Abstract

We consider the problem of implementing mutually unbiased bases (MUB) for a polarization qubit with only one wave plate, the minimum number of wave plates. We show that one wave plate is sufficient to realize two MUB as long as its phase shift (modulo 360°) ranges between 45° and 315°. It can realize three MUB (a complete set of MUB for a qubit) if the phase shift of the wave plate is within [111.5°, 141.7°] or its symmetric range with respect to 180°. The systematic error of the realized MUB using a third-wave plate (TWP) with 120° phase is calculated to be a half of that using the combination of a quarter-wave plate (QWP) and a half-wave plate (HWP). As experimental applications, TWPs are used in single-qubit and two-qubit quantum state tomography experiments and the results show a systematic error reduction by 50%. This technique not only saves one wave plate but also reduces the systematic error, which can be applied to quantum state tomography and other applications involving MUB. The proposed TWP may become a useful instrument in optical experiments, replacing multiple elements like QWP and HWP.

© 2015 Optical Society of America

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References

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2014 (1)

M. Mohammadi and A. M. Brańczyk, “Optimization of quantum state tomography in the presence of experimental constraints,” Phys. Rev. A 89, 012113 (2014).
[Crossref]

2013 (5)

M. Mohammadi, A. M. Brańczyk, and D. F. V. James, “Fourier-transform quantum state tomography,” Phys. Rev. A 87, 012117 (2013).
[Crossref]

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

2012 (3)

B. N. Simon, C. M. Chandrashekar, and S. Simon, “Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates,” Phys. Rev. A 85, 022323 (2012).
[Crossref]

F. De Zela, “Two-component gadget for transforming any two nonorthogonal polarization states into one another,” Phys. Lett. A 376, 1664–1668 (2012).
[Crossref]

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

2011 (2)

2010 (4)

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

S. Wehner and A. Winter, “Entropic uncertainty relations–a survey,” New J. Phys. 12, 025009 (2010).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[Crossref] [PubMed]

2009 (2)

S. Brierley and S. Weigert, “Constructing mutually unbiased bases in dimension six,” Phys. Rev. A 79, 052316 (2009).
[Crossref]

S. J. Wu, S. X. Yu, and K. Mølmer, “Entropic uncertainty relation for mutually unbiased bases,” Phys. Rev. A 79, 022104 (2009).
[Crossref]

2008 (1)

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

2007 (3)

A. Roy and A. J. Scott, “Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,” J. Math. Phys. 48, 072110 (2007).
[Crossref]

M. A. Ballester and S. Wehner, “Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases,” Phys. Rev. A 75, 022319 (2007).
[Crossref]

J. Fan, M. D. Eisaman, and A. Migdall, “Quantum state tomography of a fiber-based source of polarization-entangled photon pairs,” Opt. Express 15, 18339–18344 (2007).
[Crossref] [PubMed]

2006 (1)

A. J. Scott, “Tight informationally complete quantum measurements,” J. Phys. A Math. Gen. 39, 13507–13530 (2006).
[Crossref]

2004 (1)

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

2003 (1)

J. J. Benedetto and M. Fickus, “Finite normalized tight frames,” Adv. Comput. Math. 18, 357–385 (2003).
[Crossref]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

2001 (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

1990 (1)

R. Simon and N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

1989 (1)

W. K. Wootters and B. D. Fields, “Optimal state–determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[Crossref]

Adamson, R. B. A.

R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[Crossref] [PubMed]

Agnew, M.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Altepeter, J. B.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006).

Ballester, M. A.

M. A. Ballester and S. Wehner, “Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases,” Phys. Rev. A 75, 022319 (2007).
[Crossref]

Bancal, J.-D.

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

Bartlett, S. D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Benedetto, J. J.

J. J. Benedetto and M. Fickus, “Finite normalized tight frames,” Adv. Comput. Math. 18, 357–385 (2003).
[Crossref]

Bengtsson, I.

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 175 (1984).

Blume-Kohout, R.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

Bolduc, E.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Boyd, R. W.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Branczyk, A. M.

M. Mohammadi and A. M. Brańczyk, “Optimization of quantum state tomography in the presence of experimental constraints,” Phys. Rev. A 89, 012113 (2014).
[Crossref]

M. Mohammadi, A. M. Brańczyk, and D. F. V. James, “Fourier-transform quantum state tomography,” Phys. Rev. A 87, 012117 (2013).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 175 (1984).

