Quantum state tomography of a single qubit: comparison of methods

Abstract

The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods, we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.

Keywords:

• Quantum state reconstruction
• quantum state tomography

Acknowledgements

The author would like to thank Philipp Treutlein, Andreas Nunnenkamp and Christoph Bruder for valuable discussions and criticism. The Centro de Ciencias de Benasque Pedro Pascual has provided a very stimulating environment for finishing this manuscript. This work was supported by the Swiss National Science Foundation and by the EU project QIBEC.

Notes

No potential conflict of interest was reported by the author.

1 Even though the Kullback–Leibler divergence is not a distance because it is not symmetric in its arguments, we can still minimize it with respect to one of its arguments since it is a premetric.

2 The Bures measure for qubits (20(20) $$\begin{array}{c}\hfill {\mathcal{C}}_{\text{B}}(\mathit{r})={\mathcal{C}}_{\frac{3}{2}}(\mathit{r})=\left\{\begin{array}{cc}\frac{1}{{\mathit{\pi }}^2\sqrt{1-{\Vert \mathit{r}\Vert }^2}}\hfill & \text{if}\Vert \mathit{r}\Vert <1\text{,}\hfill \\ 0\hfill & \text{if}\Vert \mathit{r}\Vert >1\text{,}\hfill \end{array}\right.\end{array}$$(20) ) is the spherical equivalent of the Jeffreys prior of the Bernoulli trial, $\mathcal{C}(p)=1/[\mathit{\pi }\sqrt{p(1-p)}]$.

3 $J_u(t)$ has a branch cut discontinuity at $t=0$ for $u<-1$, and any results based on these values are ill-defined.