Editing Full Quantum state tomography with Maximum Likelihood Estimation

Quantum state tomography is the process by which a quantum state is reconstructed using a series of measurements in different basis on an ensemble of identical quantum states. Because measurement of a quantum state typically changes the state being measured, getting a complete picture of that state requires measurements on many state copies. The method described here perform quantum state tomography using maximum likelihood estimation.

'''Tags:''' [[Certification protocol]], [[Tomography]], [[Quantum State density matrix reconstruction]], [[Maximum Likelihood Estimation]]

==Assumptions==
* The selected measurement basis should be tomographically complete. 
* For the calculation of the likelihood function we assume that the noise on the coincidence measurements has a Gaussian probability distribution. We also assume that each of our measurements is taken for the same amount of time

==Outline==
Quantum state tomography attempts to characterize an unknown state ρ by measuring its components, usually in the Pauli basis or a different selected basis of measurement operators. Multiple identical copies of the quantum state are required as different measurements need to be performed on each copy. To fully reconstruct the density matrix for a mixed state, this method is used. This procedure can be used to completely characterize an unknown apparatus.

Instead of using [[matrix inversion]] technique, which can produce results that violate important basic properties such as positivity leading to a density matrix which is not valid, MLE is used. To avoid this problem, the maximum likelihood estimation of density matrices is employed. In practice, analytically calculating this maximally likely state is prohibitively difficult, and a numerical search is necessary. Three elements are required: a manifestly legal parametrization of a density matrix, a likelihood function which can be maximized, and a technique for numerically finding this maximum over a search of the density matrix’s parameters.

This method consists of the following steps:

* An unknown state is prepared multiple times so as to create many copies.
* The experimenter picks a basis of measurement operators.
* The measurement corresponding to each measurement operator in the selected basis is taken multiple times. This is used to approximate the probability of measuring the state in the value corresponding to the measurement operator.
* Generate a formula for an explicitly physical and valid density matrix, i.e., a matrix that has the three important properties of normalization, Hermiticity, and positivity.
* Introduce a likelihood function which quantifies how good the final density matrix is in relation to the experimental probability data. This likelihood function is a function of the certain real parameters according to the described formula and of the experimental probability data. This, in general, will depend on the specific measurement apparatus used and the physical implementation of the qubit (as these will determine the statistical distributions of counts, and therefore their relative weightings). To provide a likelihood function here, we assume Gaussian counting statistics and that each of our measurements is taken for the same amount of time.
* Using standard numerical optimization techniques, find the optimum set of variables for which the likelihood function has its maximum value. The best estimate for the density matrix is then determined using this function.

This method is generally preferred over the quantum state tomography using linear matrix inversion.

==Hardware Requirements==
* Trusted Measurement device.

==Notation==
* <math>d</math>: Dimension of Hilbert space
* <math>\rho</math>: Density matrix of the prepared quantum state
* <math>E_j</math>: Different measurement operators of a selected basis, where <math>j = 1, ..., d^2</math>
* <math>A</math>: Select basis of measurement with the different measurement operators <math>E_j</math>
* <math>p_j</math>: Probability of the state corresponding to <math>E_j</math>
* <math>\bar{p_j}</math>: Expected value of <math>p_j</math>. <math>\bar{p_j} = N\langle\psi_j|\rho|\psi_j\rangle</math>
* <math>N</math>: normalization parameter which can be determined from the data. 
* <math>n</math>: Number of single shots corresponding to a measurement operator. This is used to calculate the probability
* <math>m_j</math>: Sample average of <math>N</math> single shot measurements of <math>E_j</math>. 
* <math>m_{ij}</math>: Outcome of the <math>i^{th}</math> measurement from <math>N</math> single shot measurements. <math>m_{ij} \in \{0,1\}</math>
* <math>E(m_j)</math>: Expected value of <math>m_j</math>
* <math>E(m_{ij})</math>: Expected value of <math>m_{ij}</math>
* <math>\hat{\rho_p}</math>: Manifestly physical density matrix of the prepared quantum sate. This is given by the formula <math>\hat{\rho_p}</math>, <math>\hat{\rho_p}(t) = \hat{T^{\dagger}}(t) \hat{T}(t) / tr\{ \hat{T}^{\dagger}(t) \hat{T}(t)\}</math>
* <math>\hat{T}(t)</math>: Formula for multiple qubits which is used in reconstructing <math>\hat{\rho_p}</math>
* <math>t</math>: Short form of <math>t_i, i=1, .., d^2</math>
* <math>L(t_1, t_2, ..., t_{n^2})</math>: Likelihood function