Brierley, S.

S. Brierley and S. Weigert, “Constructing mutually unbiased bases in dimension six,” Phys. Rev. A 79, 052316 (2009).
[Crossref]

Caves, C. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

Chandrashekar, C. M.

B. N. Simon, C. M. Chandrashekar, and S. Simon, “Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates,” Phys. Rev. A 85, 022323 (2012).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge university, 2010).
[Crossref]

Cramer, M.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Damask, J. N.

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2004).

Darabi, A.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

de Burgh, M. D.

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

De Zela, F.

F. De Zela, “Two-component gadget for transforming any two nonorthogonal polarization states into one another,” Phys. Lett. A 376, 1664–1668 (2012).
[Crossref]

Delgado, A.

Doherty, A. C.

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

Dong, D.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Durt, T.

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

Eisaman, M. D.

Englert, B-G

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

Fan, J.

Ferretti-Schöbitz, R.

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

Ferrie, C.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

Fickus, M.

J. J. Benedetto and M. Fickus, “Finite normalized tight frames,” Adv. Comput. Math. 18, 357–385 (2003).
[Crossref]

Fields, B. D.

W. K. Wootters and B. D. Fields, “Optimal state–determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[Crossref]

Flammia, S. T.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Gilchrist, A.

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

Gisin, N.

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Gómez, E. S.

Gross, D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Guo, G.-C.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Guzmán, R.

Hou, Z.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Z. Hou, in preparation.

Houlsby, N. M. T.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

Huszár, F.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

James, D. F. V.

M. Mohammadi, A. M. Brańczyk, and D. F. V. James, “Fourier-transform quantum state tomography,” Phys. Rev. A 87, 012117 (2013).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Jeffrey, E. R.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006).

Johnson, A. S.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Kravtsov, K. S.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

Kulik, S. P.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006).

Landon-Cardinal, O.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Langford, N. K.

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

Leach, J.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Li, L.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Liang, Y.-C.

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

Lima, G.

Liu, Y. K.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Mahler, D. H.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

Migdall, A.

Mohammadi, M.

M. Mohammadi and A. M. Brańczyk, “Optimization of quantum state tomography in the presence of experimental constraints,” Phys. Rev. A 89, 012113 (2014).
[Crossref]

M. Mohammadi, A. M. Brańczyk, and D. F. V. James, “Fourier-transform quantum state tomography,” Phys. Rev. A 87, 012117 (2013).
[Crossref]

Mølmer, K.

S. J. Wu, S. X. Yu, and K. Mølmer, “Entropic uncertainty relation for mutually unbiased bases,” Phys. Rev. A 79, 022104 (2009).
[Crossref]

Mukunda, N.

R. Simon and N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Neves, L.

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge university, 2010).
[Crossref]

Nogueira, W. A. T.

Paris, M.

M. Paris and J. Řeháček, Quantum State Estimation (Springer, 2004).
[Crossref]

Plenio, M. B.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Poulin, D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Qi, B.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Radchenko, I. V.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

Rehácek, J.

M. Paris and J. Řeháček, Quantum State Estimation (Springer, 2004).
[Crossref]

Renes, J. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Rosset, D.

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

Roy, A.

A. Roy and A. J. Scott, “Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,” J. Math. Phys. 48, 072110 (2007).
[Crossref]

Rozema, L. A.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

Saavedra, C.

Salvail, J. Z.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

Scott, A. J.

A. Roy and A. J. Scott, “Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,” J. Math. Phys. 48, 072110 (2007).
[Crossref]

A. J. Scott, “Tight informationally complete quantum measurements,” J. Phys. A Math. Gen. 39, 13507–13530 (2006).
[Crossref]

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

Simon, B. N.

B. N. Simon, C. M. Chandrashekar, and S. Simon, “Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates,” Phys. Rev. A 85, 022323 (2012).
[Crossref]

Simon, R.

R. Simon and N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Simon, S.

B. N. Simon, C. M. Chandrashekar, and S. Simon, “Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates,” Phys. Rev. A 85, 022323 (2012).
[Crossref]

Somma, R.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Steinberg, A. M.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[Crossref] [PubMed]

Straupe, S. S.

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Vargas, A.

Wehner, S.

S. Wehner and A. Winter, “Entropic uncertainty relations–a survey,” New J. Phys. 12, 025009 (2010).
[Crossref]

M. A. Ballester and S. Wehner, “Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases,” Phys. Rev. A 75, 022319 (2007).
[Crossref]

Weigert, S.

