 

Sarah Plosker and
Christopher Ramsey
Bistochastic operators and quantum random variables
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Published: 
March 9, 2022. 
Keywords: 
positive operator valued measure (POVM), quantum probability measure, quantum random variable, RadonNikodým derivative, Bistochastic operator, majorization. 
Subject: 
46B22, 46C50, 46G10, 47G10, 81P15. 


Abstract
Given a positive operatorvalued measure ν acting on the Borel sets of a locally compact Hausdorff space X, with outcomes in the algebra B(H) of all bounded operators on a (possibly infinitedimensional) Hilbert space H, one can consider νintegrable functions X → B(H) that are positive quantum random variables.
We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.


Acknowledgements
S.P. was supported by NSERC Discovery Grant number 1174582, the Canada Foundation for Innovation (CFI) grant number 35711, and the Canada Research Chairs (CRC) Program grant number 231250. C.R. was supported by NSERC Discovery Grant 201905430.


Author information
Sarah Plosker:
Department of Mathematics and Computer Science
Brandon University
Brandon, MB R7A 6A9, Canada
ploskers@brandonu.ca
Christopher Ramsey:
Department of Mathematics and Computer Science
Brandon University
Brandon, MB R7A 6A9, Canada, and
Department of Mathematics and Statistics
MacEwan University
Edmonton, AB T5J 4S2, Canada
ramseyc5@macewan.ca

