Authors: Artur Czerwiński
Publish Date: 2015/06/11
Volume: 55, Issue: 2, Pages: 658-668
Abstract
In the paper we discuss possible applications of the socalled stroboscopic tomography stroboscopic observability to selected decoherence models of 2level 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 = T rQ i ρt j of certain observables Q i i=1r measured at different time instants t j j=1p The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system ie minimal value of r and p as well as the properties of the observables Q i i=1r 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 computingTherefore in this paper we follow the stroboscopic approach to quantum tomography which was proposed in 4 and then developed in 5 6 In the stroboscopic approach we consider a set of observables Q i i=1r where r N 2 − 1 and each of them can be measured at time instants t j j=1p Every measurement provides a result that shall be denoted by m i t j and can be represented as m i t j = T rQ i ρt j Because in this approach the measurements are performed at different time instants it is necessary to assume that the knowledge about the character of evolution is available eg the KossakowskiLindblad master equation 7 is known or equivalently the collection of Kraus operators Knowledge about the evolution makes it possible to determine not only the initial density matrix but also the complete trajectory of the state To make this issue clearer from now on we assume the following definition 6Other questions that arise in this approach concern the minimal number of observables for a given master equation and their properties as well as the minimal number of time instants and their choice The general conditions for observability have been determined and will be presented here as theorems and the proofs can be found in papers 4 5 6According to theorem 1 for every generator of evolution there always exists a set of η observables such that the system is Q 1Q η reconstructible Moreover if the system is also Q 1Q eta prime reconstructible then η ′ ≥ η 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 The index of cyclicity is a natural number from the set 1 2…N 2 − 1 where N= dimmathcal H and the lower the number the more advantageous it is to employ the stroboscopic approach instead of the standard tomography Moreover one can notice that the index of cyclicity can be understood as the greatest geometric multiplicity of eigenvalues of the generator of evolution Thus one can conclude that the index of cyclicity has physical interpretation which is important from experimental point of view but its value depends on the algebraic properties of the generator of evolution Therefore the question whether the stroboscopic tomography is worth employing or not depends primarily on the character of evolution of the quantum systemIt can be proved that the functions α k t are mutually linearly independent and can be computed from a system of differential equations 6 Therefore the data provided by the experiment allows us to calculate the projections langle mathbb Lk Q i rho 0 rangle for k = 01⋯ μ − 1 and i = 1 2⋯ r It can be observed that the initial state ρ0 and consequently the trajectory expmathbb L t rho 0 can be uniquely determined if and only if the operators mathbb Lk Q i span the vector space of all selfadjoint operators on mathcal H This space shall be denoted by B mathcal H and will be referred to as the HilbertSchmidt space Now if the evolution of the system is given by 4 the conclusion can be presented as a formal theorem
Keywords: