State estimation: A comparison between direct state measurement and tomography

We compare direct state measurement (DSM, or weak state tomography) to conventional state reconstruction (tomography) through accurate Monte Carlo simulations. We find that DSM is much less precise than tomography, but it is surprisingly robust to its inherent bias, which introduces an unavoidable error in the reconstruction. In addition to studying DSM and proposing a method to estimate such bias from the experimental data, we show that the number of DSM measurements can be optimized and that the dimensionality of the DSM pointer is irrelevant: Both finite- and infinite-dimensional pointers provide comparable reconstructions.

Figure 1

Comparison between pure-state DSM (4) and tomography (7) for qubits (left) and for qudits (right). The trace distance $D$ between the reconstructed ${\rho }^r$ and the true ${\rho }^t$ states is plotted as a function of the total number of copies $N$ of the system employed in the reconstructions. (a) Circles, $D$ calculated from DSM with $\phi =0.05$; triangles, $D$ calculated from DSM with $\phi =0.1$ (in DSM, $N$ counts also the systems discarded by postselection); stars, $D$ using tomography with a basis of Pauli matrices through Eq. (8); squares, $D$ using tomography with a continuous basis taken from a unitary representation of $\text{SU}\left(2\right)$ (as evident from the graph, these two bases are equivalent for tomography). As expected, tomography has smaller trace distance than DSM. Moreover, DSM saturates to the bias, indicated by the horizontal dashed lines at $D^{\prime }$ of Eq. (9), estimated from the experimental data: The arrows indicate the locations of the elbows where the trace distance starts to saturate as a function of $N$ (they correspond to the optimal number of measurements for the $\phi$ employed). Here we simulate a qubit spin-coherent state [13] with parameter $\alpha =2$. (b) Same as previous ($\phi =0.05,0.1$) but for a qudit with $d=10$. The postselection of the pure-state DSM drastically reduces the efficiency over tomography. Here we simulate a $d=10$ spin-coherent state with $\alpha =2$.

Figure 2

Same as Fig. 1, but using the general DSM (3), with no postselection: All $N$ copies of the system provide data that is employed in the reconstruction. While there is no substantial gain in efficiency over the pure DSM (4) for qubits (left), there is a substantial gain for qudits $d=10$ (right). Note that the trace distance can exceed 1 because the statistical fluctuations in the reconstructed state ${\rho }^r$ through Eq. (3) may yield an invalid quantum state (with negative eigenvalues). The bias is estimated (horizontal dashed lines) using the formulas for pure-state DSM (9), which overestimates the bias of general DSM (it has only indicative value here).

Figure 3

Comparison between the average statistical signal-to-noise (SNR) ratio of tomography (stars, squares) and DSM (circles, crosses), as a function of the number of copies $N$ employed. For DSM, the circles and crosses refer to $\phi =0.1$ and $\phi =0.01$ respectively: Even though a smaller $\phi$ gives a better quality reconstruction (e.g. Fig. 1), it has higher statistical fluctuations and hence worse SNR. The dashed lines refer to a qubit pointer; the continuous ones refer to a Gaussian pointer (the two have substantially same performance). For tomography, the stars and squares refer to Pauli tomography of Eq. (8) and to SU(2) tomography [13], respectively (the two methods have substantially same performance). Tomography also gives better average SNR than DSM. Here we simulate an $\alpha =2$ qubit coherent state and calculate the statistical error by dividing the data into 20 statistical blocks.

Figure 4

Estimation of the trace distance from the experimental data for the qubit pointer (dashed lines, filled symbols) and the Gaussian pointer (continuous lines, empty symbols) as a function of the coupling $\phi$. Squares, trace distance $D({\rho }^t,{\rho }^r)$; stars, experimentally accessible trace distance $D^{\prime }=D({\rho }^r,{\rho }^e)$; circles, trace distance $D({\rho }^t,{\rho }^a)$. All these trace distances approximate one another in the physically significant region of small coupling $\phi$, so we can approximate $D$ with the experimentally accessible $D^{\prime }$. Moreover, the small gap between the curves with empty and filled stars for small $\phi$ shows that using a Gaussian or a qubit pointer is equivalent. Here we simulate the reconstruction of a qubit coherent state with $\alpha =\pm 2$ (left, $\alpha =2$ with $N=49\times {10}^6$ total repetitions; right, $\alpha =-2$ with $N=159\times {10}^6$). Note the dependence of the reconstruction on the overlap with the postselected state: The (theoretical and experimental) trace distances are smaller for the $\alpha =-2$ state that has a larger overlap with the postselected state $|c_0\rangle$ than the $\alpha =2$ state.