Abstract
In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the form $\dot{\rho} = \mathbb{L} \rho$ and the macroscopic information about the system is provided by the mean values $m_i (t_j) = Tr(Q_i \rho(t_j))$ of certain observables $\{Q_i\}_{i=1} ^r $ measured at different time instants $\{t_j\}_{j=1}^p$. The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system, i.e. minimal value of $r$ and $p$ as well as the properties of the observables $\{Q_i\}_{i=1} ^r $.
Highlights
According to one of the most fundamental assumptions of quantum theory, the density matrix carries the achievable information about the quantum state of a physical system
In recent years the determination of the trajectory of the state based on the results of measurements has gained new relevance because the ability to create, control and manipulate quantum states has found applications in other areas of science, such as: quantum information theory, quantum communication and computing
One can quickly check that there are three linearly independent eigenvectors of L that correspond to the eigenvalue λ1. It means that the index of cyclicity for the generator of evolution given by (41) is equal 3, which implies that we need 3 different observables to perform quantum tomography on the system
Summary
According to one of the most fundamental assumptions of quantum theory, the density matrix carries the achievable information about the quantum state of a physical system. The index of cyclicity seems the most important factor when one is considering the usefulness of the stroboscopic approach to quantum tomography This figure indicates how many distinct experimental setups one would have to prepare to reconstruct the initial density matrix in an experiment. Theorem 3 The determination of the initial state of the quantum system with evolution given by (4) and which is (Q1, ...Qr )-reconstructible is possible if the time instants {tj }μj=1 satisfy the condition [6]. In that section we introduce a parametric-dependent family of Kraus operators for which the generator of evolution has no degenerate eigenvalues, i.e. in that case there exists one observable the measurement of which performed at three different instants is sufficient to reconstruct the initial density matrix
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