==Properties==

* '''Figure of merit''': Density Matrix of the quantum state
* Multiple copies of the quantum state are required in this method.
* <math>E_j</math> should be tomographically complete.
* This method is preferred over [matrix inversion] as that method in some cases produces results which are not valid. This issue occurs when not enough measurements are made. 
* This method is extremely resource intensive.
* State preparation and measurement errors are taken into consideration in this method. The measurement errors are tackled by increasing the accuracy of the measuring apparatus.
* Error from counting statistics can also occur which are reduced by performing the measurements on a larger ensemble.
* Drift error can occur either in the state produced or the efficiency of the detection system and it can constrain data collection time.
* In addition to the mentioned errors, each experimental implementation can have its own unique errors.
* We assume that the noise on the coincidence measurements has a Gaussian probability distribution.

==Procedure Description==

'''Input''': copies of the unknown quantum state

'''Output''': Density matrix of the quantum state, <math>\rho</math>

* Pick the measurement basis <math>A</math> where, <math>
       A = \begin{bmatrix}
        \vec{E_1} \\ \vec{E_2} \\ . \\. \\ \vec{E_{d^2}}
       \end{bmatrix}
    </math>
* For <math>j = 1, 2, ..., d^2</math>:
** For <math>i = 1, 2, ..., n</math>:
*** Measure <math>\rho</math> with measurement operator <math>E_j</math>
*** Get measurement result <math>m_{ij}</math>
** Calculate <math>m_{j} = \sum^{n}_{i=1} m_{ij}/n</math>
** Estimate <math>p_j, p_j = E(m_j) = \sum^{n}_{i=1} E_(m_{ij})/n</math>
* Formula for <math>\hat{\rho_p}</math> is <math>\hat{\rho_p}(t) = \hat{T^{\dagger}}(t) \hat{T}(t) / tr\{ \hat{T}^{\dagger}(t) \hat{T}(t)\}</math>. Here <math>\hat{T}(t)</math> is,<math> \hat{T}(t) = \begin{bmatrix}
             t_1 & 0 & ... & 0 \\
             t_{2^d + 1} + it_{2^d+2} & t_2 & ... & 0 \\
             ... & ... & ... & 0 \\ 
             t_{4^d -1} + it_{4^d} &  t_{4^d -3} + it_{4^d - 2} &  t_{4^d - 5} + it_{4^d - 4} &  t_{2^d}
           \end{bmatrix}
     </math>
* Find the minimum of the <math>L(t_1, t_2, ..., t_{d^2})</math> using the formula  <math>
        L(t_1, t_2, ..., t_{d^2}) = \sum_j \frac{(N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{d^2})|E_j\rangle - p_j)^2}{2N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{d^2})|E_j\rangle}
    </math>
* <math>\hat{\rho_p}</math> can be reconstructed from the values of <math>t_i</math>.

==Further Information==

* [Bayesian mean estimation (BME)] is another approach which addresses the problems of maximum likelihood estimation
* [https://arxiv.org/abs/quant-ph/0507078 Homodyne tomography] focuses on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.

==Related Papers==
* Daniel F. V. James et al PhysRevA.64.052312: Measurement of qubits
* J. B. Altepeter et al, Quantum State Tomography
* D. Leibfried et al Phys. Rev. Lett. 77, 4281 (1996)
* J. Rehacek et al arXiv:quant-ph/0009093: Iterative algorithm for reconstruction of entangled states
* Robin Blume-Kohout arXiv:quant-ph/0611080v1: Optimal, reliable estimation of quantum states
* G. M. D'Ariano et al arXiv:quant-ph/0507078: Homodyne tomography and the reconstruction of quantum states of light

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>