigroups Related Systems 15 141-168 (2007). (2007) Zbl1147.20055MR2379129
  25. Ungar A.A., The hyperbolic square and Möbius transformations, Banach J. Math. Anal. 1 1 101-116 (2007). (2007) Zbl1129.30027MR2350199
  26. Ungar A.A., Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Zbl1147.83004MR2169236
  27. Ungar A.A., From Möbius to gyrogroups, Amer. Math. Monthly 115 2 138-144 (2008). (2008) Zbl1152.30045MR2384266
  28. Urbantke H., 10.1119/1.16809, Amer. J. Phys. 59 6 503-509 (1991). (1991) DOI10.1119/1.16809

NotesEmbed ?

top

You must be logged in to post comments.