Full Quantum state tomography with Linear Inversion

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 linear inversion.

Tags: Certification protocol, Tomography, Quantum State density matrix reconstruction, Linear inversion

Assumptions

• Noise is assumed to be independent and identically distributed.
• The selected measurement basis should be tomographically complete.

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.

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.
• With the knowledge of the probabilities and the chosen measurement basis, the density matrix of the quantum state is calculated using matrix inversion.

Hardware Requirements

• Trusted Measurement device.

Notation

• $d$: Dimension of Hilbert space
• $\rho$: Density matrix of the prepared quantum state
• $E_{j}$: Different measurement operators of a selected basis, where $j=1,...,d^{2}$
• $A$: Select basis of measurement with the different measurement operators $E_{j}$
• $p_{j}$: Probability of the state corresponding to $E_{j}$
• $N$: Number of single shots corresponding to a measurement operator. This is used to calculate probability
• $m_{j}$: Sample average of $N$ single shot measurements of $E_{j}$.
• $m_{ij}$: Outcome of the $i^{th}$ measurement from $N$ single shot measurements. $m_{ij}\in \{0,1\}$
• $E(m_{j})$: Expected value of $m_{j}$
• $E(m_{ij})$: Expected value of $m_{ij}$

Properties

• The figure of merit: Density Matrix of the quantum state
• Multiple copies of the quantum state are required in this method.
• The measurement basis $E_{j}$ should be chosen such that $A_{jk}$ is invertible to calculate the density matrix. Hence, $E_{j}$ should be tomographically complete.
• This method is extremely resource-intensive.
• The main drawback of this technique is that sometimes the computed solution of the density matrix will not be a valid density matrix. This issue occurs when not enough measurements are made. Due to this drawback, the maximum likelihood estimation technique is preferred over this linear inversion technique.
• Another drawback of this method is that an infinite number of measurement outcomes would be required to give the exact solution.
• 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.

Protocol Description

Input: copies of unknown quantum state

Output: Density matrix of the quantum state, $\rho$

• Pick the measurement basis $A$, where $A={\begin{bmatrix}{\vec {E_{1}}}\\{\vec {E_{2}}}\\.\\.\\{\vec {E_{d^{2}}}}\end{bmatrix}}$
• For $j=1,2,...,d^{2}$:
  • For $i=1,2,...,N$:
    • Measure $\rho$ with measurement operator $E_{j}$
    • Get measurement result $m_{ij}$
  • Calculate $m_{j}=\sum _{i=1}^{N}m_{ij}/N$
  • Estimate $p_{j},p_{j}=E(m_{j})=\sum _{i=1}^{N}E_{(}m_{ij})/N$
• Define ${\vec {p}}$, where ${\vec {p}}={\begin{bmatrix}p_{1}\\p_{2}\\.\\.\\p_{d^{2}}\end{bmatrix}}$
• Form relation $A{\vec {\rho }}={\begin{bmatrix}{\vec {E_{1}}}{\vec {\rho }}\\{\vec {E_{2}}}{\vec {\rho }}\\.\\.\\{\vec {E_{d^{2}}}}{\vec {\rho }}\end{bmatrix}}={\begin{bmatrix}E_{1}\cdot \rho \\E_{2}\cdot \rho \\.\\.\\E_{d^{2}}\cdot \rho \end{bmatrix}}={\begin{bmatrix}P(E_{1}|\rho )\\P(E_{2}|\rho )\\.\\.\\P(E_{d^{2}}|\rho )\end{bmatrix}}={\begin{bmatrix}p_{1}\\p_{2}\\.\\.\\p_{d^{2}}\end{bmatrix}}={\vec {p}}$
• Use matrix inversion to get $\rho$, where ${\vec {\rho }}=(A^{T}A)^{-1}A^{T}{\vec {p}}$

Further Information

• Maximum likelihood estimation(also known as MLE or MaxLik) is a popular technique for dealing with the problems of linear inversion.
• Bayesian mean estimation (BME) is another approach which addresses the problems of maximum likelihood estimation
• Homodyne tomography focuses on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.

Related Papers

• 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

*contributed by Rhea Parekh