Abstract
We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued \(L^p\) space that is shown to be complete. Various other norm candidates are considered as well as generalizations of Hölder’s inequality to this new context.
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References
Ando, T.: Matrix young inequalities, operator theory: advances and applications 75. Birkhauser Basel 75, 33–38 (1995)
Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics, vol. 2. Springer, Berlin (1996)
Dell’Antonio, G.: Lectures on the mathematics of quantum mechanics II: selected topics. Atlantis Studies in Mathematical Physics: Theory and Applications 2,(2016)
Erlijman, J., Farenick, D., Zeng, R.: Young’s inequality in compact operators, linear operators and matrices. Operator Theory: Advances and Applications. vol. 130, pp. 171–184. Springer, Berlin (2001)
Farenick, D., Floricel, R., Plosker, S.: Approximately clean quantum probability measures. J. Math. Phys. 54, 052201 (2013)
Farenick, D., Kozdron, M.: Conditional expectation and Bayes rule for quantum random variables and positive operator valued measures. J. Math. Phys. 53, 042201 (2012)
Farenick, D., Kozdron, M., Plosker, S.: Spectra and variance of quantum random variables. J. Math. Anal. Appl. 434, 1106–1122 (2016)
Farenick, D., Plosker, S., Smith, J.: Classical and nonclassical randomness in quantum measurements. J. Math. Phys. 52, 122204 (2011)
Haagerup, U.: Injectivity and decomposition of completely bounded maps, Operator algebras and their connections with topology and ergodic theory. Lecture Notes in Mathematics. vol. 1132, pp. 170–222 Springer, Berlin (1985)
Han, D., Larson, D., Liu, B., Liu, R.: Operator-valued measures, dilations, and the theory of frames. Mem. Am. Math. Soc. 1075, viii+84 229 (2014)
Junge, M., Ruan, Z.-J.: Decomposable maps on non-commutative \(L_p\)-spaces, Operator algebras, quantization, and noncommutative geometry. Contemp. Math. Am. Math. Soc. 365, 355–381 (2004)
McLaren, D., Plosker, S., Ramsey, C.: On operator valued measures. Houst. J. Math. 46, 201–226 (2020)
Pisier, G.: Tensor Products of \(\text{ C}^*\)-Algebras and Operator Spaces: The Connes-Kirchberg Problem. Cambridge University Press, Cambridge (2020)
Plosker, S., Ramsey, C.: An operator-valued Lyapunov theorem. J. Math. Anal. App. 469, 117–125 (2019)
Plosker, S., Ramsey, C.: Bistochastic operators and quantum random variables, preprint arXiv:2005.00005 (2020)
Acknowledgements
The first author was supported by NSERC Discovery Grant 2019-05430. The second author was partially supported by MacEwan University USRI-Project grants. Both authors would like to thank the reviewer for his/her insightful comments and corrections.
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Ramsey, C., Reeves, A. \(\hbox {L}^p\) Spaces of Operator-Valued Functions. Integr. Equ. Oper. Theory 93, 54 (2021). https://doi.org/10.1007/s00020-021-02669-x
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DOI: https://doi.org/10.1007/s00020-021-02669-x
Keywords
- Positive operator-valued measure
- POVM
- Quantum random variable
- Operator-valued integration
- Schatten norm
- Decomposable norm