S. Brierley and S. Weigert, “Constructing mutually unbiased bases in dimension six,” Phys. Rev. A 79, 052316 (2009).
[Crossref]

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Winter, A.

S. Wehner and A. Winter, “Entropic uncertainty relations–a survey,” New J. Phys. 12, 025009 (2010).
[Crossref]

Wootters, W. K.

W. K. Wootters and B. D. Fields, “Optimal state–determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[Crossref]

Wu, S. J.

S. J. Wu, S. X. Yu, and K. Mølmer, “Entropic uncertainty relation for mutually unbiased bases,” Phys. Rev. A 79, 022104 (2009).
[Crossref]

Xiang, G.

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Yu, S. X.

S. J. Wu, S. X. Yu, and K. Mølmer, “Entropic uncertainty relation for mutually unbiased bases,” Phys. Rev. A 79, 022104 (2009).
[Crossref]

Zauner, G.

G. Zauner, “Quantum designs: Foundations of a noncommutative design theory,” Int. J. Quantum Inf. 09, 445–507 (2011).
[Crossref]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Zhu, H.

H. Zhu, Quantum State Estimation and Symmetric Informationally Complete POMs (National University of Singapore, 2012).

Zyczkowski, K

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

Adv. Comput. Math. (1)

J. J. Benedetto and M. Fickus, “Finite normalized tight frames,” Adv. Comput. Math. 18, 357–385 (2003).
[Crossref]

Ann. Phys. (1)

W. K. Wootters and B. D. Fields, “Optimal state–determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[Crossref]

Int. J. Quantum Inf. (2)

G. Zauner, “Quantum designs: Foundations of a noncommutative design theory,” Int. J. Quantum Inf. 09, 445–507 (2011).
[Crossref]

T. Durt, B-G Englert, I. Bengtsson, and K Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 8, 535–640 (2010).
[Crossref]

J. Math. Phys. (2)

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2180 (2004).
[Crossref]

A. Roy and A. J. Scott, “Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements,” J. Math. Phys. 48, 072110 (2007).
[Crossref]

J. Phys. A Math. Gen. (1)

A. J. Scott, “Tight informationally complete quantum measurements,” J. Phys. A Math. Gen. 39, 13507–13530 (2006).
[Crossref]

Nat. Commun. (1)

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Nat. Photon. (1)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photon. 7, 316–321 (2013).
[Crossref]

New J. Phys. (1)

S. Wehner and A. Winter, “Entropic uncertainty relations–a survey,” New J. Phys. 12, 025009 (2010).
[Crossref]

Opt. Express (2)

Phys. Lett. A (2)

R. Simon and N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

F. De Zela, “Two-component gadget for transforming any two nonorthogonal polarization states into one another,” Phys. Lett. A 376, 1664–1668 (2012).
[Crossref]

Phys. Rev. A (10)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, “Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses,” Phys. Rev. A 86, 062325 (2012).
[Crossref]

M. Mohammadi, A. M. Brańczyk, and D. F. V. James, “Fourier-transform quantum state tomography,” Phys. Rev. A 87, 012117 (2013).
[Crossref]

M. Mohammadi and A. M. Brańczyk, “Optimization of quantum state tomography in the presence of experimental constraints,” Phys. Rev. A 89, 012113 (2014).
[Crossref]

S. Brierley and S. Weigert, “Constructing mutually unbiased bases in dimension six,” Phys. Rev. A 79, 052316 (2009).
[Crossref]

B. N. Simon, C. M. Chandrashekar, and S. Simon, “Hamilton’s turns as a visual tool kit for designing single-qubit unitary gates,” Phys. Rev. A 85, 022323 (2012).
[Crossref]

K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, “Experimental adaptive Bayesian tomography,” Phys. Rev. A 87, 062122 (2013).
[Crossref]

S. J. Wu, S. X. Yu, and K. Mølmer, “Entropic uncertainty relation for mutually unbiased bases,” Phys. Rev. A 79, 022104 (2009).
[Crossref]

M. A. Ballester and S. Wehner, “Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases,” Phys. Rev. A 75, 022319 (2007).
[Crossref]

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122 (2008).
[Crossref]

Phys. Rev. Lett. (2)

R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[Crossref] [PubMed]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Sci. Rep. (1)

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G.-C. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3, 3496 (2013).
[Crossref] [PubMed]

Other (8)

M. Paris and J. Řeháček, Quantum State Estimation (Springer, 2004).
[Crossref]

H. Zhu, Quantum State Estimation and Symmetric Informationally Complete POMs (National University of Singapore, 2012).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge university, 2010).
[Crossref]

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 175 (1984).

