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Tesi etd-06272011-174249


Thesis type
Tesi di laurea specialistica
Author
DI CANDIA, ROBERTO
URN
etd-06272011-174249
Title
Quantum tomography via compressed sensing
Struttura
SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di studi
SCIENZE FISICHE
Commissione
relatore Calsamiglia, John
relatore Mannella, Riccardo
Parole chiave
  • matrix completion
  • rank-certificate
Data inizio appello
19/07/2011;
Consultabilità
parziale
Data di rilascio
19/07/2051
Riassunto analitico
The goal of tomography is to reconstruct the density matrix of a physical system through a series of measurements of some observables. In general, we need at most d^2 measurements to reconstruct the state, where d is the dimension of the Hilbert space where the state is embedded (e.g. d=2^n for an n-qubit system). But if the rank r of the matrix is low, then O(rd) measures could be sufficient. A priori it is not clear whether the matrix can be recovered from this limited set of measurements in a computationally tractable way, i.e., how to choose these measurements, or which algorithm to use.<br><br>We describe a method, introduced by Candes et al. and developed by Gross et al. for the application in quantum tomography, under the label of &#34;compressed sensing&#34;, able to reconstruct the unknown density matrix using O(rd log(d)) measures. Compressed sensing provides techniques for recovering a sparse vector (a vector contains only a few non-zero entries) from a small number of measurements. Matrix completion is a generalization of compressed sensing from vectors to matrices. Here, one recovers certain low-rank matrices X from a small number of matrix elements X_{ij}. The recovering is based on solving a semidefinite program, a class of convex optimization problems for which solving algorithms are particularly fast. The same holds in the quantum case, where we measure the coefficients of the matrix in a basis which differs by the standard basis.<br>In a real experiment, the measurement are noisy (e.g. statistical noise), and the true state is only approximately low-rank. We consider also this situation, and we accompany the discussion with useful examples able to elucidate what we are doing.
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