Pure-state tomography with parallel unentangled measurements

Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of that state. Most quantum algorithms rely on QST at some point and it is a well-explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, and their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the user to measure each qubit individually, e.g., using one-qubit Pauli measurements. We propose two sets of quantum measurements that one can make on a pure state as well as the algorithms that use the measurement outcomes in order to identify the state. We also discuss how those estimates can be fine tuned by finding the state that maximizes the likelihood of the measurements with different variants of the likelihood. The performances of the proposed three types of QST methods are validated by means of detailed numerical tests, including for mixed states that are close to being pure.

Figure 1

Estimation errors [defined in (23)] of the initialization algorithms. The bold red (upper) and green (lower) horizontal lines are the worst and best errors for the recursive algorithm (only available with 15 measurement types) on the 50 randomly generated pure states. The other curves represent the evolution of the error on the PhaseCut estimates with the 50 states. Each iteration of PhaseCut is costly. In Sec. 6c, we set its number of iterations to 5000; we also decide to use the recursive algorithm whenever possible.

Figure 2

Empirical cdf of the errors of the three maximum likelihood estimators.

Figure 3

Convergence of the different likelihood algorithms in the presence of initialization errors.

Figure 4

Empirical cdf of the QST error. In the four plots of the top row, ${\mathcal{L}}_{({\mathit{x}}^{\prime },{\mathit{y}}^{\prime })}^{\mathrm{exact}}$ is minimized. For the middle row, it is ${\mathcal{L}}_{({\mathit{x}}^{\prime },{\mathit{y}}^{\prime })}^{\mathrm{Gauss}}$. The lowest row corresponds to the mixed algorithm. The blue solid curve (the one on the right in every plot) is the cdf of the error of the initialization algorithm (PhaseCut for four measurement types and the recursive algorithm for 15 measurement types); it does not depend on the likelihood maximization algorithm and is the same in every row. The legends of the second and fourth columns are not displayed in order to keep the curves visible; they would be the same as the legends of the first and third columns, respectively.

Figure 5

Empirical cdf of the error of $\widehat{{\mathit{v}}_{\mathrm{ML}}}$ computed with the mixed algorithm. The input states are mixed states. The curves with the smaller $p_0$ are on the left; those with the highest $p_0$ are on the right.