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arXiv:1507.00715

A Dynamic Approach to Complex Vector Reconstruction from Intensity Measurements

Artur Czerwiński

Published Jul 2, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1507.00715

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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.
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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.
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This letter is devoted to the cross-fertilization between the fields of phase retrieval and quantum tomography. In the first part we discuss topological aspects of quantum tomography which turn out to imply lower bounds on the number of frame vectors necessary to reconstruct signals modulo phase. In the second part we generalize the approach of Balan, Casazza and Edidin to construct frames for phase retrieval to the context of quantum tomography. In this process we obtain an extension of their result to Parseval frames.