Coherent Quantum Tomography (bibtex)

by Joonas Ilmavirta

Abstract:

We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previously described in the physical literature.

Reference:

Coherent Quantum Tomography (Joonas Ilmavirta), SIAM Journal on Mathematical Analysis, volume 48, number 5, pp. 3039–3064, 2016.
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We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previously described in the physical literature.
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Bibtex Entry:

@article{qm-tomography, author = {Joonas Ilmavirta}, title = {{Coherent Quantum Tomography}}, journal = {SIAM Journal on Mathematical Analysis}, month = sep, year = {2016}, volume = {48}, number = {5}, pages = {3039--3064}, doi = {10.1137/15M1026821}, arxiv = {1507.00558}, gsid = {6561154729300318202}, url={http://users.jyu.fi/~jojapeil/pub/matrix-tomography.pdf}, abstract = {We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previously described in the physical literature.} }

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