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An Effective Approach to Complex Vector Reconstruction from Intensity Measurements

Artur Czerwiński

Published Jul 2, 2015• 0 citations
OpenAlex

Abstract

In this article we propose a stroboscopic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schr\odinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval - recovering a complex vector from modulus of inner product with frame vectors. We believe that the stroboscopic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary model.

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In this article we propose a stroboscopic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schr\odinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval - recovering a complex vector from modulus of inner product with frame vectors. We believe that the stroboscopic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary model.
In this article we propose a dynamic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schrödinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval -- recovering a complex vector from modulus of inner product with frame vectors. Phase retrieval is widely studied in many areas of science but still there is a number of problems that remain to be answered. We believe that the dynamic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary evolution model.
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 ) = T r(Q i ρ(t j )) of certain observables $\{Q_{i}\}_{i=1}^{r} $ measured at different time instants {tj}j=1p\{t_{j}\}_{j=1}^{p}{tj​}j=1p​ . 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} $ .
Quantum state tomography (QST) refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. In this paper, we concentrate on theoretical methods of quantum tomography, but some significant experimental results are also presented. Due to a considerable body of literature and intensive ongoing research activity in the field of QST, this overview is restricted to presenting selected ideas, methods, and results. First, we discuss tomography of pure states by distinguishing two aspects—complex vector reconstruction and wavefunction measurement. Then, we move on to the Wigner function reconstruction. Finally, the core section of the article is devoted to the stroboscopic tomography, which provides the optimal criteria for state recovery by including the dynamics in the scheme. Throughout the paper, we pay particular attention to photonic tomography, since multiple protocols in quantum optics require well-defined states of light.
Abstract The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal to 1, which physically means that there exists one observable the measurement of which performed a sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. The necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we introduce some remarks concerning optimal evolution models in the case of n -dimensional Hilbert space.
In this article we propose a new approach to quantum measurement in reference to the stroboscopic tomography. Generally, in the stroboscopic approach it is assumed that the information about the quantum system is encoded in the mean values of certain Hermitian operators Q1,...,QrQ_1, ..., Q_rQ1​,...,Qr​ and each of them can be measured more than once. The main goal of the stroboscopic tomography is to determine when one can reconstruct the initial density matrix ρ(0)ρ(0)ρ(0) on the basis of the measurement results ⟨Qi⟩tj\langle Q_i \rangle_{t_j}⟨Qi​⟩tj​​. In this paper we propose to treat every complex matrix as a measurable operator. This generalized approach to quantum measurement may bring some improvement into the models of stroboscopic tomography.
The article undertakes the problem of pure state estimation from projective measurements based on photon counting.Two generic frames for qubit tomography are considered -one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases.Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements.The accuracy of each frame is quantified by the average fidelity and purity.The efficiency of the frames is compared and discussed.The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.