The method of quantum tomography, which allows us to track with high accuracy the evolution of multilevel quantum
systems (qudits) in Hilbert spaces of various dimensions is presented. The developed algorithms for quantum control are
based on the use of the spinor representation of the Lorentz transformation group. In the simplest case of one-qubit states,
it turns out that, in addition to three-dimensional rotations on the Bloch sphere, one can introduce four-dimensional Lorentz
pseudorotations, similar to the transformations of the special theory of relativity. We show that feedback through weakly
perturbing adaptive quantum measurements turns out to be capable of providing high-precision control of the quantum
system, while introducing only weak perturbations into the initial quantum state. It turns out that, together with the control
of a quantum system through its weak perturbation, the developed algorithms for controlling the evolution of the state of
a quantum system can be super-efficient, providing a higher measurement accuracy than any standard POVM (PositiveOperator Valued Measure) protocols. The results of the study are important for the development of optimal adaptive
methods for quantum states and operations controlling.
The quantum measurement procedure based on the Lorentz transformation formalism and weak perturbation of the system is considered. In the simple case of a single-qubit it turns out that one can perform 4-dimension pseudo-rotation along with ordinary 3-dimension rotations on the Bloch sphere. These pseudo-rotations are similar to the Lorentz transformation in special relativity theory. The extension of the Lorentz transformation for many-qubit systems is also considered. The quantum measurement protocols based on the Lorentz transformation are proposed. It has been shown that these protocols cease to form the decomposition of unity and could be superefficient providing the fidelity higher than any POVM-measurement protocol. However, one can perform the complement of the Lorentz protocol to POVMprotocol by an additional measurement operator. If the initial mixed state is close to the pure one this operator corresponds to weak perturbation of the state while the original Lorentz protocol sets the strong perturbations. As the result, the feedback provides an effective control of a quantum system introducing weak perturbations to the quantum state.
Reliable generation and measurement of triphoton states has yet to be achieved in laboratory. We give an overview of the problems in generating and measuring triphoton quantum states and analyze several protocols of quantum measurements, which allow for high precision of reconstruction when sizes of available statistical data samples are limited. The tomography procedure under investigation is based on root approach to state estimation. In particular, we use the generalized Fisher information matrix to assess the accuracy of the quantum state parameters measurement. We use tomographic protocols, based on the symmetry of the Platonic solids. We demonstrate the capability to reconstruct triphoton quantum states with precision close to the maximum achievable value allowed by quantum mechanics.
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