Research PaperNo Access

Infinite mode quantum Gaussian states

B. V. Rajarama Bhat

Indian Statistical Institute, Stat-Math Unit, 8th Mile, Mysore Road, R. V. College Post, Bengaluru-560059, India

E-mail Address: bhat@isibang.ac.in

Search for more papers by this author

Tiju Cherian John

http://orcid.org/0000-0002-6806-9077

Indian Statistical Institute, Stat-Math Unit, 8th Mile, Mysore Road, R. V. College Post, Bengaluru-560059, India

E-mail Address: tijucherian@gmail.com

Search for more papers by this author

, and

R. Srinivasan

Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai-603103, India

E-mail Address: vasanth@cmi.ac.in

Search for more papers by this author

https://doi.org/10.1142/S0129055X19500302Cited by:8 (Source: Crossref)

Abstract

Quantum Gaussian states on Boson Fock spaces are quantum versions of Gaussian distributions. In this paper, we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in [14, 16] to the infinite mode case, which includes various characterizations, convexity and symmetry properties.

Keywords:

AMSC: 81S05, 46L60

References

1. H. Araki, On quasifree states of the canonical commutation relations. II, Publ. Res. Inst. Math. Sci. 7 (1971/72) 121–152; MR 0313834. Google Scholar
2. G. Adesso, S. Ragy and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21(1–2) (2014) 1440001, 47 pp.; MR 3190189. Web of Science, Google Scholar
3. H. Araki and M. Shiraishi, On quasifree states of the canonical commutation relations. I, Publ. Res. Inst. Math. Sci. 7 (1971/72) 105–120. Google Scholar
4. B. V. Rajarama Bhat and T. C. John, Real normal operators and williamson’s normal form, to appear in Acta Sci. Math. (Szeged). Google Scholar
5. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, 2nd edn., Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997); Equilibrium states. Models in quantum statistical mechanics. Google Scholar
6. B. V. Rajarama Bhat and R. Srinivasan, On product systems arising from sum systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(1) (2005) 1–31; MR 2126876. Link, Web of Science, Google Scholar
7. J. Dereziński and C. Gérard, Mathematics of Quantization and Quantum Fields, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2013); MR 3060648. Google Scholar
8. G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, Vol. 122 (Princeton University Press, Princeton, NJ, 1989); MR 983366. Google Scholar
9. A. Ferraro, S. Olivares and M. Paris, Gaussian States in Quantum Information, Napoli Series on Physics and Astrophysics (Bibliopolis, 2005). Google Scholar
10. A. S. Holevo, Quasi-free states on the $c *$ $c *$ algebra of CCR, Theoret. Math. Phys. 6(1) (1971) 1–12. Google Scholar
11. A. S. Holevo, Quasi-free states on the $C^{*}$ $C^{*}$ -algebra of commutation relations. II, Teoret. Mat. Fiz. 6(2) (1971) 145–150; MR 0479144. Google Scholar
12. R. Honegger and A. Rieckers, Squeezing Bogoliubov transformations on the infinite mode CCR-algebra, J. Math. Phys. 37(9) (1996) 4292–4309; MR 1408093. Web of Science, ADS, Google Scholar
13. T. Matsui and Y. Shimada, On quasifree representations of infinite dimensional symplectic group, J. Funct. Anal. 215(1) (2004) 67–102. Web of Science, Google Scholar
14. K. R. Parthasarathy, What is a Gaussian state? Commun. Stoch. Anal. 4(2) (2010) 143–160; MR 2662722. Google Scholar
15. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Modern Birkhäuser Classics (Birkhäuser, 2012); 2012 reprint of the 1992 original. Google Scholar
16. K. R. Parthasarathy, The symmetry group of Gaussian states in $l^{2} (ℝ^{n})$ , in Prokhorov and Contemporary Probability Theory, Springer Proc. Math. Stat., Vol. 33 (Springer, Heidelberg, 2013), pp. 349–369; MR 3070484. Google Scholar
17. K. R. Parthasarathy, Symplectic dilations, Gaussian states and Gaussian channels, Indian J. Pure Appl. Math. 46(4) (2015) 419–439. Web of Science, Google Scholar
18. D. Petz, An Invitation to the Algebra of Canonical Commutation Relations, Leuven Notes in Mathematical and Theoretical Physics. Series A: Mathematical Physics, Vol. 2 (Leuven University Press, Leuven, 1990); MR 1057180. Google Scholar
19. K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Lecture Notes in Mathematics, Vol. 272 (Springer-Verlag, 1972); MR 0622034. Google Scholar
20. D. Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962) 149–167; MR 0137504. Google Scholar
21. I. D. Tveritinov, Several remarks on the representation of the infinite-dimensional symplectic group and on the construction of the metaplectic group, Math. Notes 75(5) (2004) 805–818. Web of Science, Google Scholar
22. A. van Daele, Quasi-equivalence of quasi-free states on the Weyl algebra, Comm. Math. Phys. 21 (1971) 171–191; MR 0287844. Web of Science, ADS, Google Scholar
23. M. M. Wolf, G. Giedke and J. Ignacio Cirac, Extremality of gaussian quantum states, Phys. Rev. Lett. 96 (2006) 080502. Web of Science, ADS, Google Scholar
24. X.-B. Wang, T. Hiroshima, A. Tomita and M. Hayashi, Quantum information with Gaussian states, Phys. Rep. 448(1) (2007) 1–111. Web of Science, ADS, Google Scholar
25. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621–669. Web of Science, ADS, Google Scholar