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On the generalization of linear least mean squares estimation to quantum systems with non-commutative outputs
The purpose of this paper is to study the problem of generalizing the
Belavkin-Kalman filter to the case where the classical measurement signal is
replaced by a fully quantum non-commutative output signal. We formulate a least
mean squares estimation problem that involves a non-commutative system as the
filter processing the non-commutative output signal. We solve this estimation
problem within the framework of non-commutative probability. Also, we find the
necessary and sufficient conditions which make these non-commutative estimators
physically realizable. These conditions are restrictive in practice.Comment: 31 page
Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem
Quantum control could be implemented by varying the system Hamiltonian.
According to adiabatic theorem, a slowly changing Hamiltonian can approximately
keep the system at the ground state during the evolution if the initial state
is a ground state. In this paper we consider this process as an interpolation
between the initial and final Hamiltonians. We use the mean value of a single
operator to measure the distance between the final state and the ideal ground
state. This measure could be taken as the error of adiabatic approximation. We
prove under certain conditions, this error can be precisely estimated for an
arbitrarily given interpolating function. This error estimation could be used
as guideline to induce adiabatic evolution. According to our calculation, the
adiabatic approximation error is not proportional to the average speed of the
variation of the system Hamiltonian and the inverse of the energy gaps in many
cases. In particular, we apply this analysis to an example on which the
applicability of the adiabatic theorem is questionable.Comment: 12 pages, to appear in EPJ Quantum Technolog
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