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Quantum Information Processing

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01-10-2019

Non-Hermitian extensions of uncertainty relations with generalized metric adjusted skew information

Authors: Yajing Fan, Huaixin Cao, Wenhua Wang, Huixian Meng, Liang Chen

Published in: Quantum Information Processing | Issue 10/2019

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Abstract

In quantum mechanics, it is well known that the Heisenberg–Schrödinger uncertainty relations hold for two non-commutative observables and density operator. Recently some people start to focus on the uncertainty relations for two non-commutative non-Hermitian operators and density operator. In this paper, we introduce the generalized metric adjusted skew information, generalized metric adjusted correlation measure and the related quantities for non-Hermitian operators. Various properties of them are discussed. Finally, we establish several generalizations of uncertainty relation expressed in terms of the generalized metric adjusted skew information and obtain several results including previous results which can be given as corollaries of our non-Hermitian extensions of Heisenberg-type or Schrödinger-type uncertainty relations.

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Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. A 43, 172–198 (1927)CrossRef

Schrödinger, E.: About Heisenberg uncertainty relation. Proc. Prussian Acad. Sci. Phys. Math. Sec. 19, 296–303 (1930)

Wigner, E.P., Yanase, M.M.: Information contents of distributions. Proc. Natl. Acad. Sci. USA 49, 910–918 (1963)ADSMathSciNetCrossRef

Luo, S., Zhang, Q.: On skew information. IEEE Trans. Inf. Theory 50, 1778–1782 (2004)MathSciNetCrossRef

Luo, S.: Heisenberg uncertainty relation for mixed states. Phys. Rev. A 72, 042110 (2005)ADSCrossRef

Furuichi, S., Yanagi, K., Kuriyama, K.: Trace inequalities on a generalized Wigner–Yanase skew information. J. Math. Anal. Appl. 356, 179–185 (2009)MathSciNetCrossRef

Furuichi, S.: Schrödinger uncertainty relation with Wigner–Yanase skew information. Phys. Rev. A 82, 034101 (2010)ADSCrossRef

Yanagi, K.: Uncertainty relation on Wigner–Yanase–Dyson skew information. J. Math. Anal. Appl. 365, 12–18 (2010)MathSciNetCrossRef

Yanagi, K.: Wigner–Yanase–Dyson skew information and uncertainty relation. J. Phys. Conf. Ser. 201, 012015 (2010)CrossRef

10.

Ko, C.K., Yoo, H.J.: Uncertainty relation associated with a monotone pair skew information. J. Math. Anal. Appl. 383, 208–214 (2011)MathSciNetCrossRef

11.

Furuichi, S., Yanagi, K.: Schrödinger uncertainty relation, Wigner–Yanase–Dyson skew information and metric adjusted correlation measure. J. Math. Anal. Appl. 388, 1147–1156 (2012)MathSciNetCrossRef

12.

Hansen, F.: Metric adjusted skew information. Proc. Natl. Acad. Sci. USA 105, 9909–9916 (2008)ADSMathSciNetCrossRef

13.

Gibilisco, P., Imparato, D., Isola, T.: Uncertainty principle and quantum Fisher information-II. J. Math. Phys. 48, 072109 (2007)ADSMathSciNetCrossRef

14.

Gibilisco, P., Hansen, F., Isola, T.: On a correspondence between regular and non-regular operator monotone functions. Linear Algebra Appl. 430, 2225–2232 (2009)MathSciNetCrossRef

15.

Petz, D.: Monotone metrics on matrix spaces. Linear Algebra Appl. 244, 81–96 (1996)MathSciNetCrossRef

16.

Yanagi, K.: Metric adjusted skew information and uncertainty relation. J. Math. Anal. Appl. 380, 888–892 (2011)MathSciNetCrossRef

17.

Gibilisco, P., Isola, T.: On a refinement of Heisenberg uncertainty relation by means of quantum Fisher information. J. Math. Anal. Appl. 375, 270–275 (2011)MathSciNetCrossRef

18.

Yanagi, K., Furuichi, S., Kuriyama, K.: Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. J. Uncertain. Anal. Appl. 1, 12 (2013)CrossRef

19.

Müller, M., Rotter, I.: Phase lapses in open quantum systems and the non-Hermitian Hamilton operator. Phys. Rev. A 80(4), 042705 (2009)ADSCrossRef

20.

Rotter, I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 42, 153001 (2009)ADSMathSciNetCrossRef

21.

Alber, G., Delgado, A., Gisin, N., Jex, I.: Generalized quantum XOR-gate for quantum teleportation and state purification in arbitrary dimensional Hilbert spaces. Quantum Phys. arXiv:quant-ph/0008022v1 (2000)

22.

Long, G.L.: General quantum interference principle and duality computer. Commun. Theor. Phys. 45, 825 (2006)ADSMathSciNetCrossRef

23.

Guo, Z.H., Cao, H.X., Chen, Z.L., Yin, J.C.: Operational properties and matrix representations of quantum measures. Chin. Sci. Bull. 56, 1671 (2011)CrossRef

24.

Guo, Z.H., Cao, H.X.: Existence and construction of a quantum channel with given inputs and outputs. Chin. Sci. Bull. 57, 4346–4350 (2012)CrossRef

25.

Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)ADSMathSciNetCrossRef

26.

Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)ADSMathSciNetCrossRef

27.

Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)CrossRef

28.

Dou, Y.N., Du, H.K.: Generalizations of the Heisenberg and Schrödinger uncertainty relations. J. Math. Phys. 54, 103508 (2013)ADSMathSciNetCrossRef

29.

Dou, Y.N., Du, H.K.: Note on the Wigner–Yanase–Dyson skew information. Int. J. Theor. Phys. 53, 952–958 (2014)MathSciNetCrossRef

30.

Li, Q., Cao, H.X., Du, H.K.: A generalization of Schrödinger’s uncertainty relation described by the Wigner–Yanase skew information. Quantum Inf. Process. 14, 1513–1522 (2015)ADSMathSciNetCrossRef

31.

Chen, Z.L., Liang, L.L., Li, H.J., Wang, W.H.: A generalized uncertainty relation. Int. J. Theor. Phys. 54, 2644–2651 (2015)MathSciNetCrossRef

32.

Chen, Z.L., Liang, L.L., Li, H.J., Wang, W.H.: Two generalized Wigner–Yanase skew information and their uncertainty relations. Quantum Inf. Process. 15, 5107–5118 (2016)ADSMathSciNetCrossRef

33.

Yanagi, K., Sekikawa, K.: Non-hermitian extensions of Heisenberg type and Schrödinger type uncertainty relations. J. Inequal. Appl. 2015, 381 (2015)CrossRef

34.

Bender, C.M., Brody, D.C., Jones, H.F., Meister, B.K.: Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)ADSMathSciNetCrossRef

Title: Non-Hermitian extensions of uncertainty relations with generalized metric adjusted skew information
Authors: Yajing Fan
Huaixin Cao
Wenhua Wang
Huixian Meng
Liang Chen
Publication date: 01-10-2019
Publisher: Springer US
Published in: Quantum Information Processing / Issue 10/2019
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-019-2415-2

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