SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 59 , Issue 10 : 100313(2016) https://doi.org/10.1007/s11433-016-0287-y

Quantum state and process tomography via adaptive measurements

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  • ReceivedJul 18, 2016
  • AcceptedJul 28, 2016
  • PublishedAug 18, 2016
PACS numbers


We investigate quantum state tomography (QST) for pure states and quantum process tomography (QPT) for unitary channels via adaptive measurements. For a quantum system with a $d$-dimensional Hilbert space, we first propose an adaptive protocol where only $2d-1$ measurement outcomes are used to accomplish the QST for all pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography (AUPT) protocol of $d^2+d-1$ measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic resonance system. We examine the performance of the AUPT protocol when applied to Hadamard gate, $T$ gate ($\pi/8$ phase gate), and controlled-NOT gate, respectively, as these gates form the universal gate set for quantum information processing purpose. As a comparison, standard QPT is also implemented for each gate. Our experimental results show that the AUPT protocol that reconstructing unitary channels via adaptive measurements significantly reduce the number of experiments required by standard QPT without considerable loss of fidelity.

Funded by

National Natural Science Foundation of China(11175094)

National Natural Science Foundation of China(91221205)

National Natural Science Foundation of China(11375167)

National Natural Science Foundation of China(11227901)

National Natural Science Foundation of China(91021005)





This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Institute for Advanced Research (CIFAR), the National Natural Science Foundation of China (Grant Nos. 11175094, 91221205, 11375167, 11227901 and 91021005), the National Basic Research Program of China (Grant No. 2015CB921002), the National Key Basic Research Program (NKBRP) (Grant Nos. 2013CB921800 and 2014CB848700) and the National Science Fund for Distinguished Young Scholars (Grant No. 11425523).


[1] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik (Springer, Berlin, Heidelberg, 1933).. Google Scholar

[2] Weigert S.. Phys. Rev. A, 1992, 45: 7688 CrossRef Google Scholar

[3] Amiet J.-P., Weigert S.. J. Phys. A: Math. Gen., 1999, 32: 2777 CrossRef Google Scholar

[4] Flammia S. T., Silberfarb A., Caves C. M.. Found. Phys., 2005, 35: 1985 CrossRef Google Scholar

[5] Liu Y., Zhang F. H.. Sci. China-Phys. Mech. Astron., 2015, 58: 070301 Google Scholar

[6] Finkelstein J.. Phys. Rev. A, 2004, 70: 052107 CrossRef Google Scholar

[7] Gross D., Liu Y.-K., Flammia S. T., Becker S., Eisert J.. Phys. Rev. Lett., 2010, 105: 150401 CrossRef Google Scholar

[8] Long Y., Feng G. R., Pearson J., Long G. L.. Sci. China-Phys. Mech. Astron., 2014, 57: 1256 CrossRef Google Scholar

[9] Cramer M., Plenio M. B., Flammia S. T., Somma R., Gross D., Bartlett S. D., Landon-Cardinal O., Poulin D., Liu Y.-K.. Nat. Commun., 2010, 1: 149 CrossRef Google Scholar

[10] Heinosaari T., Mazzarella L., Wolf M. M.. Commun. Math. Phys., 2013, 318: 355 CrossRef Google Scholar

[11] Chen J., Dawkins H., Ji Z., Johnston N., Kribs D., Shultz F., Zeng B.. Phys. Rev. A, 2013, 88: 012109 CrossRef Google Scholar

[12] Baldwin C. H., Deutsch I. H., Kalev A.. Phys. Rev. A, 2016, 93: 052105 CrossRef Google Scholar

[13] Li N., Ferrie C., Gross J. A., Kalev A., Caves C. M.. Phys. Rev. Lett., 2016, 116: 180402 CrossRef Google Scholar

[14] Ma X., Jackson T., Zhou H., Chen J., Lu D., Mazurek M. D., Fisher K. A., Peng X., Kribs D., Resch K. J., Ji Z. F., Zeng B., Laflamme R.. Phys. Rev. A, 2016, 93: 032140 CrossRef Google Scholar

[15] T. Xin, D. Lu, J. Klassen, N. Yu, Z. Ji, J. Chen, X. Ma, G. Long, B. Zeng, and R. Laflamme, arXiv: 1604.02046.. Google Scholar

[16] C. Carmeli, T. Heinosaari, M. Kech, J. Schultz, and A. Toigo, arXiv: 1604.02970.. Google Scholar

[17] Gutoski G., Johnston N.. J. Math. Phys., 2014, 55: 032201 CrossRef Google Scholar

[18] Baldwin C. H., Kalev A., Deutsch I. H.. Phys. Rev. A, 2014, 90: 012110 CrossRef Google Scholar

[19] Goyeneche D., nas G. Ca\, Etcheverry S., Gómez E. S., Xavier G. B., Lima G., Delgado A.. Phys. Rev. Lett., 2015, 115: 090401 CrossRef Google Scholar

[20] Chow J. M., Gambetta J. M., C{ó}rcoles A., Merkel S. T., Smolin J. A., Rigetti C., Poletto S., Keefe G. A., Rothwell M. B., Rozen J. R., Ketchen M. B., Steffen M.. Phys. Rev. Lett., 2012, 109: 060501 CrossRef Google Scholar

[21] M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).. Google Scholar

[22] Ryan C., Negrevergne C., Laforest M., Knill E., Laflamme R.. Phys. Rev. A, 2008, 78: 012328 CrossRef Google Scholar

[23] Cory D. G.. A. F., 1997, Fahmy: T. F. Havel, Proc. Natl. Acad. Sci. {\bf 94}, 1634 Google Scholar

[24] Lu D., Xu N., Xu R., Chen H., Gong J., Peng X., Du J.. Phys. Rev. Lett., 2011, 107: 020501 CrossRef Google Scholar

[25] Benenti G., Strini G.. J. Phys. B: At. Mol. Opt. Phys., 2010, 43: 215508 CrossRef Google Scholar

[26] Emerson J., Alicki R., , {{\.Z}yczkowski} K.. J. Opt. B: Quantum Semiclass. Opt., 2005, 7: S347 CrossRef Google Scholar

[27] Emerson J., Silva M., Moussa O., Ryan C., Laforest M., Baugh J., Cory D. G., Laflamme R.. Science, 2007, 317: 1983 Google Scholar

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