Systematic Errors in Current Quantum State Tomography Tools

Common tools for obtaining physical density matrices in experimental quantum state tomography are shown here to cause systematic errors. For example, using maximum likelihood or least squares optimization to obtain physical estimates for the quantum state, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. Such strongly biased estimates can be avoided using linear evaluation of the data or by linearizing measurement operators yielding reliable and computational simple error bounds.

Figure 1

Histogram of the fidelity estimates of 500 independent simulations of QST of a noisy four-party GHZ state for three different reconstruction schemes. The values obtained via ML (blue) or LS (red) fluctuate around a value that is lower than the initial fidelity of 80% (dashed line). For comparison, we also show the result using LIN (green), which does not have this systematic error called bias.

Figure 2

The performance of ML, LS, and LIN methods depending on the number of events $N_s$ per setting and for four different noisy initial states ${\varrho }_0$. Note that the fidelity can only be calculated linearly if the reference state is pure which is not the case for the Smolin state [25]. Therefore, only the curves for ML and LS are plotted for the Smolin state.

Figure 3

The behavior of ML, LS, and LIN depending on the number of qubits $n$ (left) and the fidelity of ${\varrho }_0$ (right).

Figure 4

Lower bound obtained by linearizing bipartite negativity ($\text{LBL}$) for the GHZ (left) and a four-qubit product state (right) both mixed with white noise resulting in 80% fidelity. The ML and LS reconstruction leads to a systematic overestimation of the negativity, while the lower bound yields a valid estimate.