Editing Compressed Sensing Tomography (section)

==Outline==

This technique concentrates on the states that are well approximated by density matrices of rank less than the dimension. This approach can be applied to many realistic experimental situations, where the ideal state of the system is pure, and physical constraints ensure that the actual noisy state still has low entropy. Compressed sensing tomography involves two steps, measuring an incomplete set of observables and using trace minimization or regularization to reconstruct low-rank solutions. The goal of this method is to reconstruct a low-rank state using as few samples as possible.

There are two different methods used to determine the density matrix of the unknown quantum state. The first estimator is obtained by constrained trace minimisation and the second estimator is obtained by least-squares linear regression with trace-norm regularization. 

The method describes the measurement procedure first and then the density matrix reconstruction process is described. This method consists of the following steps:

* Consider a system of <math>n</math> qubits, with the dimension to be <math>d = 2^n</math>
* Form a set of Pauli operators using [[Pauli matrices]].
* Choose a particular number of Pauli operators by sampling uniformly and independently at random from the set formed in the set above. (Alternatively, one can choose these Pauli operators randomly without replacement)
* For each of the Pauli operator selected in step 3, use a particular number of copies of the unknown quantum state, and for each of this quantum state measure the selected Pauli operator. Average the measurement outcomes over all copies of the unknown quantum state to obtain an estimate of the expectation value. The number of selected copies of the quantum state and the number of the selected Pauli operators from the set are dependent on the dimension and the rank of the density matrix. A sampling operator is defined here using the expectation value after normalising it.
* The output of the measurement procedure is then described as a linear vector which also takes the statistical noise due to the finite number of samples, or even due to an adversary into consideration.
*  To estimate the density matrix of the quantum state, one of the two methods: constrained trace minimisation (a.k.a. [[the matrix Dantzig selector]]) or least-squares linear regression with trace-norm regularization (a.k.a. [[the matrix Lasso]]) can be used. Both of these methods are based on the intuition of finding the density matrix which fits the measurement data while minimizing the trace norm of that matrix, which serves as a surrogate for minimising the rank of that matrix.