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Unbiased state reconstruction using modified weak measurement

Xuanmin Zhu, Yu-Xiang Zhang, Shengjun Wu

Published Dec 11, 2015• 1 citation
OpenAlex

Abstract

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.

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Quantum state tomography strategy via weak value is valid for arbitrary dimensional systems, and can be easily realized. However, the efficiency of quantum state tomography strategy via weak value is lower than that of the conventional quantum state tomography (QST), especially for high-dimensional systems. To improve the efficiency of the scheme, we propose a two-step adaptive quantum state tomography via weak value for pure states (AQSTW). In AQSTW, by using appropriate measurement strength, the efficiency of AQSTW is significantly larger than that of the conventional QST, especially for high-dimensional systems. And AQSTW can be easily implemented. These results demonstrate AQSTW may be useful in the measurements of the unknown quantum states.

Results in Physics•Nov 20, 2021•3 citations
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.
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.
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.
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.
The direct state measurement (DSM) based on the weak measurement has the advantage of simplicity, versatility, and directness. However, the weak measurement will introduce an unavoidable error in the reconstructed quantum state. We modify the DSM by replacing the weak coupling between the system and the pointer by a strong one, and present two procedures for measuring quantum states, one of which can give the wave function or the density matrix directly. We can also measure the Dirac distribution of a discrete system directly. Furthermore, we propose quantum circuits for realizing these procedures, and the main body of the circuits consists of Toffoli gates. By numerical simulation, we find that our scheme can eliminate the biased error effectively.
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.
Quantum state tomography strategy via weak value is valid for arbitrary dimensional systems, and can be easily realized. However, the efficiency of quantum state tomography strategy via weak value is lower than that of the conventional quantum state tomography (QST), especially for high-dimensional systems. To improve the efficiency of the scheme, we propose a two-step adaptive quantum state tomography via weak value for pure states (AQSTW). In AQSTW, by using appropriate measurement strength, the efficiency of AQSTW is significantly larger than that of the conventional QST, especially for high-dimensional systems. And AQSTW can be easily implemented. These results demonstrate AQSTW may be useful in the measurements of the unknown quantum states.