Editing Compressed Sensing Tomography (section)

==Notation==

* <math>n</math>: number of qubits in the system
* <math>d</math>: Dimension of the Hilbert space. <math>d = 2^n</math>
* <math>\mathcal{P}</math>: Set of all <math>d^2</math> Pauli operators.
* <math>P</math>: Pauli operator in <math>\mathcal{P}</math>. <math>P = \sigma_1 \otimes ... \otimes \sigma_n</math>
* <math>\sigma_i</math>: This belongs to the set of Pauli matrices <math>\{I, \sigma^x, \sigma^y, \sigma^z\}</math>
* <math>m</math>: Selected number of Pauli operators. <math>m = O((rd) log d)</math>
* <math>\rho</math>: unknown quantum state
* <math>t</math>: total number of copies of <math>\rho</math>. <math>t = O((\frac{rd}{\epsilon})^2 log d)</math>, <math>r</math> is the unknown rank and <math>\epsilon</math> is the accuracy in the trace distance
* <math>\mathcal{A}</math>: Sampling operator which is a linear map defined for all <math>i \in [m]</math>. Normalisation is chosen because <math>\mathbb{E}\mathcal{A}^*\mathcal{A} = I</math>
* <math>\mathbb{E}</math>: expectation value of a random variable
* <math>z</math>: statistical noise due to the finite number of samples, or even due to an adversary
* <math>y</math>: Vector to describe the measurement procedure
* <math>X</math>: Matrix that fits data <math>y</math>
* <math>\rho_{DS}</math>: Estimate for the matrix using matrix Dantzig selector
* <math>\lambda</math>:  Parameter for trace minimisation which is set according to the noise in the data
* <math>\rho_{Lasso}</math>: Estimate for the matrix using matrix Lasso
* <math>\mu</math>: regularization parameter which is set according to the noise level
* <math>C_0, C_1, C^{'}_0, C^{'}_1</math>: fixed absolute constants
* <math>\rho_c</math>:  For any quantum state <math>\rho</math>, we write <math>\rho = \rho_c + \rho_r</math> where <math>\rho_r</math> is the best rank-r approximation to <math>\rho</math> and <math>\rho_c</math> is the residual part.