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Contributions of some Indian Scientists
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
He developed analytic tools for examining such problems also in the stationary dependent case. A statistical estimator of a parameter is said to be efficient if its variance equals the CRLB, it is said to be asymptotically efficient if the ratio of its variance to the CRLB converges to unity almost surely as the number of data samples goes to infinity. It is a well known fact that if there exists an estimator that is efficient then the maximum likelihood estimator is one such. The proof of this fact is contained in the appendix. C.R.Rao’s work on the CRLB has been extended to quantum systems. The problem with quantum systems unlike classical systems is that given a state, it is not clear what observable to measure or equivalently, what commuting family of observables to measure or equivalently, what complete orthogonal family of orthogonal projections (PVM or projection valued measurement) to measure in order to get an estimator that will have the least variance. Researchers working in quantum information theory have therefore arrived at the most general kind of measurement namely POVM;s (an abbreviation for positive operator valued measurement) which contains all the above set of measurements as special cases. Such a measurement is given by M = {Mα:α ∈I} where I is a countable index set such that Mα, ≥ 0∀α ∈ I and ∑α∈I Mα = 1. If ρ is a state in the same Hilbert space on which these measurement operators are defined, then on making the measurement M, the probability of getting the outcome α ∈ I is given by PM(α)=Tr(p.Mα)
State discrimination of two pure states with a fixed rate of inconclusive answer
Published in Journal of Modern Optics, 2018
with arbitrary a prior probability ηi satisfying η1 + η2 = 1. The authors introduced three positive operator-valued measure (POVM) elements as with , where identifies the state , while corresponds to the inconclusive outcome. The total success Ps, the total error Pe and the total inconclusive probabilities Q are conventionally defined as
Dynamics super quantum discord and quantum discord teleportation in the Jaynes–Cummings model
Published in Journal of Modern Optics, 2018
Forouzan Mirmasoudi, Sodeif Ahadpour
where denotes a complete set of positive operator-valued measure (POVM) performed on the subsystem B, in such a way that . The probability for given outcome j is: