# State Tomography

## Functionality Description

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. This technique is applied on a source of systems, to determine the quantum state of the output of that source. Hence, this technique is used to certify any process by checking the exact quantum state produced by it. The figure of merit here is the density matrix of the final quantum state. This method is extremely resource-intensive.

## Protocols

- Full Quantum state tomography with Linear Inversion
- Full Quantum state tomography with Maximum Likelihood Estimation
- Compressed Sensing Tomography
- Matrix Product State tomography
- Tensor Network Tomography

## Properties

- This method is a certification technique which has a really high sample and resource complexity.
- The figure of merit in this method is the density matrix of the final quantum state
- All of the protocols under this technique broadly have two methods of state estimation from measurements: Linear inversion and Maximum Likelihood estimation. Linear inversion process is mathematically simpler but sometimes the computed solution of the density matrix is not valid. Hence Maximum Likelihood estimation is preferred as it constructs the states which has a higher probability of being valid.

## Related Papers

- Z Hradil Physical Review A, 55(3):1561–1564, 1997: Quantum-state estimation.
- 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
- Steven T. Flammia et al arXiv:1205.2300: Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
- David Gross et al arXiv:0909.3304: Quantum state tomography via compressed sensing

**contributed by Rhea Parekh*