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arXiv:1310.6206 · Physical Review A

State estimation: A comparison between direct state measurement and tomography

Lorenzo Maccone, Cosimo C. Rusconi

Published Feb 24, 2014• 44 citations
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Abstract

We compare direct state measurement (DST or weak state tomography) to conventional state reconstruction (tomography) through accurate Monte-Carlo simulations. We show that DST is surprisingly robust to its inherent bias. We propose a method to estimate such bias (which introduces an unavoidable error in the reconstruction) from the experimental data. As expected we find that DST is much less precise than tomography. We consider both finite and infinite-dimensional states of the DST pointer, showing that they provide comparable reconstructions.

DOI
10.1103/physreva.89.022122

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We compare direct state measurement (DST or weak state tomography) to conventional state reconstruction (tomography) through accurate Monte-Carlo simulations. We show that DST is surprisingly robust to its inherent bias. We propose a method to estimate such bias (which introduces an unavoidable error in the reconstruction) from the experimental data. As expected we find that DST is much less precise than tomography. We consider both finite and infinite-dimensional states of the DST pointer, showing that they provide comparable reconstructions.
Direct state tomography (DST) using weak measurements has received wide attention. Based on the concept of coupling-deformed pointer observables presented by Zhang et al. [Y.-X. Zhang, S. Wu, and Z.-B. Chen, Phys. Rev. A 93, 032128 (2016)], a modified direct state tomography (MDST) is proposed, examined, and compared with other typical state tomography schemes. MDST has exact validity for measurements of any strength. We identify the strength needed to attain the highest efficiency level of MDST by using statistical theory. MDST is much more efficient than DST in the sense that far fewer samples are needed to reach DST's level of reconstruction accuracy. Moreover, MDST has no inherent bias when compared to DST.
We show that reformulating the Direct State Tomography (DST) protocol in terms of projections into a set of non-orthogonal bases one can perform an accuracy analysis of DST in a similar way as in the standard projection-based reconstruction schemes, i.e. , in terms of the Hilbert-Schmidt distance between estimated and true states. This allows us to determine the estimation error for any measurement strength, including the weak measurement case, and to obtain an explicit analytic form for the average minimum square errors.
Based on the modified weak measurement scheme in [Y-X Zhang, S Wu and Z-B Chen, arXiv: 1504.07130], a modified direct state measurement (MDSM) is proposed without inherent bias. Compared with the existing direct state measurement (DSM), MDSM is more efficient and valid for any coupling strength. Through accurate Monte Carlo simulations, we get the optimal coupling strength to attain high efficiency. For the same reconstruction precision, we compare the number of copies needed in MDSM and the one in conventional quantum state tomography (QST) by simulations. We find MDSM is less efficient than conventional QST. However, MDSM is much easier to be realized in actual experiments, and might be more useful.
Compared with the conventional quantum state tomography (QST), the direct state tomography (DST) using weak value is easily manipulated in experiments. However, the efficiency of the DST is lower than that of the conventional QST, especially for high-dimensional systems. For a pure state, the DST is revised to improve the efficiency. In the revised DST, the real or imaginary parts of the weak values can be obtained by measuring only one system observable. We constructed a hybrid DST by combining the original DST and the revised DST. By using the appropriate measurement strength, the efficiency of the hybrid DST is significantly larger than that of the conventional QST, especially for high-dimensional systems. The state reconstruction strategy investigated in this paper may be useful in actual experiments.
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.
New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only O(d) different operations, compared to d^{2} linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.