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006).

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2004).

Z. Hou, in preparation.

Z. Hou, H. Zhu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Error-compensation measurements on polarization qubits,” (2015). http://arxiv.org/abs/1503.00263 .

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Figures (7)

Fig. 1
Fig. 1 MUB measurement with two different settings: (a) MUB measurement with QWP-HWP setting consisting of a QWP, a HWP and a PBS. (b) MUB measurement with a wave plate (WP) and a PBS.
Fig. 2
Fig. 2 Frame potential Φ(δ) with respect to δ. The frame potential is symmetric about 180°. The inset is the amplified interval of δ where the frame potential vanishes to zero. Only if δ is in the range [111.5°, 141.7°] or its symmetric range about 180° can a complete set of MUB be realized by one wave plate.
Fig. 3
Fig. 3 The numerical solutions of rotation angles in Eq. (6) with respect to the phase δ. There are two classes of solutions with δ between 111.5° and 126.3°, represented by solid and dotted lines, respectively, and only one class of solutions in (126.3°, 141.7°]. Each class of solutions contains four complete sets of MUB, denoted by four different colors.
Fig. 4
Fig. 4 Experimental setup for qubit tomography. The apparatus consists of two parts: state preparation (green) and MUB measurement (pink). The MUB measurement part consists of a polarizing beam splitter and a wave-plate combination which has two choices: (a) TWP and (b) QWP-HWP combination.
Fig. 5
Fig. 5 Systematic error in TWP and QWP-HWP setting for qubit tomography. The dependence of the systematic error in the TWP setting (Fig. 4(a)) and the QWP-HWP setting (Fig. 4(b)) is experimentally measured with respect to the angle errors of TWP (red), QWP (black) and HWP (blue) for three states in Figs. 5(a)–5(c) at p = 0.92. The total systematic error (green) in the QWP-HWP setting is the sum of that due to QWP and HWP. The experimental results denoted as dots coincide with the theoretical calculations (solid lines). Fig. 5(d) plots the total systematic error for all these three states in the two settings and shows that the TWP setting beats the QWP-HWP setting by a factor of two. Error bars are the standard deviation of 100 trials in Monte Carlo simulation with binomial distribution of counting statistics.
Fig. 6
Fig. 6 Experimental setup for two-qubit tomography. A singlet state is prepared via SPDC process with a fidelity of 98%. In quantum state tomography, single-qubit MUB measurements are implemented on both photons with TWP or the combination of QWP and HWP in Figs. 4 (a) and 4 (b). Coincidence events are recorded by a coincidence circuit.
Fig. 7
Fig. 7 Systematic error in TWP and QWP-HWP setting for two-qubit tomography. In (a), systematic errors due to angle errors of QWP (black), HWP (blue), both of them (green) and TWP (red) are numerically simulated and denoted as solid lines. The experimental results due to angle errors of wave plates are plotted as dots for photon 1 and circles for photon 2. The total systematic error for the two photons is plotted in (b). From (b), The TWP setting beats the QWP-HWP setting by about a factor of two. Error bars are the standard deviation of 100 Monte Carlo simulations with multinomial distribution of the counting statistics.

Equations (30)

