Shiv Kumar Kaushik is associate professor at the Department of Mathematics of the Kirori Mal College, University of Delhi, where he obtained his PhD in 1991. He is member of the Indian Mathematical Society, the American Mathematical Society, the Society for History of Mathematics, and the Ramanujan Mathematical Society. His field of specialization is functional analysis, especially wavelets, frames, and bases in Banach spaces.
Operator Value Frames and Applications to Quantum Information Theory
Shiv Kumar Kaushik
(co-authored with Khole Timothy Poumai)
Kirori Mal College, University of Delhi,Delhi-110007. India
The notion of operator value frame (OPV frame) was introduced and studied in (3, 4, 6). We gave a necessary and sufficient condition for the existence of an OPV frame has been given and obtain conditions under which an OPV frame is a Riesz (orthonormal) OPV frame. Also, we show that an OPV frame is a compression of Riesz OPV frame and Parseval OPV frame is a compression of orthonormal OPV frame. A characterization of orthonormal OPV frame has been given in terms of the analysis operator. Also, various necessary and sufficient conditions for an OPV frame to be a Riesz OPV frame (orthonormal OPV frame) have been obtained. Further, perturbation of OPV frames has been discussed. As applications of OPV frames, we obtain Choi Kraus representations of quantum channels and representations of dual quantum channels by using OPV frames and give Stinesprings Theorem of quantum channel by using OPV frames. Parseval OPV frame can represent positive value measure (POVM) in quantum measurement and give the average probability of an incorrect measurement by using Parseval OPV frame. Finally, we show that the orthonormal OPV frame represents projection value measure (PVM).
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