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| ψ = U ( q , π 2 ) U ( h , π ) | H = ( cos q cos ( q 2 h ) + i sin q sin ( q 2 h ) sin q cos ( q 2 h ) i cos q sin ( q 2 h ) ) ,
U ( θ , δ ) = ( cos 2 θ + e i δ sin 2 θ 1 2 ( 1 e i δ ) sin 2 θ 1 2 ( 1 e i δ ) sin 2 θ sin 2 θ + e i δ cos 2 θ ) .
r ( q , h ) = ( sin 2 q cos ( 4 h 2 q ) , sin ( 4 h 2 q ) , cos 2 q cos ( 4 h 2 q ) ) T
r 1 r 2 = 0 .
r ( θ ) = ( sin 2 δ 2 sin 4 θ sin δ sin 2 θ , 1 2 sin 2 δ 2 sin 2 2 θ ) T .
r ( θ i ) r ( θ j ) = 0 ,
Φ ( δ ) = min θ 1 , θ 2 , θ 3 [ r ( θ 1 ) r ( θ 2 ) ] 2 + [ r ( θ 2 ) r ( θ 3 ) ] 2 + [ r ( θ 1 ) r ( θ 3 ) ] 2 .
r ( t ) = ( 3 4 sin 4 t , 3 2 sin 2 t , 3 4 cos 4 t + 1 4 ) T ,
( Δ r ) 2 = ξ r ξ 2 ( Δ ξ ) 2 ,
r δ 2 = sin 2 2 θ = ( r 2 ) 2 sin 2 δ , r θ 2 = 16 sin 2 δ 2 4 sin 2 δ sin 2 2 θ = 16 sin 2 δ 2 4 ( r 2 ) 2 .
ε 2 = j = 1 3 ( Δ r j ) 2 = j = 1 3 ξ r ξ j 2 ( Δ ξ ) 2 = ξ ( ε ξ ) 2 ( Δ ξ ) 2 , ( ε ξ ) 2 = j = 1 3 r ξ j 2 .
( ε δ ) 2 = j = 1 3 ( r 2 j ) 2 sin 2 δ = 1 sin 2 δ , ( ε θ ) 2 = j = 1 3 16 sin 2 δ 2 4 ( r 2 j ) 2 = 48 sin 2 δ 2 4 .
ε 2 = 1 sin 2 δ ( Δ δ ) 2 + ( 48 sin 2 δ 2 4 ) ( Δ θ ) 2 .
ε 2 = 1.33 ( Δ δ t ) 2 + 32 ( Δ t ) 2 .
tr ( ρ ρ ^ ) 2 = 1 4 p 2 ( ε 1 2 + ε 2 2 ) ,
90 ° θ 1 = 90 ° + θ 1 , θ 2 = θ 3 90 ° .
cos δ sin 2 2 θ 2 + cos 2 2 θ 2 = 0 , sin 2 δ sin 2 2 θ 2 = 1 2 .
θ 1 + θ 2 = 90 ° , θ 3 90 ° = 180 ° θ 3 .
θ 1 + θ 3 = 90 ° , 90 ° θ 2 = θ 2 .
cos 2 δ sin 2 2 θ 1 + cos δ cos 2 2 θ 1 sin 2 δ sin 2 θ 1 = 0 , ( 1 cos δ ) 2 sin 2 2 θ 1 cos 2 2 θ 1 = sin 2 δ sin 2 2 θ 1 + ( cos δ sin 2 2 θ 1 + cos 2 2 θ 1 ) 2 .
x ( x 1 ) y 2 + x = ( 1 x 2 ) y , ( x 1 ) 2 y 4 = ( x 1 ) 2 y 2 1 2
x 2 ( x 1 ) 2 y 4 + 2 x 2 ( x 1 ) y 2 + x 2 = ( x 1 ) 2 ( x + 1 ) 2 y 2 .
y 2 = x 2 2 ( 1 x 2 ) .
3 x 4 + 4 x + 2 = 0 .
r δ q 2 = sin 2 ( 4 h 2 q ) = ( r 2 ) 2 , r δ h 2 = sin 2 2 h , r q 2 = 4 + 4 cos 2 ( 4 h 2 q ) = 8 4 ( r 2 ) 2 , r h 2 = 16 .
( ε δ q ) 2 = j = 1 3 ( r 2 j ) 2 = 1 , ( ε q ) 2 = j = 1 3 8 4 ( r 2 j ) 2 = 20 , ( ε h ) 2 = 48 m ( ε δ h ) 2 = j = 1 3 sin 2 2 h j .
ε 2 = 48 ( Δ h ) 2 + 20 ( Δ q ) 2 + ( Δ δ h ) 2 + ( Δ δ q ) 2 .
ε 2 = 48 ( Δ h ) 2 + 20 ( Δ q ) 2 + 1.5 ( Δ δ h ) 2 + ( Δ δ q ) 2 .
tr ( ρ ρ ^ ) 2 = 1 2 ξ R T ξ s 2 ( Δ ξ ) 2 ,
R t = ( 3 1 1 3 1 1 0 2 2 2 2 ) , R δ t = ( 0 6 6 6 6 0 6 6 6 6 0 3 3 3 3 ) .

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